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Essays on bundling and discounts

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Essays on bundling and discounts

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Content ESSAYSONBUNDLINGANDDISCOUNTS by YongChao ADissertationPresentedtothe FACULTYOFTHEUSCGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulllmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (ECONOMICS) August2010 Copyright2010 YongChao ii Dedication Tomybelovedfamily whichhasalwaysbeenand willalwaysbethewindbeneathmywings. iii Acknowledgments Thisworkowesagreatdealtomanypeople,withoutwhomthisdissertationwould never be possible. And this acknowledgement is the happiest part of writing the dissertation, since it brings back the wonderful times with my professors, friends andfamily. First and foremost, I would like to express my greatest gratitude to Profes- sorGuofuTan, mysupervisorandthechairofmydissertationcommittee, forhis encouragement, guidance, support and patience. He spent countless hours dis- cussing with me about research ideas, working through proofs, and pinpointing directions I could improve my research. Guofu inspired me not just to explore in IO and antitrust area as an economist, but also to grow as an independent thinker andasensibleman. IamsurethatIwillmissthetimewehavespenttogetherand manyintellectualdiscussionswithhim. Itishardtogetabetterrolemodel. Iamalsoextremelygratefultomydissertationcommitteeaswellasqualifying examination members Professors Juan Carrillo, Harrison Cheng, Anthony Dukes and Simon Wilkie. A special word of thanks goes to Juan for leading me to the Game Theory world, and providing me the great opportunity to practice my job talk before my going on the job market. I also beneted from a great many inter- actions with Harrison during my years at USC. He offered me a lot of chances to attendmanyhighqualityconferencesandmademyresearchideaclearerthrough many valuable discussions. Anthony, from the Marshall School of Business, is always there to help me. He even spared a particular time to assist me in rening iv my presentation slides. I am especially indebted to Simon for supporting me in variouswaysduringmyjobsearch,especiallyforthosephonecallsandemails. MythanksalsoextendtoanumberoffacultiesandstaffsatourUSC.Iwantto thankProfessorsIsabelleBrocas,ChengHsiao,ShmuelLeshem,MichaelMagill, RogerMoon,JeffreyNugent,GeertRidder,andJohnStraussfortheirencourage- mentandadvice. IamindebtedtomyfriendsfrommytimeatUSC.Inparticular, I want to thank all the members of our Econ Soccer Group for providing me so muchfunduringmystressfuljobseekingyear. IoweaspecialnoteofgratitudetomyfriendTimothyP.DerdengeratTepper SchoolofBusiness,CMUforourfruitfulcollaborationsandasagreatco-author. Financial support from the USC Summer Dissertation Fellowship, the Mi- crosoft Corporation Research Grant, the NET Institute Grant and the USC Grad- uateMeritAwardFellowshipisgratefullyacknowledged. Lastbutnotleast,Iwanttoexpressmyappreciationtomyfamily,whoalways believe in me and have always been the source of unconditional love and endless support. v TableofContents Dedication ii Acknowledgments iii ListofTables vi ListofFigures vii Abstract ix ChapterI:Introduction 1 ChapterII:StrategicEffectsofThree-PartTariffsunderOligopoly 4 1Introduction 4 2TheModel 11 3TheCaseofPerfectSubstitutes 15 4TheCaseofImperfectSubstitutes 23 5ComparativeStatics 32 6Conclusion 48 ChapterIII:MixedBundlinginTwo-SidedMarkets: TheoryandEvidence 54 1Introduction 54 2PortableVideoGameConsoleIndustry 61 3ModelSettings 64 4One-SidedModel: WithoutThird-PartyGames 67 5Two-SidedModel: WithThird-PartyGames 73 6HypothesesandData 89 7ReducedFormTests 93 8Conclusion 102 Bibliography 103 Appendices 107 AppendixII.A:EquilibriumAnalysis: TheCaseofPerfectSubstitutes 107 AppendixII.B:EquilibriumAnalysis: TheCaseofImperfectSubstitutes 122 AppendixII.C:EquilibriumforGeneralDifferentiated LinearDemands 157 AppendixIII.A:TheoreticalProofs 161 AppendixIII.B:EmpiricalTests 165 vi ListofTables TableIII.1: PredictedandObservedCorrelations BetweenBundlingandComponentPrices 57 TableIII.2: PortableConsoleMarketStatistics 91 TableIII.3: PortableConsoleandBundlePrices 92 TableIII.4: StandaloneConsolePriceRegression 96 TableIII.5: StandaloneBundledSoftwarePriceRegression 99 TableIII.6: IndependentSoftwarePriceRegression 101 TableII.C-1: LP,2PTand3PTEquilibriumTariffs 158 TableII.C-2: LP,2PTand3PTEquilibriumProts 159 TableII.C-3: LP,2PTand3PTEquilibriumTotalSurpluses 160 TableIII.B-1: Non-NestedModelSelectionTest AnEncompassingTest 165 TableIII.B-2: Box-CoxModel(Non)LinearSpecicationTest 166 TableIII.B-3: Log-LogModelResidualNormalityTest 166 TableIII.B-4: Box-CoxModel(Non)LinearSpecicationTest 167 TableIII.B-5: LinearModelResidualNormalityTest 167 TableIII.B-6: Box-CoxModel(Non)LinearSpecicationTest 168 TableIII.B-7: NonlinearModelResidualNormalityTest 168 vii ListofFigures FigureII.1: 3PTTotalPaymentSchedule 12 FigureII.2: Game'sTimeline 13 FigureII.3: TheRetailer'sOptimalPurchaseDecisionin(AA) 19 FigureII.4: ManufacturerB'sProtCurveinEquilibrium 20 FigureII.5: TheTypicalTwoCurvesR 1 (q A ;q B ) =w A and R 2 (q A ;q B ) =w B inq A q B Plane,withNotationsfor theCaseofImperfectSubstitutes 29 FigureII.6: TheRetailer'sOptimalPurchaseDecisionin(AA) 31 FigureII.7: ManufacturerA'sProtunderLP,2PTand 3PTEquilibrium 34 FigureII.8: ManufacturerA'sTariffs 35 FigureII.9: LowerBoundofw A andtheAveragePrice To Qo under3PTEquilibrium 37 FigureII.10: ManufacturerB'sProtunderLP,2PTand 3PTEquilibrium 38 FigureII.11: ManufacturerB'sQuantitySaleandMarketShare underLP,2PTand3PTEquilibrium 40 FigureII.12: RetailerR'sProtunderLP,2PTand 3PTEquilibrium 41 FigureII.13: TotalSurplusunderLP,2PTand3PTEquilibrium 43 FigureII.14: TotalSurplusunder3PTExclusionEquilibrium, andLowerBoundandUpperBoundof TotalSurplusunderLPEquilibrium 47 FigureII.15: Two-BlockTariffandAll-UnitsDiscounts 50 FigureII.16: 3PTvs3PT 52 viii FigureIII.1: VideoGameMarketStructure 63 FigureIII.2: NewGamers'DemandunderIPinOne-SidedMarket 69 FigureIII.3: NewGamers'DemandunderIPinTwo-SidedMarkets 75 FigureIII.4: NewGamers'DemandunderMixedBundling inTwo-SidedMarkets 80 FigureIII.5: ConsoleandBundleSales 93 FigureII.A-1: TheRetailer'sProtCurvesin(AA)and(NA) 110 FigureII.A-2: TwoPossibleCasesofCrossings 112 FigureII.A-3: InEquilibrium,c< b w B <w e A 115 FigureII.A-4: ExistenceandUniqueness 121 FigureII.B-1: ThePhaseDiagramsofThreePossibleCases when w A <p A (Q o ;0) w B <p B (Q o ;0) 124 FigureII.B-2: TheRetailer'sProtCurvesin(AA)and(NA) 130 FigureII.B-3: (B1)w B b w B 132 FigureII.B-4: (B2i-a) ( w A <p A (Q o ;0) p B (Q o ;(w A ;Q o )) b w B <p B (q m A (w A );0) 133 FigureII.B-5: (B2-i-b) w A <p A (Q o ;0) b w B <p B (Q o ;(w A ;Q o )) 134 FigureII.B-6: (B2-ii) b w B <p B (Q o ;0) p A (Q o ;0)w A 135 FigureII.B-7: r B (w B ;Q o )and B (w A ;w B ) 150 FigureII.B-8: ManufacturerB'sProtCurvein(B2-i-b) 150 FigureII.B-9: ManufacturerB'sProtCurvein(B2-ii) 154 ix Abstract This dissertation studies a variety of bundling and discount strategies adopted by dominant rms, analyze their corresponding competitive effects in the upstream- and-downstreamcontext,aswellasintwo-sidedmarkets;anddiscusstheirimpli- cationsforregulationandantitrustlawenforcement. Therstessayinvestigatesstrategiceffectsofthree-parttariffsinasequential- move game and offers an equilibrium theory of three-part tariffs in a competitive context. Ishowthat,comparedwithlinearpricingequilibriumandtwo-parttariff equilibrium, a three-part tariff always strictly increases the dominant rm's (the leader's)protwhencompetingagainstarival(thefollower)withsubstituteprod- ucts, in the absence of usual price discrimination motive. The competitive effect of a three-part tariff in contrast to linear pricing depends on the degree of sub- stitutability between products: Competition is intensied when two products are more differentiated, yet softened when two products are more substitutable. This is in stark contrast with the competitive scenario posed by a two-part tariff: A two-parttariffalwaysenhancescompetitionandgivesthehighesttotalsurplusof these three pricing schemes. My ndings offer a new perspective on three-part tariffs, a perspective which could help antitrust enforcement agencies distinguish theexclusionarythree-parttarifffromthepro-competitiveone. The second essay extends the traditional literature on bundling and the bur- geoning literature on two-sided markets by presenting a theoretical monopoly modelofmixedbundlinginthecontextoftheportablevideogameconsolemarket x aprototypicaltwo-sidedmarket. Deviatingfrombothtraditionalbundlinglitera- tureandstandardtwo-sidedmarketsliterature,wendthat,undermixedbundling, boththestandaloneconsolepriceontheconsumersideandtheroyaltyrateonthe game developer side are lower than their counterparts under independent pricing equilibrium. Inoursetting,mixedbundlingactsasapricediscriminationtoolseg- menting the market more efciently as well as functions as a coordination device helping solve "the chicken or the egg" problem in two-sided markets. We further test the model predictions with new data from the portable video game console market,andndempiricalsupportforalltheoreticalpredictions. 1 ChapterI Introduction This dissertation studies a variety of bundling and discount strategies adopted by dominant rms, analyze their corresponding competitive effects in the upstream- and-downstreamcontext,aswellasintwo-sidedmarkets;anddiscusstheirimpli- cationsforregulationandantitrustlawenforcement. ChapterIIfocusesonanonlinearpricingstrategycalledthree-parttariff(3PT). A3PTreferstoapricingschemeconsistingofaxedfee,afreeallowanceofunits uptowhichthemarginalpriceiszero,andapositiveper-unitpriceforadditional demand beyond that allowance. The three-part tariff and its variations are com- monly used in both nal-goods and intermediate-goods markets. Recently, the offering of three-part tariffs and the like by dominant rms has become a promi- nent antitrust issue (e.g., U.S. v. Microsoft Corp. and AMD v. Intel). Existing studies have focused on monopoly models, interpreting the three-part tariff as a price discrimination device. In this chapter, I investigate the strategic effects of three-part tariffs in a sequential-move game and offer an equilibrium theory of three-parttariffsinacompetitivecontext. Ishowthat,comparedwithlinearpric- ing equilibrium and two-part tariff equilibrium, a three-part tariff always strictly increases the dominant rm's (the leader's) prot when competing against a rival (the follower) with substitute products, in the absence of usual price discrimi- nation motive. To explore the effects of a three-part tariff on welfare, I further 2 perform comparative statics analysis using general differentiated linear demand system. Ishowthatthecompetitiveeffectofathree-parttariffincontrasttolinear pricing depends on the degree of substitutability between products: Competition is intensied when two products are more differentiated, yet softened when two productsaremoresubstitutable. Thisisinstarkcontrastwiththecompetitivesce- nario posed by a two-part tariff: A two-part tariff always enhances competition and gives the highest total surplus of these three pricing schemes. Moreover, the rivalrmalwaysgetshurtinbothprotandquantitysalewhenthedominantrm switches from linear pricing to a two-part tariff, yet it enjoys higher prot when thedominantrmmovesfromatwo-parttarifftothemoreornatethree-parttariff, despite the fact that its quantity and market share are decreased even further. My ndings offer a new perspective on three-part tariffs, a perspective which could help antitrust enforcement agencies distinguish the exclusionary three-part tariff fromthepro-competitiveone. Chapter III extends the traditional literature on bundling and the burgeoning literature on two-sided markets by presenting a theoretical monopoly model of mixed bundling in the context of the portable video game console marketa prototypical two-sided market. It is shown that the monopoly platform's dom- inant strategy is to offer a mixed bundle rather than pure bundle or no bundle. Deviating from both traditional bundling literature and standard two-sided mar- kets literature, we nd that, under mixed bundling, both the standalone console price on the consumer side and the royalty rate on the game developer side are lower than their counterparts under independent pricing equilibrium. In our set- 3 ting, mixed bundling acts as a price discrimination tool segmenting the market more efciently as well as functions as a coordination device helping solve "the chicken or the egg" problem in two-sided markets. After theoretically evaluating the impact mixed bundling has on prices and welfare, we further test the model predictions with new data from the portable video game console market in the early to middle 2000s, during which Nintendo was a monopolist. We employ a reducedformapproach,andndempiricalsupportforalltheoreticalpredictions. 4 ChapterII StrategicEffectsofThree-Part TariffsunderOligopoly 1 Introduction Athree-parttariff(3PT)referstoapricingschemeconsistingofaxedfee,afree allowance of units up to which the marginal price is zero, and a positive per-unit priceforadditionaldemandbeyondthatallowance. The3PTanditsvariationsare prevalentinmanycontexts,includingbothnal-goodsmarketsandintermediate- goods markets. At the end user level, 3PTs have become popular recently in in- formationindustries. Examplesof3PTsincludethepricingstructuresforcellular phoneplans,Internetaccessservice,datacenterhosting,andcloudcomputing. 1 Asforthebusinesslevel,3PTcontractsarecommonlyusedbydominantrmsand theyhaveraisedmanyantitrustconcernsregardingtheirputativeexclusionaryef- fects. 2 1 For example, a typical cell phone plan from AT&T is $39.99/month for 450 mins, with $0.45/min for overtime calling. And in many European countries, Internet subscription pricing is a 3PT (See Lambrecht, Seim and Skiera (2007)[23]). As for online data storage, RimuHosting charges $20/month for 30GB, with $1/GB for additional storage. In addition, RackSpace adopts a3PTforcloudcomputingservice$100/monthfor50GBdiskspace,500GBbandwidthand 3millionWebrequests,with$0.50/GBadditionaldiskspace,$0.25/GBadditionalbandwidthand $0.03/1000Webrequests. 2 For instance, the CPU license in U.S. v. Microsoft Corp. (See Baseman, Warren-Boulton and Woroch (1995)[9]), and rst-dollar rebate in AMD v. In- tel (See Executive Summary of AMD Complaint at http://www.amd.com/us- 5 Although 3PTs have been widely used for quite some time, the economics andbusinessliteratureconcerningthemhasbeenrathersparseuntilrecently. And withinthelimitedliterature,3PTsareofteneitheranalogizedtothecanonicaltwo- part tariff (2PT), or are interpreted as a market segmenting tool when demand is uncertain. 3 Oi(1971)[27]mentionedIBM's3PTcontractsofmachineleasingfor the rst time in his classical Disneyland Dilemma paper, and interpreted it as a surplus extraction device for the monopoly, in the same spirit as the 2PT. Lam- brecht,SeimandSkiera(2007)[23]developedadiscrete/continuouschoicemodel forempiricalestimationusingInternetusagedata,andshowedthatdemanduncer- taintyplaysakeyroleinconsumer'sbehaviorunder3PTs. Theymodeledthe3PT as a monopoly's optimal nonlinear pricing. Along the line of a monopoly's opti- malpricingwithdemanduncertainty,Grubb(2009)[17]incorporatedconsumers' overcondenceintoasequentialscreeningmodelinwhichconsumershavetosign a contract when they are uncertain about their eventual demand. 4 He used U.S. cellular phone data to support the superiority of the overcondence assumption over the common priors assumption that both rm and consumers agree on the distributionoffuturedemandwhenexplaininga3PTasacustomerscreeningde- vice. Bagh and Bhargava (2008)[6] showed that under certain circumstances, a monopoly with a more ornate menu of 2PTs can be outperformed by a smaller menuof3PTs. en/assets/content_type/DownloadableAssets/Complaint_summary.pdf). 3 Forthe2PT,seeOi(1971)[27]. 4 For the sequential screening model, see Baron and Besanko (1984)[8] and Courty and Li (2000)[13]. 6 Nevertheless, all these papers explain the 3PT as a price discrimination tool from a monopoly perspective, and competition is assumed away in these models. Moreover, all the 3PTs discussed in the above papers are targeted to nal con- sumers, who often buy from only one 3PT seller. Rather, in intermediate-goods markets, the downstream rms typically carry products from more than one up- stream supplier. In fact, this is the level at which antitrust issues primarily arise due to limited competition. In the landmark case, U.S. v. Microsoft Corp., one of the alleged antitrust practices of Microsoft was with regard to its 3PT pricing structure for its CPU license. Under the CPU license, an OEM usually had to also commit to a minimum `requirement' (X) that approximates its annual ship- ments.(O)ncethecontractisinplace,themarginalpriceis0uptoX unitsand f for additional units. (See Baseman, Warren-Boulton and Woroch (1995)[9]). 5 Whatmakesthe3PTappearanticompetitiveisthelargequantitythresholdwithin which the marginal price is zero, a stipulation which may reduce the demand for therivals'productstolevelssolowthattherivalsareforcedtoexitthemarket,de- spite the fact that accepting the 3PT from the dominant rm does not necessarily orexplicitlyexcludetheretailer'srighttopurchasefromothersuppliers. In this chapter, I develop a game theoretical model to study the strategic rea- sonswhyadominantrmunderoligopolyofferssucha3PTtodownstreamrms and the implications of this for antitrust concerns. I consider a sequential-move 5 IntherecenthotbuttoncaseAMDv. Intel,theEuropeanCommissionunveiledonSeptem- ber 21, 2009, that Intel rebates to HP from November 2002 to May 2005 were conditioned in particularonHPpurchasingnolessthan95%ofitsCPUneedsforbusinessdesktopsfromIntel (theremaining5%thatHPcouldpurchasefromAMD...). Thisisnotexactlya3PT,butitisquite similarintermsofthequantitytarget. And,IshowinSection6thatitcanbeconvertedtoa3PT. 7 settingoftwocompetingupstreammanufacturerssellingtheirsubstituteproducts to a single downstream retailer, with the dominant rm moving rst and the rival rm as a follower in offering contracts to the retailer. To rule out price discrimi- nationasapossiblemotivefortheupstreamrmaprior,Irestrictattentiontothe case of a single buyer with complete information in which the demand and costs are common knowledge to all parties. Hence, it can be viewed as an extension of the classical Stackelberg model. 6 To assess the welfare implication, I compare 3PT equilibrium with the classical Stackelberg linear pricing (LP) equilibrium as wellas2PTequilibrium. The main nding is that the manner in which a 3PT inuences competition is strikingly different from that of LP or a 2PT: it could be either a Top Dog Ploy or a Puppy Dog Ploy. 7 I nd that a 3PT is always a strictly protable tool over LP or a 2PT for the dominant rm to compete against its rival, both for cases of perfect substitutes and imperfect substitutes. I further perform comparative stat- ics analysis to explore the effects of a three-part tariff on welfare, using general differentiated linear demand system. Compared with LP, a 2PT always intensi- es competition in terms of offering higher total surplus and lowering the rival rm's prot, but a 3PT may, contrastingly, lower the total surplus and increase the rival rm's prot. Interestingly, the competitive effect of a 3PT in contrast to LP depends on the degree of substitutability between products: competition is 6 ForStackelbergmodel,seeStackelberg(1952)[34]. 7 Top Dog Ploy refers to a strategy of being tough and aggressive to compete ecely against the rival, and Puppy Dog Ploy is a metaphor for being small and friendly to induce accommo- dationfromtherival. Fordiscussionsonthesestrategies,seeFudenbergandTirole(1984)[15]. 8 intensied (Top Dog) when two products are more differentiated, but softened (PuppyDog)whentwoproductsaremoresubstitutable. Also,italwaysreduces therivalrm'squantityaswellasitsmarketshare. Thus,althoughitmayhavethe exclusionary effect of driving the rival's prot low under certain circumstances, it is very well possible that a 3PT can serve as a strategic tool to soften competi- tion over LP. Remarkably, although a 3PT is more ornate than a 2PT, it is always the case that a 3PT lessens competition over a 2PT. Furthermore, I show that the welfare in 3PT equilibrium may be higher or lower than in LP equilibrium, de- pendinguponthedegreeofsubstitutabilitybetweenproducts,forbothcaseswhen the exclusion of the rival rm is possible or not. It is also important to note that thecutoffsforthedirectionofthechangeinwelfareareuniquelydetermined. The central idea of the chapter is that the 3PT can be a credible commitment tool for the dominant rm to set the tone for competition, and induce the rival rm'soptimalresponsestowardtheinterestsofthedominantrmthroughacare- fullydesigned3PT.Interestingly,a3PThasimpactswhicharedistinctfromthose ofa2PToncompetitionoverLP.Inparticular,a2PTisalwaysaTopDogPloy whichstrengthenscompetitionandincreasestotalsurplus. Onthecontrary,a3PT will switch from a Top Dog Ploy to a Puppy Dog Ploy as two products be- comesmoresubstitutable, althoughitalwaysimprovesthedominantrm'sprot overeitherLPora2PT. The possibility of the Puppy Dog Ploy can be best understood in the po- lar case, when two products are homogeneous. When two products are perfect substitutes, the Stackelberg LP equilibrium outcome is reduced to simultaneous 9 Bertrandoutcomebothmanufacturerssellatmarginalcostandearnzeroprof- its. This is because in price competition, the follower enjoys the second-mover advantagethatpermitshimtoalwaysundercuttheleader'sofferslightlyandcap- ture the whole market. 8 Such an advantage backres because the leader can an- ticipate it, and hence a price war will ensue. This Bertrand outcome remains the caseevenifa2PTisavailable,sincetheundercuttingfromtherivalrmstillpre- vails. Note that under either LP or a 2PT, a uniform per-unit price is applied to all units purchased from the leader. It is this very applying-to-all-units feature of a single per-unit price that results in an all-or-nothing purchase decision from the downstream retailerit will buy all of its requirements from the supplier with the higher surplus offer only as two products are identical. Such an all-or- nothing purchase decision creates a pressure on prices, which then dwindle away tothemarginalcost. However,bycrediblycommittingitselftoalimitedquantity throughthequantitythresholdandtheper-unitpriceafterwardsina3PT,thelead- ingrmmakesitselflessoffensivetothefollowerbyleavingthefollowerwiththe remaining demand, thus improving its strategic position. Recall that under either LP or a 2PT, it is impossible for the leader to leave such a residual demand for thefollower,sinceitsper-unitpricewillbeuniformlyappliedtoallunits. Instead of undercutting and seizing the whole market, the rational follower is induced to accommodate the leader's limited supply and serve the residual demand of the marketatahigherprice. Inthisway,a3PTresolvesBertrandparadox. 8 For discussion of rst-mover and second-mover advantage in Stackelberg model, see Gal-Or (1985)[16],AmirandStepanova(2006)[4]. 10 More interestingly, the basic idea of utilizing a 3PT as a credible commit- ment to induce the competition toward the leader's interests generalizes to any degrees of substitutability, but with a new twist on competitionnow the de- greeofproductdifferentiationinteractingwiththesecond-moveradvantageplays a key role in the switch of a 3PT's competitive effects. Due to the presence of a xedfee,a2PTtendstoreducetheper-unitpriceinordertoextractsurplusmore efciently. This downward pressure on the per-unit price will make the compe- tition more severe and thus improve the total surplus all the time. Nonetheless, this ercer competition may not be in the best interests of the leader, especially when the second-mover advantage is very strong, as explained in the case of ho- mogeneous products above. For this reason, the new element from a 3PTa quantitythresholdcanmitigatethesecond-moveradvantageandthusimprove over a 2PT under competition. By breaking its pricing scheme into two blocks, with zero marginal price for the rst block ranging from zero unit to the quan- tity target, and with a positive marginal price afterwards, the leader becomes free from the applying-to-all-units pricing feature of a single per-unit price. This quantity target instrument enriches the leader's strategy set in dictating the com- petition to a specic level. As the intuition from the case of perfect substitutes suggests,whentwoproductsaremoresubstitutable,thesecond-moveradvantage is so strong that the 3PT would be used to restrict competition. Instead, as two products become more and more differentiated, the second-mover advantage is diluted, andthe3PTcanfunctionmoreasasurplusextractiontooltowarda2PT. This can be seen from the other extreme casewhen two products are indepen- 11 dent, the 3PT equilibrium outcome will converge to the 2PT's efcient surplus extraction equilibrium outcome. From the continuity, we know that a 3PT must switch from a competition-softening tool to a competition-intensifying device as thedegreeofproductdifferentiationincreases. Theremainderofthechapterisorganizedasfollows. InSection2,Isetupthe model and describe the game. Section 3 presents and analyzes the perfect substi- tutesmodel. Section4extendstheanalysistotheimperfectsubstitutesmodel. In Section5,Iperformcomparativestaticsanddiscusspropertiesoftheequilibrium under all three pricing regimes, as well as their implications on competition. The chapter closesin Section6 with someconcluding remarks anddiscussions onfu- turedirectionsofthislineofresearch. AllproofsarerelegatedtotheAppendices. 2 TheModel In this section, I set up a model to analyze a 3PT in a competitive environment, andIdescribehowthegameproceeds. Generally, a 3PT consists of a triple (T o ;Q o ;w), whereT o is the xed fee for the right to stock supplier's product, Q o is the quantity threshold within which it is free of charge, and w is the per-unit price for the quantities exceeding the thresholdQ o . Specically,the3PTtotalpaymentscheduleis T(q) = 8 > < > : T o ifq <Q o T o +w(qQ o ) ifqQ o : 12 AnditisillustratedinFigureII.1below. FigureII.1: 3PTTotalPaymentSchedule Note that whenQ o = 0, the 3PT is reduced to the classical 2PT. Furthermore, if wehaveT o =Q o = 0,thenitbecomesauniformlinearpriceschedule. Themodelconsistsoftwoclassesofagents. First,twomanufacturersAandB arelocatedintheupstreammarketandproducesubstituteproductswiththesame marginal cost c. In this set-up, the two products manufactured by A and B are allowed to be general substitutes, including both cases of homogeneous products and differentiated products. Second, there are a large number of retailers, each of whom is a local monopoly in selling to nal consumers. 9 I assume complete information about demands in every retailing market here, and two manufactur- ers make customized offers to each local monopoly retailer. Hence, it is without lossofgeneralitytoconsiderarepresentativeretailerR inthedownstreamwitha 9 For the justication and detailed discussions on this assumption, see Mathewson and Winter (1987)[24]. 13 retailing revenue function denoted as R(q A ;q B ). For simplicity, we assume that retailers have no cost other than the wholesale prices charged by the manufactur- ers. The game is a sequential-move game involving three stages. At date 1, man- ufacturer A offers a contract to the retailer. The contract is, in general, a 3PT contract(T o ;Q o ;w A ). AfterobservingthecontractofferfrommanufacturerA,at date2,manufacturerBsetsitsper-unitwholesalepricew B fortheretailer. Aswill bediscussedinSection6,themainresultsstillholdwhenweallowmanufacturer Btooffera3PT.HerewerestrictthemodeltolinearpricingfrommanufacturerB onlyforthepurposeofillustratingtheinsightsinaconciseway. Atdate3,there- tailerRdecideswhatandfromwhomtobuy. Rememberthattherearemanylocal monopoly retailers here, but only two manufacturers. Therefore, it is reasonable to assume that those retailers have no monopsony power, and hence they take the bestcontractofferedinthewholesalemarket,ratherthansettingthecontracttobe just acceptable to the manufacturers. The game's timeline is described below, in FigureII.2. FigureII.2: Game'sTimeline Sincemyobjectivehereistoseeifa3PTcanhaveanystrategiceffectspurely coming from competition, I want to rule out any other motives as best as I can. The complete information assumption in the model prevents price discrimination 14 from being a plausible explanation. In addition, the local monopoly retailer as- sumption helps us abstract away from strategic interactions resulting from down- stream competition. As will be illustrated soon, even in this simple framework, a 3PThassomebiteoncompetitioninastrikinglydifferentwayfromLPora2PT. AndinSection6,Icommentoncomplicatingthebasicmodelbyintroducingcost differentialanddemanduncertainty. Forthetimingofthegame,inpractice,theremustbeonermthatmovesrst, and the number of moves is small under oligopoly 10 ; in theory, more sequential plays involve more transaction costs and the possibility of renegotiation could dissipateprots. Moreover,ascanbeseenintheanalysislater,a3PTwouldhave nobitewithsimultaneousmoves,instead. Italsomustbethecasethatrmswant toresorttoa3PTonlywhentheycanpairthispricingwithaStackelberggame. Inthissetting,Ideterminesubgameperfectequilibriumoutcomesandanalyze the equilibrium as the degree of product differentiation varies. In the following study, I divide the analysis into two categoriesthe case of perfect substitutes, andthecaseofimperfectsubstitutes. 11 10 Forinstance,historically,GMisusuallyacknowledgedasapriceleaderandChrysleriswidely recognizedasafollower(SeeGMtoIncrease'74-ModelPricesAnAverage$97onWallStreet Journal(Sep. 13th,1973)). Moreover, the leader doesn't change the price that often once it is settled. The latest Chrysler priceboostsfollowedincreasesbyGM,theindustry'straditionalpriceleader,andAmericanMo- tors Corp. No. 2 Ford Motor Co. has said it doesn't expect any further price increases in the remaining three months of the 1980-model year (See Chrysler, Following Other Auto Firms, RaisesPrices2.2%onWallStreetJournal(Jul. 10th,1980)). 11 Ascanbeseenbelow,theybothhavethesameavorinequilibriumcharacterization,although thelatterismoretechnicallyinvolved. 15 3 TheCaseofPerfectSubstitutes I begin with the case of perfect substitutes. It illustrates the basic idea of how a 3PT affects competition in a simple setting, and thus helps us understand the distinctionsbetweena3PTandLPora2PT. When two products are homogeneous, the retailing revenue function depends only on the sum of the quantities. That is, R(q A ;q B ) = R(q A +q B ). And we denote the optimal quantity demanded by the retailer when he buys at per-unit price w as q m (w) argmax x0 [R(x) w x]. Following that, I make these standardassumptions. AssumptionII.1 (MonotonicityandConcavity)R 0 (q)> 0;R 00 (q)< 0: AssumptionII.2 (EfciencyRequiresPositiveSales)q m (c)> 0: As a benchmark, we rst look at the situation in which the leading rm can onlyofferalinearpriceora2PT,wheretheBertrandparadoxprevails. Afterthat, we will see how the dominant rm can gain strictly positive prot through a 3PT incaseswhereneitherLPnora2PTcouldwork. In the case of homogeneous products, no matter whether the leader adopts LP or a 2PT, a uniform per-unit price will be applied to all units purchased from him. Let v(w) max x0 [R(x) w x] be the retailer's prot under a per- unit pricew. Denotew A andw B as the per-unit price from manufacturers A and B, respectively, and T and T o as the xed fees for a 2PT and a 3PT. Since two productsare identical, the retailerwillbuy all ofits demandfromsupplieri only, 16 with surplus as maxfv(w A );v(w B )g under LP, or as maxfv(w A ) T;v(w B )g under a 2PT. It is this very all-or-nothing feature of the retailer's decision that drives the follower to always undercut the leader's offer slightly and capture the whole market, as long as the per-unit price is above cost. Consequently, when two products are perfect substitutes, no matter whether it is LP or a 2PT adopted bytheleader,theequilibriumofthisStackelberggameisthatbothmanufacturers earnzeroprotfrommarginalcostpricing. PropositionII.1 (LPand2PTEquilibriumfortheCaseofHomogeneousProd- ucts)Whentwoproductsareperfectsubstitutes,theLPand2PTequilibriumout- comesarethesameasthesimultaneousBertrandoutcomebothmanufacturers setpricesatmarginalcostandearnzeroprots. Asexplainedabove,thisBertrandparadoxisadirectresultfromthe"applying- to-all-units"featureofasingleper-unitpriceinLPora2PT.Comparedwitheither linear pricew A or a 2PT (T;w A ), the distinct element of a 3PT (T o ;Q o ;w o ) is its quantitythresholdQ o . Theintroductionofsuchaquantitythresholdmakessetting twodifferentpricesfordifferentquantityrangespossible,asthemarginalpriceis zeroforquantitywithinthethresholdandpositiveforthatexceedingit. Asshown in the next proposition, with a 3PT, both manufacturers set above-cost prices and earnstrictlypositiveprots. ThisisinstarkcontrastwithLPor2PTequilibrium. Before presenting our rst main result, Proposition II.2, I make an additional assumption ensuring the existence and uniqueness of the equilibrium. Denote the manufacturer's prot as being a monopoly supplier when charging a per-unit 17 price w only as m (w) (w c)q m (w), and the optimal monopoly price as w m argmax w m (w). AssumptionII.3 (Single-Peakedness)Assumeh(w) (wc) m0 (w)issingle- peakedin[c;w m ]. This assumption says h(w) has a unique peak in the range between cost and monopoly price. This single-peakedness assumption is rather mild. As can be easily checked, both general linear demands and general CES demands satisfy thisassumption. With this singled-peakedness assumption, we can show that in the uniquely determined3PTequilibrium,theBertrandparadoxisresolved. PropositionII.2 (3PT Equilibrium for the Case of Homogeneous Products) When two products are perfect substitutes, under Assumption II.3, 3PT equilib- rium(T o ;Q o ;w A ;w B )existsandisuniquelycharacterizedby T o = b w B Q o (1) w B w A (2) Q o = m0 (w B ); (3) where (w B ;b w B ) = argmax (x;y) 8 > < > : (yc) m0 (x) s:t: m (y) = m (x)(xc) m0 (x) 9 > = > ; : (4) 18 In this equilibrium, both manufacturers earn strictly positive prots in equi- librium. Recall that in either LP or 2PT model, the uniform per-unit price applied to every unit from each manufacturer leads to an all-or-nothing purchase decision from the retailer, which provokes the undercutting from the follower. Since two products are identical here, the only way for the leader to earn positive prot is to avoid the undercutting from the follower, which is impossible under LP or a 2PT. However, with the quantity threshold Q o and per-unit price for incremental demandw A ,theleadernowcancrediblycommititselftothelevelofsupplyatQ o . Inequilibrium,thisisrealizedbysettingitsper-unitpriceforincrementaldemand w A higher than manufacturer B's per-unit price w B , as indicated by inequality (2) above. Facing such a 3PT, the follower has two optionsundercutting and seizing the whole market with a lower price, or accommodating and serving the residualdemandatahigherprice. Sincetheleaderwillhavenosaleifthefollower undercuts its offer, a leader will choose Q o and T o appropriately to induce an accommodation from a rational follower. This is represented by constraint in (4) in Proposition II.2, where the left hand side is the maximal monopoly prot from undercutting while the right hand side is the maximal prot from supplying residualdemand. Formally, this is a sequential-move game with complete information, we can solve the game by backward induction. It also turns out that the determination of the leader's optimal 3PT can eventually be reduced to a mechanism design problem. In particular, by carefully choosing the quantity threshold along with 19 theper-unitpriceforoverage,theleadingrmcancrediblycommittothemarket coverageatwhichitwantstoserve,andleavesufcientresidualdemandandprot tothefollower,whichpreventsitfromundercuttingandtriggeringapricewar. Here I only sketch the proof, leaving the complete analysis to the Appendix II.A. I rst examine the monopoly retailer's problem in the last stage of the game. Withtwooffersonthetable,theretailerhastwooptions(AA):acceptA's3PT, or (NA): reject A's 3PT. Let p(q) R 0 (q) denote the marginal price implied by R(q). We next write the highest price manufacturer B can charge for a positive sale when the retailer accepts manufacturer A's 3PT as w e A minfw A ;p(Q o )g. The residual demand for manufacturer B's products after buying Q o units from A is then written as q r (w;Q o ) = argmax y0 [R(Q o + y) w y] = 1fw < p(Q o )g [q m (w) Q o ]. Under (AA), the manufacturer B must set w B < w e A in order to have a positive sale. The retailer's optimal purchase decision can be summarizedbelow,inFigureII.3. FigureII.3: TheRetailer'sOptimalPurchaseDecisionin(AA) 20 Afterexploitingthepropertiesoftheretailer'sprotcurvesunder(AA)and(NA), wecanprovethat,inequilibrium,theremustexistauniquecutoff b w B = To Qo such thattheretailerwillrejectA'sofferandbuyexclusivelyfromBifw B < b w B ,while acceptingA's3PTifw B b w B . ThisbasicallytellsusthatmanufacturerBcanchoosetobeeitheramonopoly supplierandearn m (w B )bysettingitsw B belowthecutoff,oraresidualdemand supplierandearnresidualprot r (w B ;Q o ) (w B c)q r (w B ;Q o )bysettingits w B above the cutoff. From the properties of m (w B ) and r (w B ;Q o ), we know thatthereisadiscontinuousdropat b w B inmanufacturerB'sprotcurve,whichis shownbelowastheredlineinFigureII.4. FigureII.4: ManufacturerB'sProtCurveinEquilibrium. Fromitsprotcurve,wecaneasilyseethetrade-offmanufacturerBfaces: Under- cuttingA'saveragepriceb w B = To Qo forQ o unitscanallowBtograbtheQ o demand andmakeitselfamonopolysupplier,butatalowerprice;accommodatingallows B to charge a higher price b w B w B < w e A , but at the cost of leavingQ o units to manufacturerA.Letw B bemanufacturerB'soptimalpriceinequilibrium. 21 WenowturntomanufacturerA'schoiceofa3PT.ManufacturerA'sprotis A = 8 > > > > < > > > > : 0 ifw B < b w B (b w B c)Q o if b w B w B <w e A b w B Q o w e A Q o +(w e A c)q m (w e A ) ifw e A w B : WeknowthatmanufacturerBwouldneverchoosew e A w B ;sinceitwouldearn zero in that case. Thus, for possible positive prot, manufacturer A must ensure b w B w B < w e A . And this is equivalent to r (w B ;Q o ) max x<b w B m (x), which says being a residual demand supplier is at least as protable as being a undercuttingmonopoly. SomanufacturerA'sproblemcanbewrittenas max To;Qo;w A (b w B c)Q o s:t: r (w B ;Q o ) max x<b w B m (x) (5) w B = argmax b w B x<w e A r (x;Q o ) (6) b w B = T o Q o : (7) Note that the whole game now is reduced to a mechanism design problem frommanufacturerA'spointofview. Constraint(5)isequivalenttoanincentive- compatibility constraint in the standard mechanism design problem. Constraint (6) is the denition ofw B , and constraint (7) is the characterization of the cross- ingpointconditionfromretailer'soptimalchoice. Attheoptimum,theincentive- compatibility constraint holds with equality. By substituting the constraint into 22 manufacturerA'sprotfunction,itisequivalentformanufacturerAtoset(w B ;b w B ); andtheproblemischaracterizedbytheprogram(4)inPropositionII.2. By restricting its own supply toQ o , it makes accommodation more protable for the follower than undercutting for the follower. First, a small average price b w B = To Qo for Q o units from A makes the cost of undercutting large. Second, a small Q o limits the loss of demand from accommodation. These two forces resolvetheBertrandparadoxandsoftencompetition. An immediate conclusion from the characterization of the equilibrium is that thesecond-moveradvantageissostrongthatthesecond-moverearnsmoreprot. CorollaryII.1 When two products are perfect substitutes, in 3PT equilibrium, thefollowerearnsmoreprotthantheleaderdoes. Notice that with homogeneous products case, even with the 3PTs lessening competition, the second-mover advantage is still strong enough that the follower earnsmoreprotthantheleader. LinearDemandExampleForgenerallineardemandq(w) =w gener- ated fromR(q) = q( 1 2 q) with costc, denote the term standing for efciency levelasc,theequilibriumisb w B = 2 (1 p 2 2 )+c;w B = 2 p 2 2 +c;Q o = (1 p 2 2 );q r (w B ;Q o ) = p 2 4 ;T o = [ 2 (1 p 2 2 )+c](1 p 2 2 ); and manufacturers'protsare A = 2 2 32 p 2 2 ; B = 2 2 1 4 : This linear demand example conrms our corollary above that the follower earnsmoreprotthantheleaderdoes. Moreover,noticethattheaveragepricefor Q o unitsfrommanufacturerA b w B isstrictlylowerthanw B thelowerboundof 23 its per-unit price for incremental demand. This reects that the equilibrium 3PT involves a quantity premium, such that it would cost more per unit for purchases exceedingQ o unitsthanthosewithinQ o units. 4 TheCaseofImperfectSubstitutes Inowextend themodeltothe caseinwhichtwo competingproductsfromAand Bareimperfectsubstitutes. Thisisanimportantcasetoconsiderbecauseproduct differentiation is a feature of real life markets. 12 In addition, the homogeneity of themanufacturers'productsiscrucialfortheBertrandparadoxinthelastsection. One may wonder how general our results on a 3PT can be when products are differentiated. As we shall see, with any degree of product substitutability, a 3PT always helps the leader in terms of quantity sale and prot, but it may not soften competition and benet the rival rm, as in the case of homogeneous products. Rather, a 3PT could intensify competition and have some exclusionary effects on the rival as two manufacturers' products become less substitutable. I will rst characterizethe3PTequilibriuminthissection,andthenIwillexploreitswelfare effectsinthenextsection. With product differentiation, now the retailing revenue function is a general functionR(q A ;q B ). Writetheretailer'sprotwhenfacingper-unitprices(w A ;w B ) asv(w A ;w B ) max x0;y0 [R(x;y)w A xw B y],anditsoptimalquantities 12 (I)t is evident that virtually all products are differentiated, at least slightly, and that, over a wide range of economic activity, differentiation is of considerable importance (See Chamberlin, 1962[11]). 24 as(q A (w A ;w B );q B (w A ;w B )) argmax x0;y0 [R(x;y)w A xw B y]. Cor- responding to the two assumptions in the case of homogeneous products, I make anothertwoassumptionsparallelhere. AssumptionII.4 (MonotonicityandDiagonalDominance)R 1 > 0;R 2 > 0;R 11 < R 12 < 0;R 22 <R 21 < 0: AssumptionII.5 (EfciencyRequiresCarryingBothProducts)q i (c;c)> 0;i = A;B. AssumptionII.4saystheretailingrevenuefunctionisincreasingineachargu- ment, and the own effect on marginal revenue is larger than the cross effect on it. Thisguaranteesthestrictquasiconcavity. AndAssumptionII.5ensuresthatthere isnoexclusioninefciency. As before, we begin with the benchmark case where the leader can only offer a linear price. Next, we look at the 2PT equilibrium as an improvement over LP, andafterthat,wecharacterizethe3PTequilibriumasafurtherimprovementover a2PT. When two products are differentiated, the retailer does not simply buy the product from the cheaper source as in the case of homogeneous products. We use i (w A ;w B ) (w i c)q i (w A ;w B ), i = A;B to denote the manufacturer i's prot from a per-unit price charge under (w A ;w B ). Under LP, the retailer will decide whether to carry both products and earn v(w A ;w B ) or buy exclusively frommanufacturerBandearnv(1;w B ). Itiseasytoseethattheformerweakly dominatesthelatter. Andtheequilibriumisasfollow: 25 PropositionII.3 (LPEquilibriumfortheCaseofDifferentiatedProducts)When twoproductsareimperfectsubstitutes,theLPequilibriumis(w LP A ;w LP B ),where (i)w LP B =B(w LP A ) = argmax w B (w LP A ;w); (ii)w LP A = argmax w A (w;B(w)): This is the classical Stackelberg price-setting equilibrium, withB(w A ) as the follower'soptimalresponsefunctionforanygivenw A . ItcanbefoundinGal-Or (1985)[16]. CorollaryII.2 (Sequential-moveHelpsBothManufacturersOverSimultaneous- move) LP B > LP A > Simultaneous A = Simultaneous B . First, in such a price-setting game, the follower enjoys second-mover advan- tage and earns more prot than the leader. Second, even if the leader earns less than the follower, both manufacturers are still better off than that in the simultaneous-move case. Consequently, if we allow manufacturers choose to movesimultaneouslyorsequentially,thentheybothwillagreewiththelatterone, althoughbothofthempreferhimselftobethefollower. When 2PT is available for the leading rm, the xed feeT can now help the leading rm to extract more surplus. Now, the retailer will earnv(w A ;w B )T o if buying both, but v(1;w B ) if buying from B only. Then, parallel to the rea- soning process in the case of homogeneous products, we can show that, in equi- librium, they must cross each other only once at b w 2PT B and T = v(w A ;b w 2PT B ) v(1;b w 2PT B ). Inthatcase,theequilibriumisasfollows. 26 PropositionII.4 (2PTEquilibriumfortheCaseofDifferentiatedProducts)When twoproductsareimperfectsubstitutes,the2PTequilibriumis((T 2PT ;w 2PT A );w 2PT B ), where (i)w 2PT B =B(w 2PT A ) = argmax w B (w 2PT A ;w); (ii)(w 2PT A ;b w 2PT B ) = argmax w;b w 8 > < > : A (w;B(w))+v(w;b w)v(1;b w) s:t: B (w;B(w)) = m B (b w) 9 > = > ; ; (iii)T 2PT =v(w 2PT A ;b w 2PT B )v(1;b w 2PT B ): Inthisequilibrium,theleaderearnshigherprotthanthatunderLP. The xed fee T in a 2PT now provides an extra channel to extract surplus, increasing the leader's prot over that from the LP case. Moreover, comparing the objective function under LP and a 2PT, due to the presence of the extra term under a 2PT, the xed fee v(w 2PT A ;b w 2PT B ) v(1;b w 2PT B ), there is a downward pressure on the per-unit price, since it is more efcient to extract surplus through this xed fee rather than through the per-unit price. Therefore,w 2PT A < w LP A and wehavew 2PT B =B(w 2PT A )<B(w LP A ) =w LP B . Oneimmediateimplicationisthat manufacturer B earns less prot now. In addition, since the per-unit prices from both manufacturers are lower than those in the LP case, we must have a higher totalsurplussincewhatmattersfortheefciencyistheper-unitprices. CorollaryII.3 Whentwoproductsareimperfectsubstitutes, (i) 2PT A > LP A ; 2PT B < 2PT B : (ii)TS 2PT >TS LP : Parallel to the case of homogeneous products, we will see how a 3PT can furtherimprovetheleader'sprotoverLPora2PT. 27 We dene the monopoly quantity faced by manufacturer B as q m B (w B ) argmax y0 [R(0;y)w B y],andsimilarlyforq m A (w A ). IfmanufacturerBisthe solesupplierfortheretailer,itwillearnmonopolyprot m B (w) (wc)q m B (w). Denote the residual demand for manufacturer B's products after buyingQ o units fromAasq r B (w B ;Q o ) argmax y0 [R(Q o ;y)w B y],andwriteitsprotfrom beingaresidualdemandsupplieras r B (w;Q o ) (wc)q B (w B ;Q o ). PropositionII.5 (3PTEquilibriumfortheCaseofDifferentiatedProducts)When twoproductsareimperfectsubstitutes,3PTequilibrium(T o ;Q o ;w A ;w B )ischar- acterizedby T o = max y [R(Q o ;y) b w B y]v(1;b w B ) (8) w A R 1 (Q o ;q r B (w B ;Q o ))w A (9) r0 B (w B ;Q o ) = 0; (10) where (Q o ;b w B ) = argmax q;w 8 > < > : max y [R(q;y)wy]v(1;w)cq s:t: r B (w B ;q) = m B (w) 9 > = > ; : (11) Inthisequilibrium,theleaderearnshigherprotthanthatunderLPora2PT. Proposition II.5 and Proposition II.2 are parallel. Actually, the basic idea of utilizing the quantity target as a commitment to set the tone for competition by specifying the supply level can still be applied to the case of differentiated prod- 28 ucts. From manufacturer A's objective function in program (11), we can see that, in equilibrium, it is in manufacturer A's best interest to supply exactly Q o units. This advantage is unique with a 3PT because the quantity target is missing in LP ora2PT.Thereasonwhya3PTcanfurtherimprovetheleader'sprotwhentwo products are imperfect substitutes can be understood in the following way: Start- ing from LP, a 2PT can improve manufacturer A's prot by the xed fee, since it is more efcient in surplus extraction than a linear price. In the meantime, this sameinstrumentputsadownwardpressureontheper-unitprice,whichintensies competition. When two products are more homogeneous, the second-mover ad- vantageisstrongenoughthatthe2PTmaynotsignicantlyimprovemanufacturer A'sprot,becausethemoreseverecompetitionoffsetsthebenetsfromthemore instrument. In such a case, it would be better if the leader could soften the com- petition while maintaining the more efcient surplus benets from the xed fee. This is implementable by a 3PT. By directly specifying the level of supply, along with a high per-unit price afterwards (equation (9) indicates a lower bound for w A ),themanufacturerAcandictatetheretailer'sdemandforitsproduct. Rather, under either LP or 2PT equilibrium, the retailer is allowed to freely choose its quantity. It is such the 3PT's quantity forcing power that gives the leader more leverage to control competition. When two products are quite differentiated, the second-mover advantage is diluted, and the 3PT works more as a surplus extrac- tiondeviceinthesamespiritasa2PT.Whentwoproductsareverysubstitutable, thesecond-moveradvantageissignicant,sothe3PTtendstorestrictcompetition inordertomitigatetheaggressiveresponsefromthefollower. 29 As in the case of perfect substitutes, I will only sketch the proof herethe completeanalysisisavailableinAppendixII.B. Priortotheanalysis,letusrststudythepropertiesoftwocurvesR 1 (q A ;q B ) = w A andR 2 (q A ;q B ) = w B inq A q B plane. In order to describe two curves, we use(w A ;Q o ) to denote the intercept of the curveR 1 (q A ;q B ) = w A onq A = Q o when R 1 (Q o ;0) > w A . That is, R 1 (Q o ;) = w A . And we write the intercept of the curve R 2 (q A ;q B ) = w B on q B = 0 when R 2 (Q o ;0) > w B by (w B ) That is, R 2 (;0) = w B . It can be shown that in q A q B plane, for any given (w A ;w B ), both curves R 1 (q A ;q B ) = w A and R 2 (q A ;q B ) = w B are downward sloping. Moreover,theslopeofR 1 (q A ;q B ) =w A isalwayssteeperthantheslope ofR 2 (q A ;q B ) = w B . The two typical curves are shown with notations below, in FigureII.5. FigureII.5: TheTypicalTwoCurvesR 1 (q A ;q B ) =w A andR 2 (q A ;q B ) =w B in q A q B Plane,withNotationsfortheCaseofImperfectSubstitutes. 30 Werstlookatthelaststageofthegame. Paralleltothecaseofhomogeneous products, the retailer has the same two options(AA): accept A's offer, and (NA): reject A's offer. Let p i (q A ;q B ) R i (q A ;q B )(i = A;B) be the marginal price for product i implied by R(q A ;q B ). But now, the highest price manufac- turer B can charge for a positive sale under (AA) is no longer w e A , because the two products are differentiated, and B does not need to undercut below w e A for a sale. Instead, it is w B minfR 2 (q m A (w A );0);R 2 (Q o ;0)g = p B (q m A (w e A );0). Under (AA), manufacturer B must set w B < w B in order to have a positive sale. Moreover, under (AA), when the retailer buys from both manufacturers, there is the extra possibility that it will carry both products even if the quan- tity requirement is not binding. In particular, now it is possible that demands (q A (w A ;w B );q B (w A ;w B )) both could be positive. Instead, when two products arehomogeneousandtheretailerisnotrestrictedbythequantitytarget,itwillbuy exclusively from a sole supplier and thus only one of (q A (w A ;w B );q B (w A ;w B )) could be positive. The retailer's optimal purchase decision can be summarized in Figure II.6 below. Recall that in Figure II.3, the divergent red line and blue line here,beforereaching(p A (Q o ;0);p B (Q o ;o)),coincideonthe45 o linethere. 31 FigureII.6: TheRetailer'sOptimalPurchaseDecisionin(AA) With the properties of the retailer's prot curves under (AA) and (NA), we can provethat,inequilibrium,theremustexistauniquecutoffb w B suchthattheretailer will reject A's offer and buy exclusively from B ifw B < b w B , but accept A's 3PT ifw B b w B . And b w B isdeterminedbyv(1;b w B ) = max y [R(Q o ;y)b w B y]T o . Whentwoproductsarehomogeneous,thisisreducedto b w B = To Qo . Althoughitistechnicallymorecomplicatedbecausemanymorepossiblecases mustbeconsidered,inthesamevein,thewholeproblemcanbetransformedinto a mechanism design problem from the perspective of manufacturer A. The con- straint in (11) is essentially an incentive-compatibility constraint, saying that ac- commodation is at least as protable as undercutting. Also, equation (10) is the rst order condition showing that manufacturer B's optimal price is an interior solution on the residual prot curve. From equation (8), the xed fee is used to extract all the incremental surplus from carrying product A over buying B only. Compared with a 2PT, the surplus extraction function from a xed fee is main- 32 tained in a 3PT. Further, instead of allowing the retailer to self-select its demand accordingtotheapplying-to-all-unitsper-unitprice,a3PTcancontrolcompeti- tionmore accuratelybysettingitsquantity leveltosupply. Throughthisquantity forcing style contract, manufacturer A can increase its prot even further over a 2PT.Aswillbeshowninthenextsection,thisincreasedauthorityoverthequan- titytargetistwo-fold. Whenthesecond-moveradvantageenjoyedbythefollower issignicant,itcansoftencompetitionbyrestrictingitssupply;whenproductdif- ferentiationbecomeslargerinoffsettingthesecond-moveradvantage,itfunctions moreasasurplusextractiondevice. 5 ComparativeStatics In the sections above, I have characterized the 3PT equilibrium for general re- tailingrevenuefunctionR(q A ;q B ), andIhaveshownthatitimprovestheleading rm's prot over either LP or 2PT equilibrium. To illustrate my analysis above andgainmoreinsightsonhowtheproductdifferentiationaffectstheequilibrium, inthissection,Iperformthecomparativestaticstoinvestigatetheeffectsofprod- uctdifferentiationonequilibrium. Weconsiderageneraldifferentiatedlineardemandsystem,whichisgenerated bytheretailingrevenuefunction R(q A ;q B ) =(q A +q B ) 1 2 (q 2 A +q 2 B +2q A q B ); 33 where > 0, 1 0. The parameter measures the degree of substitution between products A and B. The larger is, the more substitutable two products are. When = 1,twoproductsarehomogeneous. When = 0,twoproductsare independent. When1> > 0,twoproductsareimperfectsubstitutes. WithR(q A ;q B ) = (q A +q B ) 1 2 (q 2 A +q 2 B +2q A q B ), now we can directly apply Propositions II.3, II.4 and II.5 to compute the corresponding equilibrium. TheresultsarelistedinAppendixII.C. A very nice property from those computed equilibria is that all the surplus functions (i.e. producer surpluses and total surpluses) can be written as products ofatermstandingforefciencylevel,say(c) 2 ,andatermonlydependingon the degree of substitution parameter . Utilizing this property, we can compare allthesesurpluseswithoutworryingabouttheefciencylevel(c) 2 ,whichcan be cancelled out when comparing. As will be shown below, all the cutoffs in this model can be uniquely identied based only on, and they will not change with theefciencylevel(c) 2 . 13 ManufacturerA'sProt AsprovedinPropositionII.5inthelastsection,wehave 3PT A > 2PT A > LP A for any 0 < < 1, with 3PT A = 2PT A only when = 0, and 2PT A = LP A only when = 1. And this is conrmed by general differentiated linear demand systemhere,inFigureII.7. 13 Consequently,wecandroptheefciencylevelterm(c) 2 whencomparingthesesurpluses' relative magnitude. All the surplus functions shown in the gures below are only a function of aftercancellingout(c) 2 . 34 FigureII.7: ManufacturerA'sProtunderLP,2PTand3PTEquilibrium One point worth noticing is that, when two products are less substitutable, a 2PTmimicsa3PTquitewellintermsofmanufacturerA'sprot. Thisisbecause, whentwoproductsaremoredifferentiated,both3PTand2PTwillfunctionmore asasurplusextractiontool. Attheextremecasewheretwoproductsareindepen- dent, they will converge to the same outcomethe full surplus from market A can be extracted by either pricing scheme. However, they achieve the same out- come through distinctive ways. A 3PT extracts surplus through its quantity forc- ingpowerbydictatingitssupplyleveldirectly. Instead,a2PTleavesthequantity choicetothebuyerandextractssurplusbyitsxedfeealongwithaper-unitprice. ManufacturerA's3PT The equilibrium manufacturer A's quantity sales under three regimes, and xedfeesunder2PTand3PTequilibriumareshowninFigureII.8. 35 (i)ManufacturerA'sQuantitySale (ii)ManufacturerA'sFixedFee underLP,2PTand3PTEquilibrium under2PTand3PTEquilibrium FigureII.8: ManufacturerA'sTariffs PropositionII.6 (i)QuantitySales: (a) q 2PT A > maxfQ o ;q LP A g,withq 2PT A =Q o >q LP A when = 0; (b) There exists a unique 1 = 0:91 such that Q o > q LP A when < 1 ; and Q o <q LP A when > 1 . (ii)FixedFees: T 3PT >T 2PT : First of all, manufacturer A has the largest quantity sale under 2PT equilib- rium. As we will see soon, this is one aspect consistent with our conclusion later that a 2PT always enhances competition. The applying-to-all-units per- unit price will be reduced under a 2PT to encourage more purchases, since the xed fee is more efcient in surplus extraction. In the meantime, such a lowered 36 per-unit price will stimulate more aggressive response from the follower. Hence, a 2PT will strengthen competition and increase supply from both manufacturers. Because of this, under 3PT equilibrium, the leader will have incentive to restrict itssupplyinordertoinduceamorefavorableresponsefromthefollowerthanthat in 2PT equilibrium. This explains whyq 2PT A > Q o . And only when two products areindependent,both3PTand2PTachievethesameefcientlevelofsupplyand fullsurplusextractionoutcome. Thepart(b)of(i)compares3PTequilibriumwithLPequilibrium. Whentwo products are homogeneous, it is more important for the leader to limit its supply, through committing to a small Q o ; to avoid a price war. When two products are more differentiated, the surplus extraction is more feasible since the second- moveradvantagebecomesweaker. AnditismoreprotableformanufacturerAto extract surplus through the xed fee by offering more free allowance. Therefore, wecanseethefunctionalswitchofa3PTfromacompetition-softeningdeviceto asurplusextractiontool. Part(ii)directlyfollowsfromthefactthatmanufacturerAcanearnextraprof- itsonsalesfromthepositiveper-unitprice,ontopofthexedfee. QuantityPremiumorDiscount? Recall that, in the equilibrium 3PT, manufacturer A commits to Q o units of supply by setting its per-unit price for incremental demand w A above the lower boundw A . Hence,settingw A =1isonlyjustoneoptiontoimplementourequi- librium3PT.Inpractice,itisveryraretoobservesucharefusaltodealforbuying more from the manufacturer. Rather, w A is nite almost all the time, although it 37 can be higher or lower than the average price for the rst few units. So we com- pare the lower bound w A with the average price at the free allowance To Qo to see whenthe3PTwillinvolveaquantitypremium,orwhenthe3PTcanbeaquantity discount. The equilibrium manufacturer A's lower bound of w A and the average price To Qo under3PTequilibriumareshowninFigureII.9. FigureII.9: LowerBoundofw A andtheAveragePrice To Qo under3PT Equilibrium PropositionII.7 (Quantity Premium or Discount) There exists a unique 2 = 0:69suchthatw A < To Qo when < 2 ;andw A > To Qo when > 2 . This proposition says, the more substitutable the two products are, the more likely the 3PT will be a quantity premium. The intuition for this is: when two products are quite close substitutes, the strong second-mover advantage imposes anupwardpressureontheleader'sper-unitpriceforincrementaldemand,because 38 otherwise the commitment to limit supply withinQ o units would be less credible andaccommodationwouldbethuslessprotable. Sincequantitypremiumordis- count is easy to observe in practice, this proposition can actually help us identify which function a 3PT performs, and infer the degree of substitution between two productsreversely. ManufacturerB'sProt The equilibrium manufacturer B's prots under three regimes are depicted in FigureII.10. FigureII.10: ManufacturerB'sProtunderLP,2PTand3PTEquilibrium PropositionII.8 (i) 2PT B < minf 3PT B ; LP B g; with 3PT B = 2PT B = LP B only when = 0; (ii)Thereexistsaunique 3 = 0:78suchthat 3PT B < LP B when < 3 ;and 3PT B > LP B when > 3 . Part (i) indicates that a 2PT hurts manufacturer B most. This is interesting because manufacturer B gets hurt when manufacturer A switches from LP to a 39 nonlinear pricinga 2PT, yet becomes better off when manufacturer A moves furthertoamoreornatenonlinearpricinga3PT. The reason for this is that the introduction of a xed fee from a 2PT gives manufacturerAamoreefcientsurplusextractioninstrument. Thismoreefcient surplus-extraction method puts a downward pressure on its per-unit price, which intensies competition against manufacturer B, resulting in a lower prot for B thantheoneinLPequilibrium. In the meantime, under such competitive pressure, manufacturer B will price moreaggressivelyasafollower. Thisisnotinthebestinterestsoftheleader. Un- der a 3PT, the extra toolthe quantity targetactually allows manufacturer A to restrict its supply and mitigate the more aggressive response from the follower when its second-mover advantage is signicant. This can be seen from the polar case,wheretwoproductsarehomogeneous. Thus,thequantitytargetina3PTcan not only extract surplus in a similar vein as a 2PT, but can also be a commitment tomitigatethesecond-moveradvantageifneeded. Thisexplainsbothpart(i)and (ii). Notethat, whentwoproductsarelesssubstitutable( < 3 ), manufacturerB earns less prot under a 3PT than under LP. If there is any sunk cost which lies between 3PT B and LP B (i.e. theareabetweentheredlineandtheblacklineinFig- ureII.10),thenmanufactureBwillbeexcludedundera3PT,whileitwillsurvive underLP.Iwilldiscussthispossibilityinthenextsubsection,whichconcernsthe implicationsonexclusion. 40 ManufacturerB'sQuantitySaleandMarketShare TheequilibriummanufacturerB'squantitysalesunderthreeregimesareshown inFigureII.11. FigureII.11: ManufacturerB'sQuantitySaleandMarketShare underLP,2PTand3PTEquilibrium PropositionII.9 q 3PT B < q 2PT B < q LP B ; with q 3PT B = q 2PT B = q LP B only when = 0;ands 3PT B <s 2PT B <s LP B ;withs 3PT B =s 2PT B onlywhen = 0. This proposition tells us that the more ornate tariff adopted by the leader will hurt manufacturer B more in terms of quantity sale and market share, although it ispossibletoincreasemanufacturerB'sprot,asshowninPropositionII.8. 41 RetailerR'sProt The equilibrium retailer R's prots under three regimes are shown in Figure II.12. FigureII.12: RetailerR'sProtunderLP,2PTand3PTEquilibrium PropositionII.10 (i) 2PT R > 3PT R ; (ii)Thereexistsaunique 4 = 0:37suchthat 2PT R < LP R when < 4 ;and 2PT R > LP R when > 4 . (iii) There exists a unique 5 = 0:44 and 6 = 0:62 such that 3PT R < LP R when < 5 ; 3PT R > LP R when 5 < < 6 ;and 3PT R < LP R when 6 <. Part (i) is consistent with our explanation above that a 2PT, when compared with a 3PT, will always intensify competition. To be more specic, the lowered applying-to-all-unitsper-unitprice,duetotheavailabilityofaxedfee,always stimulates a more aggressive response from manufacturer B, while the lack of a quantity target in a 2PT makes the leader unable to commit to restrict supply 42 in order to mitigate the second-mover advantage enjoyed by the follower. As a result, compared with a 3PT, a 2PT will always strengthen competition. This more severe competition in the upstream market then benets the downstream monopolyretailer. Compared with LP, although a 2PT will push manufacturer B to lower its per-unitpriceoffer,thexedfeecanextractthesurplusfromtheretailermoreef- ciently,andthismayoffsetthecompetitivegainfortheretailerfromthelowered per-unit prices. Part (ii) shows how the two forces balance. When two prod- uctsaremorehomogeneous,thecompetitioneffectdominatesthexedfeeeffect. Nonetheless, when two products are more differentiated, the xed fee extraction willoffsetthecompetitivegainfromreducedprices. The case for a 3PT is more complicated. First of all, it has the feature of a 2PT, in extracting surplus more efciently through the xed fee, as well as the zero marginal price within the quantity threshold, which will provoke more ag- gressiveresponsefromthefollower. Second,itscommitmentpowerofrestricting its supply level by setting the quantity target low helps it to soften the rival's second-mover advantage. Therefore, when two products are quite differentiated, restricting supply becomes secondary because the second-mover advantage is di- luted. A 3PT will then work more as a 2PT and extract surplus more efciently from the retailer by the xed fee. In this way, the retailer is worse off than in LP equilibrium. At the same time, when two products are very close substitutes, it will lessen competition by credibly committing to a limited supply. This lim- ited supply induces the follower to accommodate rather than compete against the 43 leaderharshly. Hence,theretailerisworseoff,too. Notethattheretailergetshurt in these two end cases, but for different reasonsin the former case, its surplus is extracted more by the xed fee; in the latter case, a 3PT harms it by softening competition and preventing the follower from undercutting. Interestingly, in the middle range, when two products are neither too differentiated nor too homoge- neous, the retailer can be better off than under LP equilibrium. This is the range inwhichthexedfeehasnotbeenthatefcientinsurplusextractionbecausetwo products are not that differentiated, but the degree of substitution has not been largeenoughneithertojustifyacompetition-softeningstrategy. Inthiscase,com- petitivegainsfromthefollower'smoreaggressiveresponsedominatethexedfee extractioneffectandmaketheretailerbetteroff. TotalSurplus The equilibrium total surplus under three regimes are summarized in Figure II.13. FigureII.13: TotalSurplusunderLP,2PTand3PTEquilibrium 44 PropositionII.11 (i) TS 2PT > maxfTS 3PT ;TS LP g; with TS 3PT = TS 2PT onlywhen = 0; (ii) There exists a unique 7 = 0:64 such thatTS 3PT > TS LP when < 7 ; andTS 3PT <TS LP when > 7 . Part (i) tells us a 2PT gives highest efciency. The introduction of a xed fee has two effects reinforcing the efciency. First, the xed fee is a more efcient way of extracting surplus than a per-unit price. Second, the per-unit price from manufacturer A will be reduced because it is more favorable to encourage more purchaseandthenextractprotthroughthexedfee. Thisloweredper-unitprice willmakethefollowerpricemoreaggressivelyinresponse. Asaresult,thecom- petition is intensied under a 2PT over LP. Moreover, as can be seen from the characterizationof3PTequilibrium,theoptimal3PTwillbedesignedwithahigh per-unit price for incremental demand, so that the retailer will not buy from it. Thiscompetition-restrictingfeaturefroma3PThighlightsitsinferioritytoa2PT inefciency. ComparedwithLP,a3PTwillresultinlowertotalsurpluswhentwoproducts aremoresubstitutable,asitwillsoftencompetitionduetothesignicantsecond- moveradvantagethen. Thiscorrespondstopart(ii). AnExtension: WhenExclusionisPossible From the perspective of antitrust enforcement, the primary concern regarding a 3PTisitspotentiallyexclusionaryeffectonrivals. Asanalyzedabove,itispossi- 45 blefora3PTtoreducearival'sprot. Inparticular,asshowninPropositionII.8, 3PT B < LP B when < 3 = 0:78. In this subsection, we explore the potential risk of exclusion in more depth by introducing a sunk cost for the rival. In addi- tion, we modify the second stage of the game a little bit by assuming the rival is not present in the market, as in our basic model. Instead, at date 2, the potential entrant has to incur a sunk costF in order to enter the market and compete with the incumbent. Deviating from the traditional entry model, where the incumbent could usually sign a contract with the buyer in the absence of the entrant, I here assume the retailer will purchase only after the entrant's entry decision, as in our basicmodel. 14 Thisrulesoutthelock-inexternalityfromtheexogenouscontract- ing time. So, if we still can see an exclusion without such a lock-in externality, thenthe3PT'sexclusionaryeffectswillonlybestrengthenedwhentheincumbent hasalock-inadvantageinsigningacontractwiththebuyerbeforeentry. In antitrust economics, promoting total surplus is perceived as the nal goal of antitrust law. 15 As suggested in Proposition II.11, total surplus is higher un- der 3PT equilibrium when two products are more differentiated. However, from Proposition II.8, manufacturer B gets hurt at the same time. As can be seen from Figure II.10, this is especially true when is in the middle rangewhen two products are neither too differentiated nor too homogeneous. In other words, the 14 Forstandardentrymodelconsideringexclusionwhentheincumbentsignscontractwithbuy- ers prior to entry, see Aghion and Bolton (1987)[2], Rasmusen, Ramseyer and Wiley (1991)[28], SegalandWhinston(2000)[33]. 15 To an economist the thought of designing antitrust policy to maximize aggregate surplus comes naturally and, indeed, much of the economics literature implicitly has taken this to be the appropriateobjectiveforantitrustpolicy(SeeWhinston(2006)[36](pp. 6-7)). 46 increaseintotalsurplusinourbasicmodelisaresultconditionalonthepresence ofmanufacturerB.Butwithasunkcostforanentry,areductioninthefollower's post-entry protability may deter the entry, which would occur when a 3PT is banned. This entry-deterrent effect may lead to a decrease in total surplus as a resultofeliminatingtheentrant. Inordertoconsiderthewelfarewiththeriskofpotentialexclusionaryeffects, we have to re-evaluate the total surplus, taking into account the potential exclu- sion of the rival. To do so, I assume the sunk cost F the entrant must incur lies in [ 3PT B ; LP B ]. In this range, the entrant will enter under LP equilibrium, not un- der 3PT equilibrium. Hence, the entrant will be excluded when the incumbent adoptsa3PT.Thegameisstillasequential-moveStackelberggame,asintheba- sic model. To be more specic, the incumbent's pricing scheme is set before the entrant'sdecision,andthenitisxed. 16 Taking into account this sunk cost and its resulted exclusion under a 3PT, I re-compute the total surplus below. Note that manufacturer B will not enter in this case, and thus the total surplus from a 3PT with exclusion TS 3PT ED will not include its prot. And for LP equilibrium, since F 2 [ 3PT B ; LP B ], I compute both the lower bound TS LP L and upper bound TS LP U for total surplus under LP, corresponding toF = LP B andF = 3PT B ; respectively. The equilibrium results aredepictedbelow,inFigureII.14. 16 Nalebuff(2004)[26]makesthesameassumption,claimingthat(i)fandentrantcannotjustify entry costs at the prevailing preentry prices, then this is a persuasive argument not to enter the market. 47 FigureII.14: TotalSurplusunder3PTExclusionEquilibrium,andLowerBound andUpperBoundofTotalSurplusunderLPEquilibrium PropositionII.12 When the entrant has to incur a sunk cost F 2 [ 3PT B ; LP B ], thereexistsaunique 8 = 0:54,suchthat (i)TS LP L <TS 3PT ED <TS LP U when0< < 8 . Inthiscase,exclusionoccurs undera3PT,andtheefciencyassociatedwith3PTexclusionequilibriummaybe higher or lower, depending on whetherF is closer to the upper bound of LP B or not. (ii) TS 3PT ED < TS LP L < TS LP U when 8 < < 3 . In this case, exclusion occurs under a 3PT, and the efciency under 3PT exclusion equilibrium is lower thanunderLPequilibrium. (iii) TS 3PT < TS LP when 3 < . In this case, no exclusion occurs, and a 3PTreducesefciencyoverLP. Recall that, with the sunk cost and exclusion, total surplus under 3PT de- creases due to the reduction in retailer's prot which is resulted from the elimi- 48 nation of the potential entrant's product variety, although the sunk cost for entry is saved. Meanwhile, total surplus under LP is also reduced because of the sunk cost incurred by the entrant. Compared with Proposition II.11, where sunk cost is missing and the presence of manufacturer B is assumed, it turns out that total surplus under 3PT decreases more than the reduction of total surplus under LP. Without sunk cost and exclusion, a 3PT increases welfare when < 7 = 0:64. Nevertheless,itisonlypossiblefora3PTtoincreasewelfarewhen < 8 = 0:54 with sunk cost and exclusion. Clearly, with the consideration of sunk cost and its resultant exclusion, the set of welfare-reducing exclusion from a 3PT is enlarged signicantly. Hence, when sunk cost is large and entry is costly, it is likely that a 3PTwillbeabarriertoentryandreducewelfare. 6 Conclusion 3PTshavebeencommonlyusedforalongtime,andnowadays,theyarebecoming evenmorepopularintheinformationindustry. Inintermediate-goodsmarkets,the use of a 3PT as a vertical restraint has become a hotly debated issue in several high-proleantitrustcases,suchasU.S.v. MicrosoftCorp. andAMDv. Intel. The anticompetitive theoryof a 3PTis based onthe notion thatthe quantitythreshold in the 3PT offered by a dominant rm, within which the marginal price is zero, will deprive the rival of the opportunity to reach minimum efcient scale. The existing literature has focused on monopoly models and interpreted the 3PT as a pricediscriminationtool. 49 In this chapter, I investigate the strategic effects of 3PTs under oligopoly and offer an equilibrium theory of 3PTs in a competitive context. I show that, com- pared with LP equilibrium and 2PT equilibrium, a 3PT always strictly increases the leading rm's prot when competing against a rival with substitute products, intheabsenceofusualpricediscriminationmotive. Thedistinctfeatureofa3PT over a standard 2PT is its quantity threshold, which is the key provision being utilized to inducecompetition toward the leading rm'sinterest. Undercomplete information,theper-unitpriceforincrementaldemandissetaboveacertainlevel sothatitwillnotbepurchasedbythebuyerinequilibrium. Inthisway, thelead- ing rm can control its supply level at its own discretion, taking into account the rival'soptimalresponse. Iestablishfurtherthatproductdifferentiationisakeydeterminantofthe3PT's functionandwelfarechange. Undergeneraldifferentiatedlineardemandsystem, I show that the competitive effect of a 3PT in contrast to LP depends on the de- gree of substitutability between productscompetition is intensied when two products are more differentiated, but softened when two products are more sub- stitutable. This is in stark contrast with that of a 2PT, which always enhances competition and gives the highest total surplus of these three pricing schemes. Moreover, the rival rm always gets hurt in both prot and quantity sale when thedominantrmswitchesfromLPtoa2PT,yetitenjoyshigherprotwhenthe dominant rm moves from a 2PT to the more ornate 3PT, although its quantity andmarketsharearedecreasedevenfurther. Myndingsofferanewperspective 50 on 3PTs, which could help antitrust enforcement agencies distinguish the exclu- sionary3PTfromtheprocompetitiveone. Therearesomeotherpricingschemes,whichcouldbeconsideredasvariations of a 3PT. One example is a two-block tariff (See Dolan (1987)[14]). Another re- lated pricing scheme is called all-units discount (See Kolay, Shaffer and Ordover (2004)[22]). Given the common feature of a quantity target in all three of these pricing schemes, as well as the fact that, in practice, almost all the realized pur- chasesunderanyofthemarenearorabovethequantitythreshold,bothtwo-block tariffandall-unitsdiscountcanbeconvertedtoa3PT,asindicatedbytheboldred dashedlinesinFigureII.15below. FigureII.15: Two-BlockTariffandAll-UnitsDiscounts Therefore, our theory here can also be applied to these variations if their conver- sionsto3PTsapproximatetheoriginalcontractswell. 51 Discussion One natural extension is to allow manufacturer B to offer a 3PT as well. This moreornatetariffwouldallowmanufacturerBtoextractmoresurplus,leavingthe retailer worse off. While this possibility is not explicitly analyzed here, we can see the basic idea of this chapter still holds: the leader could use a 3PT to induce competition toward its interests as an improvement over LP or 2PT. Considering thepolarcasewhentwoproductsarehomogeneous,theleaderwillstillearnzero prot with either LP or a 2PT if the follower can offer a 3PT. Now, suppose the leader can offer a 3PT (T A ;Q A ;w A ) as the blue line in Figure II.16 below. As in our basic 3PT vs LP model, the follower has two optionsundercutting or accommodating. Notice that the most protable undercutting for the follower is to match Q A units at T A rst, and then to charge an optimal per-unit price for incremental demand with the restriction of w B < w A . This is shown as the red dashed line in Figure II.16. Instead, when the follower chooses to give up the rst Q A units, it has the exibility to set a new 3PT afterwards. This is shown as the green line there. As can be seen from the gure, it is quite possible that the follower would earn more prot from accommodation (green line) than from undercutting(redline). Thereasonhereisthat,whenundercutting,thecontinuity of the payment schedule restricts the follower's pricing scheme to extract surplus throughalargexedfeeafterQ A units. Thiscanbeseenfromthejumpbetween greenlineandtheredlineatQ A inFigureII.16. 52 FigureII.16: 3PTvs3PT Another extension is to introduce asymmetric cost into our basic model. Re- callthat,inthecaseofhomogeneousproducts,theleaderearnslessprotthanthe follower. FromPropositionII.2,theequilibriumisdeterminedby (w B ;b w B ) = argmax (x;y) (yc) m0 (x) s:t: m (y) = m (x)(xc) m0 (x): Andifweallowc 1 6=c 2 now,then A (c 1 ;c 2 ) = (b w B c 1 ) m0 (w B );and B (c 1 ;c 2 ) = m (b w B ). It is easy to see that A (c 1 ;c 2 ) is decreasing in c 1 and increasing in c 2 ; while reverse is true for B (c 1 ;c 2 ). Also, we know that, when c 1 = c 2 = c, A (c;c) B (c 1 ;c 2 ) when c 2 >> c 1 . From the continuity of these two prot functions in (c 1 ;c 2 ), we know that for any givenc 2 , there must exists a unique cutoff, say(c 2 ), such that A (c 1 ;c 2 ) > B (c 1 ;c 2 ) when c 1 < (c 2 ); and A (c 1 ;c 2 ) (c 2 ). The 53 immediate implication is that when the cost differential is out of a certain range, the leader will earn more prot than the follower and a 3PT can be an effective exclusionarytool. Inaddition,recallthat,inthe3PTequilibriumhere,thebuyerwillnotpurchase at the per-unit price from manufacturer A in equilibrium. This is a result from our complete information assumption. When there is demand uncertainty, it is possible to have realized demand exceeding the quantity threshold, and w A will matterthen. So,wecancomplicatethemodelbyintroducingdemanduncertainty into the current one. In addition, with the uncertainty, the effects of rms' risk aversionareworthfurtherinvestigation. 17 17 Cheng(2002)[? ] showedthatwithcostuncertaintyandriskaverserms,thehighestBertrand equilibrium price is always above the competitive equilibrium price, even when the number of rmsgoestoinnity. Thebasicideaisthatcostuncertaintymayresultinabiglossforundercut- tingtoomuchandthenhavingtomeetallthedemands. Thus,inmy3PTcase,itwillbeinteresting to see if this idea holds, since the quantity threshold with quantity premium can mitigate such a loss. 54 ChapterIII MixedBundlinginTwo-Sided Markets: TheoryandEvidence 18 1 Introduction The practice of mixed bundling consists of selling two or more separate prod- ucts together with a discount, in addition to selling them individually. The most obvious explanation of bundling is efciencybundling can reduce costs and improve quality. However, antitrust authorities are often concerned that bundling may be strategically used by rms which have market power in one market and wanttoleveragethatpowertoanother. StartingwithStigler's(1963)[35]classical "Block-Booking"andfollowedbyAdamsandYellen(1976)[1],strategicreasons of offering a bundle have been extensively studied to date. Although Chicago School's famous "single-monopoly-prot theorem" debunks leverage theory, the post-chicago literature renes leverage theory and identies some circumstances underwhichbundlingcouldbestrategicallyprotable,takingintoaccountChicago School's intellectual argument. Nowadays, two leading explanations of bundling are price discrimination and entry deterrence. For price discrimination 19 , it is the 18 Thischapterisco-authoredwithTimothyP.Derdenger. 19 See Stigler (1963)[35], Adams and Yellen (1976)[1], Schmalensee (1984)[32], McAfee, McMillanandWhinston(1989)[25],andBakosandBrynjolfsson(1999)[7]. 55 heterogeneityinconsumervaluationswhichfrustratesthesellerinitsabilitytoex- tractconsumersurplusthroughoneprice. Thus, bundlingwhichhelpsreducethe dispersioninvaluationswillincreasearm'sprot. Whinston(1990)[37]studies the case when the rm is a monopoly in the primary market and a differentiated duopolyinthesecondarymarket. Heproposesanotherexplanationforbundling changing the market structure by exclusion, and precommitment matters in his model. Nalebuff(2004)[26]advancestheliteraturebyshowingthatbundlingcan effectively deter entry even in the absence of precommitment. And furthermore, an entry deterrent effect is more important than the price discrimination effect in thefaceofcompetition. We extend the literature on mixed bundling as well as two-sided markets by presenting a theoretical monopoly model of mixed bundling in the context of a two-sided(ormulti-sided)market. Atwo-sidedmarketdiffersfroma"traditional" one-sided market (such as those studied above) because it involves two or more end users which interact via an intermediary. Moreover, each end user's partici- pation is determined by the participation of other types of end users. Examples ofsuchmarketsarecreditcards, media, yellowpagephonedirectories, computer operating systems and video game consoles. In this chapter, our focus is directed towards the portable video game console market, where a bundle consists of a game and console sold together for a single price. We elect to focus our study on the video game industry because it is a prototypical two-sided market with con- sumersandgamedevelopersinteractingwitheachotherthroughtheintermediary console. Furthermore,duringaperiodfrommid2001throughlate2006,thereex- 56 istedonlyoneportablevideogameconsolemanufacturer, Nintendo. Withaccess to a new data set which tracked sales and revenue of Nintendo's portable con- soles, all available software and bundles, we are able to determine whether our theoreticalmodelpredictionsholdtodata. Although mixed bundling has been widely studied, as is evident from the above literature, it has yet to be studied in the context of a two-sided (or multi- sided)market. Thisisbecausetwo-sidedmarkettheoryisquiterecentandsparse. 20 Tothebestofourknowledge,onlythreepapersRochetandTirole(2008)[30], Amelio and Jullien (2007)[3] and Choi (2007)[12]have analyzed an extreme form of bundlingtying. Rochet and Tirole (2008)[30] study the payment card industry and illustrate tying can make the pricing structure more balanced and raise social welfare. Amelio and Jullien (2007)[3] explains tying as a coordina- tion tool for a platform to increase participation on both sides yet the effect on competition is ambiguous. Choi (2007)[12] analyzes the effect of tying on two- sidedmarketcompetitionwithmulti-homingandshowsthattyingcanbewelfare enhancing. Our interest in developing a new theoretical model of mixed bundling in a two-sidedmarketsettingalsostemsfromthefactthatthereareafewpeculiardata trends which run contrary to the theory of mixed bundling in one-sided markets. For instance, Adams and Yellen (1976)[1] determine that component prices of the bundled goods should increase when a bundle is offered in order to make 20 See Caillaud and Jullien (2003)[10], Rochet and Tirole (2006)[29], Armstrong (2006)[5], CarrilloandTan(2006)[? ]. 57 the bundle more attractive. The table below presents the predicted price-bundle correlations from Adams and Yellen's analysis next to correlation statistics from our data. Yet, simply looking at the correlation between the console component price and the presence of a bundle illustrates the opposite of Adams and Yellen (A&Y)'sanalysis. TableIII.1: PredictedandObservedCorrelations BetweenBundlingandComponentPrices A&Y Data StandaloneConsolePrice + 0:6076 StandaloneVideoGamePrice + 0:0113 VideoGameRoyaltyRate ? ? Consequently,wedevelopandpresentanewtheoreticalmodelofmixed bundlinginatwo-sidedmarketsetting,whichcaptureandexplainthisinteresting trend. 21 In the traditional bundling literature, the standalone prices should go up un- der bundling in order to make the bundle more attractive. But in our data, the standalone console price goes down with the introduction of bundling. This is 21 Determining the correlation between royalty rate and bundling requires a two-sided markets model as well as assumptions given that data on royalty rate is unobserved. We develop the two- sided markets model and make such assumptions below in the econometric section of the chapter inordertorecovertherelationship. 58 due to the effects of cross-group externalities on bundling. To be more specic, inthepresenceofindirectnetworkeffects, theplatformhasanincentivetolower its standalone prices on consumer side, in order to attract more participation on the game developer side. Since the standalone price for the rst-party game is xedattheinstalledbaseconsumers'commonreservationvalue,andtheinstalled base is locked-in, the platform will only reduce its standalone price on console. Hence,alargernumberofmarginalconsumersareattractedtotheconsole,given thatthebundlepriceislessthanthesumofthecomponentprices. Theincreasein consumers consequently leads to more games being produced on the other side. And hence a further increase in the demand for the console (a consequence of the presence of cross-group externalities), which compensates the platform more for its prot loss from the lowered standalone console price. Such two-way indi- rect network effects reinforce each other, and give the platform more incentive to lowerprices,butstillincreaseitsprotsduetotheincreasedparticipationonboth sides. Additionally, we nd the royalty rate video game developers pay to the plat- form for the right to produce and sell a game declines too. This is also quite a surprising result and is counter to the standard price structure associated with two-sided markets. In standard two-sided markets literature, it is widely known thattheoptimalpricinginvolvescrosssubsidyfromtheinelasticsidetotheelastic side, which is in the same spirit as Ramsey pricing. And in the context of video gameindustry,theconsumersideisusuallyconsideredastheelasticside,andthe game developer side is acknowledged as the inelastic side. Under bundling, we 59 have shown that prices on consumer side are lower due to price discrimination. So according to the cross-subsidization rule, we should expect an inated price on game developer side. However, we see the opposite. The intuition behind this result is nonetheless quite simple. This is due to a different price discrimination effect in the two-sided markets contextit changes the relative elasticities of two sides with respect to participation. In a two-sided markets setting, the offer- ingofabundleenablesconsumerstorevealtheirtruetype. Specically,bundling generates two forms of price discrimination. The rst segments new potential customers into distinct groups, like a traditional mixed bundling case, while the secondisspecictothetwo-sidedmarketssettingandtechnologyindustries. The second form capitalizes on the fact that by offering the bundle the rm can seg- ment consumers into two additional independent groupspotential consumers and the installed base, and set segment specic pricesthe effective game price for potential console consumers (the difference between the bundle price and the standaloneconsoleprice)andthegamepricefortheinstalledbase. Consequently, with the introduction of a bundle, consumers become more inelastic with respect totheirparticipationontheplatformfromthefactthatbundlingcantargetthe consumers more accurately. Such a shift changes the relative elasticity between consumersandgamedevelopersplatformparticipation. Withrelativelymoreelas- tic game developers with respect to participation, the platform is required to shift its relative attention away from consumers to game developers. The platform, consequently,lowersitsroyaltyratetogamedevelopersinordertoattractthemto itsplatform. 60 Furthermore, we nd total surplus increases with mixed bundling. The intro- duction of a mixed bundle not only acts a price discrimination tool but also as a method to better coordinate the participation of consumers and game developers which aids in the solving of "the chicken or the egg" problem of which comes rst. As we mentioned above, Amelio and Jullien (2007)[3] nd that a platform's sole incentive for offering a tie is to coordinate each side of the platform to join, whichthenenablestheplatformtoskewpricesevenmoreheavilytotheinelastic side of the market. We nd in a more generalized form of bundling that this is not the entire story. A rm's incentive is now based upon price discrimination. The rm wants to lock-in as many consumers as possible, and perfectly price discriminate with respect to those who buy the standalone game; and it does so with lower standalone console price to consumers and lower royalty rate to game developers. Consequently, we show unambiguously that platform participation increasesoneachsideofthemarket. After theoretically evaluating the impact mixed bundling has on prices and welfare,wetestthemodelpredictionswithdatafromtheportableconsolemarket intheearlytomid2000s. WeemployareducedformapproachsimilartoJinand Rysman (2010)[19] to do so. With a reduced form methodology we look to see whethertherearecorrelationsinthedatawhichareconsistentwithourtheoretical results. We conclude that the proposed model is correct and all theoretical pre- dictionsareevidentinourdatawendstrongevidenceofnegativecorrelation betweenmixedbundlingandtwo-sidedmarketprices. 61 Thestructureofthischapterisasfollows. First,weprovideanoverviewofthe portablevideogameconsoleindustry. Next,wesetupthemodelanddescribethe game. In Section 4, we present two theoretical models in a traditional one-sided market structure, the rst does not allow for mixed bundling while the second does,toassistintheidenticationoftheimpactmixedbundlinghasinastandard one-sided market structure. In this section we also compare prices, prots and welfare between the two regimes. After introducing the one-sided market model, wepresentanddiscussourtwo-sidedmarketmodelofmixedbundlinginSection 5. Moreover,duetotheunobservabilityoftheroyaltyrateongamedeveloperside in our data, we perform the analyses with the royalty rate as both exogenously and endogenously determined. We, again, present the analysis and results of a model without bundling rst and follow with a model which introduces mixed bundling. Section6discussesourdataandpresentsindustrystatistics. Wepresent theresultsofourreducedformtestsofthetheoreticalmodelinSection7. Lastly, weconclude. 2 PortableVideoGameConsoleIndustry Duringthe early2000s throughlate2006, Nintendo wasamonopolist inthepro- ductionofportablevideogameconsoles. Specically,itwasamulti-productmo- nopolist producing two versions of its very popular Game Boy Advance (GBA) console as well as a portfolio of games to be played on its console. Each ver- sion was internally identical and played the same games, but the second version 62 dubbedtheGBASPwasreorientedwiththedisplaylyinghorizontallyratherthan vertically. Moreover, the target market of these two devices was toward younger kids, rather than teenagers or young adults for a home video game console. The portable console market most drastically differs from the traditional home video game console market in that it is extremely portable with the size of the device no larger than an adult hand. It can easily travel with a consumer and be played in a car or airplane, while a home console is restricted to a location which has a televisionandelectricity. The structure of the video game industry is a prototypical platform market where a video game console acts as a platform to two different end users consumers and game developers. 22 A portable console permits two end users to interact via its platform creating externalities for each side of the market, where thedemand-sideindirectnetworkeffectspertaintotheeffectthatagametitlehas on a console's value to the consumer as well as the benet a game developer re- ceiveswhenanadditionalconsumerjoinstheconsole'sownerbase. Determining the size of these cross group externalities depends on how well the console per- formsinattractingtheotherside. Withintheconsolemarkettherearethreeclasses ofplayers: the console, consumers, and gamedevelopers. Aconsumerpurchases a console in order to play games. Moreover, a consumer pays a xed fee for the console and a xed price for a video game. However, in order for a consumer to play a video game, the developer of the game is required to pay the console 22 See Kaiser (2002)[20], Caillaud and Jullien (2003)[10], Rochet and Tirole (2006)[29], Rys- man (2004)[31], Kaiser and Wright (2006)[21], Armstrong (2006)[5], Hagiu (2006)[18] and for generalliteratureontwo-sidedplatformmarkets. 63 a royalty rate for the rights to the code which allows the developer to make his game compatible with the console. This royalty rate is not a xed one-time fee. Rather,adeveloperpaysaroyaltyrateforeachcopyofitsgamethatisboughtbya consumer. 23 FigureIII.1presentsanillustrationofthediscussedmarketstructure. FigureIII.1: VideoGameMarketStructure The above gure describes a generalized industry structure. A more tailored structure makes a distinction between two different types of video game devel- opers. The rst is what we and the industry note as rst-party games. These games are produced by the console's (Nintendo's) own in house design studio. The second type of game is produced by independent rms not associated with the producing console (Nintendo). We denoted these developers as third-party gamedevelopers. In addition to selling two portable devices as well as a large library of rst- party games, Nintendo also sold bundles. The contents of these bundles always 23 Console manufacturers actually manufacture all video games themselves for quality control purposesandtotracksalesforroyaltycollection. 64 includedaportableconsoleandoneofitsrst-partygamesoftentherst-party gamewasahitgame. 3 ModelSettings There are three classes of players in the model: two types of agents and a plat- form. In the context of the portable video game console industry, the agents are consumers (C) and video game developers (D) while the platform (P) is the portableconsole. Weassumeinteractionsamongallthreeclassesofplayersexist and are illustrated by Figure III.1 above. In this section, I use lower-case letters to denote prices. And in the later part, lower-case letters are used specically for independent pricing (IP) regime, while upper-case letters are used to denote the correspondingpricesunderbundling. ConsolePlatform: Thereisamonopolyconsoleplatformwhichlocatesattheoriginofahorizon- tal line and produces its own video games. For simplicity, we assumeP has only one rst-party game, with marginal cost,c; while the marginal cost of producing its console is f: The platform, P; interacts with both agents by charging a xed feep c toconsumersforthepurchaseoftheconsolehardware,axedfeep g forthe rst-party game, and levying a per unit royalty rater to game developers for the righttoproduceandsellgamescompatiblewiththeconsolehardware. Likewise, consumersandgamedevelopersinteractwithconsumerspurchasingvideogames fromdevelopersattheircorrespondinggameprices. 65 Consumers: We implement a modied Hotelling model to analyze the consumers' deci- sions. Assume there are two groups of consumers (i = 1;2) with total size nor- malizedtoone. Group1,identiedastheinstalledbase(withfraction)locating attheorigin,isapre-existinggroupwhoalreadyhavepurchasedthehardwarefor platformP buthaveyettopurchasetherst-partygame. 24 Thegrossutilityacon- sumerfromgroup1garnersfrompurchasingtherst-partygameisu installed =v g : Andgroup2,acontinuumofnewgamerswithfraction1population,areuni- formly located on a horizontal line and have yet to purchase platform P 0 s hard- ware. Theutilityagroup2consumerreceivesfrompurchasingaportableconsole is dependent upon the number of games (or game developers, since we assume each developer only produces one game) x available for consumers to play, and the transportation cost equal totd. Here,t is the transportation cost per unit of length,d is the distance of the consumer from the origin, and the marginal utility of an additional game is : To be more specic, the gross utility associated with aconsumersituatedatpointdwhoelectstopurchaseonlyplatformP'shardware is(v c td)1fx> 0g+x 25 ;whilev c +v g +xtdifhepurchasesboth thehardwareandtheplatform'ssoftware,where1fgistheindicatorfunction,v c and v g are the new gamers' intrinsic values for the console and rst-party game, respectively. For simplicity, we assume thatv c ,v g are constant and known to all, 24 Oneinterpretationoftheassumptionthattheinstalledbasehasyettopurchasetherst-party gameisthattheconsolemanufacturerhasinnovatedandcreatedavideogameafterthereleaseof itsconsole. 25 When x = 0, the console only won't provide any utility to the consumer unless purchased withtherst-partygame. 66 and v g is drawn from the uniform distribution U() on [0;1]. Lastly, note that newgamersareheterogeneousintwodimensions: intheirlocationd;andintheir valuationfortherst-partygamev g . GameDevelopers: We assume that the console is essential for consumers to enjoy a game. In thecaseofgamedevelopers,theyalsomustjointheconsoleplatforminorderfor their software to be compatible with the console. Moreover, we assume there is free entry into the market for video game software and that developers are het- erogeneous. Each game developer's type can be summarized by , which is its xed cost of developing a game for platform P. For simplicity, we assume is i.i.d. according to a uniform distribution U() on [0;1]: The total number of po- tential game developers is therefore normalized to one. With the assumption of free entry into the game developer segment, developers do not set game prices. Instead,theydecidewhethertoenterthemarketandjoinP. Theproteachgame developer can receive per consumer if it elects to produce a game is . Conse- quently, a type developer will create and produce a video game for platformP if and only if (r)(q c +q both +); where q c is the quantity of consoles sold individually; q both is the number of consoles purchased with the rst-party game, and is the aggregate number of consoles previously purchased or what wedenoteastheinstalledbase. Thenumberofvideogamesavailableonconsole P isthen 67 x = U((r)(q c +q both +)) (12) = (r)(q c +q both +): The above equation therefore implies that as more consumers join the platform more games will be produced and is denoted throughout the economic literature asanindirectnetworkeffect. The timing of the game is as follows. First, the monopoly platform chooses either to bundle or not and then sets prices accordingly. Next, after observing the price offers from the platform, consumers and game developers make their purchase decisions and entry decisions, respectively. Rational expectations are assumedforthesimultaneousequilibriumoutcome. 4 One-SidedModel: WithoutThird-PartyGames Beforeweintroduceatheoreticaltwo-sidedmarketmodeltoanalyzetheimpactof mixedbundlingwerstpresentabenchmarkmodelinaone-sidedmarketsetting in order to assist in drawing comparison between the impact mixed bundling has on a traditional one-sided market and that on a two-sided market model. In our basic setting, if we eliminate all the third-party games, that is, letx = 0, then the indirect network effect disappears since there is only one game available on the platform,whichistherst-partygame. Inthiscase,themarketisreducedtoaone- 68 sided market with the console and the rst-party game as perfect complements. We begin by constructing a one-sided market model which omits the practice of mixed bundling and then modify the model to allow for its practice. After the introduction and description of the equilibrium of both models, we compare the tworegimestodeterminetheeffectsofmixedbundlingonpricesandwelfare. We thenfollowwiththeanalysisofmixedbundlinginatwo-sidedmarketsetting. IndependentPricingModel TheIPequilibriumconsistsofthemonopolyplatformsettingprices(p c ;p g ). The two groups of consumers' decisions are as follows. For the installed base, they will purchase the rst-party game fromP if and only ifu installed = v g p g 0: Hence,eachindividual'sdemandfortherst-partygameis n g = 1fv g p g g: Aggregatingacrosstheinstalledbaseyieldsanaggregatedemandof q g = n g = 1fv g p g g: Ifweassumev g islargeenoughtorepresenttheloyaltyorlock-ineffectofthe installed base, it is without loss of generality to restrict our attention to v g p g case. Thus,q g =: 69 The equilibrium number of new gamers is a bit more complicated to derive given that consumers can purchase the console and the rst-party game in con- junctionorelecttonotpurchaseeithertheconsoleorthegame. 26 Notethat,since there is no third-party game (x = 0), the console will only provide utility to con- sumers who own or purchase the rst-party game. Therefore, no one only buys theconsole. Thenewgamerswillbuyboththeconsoleandtherst-partygameif and only if (v c p c )td+(v g p g ) 0; and buy nothing otherwise. The new gamers'demandisinthegurebelow. (i)p c +p g v c (ii)p c +p g <v c FigureIII.2: NewGamers'DemandunderIPinOne-SidedMarket Therearetwopossibilities: (i)whenp c +p g v c ;and(ii)whenp c +p g <v c . In (i), the demand for only the console and the demand for both a console and a 26 Recallthattheconsoleisessential,sotheonegamedoesn'tprovideanyutilitytonewgamers, whomhavenoconsole. 70 rst-partygameare q c = 0 q both = (1) (1+v c p c p g ) 2 2t ; respectively. Thus,theplatformP 0 sprotunderIPis IP (i) = (p g c)q g +(p c +p g fc)q both = (p g c)1fv g p g g +(1)(p c +p g fc) (1+v c p c p g ) 2 2t : Similarly, in (ii), the demand for only the console and the demand for both a consoleandarst-partygameare q c = 0 q both = (1) 2(v c p c p g )+1 2t ; respectively. Thus,theplatformP 0 sprotunderIPis 71 IP (ii) = (p g c)q g +(p c +p g fc)q both = (p g c)1fv g p g g +(1)(p c +p g fc) 2(v c p c p g )+1 2t : Noticethatthesecondtermispurelyafunctionofp c +p g . Andhencep g could be freely set as long as p c +p g is kept as a certain constant. Obviously, the rst termwillbemaximizedatp g =v g . Andwehavethefollowingresult. PropositionIII.1 (One-Sided Market IP Equilibrium) When there is no third- partygame,theIPequilibriumis 27 p g = v g p c = 8 > < > : 1+vc+2(f+c) 3 v g if 3 2 <v c 1 2 (f +c) 0 vc+ 1 2 +(f+c) 2 v g if0<v c 1 2 (f +c) : BundlingEquilibrium Themonopolyplatformunderabundlingmodelcansetprices(P c ;P g ;P B ),where P B isthebundleprice. Hence,thenewgamersnowpossesstheoptionofpurchas- ing the portable console and the rst-party game bundled together. However, in this one-sided case, only the bundle will be purchased since the console does not 27 Note that if v c +1(f +c) 0, then there won't be any sale from new gamers. So we excludethisuninterestingcaseinourmodel. 72 provide any utility unless it is combined with the purchase of the only one avail- able game. As a result,P c doesn't matter, and the prices that bind areP g andP B due to the fact thatP c +P g P B . The monopoly platform's prot is essentially the same as IP in the IP equilibrium above with P g substituting for p g and P B substitutingforp c +p g . PropositionIII.2 (One-SidedMarketBundlingEquilibrium)Whenthereareno third-partygames,thebundlingequilibriumis P g = v g P c 8 > < > : 1+vc+2(f+c) 3 v g if 3 2 <v c 1 2 (f +c) 0 vc+ 1 2 +(f+c) 2 v g if0<v c 1 2 (f +c) P B = 8 > < > : 1+vc+2(f+c) 3 if 3 2 <v c 1 2 (f +c) 0 vc+ 1 2 +(f+c) 2 if0<v c 1 2 (f +c) : An immediate conclusion from the comparison between these two regimes is thatthereisnoneedtobundleinsuchaone-sidedmarketsetting. 28 PropositionIII.3 (One-Sided Market Comparison) When there are no third- party games, the IP equilibrium and the bundling equilibrium yield exactly the sameoutcome. 28 Of course, when there are more than one rst-party games, bundling will dominate indepen- dent pricing as the traditional bundling in one-sided market literature applies. However, since the maincontributionsofthispaperasshownlaterarethe newpricingstructureandclearwelfareef- fectfrombundlingintwo-sidedmarkets,wechoosetoabstractawayfromthemultiplerst-party games case. In addition, introducing more than one rst-party game would complicate the model more,becausewhichgametobundlebecomesanotherinterestingissue. 73 Such a redundancy of bundling comes from the fact that here the console and the rst-party game are perfect complements with xed proportion. The Chicago School's"single-monopoly-prottheorem"appliesforthenewgamerssincethey will demand both the console and the rst-party game. In summary, a pure tie wouldsufcefortheplatform. Yet,duetotheexistenceoftheinstalledbase,who only buy the rst-party game since they already own the console, the standalone price for the rst-party game has bite in both regimes and is necessary to further extractrentsfromtheseconsumers. As we will see next, once third-party games are introduced, the market struc- ture will switch to a two-sided market. Although the console and the rst-party game remain complements, they no longer are perfect complements with xed proportionnewgamerscouldonlybuytheconsolewithoutpurchasingtherst- party game, since third-party games are available to be played with the console. Consequently, the availability of substitutes to the rst-party game dramatically changes the market structure and invalidates the "single-monopoly-prot theo- rem". 5 Two-SidedModel: WithThird-PartyGames We now allow for third-party games to enter into our model. As indicated in Section3,thenumberofthird-partygamesavailableisendogenouslydetermined asx = (r)(q c +q both +). The presence of these third-party games has two implications: rst,therst-partygameisnolongeressentialtonewgamers,since 74 they can now enjoy the console with other third-party games; second, indirect network effects emerge in this setting, because the number of third-party games depends on the number of total console ownersq c +q both + and vice versa. As aresult,themarketstructurebecomestwo-sided. IndependentPricingEquilibrium Similarlytotheone-sidedmarketmodel,thedemandfortherst-partygamefrom theinstalledbaseis q g =1fv g p g g: The equilibrium number of new gamers is more challenging to derive given that consumers can either solely purchase the console, purchase the console and therst-partygameinconjunctionorelecttonotpurchaseeitherconsoleorgame. Wethusclassifyconsumersbasedamongtheirdifferentlocationsintotwodiffer- ent types. The rst, Type A, values the console enough on its own to purchase. Thatis,theconsumer'sutilityfromconsumptionofhardwareisgreaterthanzero, or v c + x p c td 0. Hence, these Type A new gamers will buy both the hardware and rst-party game if v g p g and only the console if v g < p g . The second, Type B, consumer does not value the console enough on its own to purchase it. That is, v c +xp c td < 0. Thus, these consumers only pur- chase the console with the rst-party game. In this case the rst-party game is a complementary product which makes the console more attractive, although it is notessentialasinourone-sidedmarketmodel. They,therefore,willbuyboththe 75 consoleandtherst-partygameif(v c +xp c td)+(v g p g ) 0;andbuy nothingotherwise. Thenewgamers'demandisinthegurebelow. FigureIII.3: NewGamers'DemandunderIPinTwo-SidedMarkets Inordertodeterminetheaggregatedemandforconsumerswhopurchaseonly a console and those who purchase both a console and rst-party game we must aggregatethedemandfrombothtypesofnewgamers. Againwecanthinkofthe consumerswhopurchaseaconsoleonlyasthosewhovaluetheconsoleseparately without any game or value a third-party game more than the game produced by theplatform. Denoteuv c +xp c thenthenumberofTypeAnewgamers buyingbothproductsis Z u t 0 (1p g )ds = (1p g ) u t ; whilethedemandforonlytheconsolefromTypeAconsumersis Z u t 0 p g ds =p g u t : 76 Likewise, the demand of Type B consumers who buy both a console and a rst- partygameis Z u t + 1pg t u t f1[p g (uts)]gds = (1p g ) 2 2t : The aggregate demand for the console as well as the demand for both a con- sole and a rst-party game under the assumption that consumers form rational expectationastothenumberofavailablegamesis q c = (1) p g t u q both = (1) 1p g t (u+ 1p g 2 ); respectively. Given the demands for each of these products, the equilibrium number of videogamesavailableonconsoleP is x = r f+(1)[ v c p c t + (1p g ) 2 2t ]g; where = 1 (1)(r) t : With equilibrium demand for consoles and rst-party games as well as the number of third-party game developers determined, in terms of hardware price p c , software price p g and royalty rate r, the portable manufacturer maximizes its prot with respect to these two strategic variables. The corresponding platform protunderindependentpricingis 77 IP (p c ;p g ;r) = rx(q c +q both +)+(p g c)q g +(p c f)q c +(p c +p g fc)q both = r(r)f+(1)[ u t + (1p g ) 2 2t ]g 2 +(p g c) +(p c +p g fc)(1) 1p g t (u+ 1p g 2 ) +(p c f)(1) p g t u Since the royalty rates are unobservable in our data, we analyze two possi- ble cases: i) when the royalty rate is exogenously determined, and ii) when it is endogenouslydetermined. Lemma1 (Two-SidedMarketIPEquilibrium)Whentherearethird-partygames, themarketstructureistwo-sided. Whentheroyaltyrater isexogenouslydetermined,theIPequilibriumis (p r c ;p r g ) = argmax x;y IP (x;y;r); whentheroyaltyrater isendogenouslydetermined,theIPequilibriumis (p c ;p g ;r ) = argmax x;y;z IP (x;y;z): 78 BundlingEquilibrium Our mixed bundling model differs slightly from the above IP model in that new consumers now possess the option of purchasing the portable console and the rst-partygamebundledtogether. Consumersstillretaintheoptionofpurchasing the hardware and software separately. Like the above IP model, the platform, P; interactswithbothagentsbychargingaxedfeeP c toconsumersforthepurchase ofthehardwareandlevyingaperunitroyaltyrate,R,togamedevelopersforthe righttoproduceandsellgamescompatiblewiththeconsolehardware. Consumers and game developers still interact with consumers purchasing video games from developersattheircorrespondinggameprices. Consumerspurchasetherst-party videogameseparatelyforaxedfee,P g :Yet,inthebundlingmodel,platformP alsosellsitsrst-partygameandhardwaredevicetogetheratpriceP B: Pricesare thusfP c ;P g ;P B ;Rg. To begin our equilibrium analysis, rst note that in order for the bundle to be effective, we must haveP c +P g > P B . By doing so, we ensure that new gamers willneversolelypurchasetherst-partygameatP g sincethisgameprovideszero utility without the ownership of the portable console. Moreover, if they elect to purchasetherst-partygametheywilldosoviathebundle. SoP g isthusspeci- callytargetedtotheinstalledbaseofuserswhoalreadyowntheconsolebuthave not purchased the console produced game. Hence, it is easy to see that the price of the rst-party game is set toP g = v g , sinceP g is directed to the installed base and its intrinsic valuation for rst-party game is known . The resulting demand fortherst-partygamefromtheinstalledbaseisQ g =: 79 Note that if we remove the installed base from our model, then there is no need to bundlefor new gamers, only two prices matter, since they either buy consoleonly,orbuybothconsoleandrst-partygametogether. Inotherwords,it isthepresenceofinstalledbasemakingthebundlenecessary. Under bundling, new consumers determine their purchase decisions on two strategic variables, the price of the console and the effective price of the rst- party game P e g P B P c : Our analysis regarding new gamer demand for the portable console and the purchase of both the console and rst-party game (the bundle) takes the same structure as the IP equilibrium if we dene the effective priceoftherst-partygamefornewgamersasP e g andbyreplacingp g therewith theeffectivegamepriceP e g . Thestandalonedemandfortheconsoleis Q c = (1) P e g t U whereU v c +XP c whilethebundledemandis Q B = (1) 1P e g t (U + 1P e g 2 ): Consumers' decisions are shown below and are consequently quite similar to theirdecisionundertheindependentpricingmodel. 80 FigureIII.4: NewGamers'DemandunderMixedBundlinginTwo-Sided Markets Likewise, assuming rational expectations, the number of gamer developers joiningtheplatformis X = (R)(Q c +Q B +) = R f+(1)[ v c P c t + (1P e g ) 2 2t ]g; where = 1 (1)(R) t . Given the demand for the rst-party game from the installed base, the de- mand from new gamers for only the console and the demand for the bundle, the 81 monopolyplatform'sprotunderbundlingis B (P c ;P e g ;R) = RX(Q c +Q B +)+(P g c)Q g +(P c f)Q c +(P B fc)Q B = R(R)f+(1)[ U t + (1P e g ) 2 2t ]g 2 +(P e g c) +(P c +P e g fc)(1) 1P e g t (U + 1P e g 2 ) +(P c f)(1) P e g t U +(v g P e g ) = IP (P c ;P e g ;R)+(v g P e g ) NoticethatthestructureofthisprotfunctionisidenticaltotheIPmodel. The platform receives prots from third-party game developers via royalties, prot from selling its rst-party game and console separately and from consumers who purchasethebundleofconsoleandgameatthebundlepriceP B . Moreover, compared with the platform's prot under IP, the only extra term is the surplus gains extracted from the installed base, that is, (v g P e g ) , if we substitute p g in IP with P e g . Consequently, we determine that bundling is a dominant strategy for the monopoly platform since offering P B and P g simul- taneously is equivalent to offer P e g and P g to new gamers and the installed base separately while retaining P c = P B P e g as the console price. Offering a bun- dle,therefore,providesthemonopolyplatformanadditionalinstrumenttoextract consumersurplus. 82 Lemma2 (MixedBundlingisProtable)Wheneverbundlingispossible,mixed bundlingisadominantstrategyovernobundlingorpurebundling. Proof. Any IP menu (p c ;p g ) can be perfectly mimicked by (P c ;P g ;P B ) with P c = p c , P g = p g and P B = p c + p g : Due to the presence of installed base, offering mixed bundling gives the platform more freedom in extracting surplus. Thus,mixedbundlingisstrictlybetterthanIP. Under pure bundling, neither the installed base nor the new gamers with low value on rst-party game would be served. Hence, pure bundling will be strictly dominated,too. Q.E.D. The above lemma is consistent with the existing literature on mixed bundling in traditional one-sided market in that mixed bundling is the optimal strategy for themonopolist. Parallel to the IP case, we perform the analyses for both cases in which the royaltyrateisexogenouslyorendogenouslydetermined. Lemma3 (Two-SidedMarketsBundlingEquilibrium)Inthetwo-sidedmarkets contextwiththird-partygames, P (R) g =v g : When the royalty rate R is exogenously determined, the bundling equilib- riumis (P R c ;P R g ) = argmax x;y B (x;y;R); 83 when the royalty rateR is endogenously determined, the bundling equilib- riumis (P c ;P e g ;R ) = argmax x;y;z B (x;y;z): Prices,ProtsandWelfareComparison In this subsection, we compare the equilibria of the two regimesIP vs Bundling. Interestingly, we nd the price and welfare effects of bundling when the royalty rate is exogenously given to differ from those when the royalty rate is endoge- nouslydetermined. WhenRoyaltyRateRisExogenouslyDetermined PropositionIII.4 When royalty rate R is exogenously determined, under mixed bundling, the standalone price for the rst-party game are higher than those un- der IP, while the effective rst-party game is lower under bundling. In addition, the bundle price is lower than the sum of console and rst-party game under IP. Specically, P eR g = P R B P R c <p r g <v g =P R g P R B = P R c +P eR g <p r c +p r g : However,thedirectionofthechangeinstandaloneconsolepriceisundetermined, dependingontheexogenousroyaltyrateR . 84 We determine from the above proposition that by offering the bundled option it allows the monopolist to increase the standalone prices of the rst-party video game. Theabovepricestructureallowsconsumerstosortintodistinctgroupsand consequently reveal their true preferences. The mixed bundle option thus acts as apricediscriminationtoolandallowsthemonopolisttoraisestandalonepricesin searchofmoreefcientandcompleteextractionofconsumersurplus. Lastly,wedeterminetheimpactontotalsurplusdependsonwhetherthetotal number of game developers increases or decreasesunder numerical simulations the effect of total surplus is not ambiguous but clearly shows that total surplus increases. PropositionIII.5 (TotalSurplus)WhenroyaltyrateRisexogenouslydetermined, totalsurplusunderbundlingishigherthanIPifandonlyiftherearemorepartic- ipationonbothsidesinequilibrium. Proof. Given the standalone price of the rst-party video game increases under a mixed bundling regime, the installed base is worse off. Yet, this decrease in surplus is a direct transfer to the console resulting in total surplus to remain un- changed. Consequently, total surplus is dependent upon the change in surplus of new gamers. The impact of new gamer surplus is ambiguous. First, there are consumers who only purchase the console. These consumers are not necessarily worse off since the standalone console price doesn't have to be higher, holding the number of games constant, while gamers who purchase the bundle are better off since they purchased at the lowered bundled price. Moreover, a new gamer's 85 utilityisnotonlyafunctionofpricebutalsothenumberofgamesontheplatform. The change in consumer surplus caused by a change in price could very well be offset or dominated by the change in surplus from the indirect network effect. Moreover, the equilibrium number of game developers is also ambiguous given that the number of game developers is decreasing in P e g and P c and P c > < p c ; P e g <p g :Q.E.D. WhenRoyaltyRateRisEndogenouslyDetermined PropositionIII.6 WhenroyaltyrateRisendogenouslydetermined,undermixed bundling, all prices except the standalone price of the rst-party game are lower thanthoseunderIP.Specically, R < r P c < p c P e g = P B P c <p g <v g =P g P B = P c +P e g <p c +p g : This is quite a surprising result! Both the standalone console price and the royalty rate are lower under the mixed bundling equilibrium than their respective counterparts in the IP equilibrium. It is widely known that in two-sided markets the optimal pricing scheme is to subsidize the more elastic side of the market and extract rents form the other, more inelastic, side. Or more generally, the 86 optimalpricestructureistoadjustpricesdownwardbytheexternalbenetacon- sole receives from attracting an additional side i user. When the console maker uses mixed bundling they are in affect offering a "subsidy" to consumers which increases demand for its console by attracting a greater number of marginal con- sumers. It is typically thought that by further subsidizing consumers, via mixed bundling in our case, the console maker is increasing the game developers' will- ingnesstoparticipateandthustheabilitytoraisetheroyaltyrateinwhichitlevies. Yet, this is not what we encounter. We nd that the royalty rate is in fact lower underthemixedbundlingequilibrium. Byofferingthemixedbundle,theconsole makerbecomesmoreeffectiveinextractingconsumersurplus,comparedtotheIP case. Consequently,byofferingthemixedbundletheconsumersidebecomesless elastictoplatformpricingsincetheconsolecanmoreefcientlyextractconsumer surplus without deterring consumer participation. The game developer therefore becomes relatively more elastic, which creates an incentive for the console plat- formtolowerRundermixedbundling. There also is an additional argument for the lowering of the royalty rate. We know that the console platform would like to increase participation on the side it can more efciently extract surplus from, since doing so will increase prots. Given that nonlinear pricing is only available to the consumer side, the console consequentlyisabletomoreeffectivelyextractrentsfromconsumers. Giventhis, the console platform has an incentive to increase demand for its console. How does the console accomplish this? It does so by reducing the game developers' royalty rate R: A reduction in royalty rate will lead to an increase in game de- 87 velopment and thus attract more consumers through the indirect network, which willconsequentlyleadtomoreentryofgamesthroughtheindirectnetworkeffect resultingineachofthesenetworkeffectstoreinforcetheother. In addition to a decrease in royalty rate, we also nd the standalone console priceislessunderamixedbundlingregime. Thissmallerstandaloneconsoleprice isaconsequenceofthemixedbundlesegmentingthemarketintonewgamersand the installed base. Under a mixed bundle regime the standalone rst-party game price is specically targeted to the installed base as oppose to a uniform price undertheindependentpricingequilibrium. Since,theinstalledbase'svalueofthe rst-partyisknowntoall,theconsolemakerisabletoperfectlypricediscriminate andsetpriceequaltov g ,whichisgreaterthanp g :Asaresult,theadditionalprot the console receives from selling its rst-party game and the payment of royalty rates from third-party developers is larger under a mixed bundling equilibrium leading to a larger discount of the standalone console price and hence a smaller price. 29 Unliketheone-sidedcaseabove,wecandeterminetheimpactmixedbundling has on the equilibrium number of game developers. We show that the number of developers increase under a regime which includes mixed bundling. We, there- fore,concludethestrategicdecisiontoofferamixedbundlealsosolvesthecoordi- nationproblemof"thechickenoregg",sincebothconsumerandgamedeveloper participationincreases. 29 Remember that the presence of installed base makes bundling have bite. If we eliminate the installed base from our model, then there won't be any bundling, and hence no such prices' drop associatedwithbundling. 88 PropositionIII.7 WhenroyaltyrateRisendogenouslydetermined,undermixed bundling,thenumberofthird-partygamedevelopersishigherthanunderIP. Proof. X = R f+(1)[ vcPc t + (1P e g ) 2 2t ]g @x @R < 0; @x @P e g < 0; @x @Pc < 0 Since (R;P c ;P e g ) is uniformly lower than (r;p c ;p g ) we nd that the number of game developers under a mixed bundling equilibrium is larger than under the IPmodel. Q.E.D. After determining that all prices are lower, with the exception of the stand- alone rst-party game price, in addition to the number of third-party developers beinggreaterunderamixedbundlingequilibrium,wendnewgamersarestrictly better off. Yet, the installed base of consumers is strictly worse off, which is a consequenceoftheinstalledbasebeinglocked-intotheconsoleandtheabilityof console manufacturer to segment the market and target the installed base with a segmentspecicsoftwarepricewhichextractsallsurplusfromthemundermixed bundling. Thisextraction,however,doesnotcausetotalsurplustochangesinceit isatransferfromconsumerstotheplatform. Moreover, fromLemma2weknow that the platform's prots are strictly higher under mixed bundling. We, thus, havethefollowingpropositionregardingthecomparisonoftotalsurplusbetween regimes. PropositionIII.8 WhenroyaltyrateRisendogenouslydetermined,totalsurplus ishigherunderbundlingthanunderIP: TS B >TS IP : 89 From our theoretical analyses, we show the effects of mixed bundling on prices, surplus and demand for the console (both consumer and game developer demands) to differ substantially under two different market structures. While, themotivationsbehindtheactofofferingabundleareconsistentacrossstructures (pricediscrimination),mixedbundlingunderatwo-sidedmarketstructureleadsto a very different and unique outcome. We determine, unlike the single-sided case, totalsurplusinatwo-sidedmarketstructureisdenitivelylargerthanthewelfare under an IP regime, even though all prices with the exception of the standalone video game price are lower. When a console producer is able to optimally set its royalty rate and offer a mix bundle, the rm's response is not to increase the standaloneconsolepricelikethecaseoftheone-sidemarket,butistolowerboth theroyaltyrateandstandaloneconsoleprice. Thedecreasesinmarginalrevenues from the decline in console price and game developer royalty rate are more than overcome by the increases in consumer and game developer demand to join the platform. 6 HypothesesandData In this section, we test the above theoretical predictions for the two-sided market model via reduced form regressions with data from the portable console market. Our model above generates three distinct price correlations between two regimes whichincludemixedbundlingandonethatdoesnot. Theyare: 90 1 : P c p g 3 : R <r Inorderforthetheoreticalmodeltoholdtothedata,allthreecorrelationsneed tobepresent. The data used in this study originates from NPD Funworld. Data from the marketing group NPD Funworld tracks sales and pricing for the video game in- dustry and is collected using point-of-sale scanners linked to over 65% of the consumerelectronicsretailstoresintheUnitedStates. NPDextrapolatesthedata to project sales for the entire country. Included in the data are quantity sold and total revenue for the two consoles and three bundles and all of their compatible video games, roughly 700. The data sets cover 45 months starting in June 2001 andcontinuethroughFebruary2005,duringwhichNintendowasamonopolistin portablevideogamemarketbeforeSONY'sPlayStationPortablecamein. In June of 2001, Nintendo launched a new generation of portable console de- vicesdubbedtheGBA.Thisgenerationofportableimprovedonthepreviousgen- eration by increasing the CPU speed, RAM, screen size and screen resolution. Nonetheless,therewereawswiththedevice,mostlyduetoitssizeandshape. In early 2003, a new GBA device called the GBA SP was launched to rectify these issues. ThisnewdeviceaugmentedtheshapeandsizeoftheGBAbuttheinternal 91 workings of the GBA remained. The GBA SP looks like a mini laptop computer and was close to half the size of the original GBA. Moreover, it is usually the case with the introduction of a new device new games are released which are not backwardscompatible. Yet,withtheintroductionoftheGBASP,thiswasnotthe case since the internal parts of both devices were identical. Consequently, both devices shared the same set of games. And, at the end of the data set there were over600uniquevideogamesproduced. Generalstatisticsoftheportablevideogameindustryareprovidedinthetables below. TableIII.2: PortableConsoleMarketStatistics ReleaseDate Units MonthsonConsoleMarket Nintendo GBA June2001 12,821,233 45 GBASP March2003 13,070,720 24 GBAw/MarioKart November2001 215,394 29 GBAw/MarioAdvance2 November2002 199,225 17 GBASPw/MarioAdvance4 November2003 149,065 4 92 TableIII.3: PortableConsoleandBundlePrices Average Price MaxPrice MinPrice IndependentGamesSold Nintendo GBA $72.00 $94.46 $52.37 GBASP $93.73 $100.30 $70.60 GBAw/MarioKart $86.17 $150.54 $61.50 2,027,636 GBAw/MarioAdvance2 $67.33 $71.73 $56.60 2,438,732 GBASPw/MarioAdvance4 $97.62 $99.85 $94.92 1,673,304 In the above two tables we present statistics regarding the release date, total units sold and the number of months on the console market, average (min and max)pricesandtotalstandaloneunitssoldofthebundlegamesfortwostandalone consoles and three bundles. An interesting fact which is clearly evident from the table is that Nintendo elected to release its bundles at the height of the holiday timeperiodtherstbeingaGBAdevicebundledwiththehitgameMarioKart in November 2001. Moreover, the bundled games were high quality hit video gameseachsellingoveroneandhalfmillionstandaloneunits. BelowwepresentFigureIII.5whichillustratesthesalesofconsolesandbun- dles over time. The video game industry exhibits a large degree of seasonality in console sales with signicant increases in the months of November and Decem- 93 ber. It,therefore,isimportanttoaccountforthelargedegreeofseasonalityinour empiricaltests. Nov 01 May 02 Nov 02 May 03 Nov 03 0 0.5 1 1.5 2 x 10 6 Total Monthly Quantity Sold (M) Month Console Sales Bundle Sales FigureIII.5: ConsoleandBundleSales 7 ReducedFormTests Thersttheoreticalpredictionwetestiswhetherthestandaloneconsolepricede- creases when a mixed bundle is offered. Accordingly, if our theoretical model is correct, there should exist a negative correlation between console price and the presenceofbundlesinthedata. Inordertoproperlydeterminewhetheranegative correlation is present we must address econometric issues such as whether unob- servedheterogeneityandomittedvariablebiasisanissue,properspecicationand 94 model selection. We rst tackle the issue of model selection given that we have two variables which capture the presence of a bundle. They are i) an indicator takingvalueequaltooneifabundleisofferedinperiodtandzerootherwiseand ii) the count of the number of bundles offered in time t: However, given that we areuncertainaboutmodelspecication,werunthreeencompassingtestsforthree different model specicationslinear, log-linear and log-log. These results are presented in Appendix B.24, and from them we determine the model which em- ploysthecountofbundlesencompassesallthefeaturesofamodelemployingthe indictor variable. We thus use the count of the number of bundles as the variable whichcapturesthepresenceofabundle. Withthemodelselectioncompleted,we turn our attention to model specication. We implement a Box-Cox regression model to determine the proper specication. In this regression, we allow nonlin- earity to enter both the dependent as well as (a few) independent variableswe thusaretestingbetweenalinearandnonlinearmodel. Again,theseresultsarepre- sented in the appendix, but from them we conclude that the best specication is a double log model. Moreover, the residuals of the preferred model are normally distributed, unlike a linear or log-linear model. Lastly, we perform two Hausman tests. Therst,testsfortheendogeneityofthelognumberofsoftwaretitles,since ourtheoreticalmodelillustratesthatthenumberofsoftwaredevelopersisendoge- nously determined. The purpose of this test is due to the fact that a simple OLS model yields inconsistent estimates of the 0 s when correlation between ct and ln(# of Software ct ) is present. Consistent estimates of the model parameters can, nonetheless, be obtained using an instrument that is correlated with ln(#of 95 Software ct ),butuncorrelatedwith ct weusethelogoft1 0 sinstalledbase as an instrument, since we illustrate above that the installed base will inuence the number of video game developers ( = IB t1 ). Likewise, in the presence of panel data the correlation can be eliminated with the use of console xed ef- fects resulting inE[ ct jX c ;c c ] = 0 and thus providing an unbiased estimate of. The use of console xed effects also corrects for any unobserved heterogeneity or omitted variable bias. We consequently implement a second test, which de- termines whether a pooled regression model is more efcient than a xed effects model. TheresultsofeachofthesemodelsinadditiontotheHausmanteststatis- ticarepresentedbelow. Fromtheseresultswendtheuseofapooledregression is more efcient than the xed effects model, and the instrument for the number ofsoftwaretitlesisnotendogenous. We analyze the impact of mixed bundles on standalone console price by re- strictingthedatatoconsistonlyofthetwostandaloneconsoles,theGBAandthe GBASP.Inadditiontoavariablewhichmeasurestheentryofabundleweinclude monthxedeffectstoaccountforlargeseasonalspikesduringtheseperiods,con- soleage,thenumberofcompatiblevideogamesandthetotalnumberofconsoles present in market t as covariates. The model we take to the data is a double log model(Model1below): ln(P ct ) = 0 + 1 Age ct + 2 ln(#of Software ct ) + 3 ln(1+#of AdditionalConsoles ct ) + 4 ln(1+#of Bundles ct )+ m=11 P m=1 4+m Month m + ct : 96 Below we present the results of three models. Models 1 and 2 implement a modelwithoutxedeffectswhilemodel3includesxedeffects. TableIII.4: StandaloneConsolePriceRegression ln(Price) Model1-OLS Model2-IV Model3-FEnoIV ln(1+#ofBundles) 0:0587 0:0578 0:0278 (0:0249) (:0252) (0:0488) ln(1+#ofAdditionalConsoles) 0:2340 0:2255 0:1336 (0:0504) (:0526) (0:0695) ln(#ofSoftwareTitles) 0:0029 0:0038 0:0807 (0:0223) (:0256) (0:0439) Age 0:0137 0:0138 0:0078 (0:0012) (:0012) (0:0031) ConsoleFE's No No Yes NumberofObs. 69 69 69 HausmanEndogeneityTest 0:71(0:4029) HausmanTestModel3vs. Model1 4:16(0:9986) AllmodelsincludeMonthxedeffectsandanunreportedconstant. **signicantat95%*signicantat90% Fromthetableabove,thecoefcientcorrespondingtothepresenceofabundle formodelone,ourpreferredmodel,isnegative. Anegativeandsignicantcorre- lationbetweenconsolepriceandthepresenceofabundlethusillustratesthatour 97 rst theoretical prediction holds to data. Other predictions from economic theory also hold. For instance, the coefcient corresponding to the total number of con- soles in the market is positive and signicantly different from zero. This positive sign is consistent with economic theory of a multiproduct monopolist, which in- ternalizestheeffectofitspricesonitsothersubstituteproductscreatingapositive price externality. Likewise, note the negative sign with regard to the log of one plus the number of software titles. This negative sign indicates price decreases as the number of titles increase, but it is insignicantly different from zero. Al- though console price should increase when games are added since doing so can extract more surplus from the new network benet, there is also a subsequent ef- fect which creates an incentive to decrease console price, thus the insignicance of the sign does not worry us. To be more specic, when an additional game is added,theconsoleacquireseithermoreroyaltiesorthepriceofthegame,depend- ing on whether the game is a third- or rst-party game. This additional revenue creates an incentive to decrease console price through the externality associated with multiproduct pricing of complementary products. Our results illustrate that these two effects offset each other and cause prices to remain unchanged with the introduction of an additional software title. Most importantly, though we do present evidence that our rst theoretical prediction holds to the data. Yet, in or- derforustoclaimthatourtheoreticalmodeliscorrect,weneedtofurthertestthe remainingtwotheoreticalpredictions. Next,wetestpredictiontwowhetherthestandalonebundledsoftwareprice increases when the bundle is introduced. Following a similar methodology to the 98 above analysis, we rst implement a Box-Cox regression to test model speci- cation and follow with Hausman tests to determine whether the number of soft- ware titles is endogenous and whether a xed effect or pooled regression is more efcient. We rst test to determine whether the number of software titles is en- dogenous. We do so with a Hausman test; rst running an OLS regression and following with an instrument variable estimator. We concluded that the number of software titles is endogenous and thus requires the use of instruments or xed effects. ItisthereforenosurprisethataHausmantestconcludesthataxedeffects modelismoreefcientthanapooledmodel. We test the second theoretical prediction by restricting the data set to include only software which was bundled with a console, and regress its price on month xed effects, software age, the number of software titles present as a measure of competition and an indicator variable which takes the value one if the software wasalsobundledwithaconsoleinagivenperiodandzerootherwise. Ifourtheo- reticalmodeliscorrectweexpectthesignontheindicatorvariabletobepositive, signifying a positive correlation between software price and bundling. From our model selection analysis and Hausman tests, we determine that the proper model toestimateisalinearinstrumentalvariablemodelwithxedeffects. P g b t = g b + 1 I[Bundle] t + 2 SoftwareTitles t + 3 Age g b t + m=11 P m=1 3+m Month m + g b t : TheresultsofthemodelwetaketothedataareinTableIII.5below,modeloneisa 99 pooled model, model two includes instrumental variables for number of software titleswhilemodelthreeincludessoftwarexedeffects. Whatisevidentfromthese results is that there is clear evidence of a positive correlation between standalone software price and whether the game was bundled with a console in period t. We also would like to note the sign corresponding to the number of software titles for model two and three is negative, and suggests that bundled games face competition from other software titles. We conclude that our second prediction fromthetheoreticalmodelalsoholdstothedata. TableIII.5: StandaloneBundledSoftwarePriceRegression Price Model1-OLS Model2-IV Model3-FEnoIV I(Bundle) 0:1708 0:3109 0:5084 (0:1501) (0:1695) (0:1616) #of SoftwareTitles 0:0006 0:0022 0:0032 (0:0007) (0:0013) (0:0022) Age 0:0206 0:0692 0:0860 (0:0147) (0:0237) (0:0330) SoftwareFE's No No Yes NumberofObs. 97 97 97 HausmanEndogeneityTest 9:74(0:0025) HausmanTestModel3vs. Model1 26:87(0:0298) AllmodelsincludeMonthxedeffectsandanunreportedconstant. **signicantat95%*signicantat90% 100 The last prediction we test is whether the royalty rate levied by Nintendo de- creaseswhenmixedbundlingisoffered. Unfortunately,royaltyrateisunobserved so we are unable to directly regress royalty rate on a set of covariates. However, weareabletodetermineifroyaltyratedecreasesindirectlywithasimpleassump- tion regarding the marginal cost of third-party games. We assume that marginal cost is constant, with the exception to the impact royalty rate has, over a soft- ware's life cycle (certainly not unrealistic). With this assumption and covariates which account for varying degrees of competition, we can infer that the royalty ratedecreasesifsoftwarepricesdecreasewiththepresenceofabundle. Alsonote that software competition does not increase as a result of the entry of a bundle sincethebundledgamehasalreadybeenonthemarketpriortothebundling. We implementthistestbyrestrictingthesetofvideogamestoonlythird-partygames and employing a regression with identical covariates as the above test with the exception of the indicator variable for whether the software was bundled in pe- riodt. Instead,weuseanindicatorvariablewhichtakesthevalueoneifabundle was offered in periodt and zero otherwise. Like the above two tests, a Box-Cox regression is rst performed to determine the correct model specication, and is thenfollowedbyHausmantests. For our third prediction as well as the entire theoretical model to hold to the data,theremustbeanevidenceofanegativecorrelationbetweenthird-partysoft- warepriceandthecorrespondingbundlemeasure. Thelastregressionweestimate 101 isanonlinearmodelwithxedeffects: P 0:5 gt = g + 1 I[Bundle] t + 2 SoftwareTitles 0:5 t + 3 Age gt + m=11 P m=1 3+m Month m + gt : TableIII.6: IndependentSoftwarePriceRegression Price Model1-OLS Model2-IV Model3-FEnoIV I(Bundle) 0:2353 0:2265 0:0469 (0:0159) (0:0159) (0:0161) #of SoftwareTitles 0:0929 0:0910 0:1492 (0:0020) (0:0020) (0:0063) Age 0:0265 0:0267 0:0018 (0:0006) (0:0006) (0:0028) SoftwareFE's No No Yes NumberofObs. 14;445 14;445 14;445 HausmanEndogeneityTest 65:49(0:0000) HausmanTestModel3vs. Model1 35:80(0:0011) AllmodelsincludeMonthxedeffectsandanunreportedconstant. **signicantat95%*signicantat90% Thepresenceofanegativecorrelation,withregardtotheimpactamixedbun- dle has on independent software price, is clearly evident from Table III.6 above. 102 In each of these models, the sign of the coefcient corresponding to whether a bundle was offered is negative with point estimates signicantly different from zero. Likewise, the measure of competition is negative and signicantlydifferent from zero. Consequently, as more games enter the software market, the prices of video games decrease. With this result in addition to the two tests run above, we show our theoretical monopoly model of mixed bundling in a two-sided market settingholdstorealworlddata. 8 Conclusion In this chapter, we further extends the traditional literature on bundling and the burgeoning literature on two-sided markets by presenting a theoretical monopoly model of mixed bundling, in the context of the portable video game console marketaprototypicaltwo-sidedmarket. Deviatingfrombothtraditionalbundling literature and standard two-sided markets literature, we nd that, under mixed bundling, both the standalone console price on the consumer side and the roy- alty rate on the game developer side are lower than their counterparts under in- dependent pricing equilibrium. 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[37] M.D.Whinston.Tying,foreclosure,andexclusion.TheAmericanEconomic Review,80(4):837859,sep1990. 107 Appendices Appendix II.A: Equilibrium Analysis: The Case of PerfectSubstitutes This is a sequential-move game with complete information, I am going to solve thegamebybackwardinduction. Date3: Retailer'sPurchaseDecision Webeginwiththelaststageofthegamethemonopolyretailer'schoiceon what to buy. With (T o ;Q o ;w A ) from manufacturer A andw B from manufacturer B on the table, the retailer could either buy from manufacturer A or not. Note that if the retailer decides to buy from manufacturer A, he will buy at least Q o units from A since the marginal price for purchase less thanQ o is zero while the marginal revenue is always positive from Assumption II.1. Thus, I denote the retailer'stwooptionsintotwocategories: (AA)whentheretaileracceptsA's3PT and(NA)whentheretailerrejectsA's3PT. Lemma4 (Retailer'sChoicein(AA))Under(AA), Ifw B <w e A ,thenq A =Q o andq B =q r (w B ;Q o ) =q m (w B )Q o > 0; Ifw e A w B ,thenq A =q A (w e A )andq B = 0. AppendixII.Acontinued 108 Hence,theretailer'sprotfunctioncanbesummarizedas r AA (w B ;T o ;Q o ;w A ) = 8 > < > : v(w B )+w B Q o T o ifw B <w e A v(w e A )+w e A Q o T o ifw e A w B : Proof of Lemma 4. In (AA) case, the retailer will in principle buy from both manufacturers. r AA = max q A Qo q B [R(q A +q B )w A (q A Q o )w B q B ]T o = max 0 q B [R(Q o ++q B )w A w B q B ]T o : Therearetwopossibilitiesforthiscase: If w B < w e A , then the retailer will stop buying from manufacturer A after fulllingQ o requirement,andbuytheextraunitsfrommanufacturerB.That is,q A =Q o andq B =q r (w B ;Q o ) =q m (w B )Q o > 0. r AA = max q A Qo q B [R(q A +q B )w A (q A Q o )w B q B ]T o = max q B [R(Q o +q B )w B q B ]T o : If w e A w B , then the retailer will buy exclusively from manufacturer A. AppendixII.Acontinued 109 Thatis,q A =q A (w e A )andq B = 0. r AA = max q A Qo q B [R(q A +q B )w A (q A Q o )]T o = 8 > < > : v(w A )+w A Q o T o ifw A <p(Q o ) R(Q o )T o ifw A p(Q o ) : Q.E.D. This lemma tells us the retailer's optimal purchase decision if accepting A's 3PT.NotethatmanufacturerBmustsetw B < w e A inordertohaveapositivesale when the retailer accepts A's 3PT. Graphically, it can be summarized in Figure II.3inChapterII. In(NA),theretailerwillbuyexclusivelyfrommanufacturerB. r NA (w B ) =v(w B ): Givensuchtwooptions(AA)or(NA),theretailerwillchoosetheonegiv- inghimhigherprot. Inotherwords,theretailerwillchoose(AA)if r AA (w B ;T o ;Q o ;w A ) r NA (w B ) and (NA) ifr AA (w B ;T o ;Q o ;w A ) < r NA (w B ). The following lemma tells us the properties of the retailer's prot curves associ- atedwiththesetwooptionswithrespectto(w.r.t.) w B . Lemma5 The properties of the retailer's prot curves for two options(AA) and(NA): (i) Slopes: In (AA), @r AA @w B = 1fw B < w e A g[q r (w B ;Q o )]; in (NA), @r NA @w B = AppendixII.Acontinued 110 1fw B <p(0)g[q m (w B )]. (ii)Cutoffs: w e A p(0). (iii)Relativesteepness: @r NA @w B @r AA @w B 0;8w B : ProofofLemma5. Part(i)followsdirectlyfrompartialdifferentiationw.r.t. w B . Part (ii) follows from the fact that w e A p(Q o ) p(0). Sinceq r (w B ;Q o ) = Q o q m (w B ) q m (w B );8w B < w e A and 0 @r NA @w B ;8w B w e A , part (iii) follows. Q.E.D. Part (i) tells us that the prot curves for (AA) and (NA) are both downward sloping before reaching a plateau w.r.t. w B . And from (ii), we know that the at partoftheprotcurvein(AA)emergesbeforethatoftheprotcurvein(NA).(iii) statesthattheprotcurveunder(AA)isneversteeperthantheprotcurveunder (NA)w.r.t. w B . AndtheprotcurvesfortwocaseisshowninFigureII.A-1. (AA) (NA) FigureII.A-1: TheRetailer'sProtCurvesin(AA)and(NA) From the properties of these prot curves, we know that there are only two possible cases: (r1) r NA is always above r AA ;8w B ; (r2) r NA crosses r AA from aboveat b w B , wherer NA (b w B ) = r AA (b w B ;T o ;Q o ;w A ). (r1)isimpossibleinequi- AppendixII.Acontinued 111 librium since we are looking for some prot improvement from LP or 2PT equi- librium and in (r1) manufacturer A would earn 0 prot for sure, which must be avoidedbymanufacturerAwhendesigningthecontract(T o ;Q o ;w A )atdate1,if possible. Thus,thepossibleprotimprovementcanonlyoccurin(r2). Lemma6 (Retailer'sChoice)Inequilibrium,9aunique b w B (T o ;Q o ;w A ), wherer NA (b w B ) = r AA (b w B ;T o ;Q o ;w A ). Andr NA (w B ) > r AA (w B ;T o ;Q o ;w A ), whenw B < b w B ;r NA (w B )r AA (w B ;T o ;Q o ;w A ),whenw B b w B . Thislemmatellsusthatin3PTequilibrium,thereexistsauniquecutoff b w B (T o ;Q o ;w A ) such that the retailer will not buy from A ifw B is lower than the cutoff,whilebuyfromAifw B ishigherthanthecutoff. Date2: ManufacturerB'sProblem From Lemma 6, we know that r NA must cross r AA uniquely from above at b w B . There are two possible cases: (B1)w e A b w B , thatis, they cross atr AA0 s at part;(B2) b w B <w e A ,thatis,theycrossatboth'sdownwardslopingparts. AppendixII.Acontinued 112 (B1)w e A b w B (B2) b w B <w e A FigureII.A-2: TwoPossibleCasesofCrossings In (B1), if w B < b w B , the retailer will buy exclusively from manufacturer B; ifw B b w B , the retailer will buy only from manufacturer A. Thus, manufacturer B'sprotis B = 1fw B < b w B g m (w B ): Inthiscase, b w B isdeterminedby v(b w B ) = v(w e A )+w e A Q o T o T o = v(w e A )+w e A Q o v(b w B ): (A.1) In (B2), if w B < b w B , the retailer will buy exclusively from manufacturer B; if b w B w B < w e A , theretailerwillbuyQ o frommanufacturerAandq r (w B ;Q o ) from manufacturer B; if w e A w B , the retailer will buy only from manufacturer AppendixII.Acontinued 113 A.Thus,manufacturerB'sprotis B = 8 > > > > < > > > > : m (w B ) ifw B < b w B r (w B ;Q o ) if b w B w B <w e A 0 ifw e A w B Inthiscase, b w B isdeterminedby v(b w B ) = v(b w B )+ b w B Q o T o T o = b w B Q o : (A.2) Ineither(B1)or(B2),manufacturerBwillchoosetheoptimalw B tomaximize B . Date1: ManufacturerA'sProblem Accordingly,bysubstitutingT o using(A.1)and(A.2),nowwecanwritedown manufacturerA'sprot. In(B1)w e A b w B , A = 8 > < > : 0 ifw B > > > < > > > > : 0 ifw B r NA (w e A ). Thatis, To Qo = b w B < w e A . Proof of Lemma 7. (By contradiction) Suppose not. That is, in equilibrium w e A b w B and hence it is case (B1). In this case, manufacturer A will denitely earn 0 prot if w B < b w B , so he must ensure manufacturer B's optimal price s.t. b w B w B . However, manufacturer B will get 0 prot when b w B w B . As a result, the only possibility for (B1) to be the equilibrium is that manufacturer A makessure b w B csothatw e A b w B w B =c. However, A =v(w e A )+(w e A c)q m (w e A )v(b w B )v(c)v(b w B ) 0,whichcontradictswithourobjectiveof lookingforprotimprovement. Sothelemmafollows. Q.E.D. This lemma shows that only (B2) is possible in equilibrium. Moreover, it showsthattheaveragepriceatthequantitythresholdislessthanthehighestprice manufacturer B can charge for a positive sale when the retailer accepts A's 3PT. Thisleavessomeroomforthefollowertosupplytheresidualdemandatahigher pricethantheaveragepricefromAatthequantitythreshold. Besides, for positive prot, manufacturer A will design its 3PT such that the averagepriceatthequantitythresholdisstrictlyhigherthanthecost. AppendixII.Acontinued 115 Lemma8 Inequilibrium,c b w B and A = (b w B w e A )Q o +(w e A c)q m (w e A )< 0,whichcontradictswithourobjective. Second,c< b w B . Suppose not. That is, c b w B . Then b w B c < w B < w e A and A = (b w B c)Q o 0. Q.E.D. Therefore, in equilibrium, the retailer's prot curves must cross as shown in FigureII.A-3. FigureII.A-3: InEquilibrium,c< b w B <w e A CharacterizationoftheEquilibrium WebeginwiththepropertiesrelatingtomanufacturerB'sprotcurves,which followsfromdirectcomputation. Lemma9 (1) m (w) r (w;Q o )with"="atw =c; (2) m0 (w) r0 (w;Q o );8Q o : AppendixII.Acontinued 116 FromLemma8c > > > < > > > > : m B (w B ) ifcw B > > > < > > > > : 0 ifw B < b w B (b w B c)Q o if b w B w B <w e A b w B Q o w e A Q o +(w e A c)q m (w e A ) ifw e A w B : We know that, manufacturer B would never choose w e A w B since it would earn zero then. Thus, for possible positive prot, manufacturer A must ensure b w B w B < w e A . And this is equivalent to r (w B ;Q o ) max x<b w B m (x) 0, wherethesecondinequalitycomesfromc< b w B . Lemma10 Inequilibrium, r (w B ;Q o ) max x<b w B m (x). AppendixII.Acontinued 117 SomanufacturerA'sproblemcanbewrittenas max To;Qo;w A (b w B c)Q o s:t: r (w B ;Q o ) max x<b w B m (x) (A.3) w B = argmax b w B x<w e A r (x;Q o ) (A.4) b w B = T o Q o : (A.5) Notethatthewholegamenowisreducedtoamechanismdesignproblemfrom manufacturer A's point of view. Constraint (A.3) is equivalent to an incentive- compatibility constraint in the standard mechanism design problem. Constraint (A.4)isthedenitionofw B . Andconstraint(A.5)isthecrossingpointcondition fromretailer'soptimalchoice. Lemma11 (i) Constraint (A.3) and (A.4) imply that m0 (b w B ) > 0 and hence max x<b w B m (x) = m (b w B ): (ii)w B mustbeaninteriorsolutions.t. b w B <w B <w e A and r0 (w B ;Q o ) = 0. Thatis, m0 (w B ) =Q o : (iii) Constraint (A.3) must be binding. That is, m (w B ) (w B c)Q o = m (b w B ): ProofofLemma11. Part(i): (Bycontradiction)Supposenot. Thatis, m0 (b w B ) 0. From Lemma 9, we know that r0 (b w B ;Q o ) m0 (b w B ) 0, which im- pliesw B = argmax b w B x<w e A r (x;Q o ) = r (b w B ;Q o ). However, r (b w B ;Q o ) m (b w B ) with "=" only when b w B = c from Lemma 9. So for (A.3) to hold, AppendixII.Acontinued 118 we must have w B = b w B = c. However, then m0 (b w B ) = m0 (c) > 0 con- tradicts with our supposition that m0 (b w B ) 0. Thus, m0 (b w B ) > 0. And max x 0 or b w B < w B . Moreover, since w e A does not enter the objective function, and we can increase it without any problem be- causew e A p(Q o ) holds by the denition ofw e A , and r (x;Q o ) x=p(Qo) = 0 and r0 (x;Q o ) x=p(Qo) < 0. Sopart(ii)ofthelemmafollows. Part(iii): (Bycontradiction)Supposenot. Noticethat @ A @b w B =Q o 0: Then we can increase A simply by increasing b w B until (A.3) becomes binding. Q.E.D. Besides, we needw B w e A = minfw A ;p(Q o )g. Andp(Q o ) > w B automat- ically holds since r (x;Q o ) x=p(Qo) = 0 and r0 (x;Q o ) x=p(Qo) < 0. Thus, we onlyneedw B w A . AppendixII.Acontinued 119 Consequently,manufacturerA'sproblemcanbewrittenas max To;Qo;w A (b w B c)Q o s:t: m (w B )(w B c)Q o = m (b w B ) m0 (w B ) =Q o b w B = T o Q o w B w A Note that w A does not enter the objective function, and we always can guaran- tee w B w A holds by setting w A sufciently high. Meanwhile, (T o ;Q o ) can be uniquely determined by b w B = To Qo and m0 (w B ) = Q o . Instead of choosing (T o ;Q o ),itisequivalentformanufacturerAtoset(w B ;b w B )s.t. (w B ;b w B ) = argmax (x;y) 8 > < > : (yc) m0 (x) s:t: m (y) = m (x)(xc) m0 (x) 9 > = > ; : RecallthatwithAssumptionII.3,weknowthat9auniquepeakw s.t. w = argmax x2[c;w m ] h(w) and h 0 (w) > 0;8w < w ; h 0 (w) < 0;8w > w . This assumptioncanhelpguaranteetheexistenceanduniquenessoftheproblem. Lemma12 (Existence and Uniqueness) Under Assumption II.3, there exists a AppendixII.Acontinued 120 uniqueinteriorsolution(w ;b w )2 (c;w m )(c;w m )totheproblem max (x;y) (yc) m0 B (x) s:t: m (y) = m (x)(xc) m0 (x): Proof of Lemma 12. Write down the Lagrangian L = (b w c) m0 (w ) + [ m (b w) m (w )+(w c) m0 (w )]: (b w) : m0 (w )+ m0 (b w) = 0 (w ) : (b wc)+(w c) = 0 () : m (b w) = m (w )(w c) m0 (w ) Eliminating,weknow(w ;b w )ischaracterizedby h(b w) = h(w ) (A.6) m (b w) = m (w )h(w ): (A.7) Noticethatatboth(c;c)and(w m ;w m ),(A.6)and(A.7)hold. From(A.7)and h(w)> 0;8c<w <w m ,weknowthat b w <w . Differentiate(A.7)w.r.t. w : db w dw = (w c)[ m00 (w )] m0 (w ) > 0 AppendixII.Acontinued 121 Differentiate(A.6)w.r.t. w : db w dw = h 0 (w ) h 0 (b w) < 0;8b w <w <w : ThisisfollowedfromAssumptionII.3. And(A.6)startsfrom (w ;w )andendsat (c;w m )when b w < w . Moreover, b w =w alsosatisfy(A.6). Hence,(A.6)and(A.7)mustbluelineandredlinerespectivelyinFigureII.A- 4below. FigureII.A-4: ExistenceandUniqueness Asaresult,theexistenceanduniquenessareeasilyfollowed. Q.E.D. From the equivalence of two optimization problems, we know that the equi- libriummustexistanditisuniquelydetermined. ThePropositionII.2follows. ProofofCorollaryII.1. Inequilibrium, b w B > c,Q o > 0andthus A = (b w B c)Q o > 0: B = r (w B ;Q o ) = m (b w B ) = (b w B c)q m (b w B )> (b w B c)Q o = A , wheretheinequalityfollowsfrom b w B <w e A p 0 (Q o ).Q.E.D. 122 Appendix II.B: Equilibrium Analysis: The Case of ImperfectSubstitutes Before proceeding, for the purpose of characterizing the demand system, let us rststudythepropertiesoftwocurvesinq A q B plane. The Properties of Two Curves inq A q B Plane: R 1 (q A ;q B ) = w A , and R 2 (q A ;q B ) =w B Lemma13 dq B dq A R 1 (q A ;q B )=w A <1< dq B dq A R 2 (q A ;q B )=w B < 0: Proof of Lemma 13. It follows directly from differentiating two curves with respecttoq A andq B respectively,andthenapplyingAssumptionII.4. Q.E.D. Thislemmatellsusthatinq A q B plane,foranygiven(w A ;w B ),bothcurves R 1 (q A ;q B ) = w A and R 2 (q A ;q B ) = w B are downward sloping. Moreover, the slope of R 1 (q A ;q B ) = w A is always steeper than the slope of R 2 (q A ;q B ) = w B . ThetypicaltwocurveswithnotationsareshowninFigureII.5inChapterII. This is a sequential-move game with complete information, I am going to solvethegamebybackwardinduction. Date3: Retailer'sPurchaseDecision Thesameasouranalysisforthecaseofperfectsubstitutes,Idividetheretailer hastwobasicoptions: (AA)whentheretaileracceptsA's3PTand(NA)whenthe retailerrejectsA's3PT. AppendixII.Bcontinued 123 Under (AA), the retailer will buy at leastQ o units because the marginal price withinthatisalwayszero. Andretailer'sprotunder(AA)is r AA = max q A Qo q B [R(q A ;q B )w A (q A Q o )w B q B ]T o = max 0 q B [R(Q o +;q B )w A w B q B ]T o : Herethetwoproductsaredifferentiatedsothattheretailerwillnotcompletely switch to the other source with lower price. The all-or-nothing purchasing rule fromtheretailerdisappearnow. Thus,thedecisionafterfulllingQ o requirement becomesmorecomplicatedthanthecaseofhomogeneousproducts. (a) Whenw A p A (Q o ;0), the retailer will stop buying from manufacturer A atQ o . TheextrademandonB,ifany,willdependonw B . Ifw B p B (Q o ;0),thenq B = 0. Ifw B <p B (Q o ;0),thenq B =q r B (w B ;Q o ). (b)Whenw B p B (Q o ;0),thentheretailerwillneverbuyfromBaftergetting Q o frommanufacturerA.Andthusq B = 0. Ifw A p A (Q o ;0),thenq A =Q o . Ifw A < > : w A <p A (Q o ;0) w B <p B (Q o ;0) , from the properties of two curves by Lemma 13, there are three possible cases. And their phase diagrams are shown in Figure AppendixII.Bcontinued 124 II.B-1below. (a)(Q o ;0) (b)(q A (w A ;w B );q B (w A ;w B )) (c)(Q 0 ;q r B (w B ;Q 0 )) FigureII.B-1: ThePhaseDiagramsofThreePossibleCases when 8 > < > : w A Q o ifandonlyifw A 0ifandonlyifw B < > : w A 1> dw B dw A w B =p B (q m A (w A );0) > 0: (iii)w A =p A (Q o ;q B (w B ;Q o ))()w B =p B (Q o ;(w A ;Q o )): Proof of Lemma 14. For the curve w A = p A (Q o ;q r B (w B ;Q o )), note that when w B = p B (Q o ;0), q r B (w B ;Q o ) = 0, and hence w A = p A (Q o ;0). Differentiate w A =R 1 (Q o ;q B (w B ;Q o ))w.r.t. w A : 1 = R 12 @q r B (w B ;Q o ) @w B dw B dw A ) dw B dw A = 1 R 12 1 @q r B (w B ;Qo) @w B = R 22 R 12 (* @q r B (w B ;Q o ) @w B = 1 R 22 ) > 1(*R 22 <R 12 < 0): For the curve w B = p B (q m A (w A );0), we can prove its corresponding part by thesamevein. *R 1 (Q o ;(w A ;Q o )) =w A =R 1 (Q o ;q r B (w B ;Q o )) AppendixII.Bcontinued 126 ) q r B (w B ;Q o ) = (w A ;Q o ): And we know that w B = R 2 (Q o ;q r B (w B ;Q o )) by q r B (w B ;Q o )'s denition. Therefore, the equivalence between two curves fol- lows. Q.E.D. This lemma says, in w A w B plane, the curve w A = p A (Q o ;q B (w B ;Q o )) passes (p A (Q o ;0);p B (Q o ;0)) with positive slope strictly greater than 1; and the curve w B = p B (q m A (w A );0) passes (p A (Q o ;0);p B (Q o ;0)) with positive slope strictly less than 1. In addition, the curve w A = p A (Q o ;q r B (w B ;Q o )) is equiv- alent to w B = p B (Q o ;(w A ;Q o )). Thus, the curve w B = p B (Q o ;(w A ;Q o )) is belowthecurvew B =p B (q m A (w A );0)when 8 > < > : w A < > : w A <p A (Q o ;0) w B <p B (Q o ;0) , Ifw B <p B (Q o ;(w A ;Q o )),thentheretailerwillbuy(Q o ;q r B (w B ;Q o )). If p B (Q o ;(w A ;Q o )) w B < p B (q m A (w A );0), then the retailer will buy (q A (w A ;w B );q B (w A ;w B )). Ifp B (q m A (w A );0)w B ,thentheretailerwillbuy(q m A (w A );0). Combining all the cases (a)~(c) under (AA), the retailer's optimal response canbesummarizedinFigureII.6inChapterII,andstatedinthelemmabelow. Lemma15(Retailer'sChoicein(AA)) Under(AA), Whenw A <p A (Q o ;0)(AA-ihereafter), AppendixII.Bcontinued 127 Ifw B <p B (Q o ;(w A ;Q o )),thentheretailerwillbuy(Q o ;q r B (w B ;Q o )). Andtheretailer'sprotis r AA = max y [R(Q o ;y)w B y]T o : If p B (Q o ;(w A ;Q o )) w B < p B (q m A (w A );0), then the retailer will buy(q A (w A ;w B );q B (w A ;w B )). Andtheretailer'sprotis r AA =v(w A ;w B )+w A Q o T o : If p B (q m A (w A );0) w B , then the retailer will buy (q m A (w A );0). And theretailer'sprotis r AA =v(w A ;1)+w A Q o T o : Whenw A p A (Q o ;0),(AA-iihereafter) If w B < p B (Q o ;0), then the retailer will buy (Q o ;q r B (w B ;Q o )). And theretailer'sprotis r AA = max y [R(Q o ;y)w B y]T o : Ifp B (Q o ;0)w B ,thentheretailerwillbuy(Q o ;0).Andtheretailer's protis r AA =R(Q o ;0)T o : AppendixII.Bcontinued 128 This lemma tells us the retailer's optimal purchase decision if accepting A's 3PT.NotethatmanufacturerBmustsetw B <w B inordertohaveapositivesale under(AA).Graphically,itcanbesummarizedinFigureII.6inChapterII. In(NA),theretailerwillbuyexclusivelyfrommanufacturerB. r NA (w B ) =v(1;w B ): Givensuchtwooptions(AA)or(NA),theretailerwillchoosetheonegiv- inghimhigherprot. Inotherwords,theretailerwillchoose (AA)ifr AA (w B ;T o ;Q o ;w A )r NA (w B ) and(NA)ifr AA (w B ;T o ;Q o ;w A )<r NA (w B ). The following lemma tells us the properties of the retailer's prot curves as- sociatedwiththesetwooptionswithrespectto(w.r.t.) w B . Lemma16 The properties of the retailer's prot curves for two options(AA) and(NA): (i)Slopes: In(AA-i), @r AA @w B = 8 > > > > < > > > > : q r B (w B ;Q o ) ifw B <p B (Q o ;(w A ;Q o )) q B (w A ;w B ) ifp B (Q o ;(w A ;Q o ))w B <p B (q m A (w A );0) 0 ifp B (q m A (w A );0)w B ; in(AA-ii), @r AA @w B = 1fw B <p B (Q o ;0)g[q r B (w B ;Q o )]; in(NA), @r NA @w B = 1fw B <p B (0;0)g[q m B (w B )]: (ii)Cutoffs: w B R 2 (0;0): AppendixII.Bcontinued 129 (iii)Relativesteepness: @r NA @w B @r AA @w B 0;8w B : Proof of Lemma 16. Part (i) follows directly from partial differentiation w.r.t. w B . Forpart(ii), * minfQ o ;q m A (w A )g 0 ) R 2 (0;0) maxfR 2 (q m A (w A );0);R 2 (Q o ;0)g fromR 21 < 0. Andpart(iii)followsfrom q m B (w B ) = q r B (w B ;0)q r B (w B ;Q o );8Q o 0 q m B (w B ) = q r B (1;w B )q B (w A ;w B );8w A : Q.E.D. (i)tellsusthattheprotcurvesfor(AA)and(NA)arebothdownwardsloping before reaching a plateau w.r.t. w B . And from (ii), we know that the at part of theprotcurvein(AA)emergesbeforethatoftheprotcurvein(NA).(iii)states that the prot curve under (AA) is never steeper than the prot curve under (NA) w.r.t. w B . Notethatin(AA-i),therearethreesegmentsratherthantwo,compared with(AA-ii). AppendixII.Bcontinued 130 AndtheprotcurvesforeachcaseisshowninFigureII.B-2. (AA-i) (AA-ii) (NA) FigureII.B-2: TheRetailer'sProtCurvesin(AA)and(NA) From the properties of these prot curves, we know that there are only two possible cases: (r1) r NA is always above r AA ;8w B ; (r2) r NA crosses r AA from aboveat b w B , wherer NA (b w B ) = r AA (b w B ;T o ;Q o ;w A ). (r1)isimpossibleinequi- AppendixII.Bcontinued 131 librium since we are looking for some prot improvement from LP or 2PT equi- librium and in (r1) manufacturer A would earn 0 prot for sure, which must be avoidedbymanufacturerAwhendesigningthecontract(T o ;Q o ;w A )atdate1,if possible. Thus,thepossibleprotimprovementcanonlyoccurin(r2).r NA (b w B ) = r AA (b w B ;T o ;Q o ;w A ). And Lemma17 (Retailer'sChoice)Inequilibrium,9auniqueb w B (T o ;Q o ;w A ),where r NA (b w B ) = r AA (b w B ;T o ;Q o ;w A ). And r NA (w B ) > r AA (w B ;T o ;Q o ;w A ), when w B < b w B ;r NA (w B )r AA (w B ;T o ;Q o ;w A ),whenw B b w B . Thislemmatellsusthatin3PTequilibrium,thereexistsauniquecutoff b w B (T o ;Q o ;w A ) such that the retailer will not buy from A ifw B is lower than the cutoff,whilebuyfromAifw B ishigherthanthecutoff. Date2: ManufacturerB'sProblem From Lemma 17, we know that r AA must cross r NA uniquely from above at b w B . Broadly speaking, there are two possible cases: (B1)w B b w B , that is, they cross at r AA0 s at part; (B2) b w B < w B , that is, they cross at both's downward sloping parts. As we will see, because of the more cutoffs in (B2) case in the case of differentiated products than in the case of homogeneous products, there aremanysubcasesin(B2),whichcomplicateouranalysishere. AppendixII.Bcontinued 132 FigureII.B-3: (B1)w B b w B In (B1), if w B < b w B , the retailer will buy exclusively from manufacturer B; if w B b w B , the retailer will buy only from manufacturer A. Thus, manufacturer B'sprotinthiscaseis B = 1fw B < b w B g m B (w B ): Inthiscase, b w B isdeterminedby T o =v(w e A ;1)+w e A Q o v(1;b w B ): (B.1) In (B2), according to the possibilities of b w B falling into different ranges, I furthercategorizetheminto3subcases: (B2-i-a): 8 > < > : w A < > : w A < > : p A (Q o ;0)w A b w B < > : w A <p A (Q o ;0) p B (Q o ;(w A ;Q o )) b w B <p B (q m A (w A );0) In (B2-i-a), if w B < b w B , the retailer will buy exclusively from manufacturer B; if b w B w B < p B (q m A (w A );0), the retailer will buyq A (w A ;w B ) andq B (w A ;w B ) from A and B, respectively; if p B (q m A (w A );0) w B , the retailer will buy only frommanufacturerA.Thus,manufacturerB'sprotinthiscaseis B = 8 > > > > < > > > > : m B (w B ) ifw B < b w B B (w A ;w B ) if b w B w B < > : w A <p A (Q o ;0) b w B <p B (Q o ;(w A ;Q o )) In(B2-i-b),ifw B < b w B ,theretailerwillbuyexclusivelyfrommanufacturerB;if b w B w B <p B (Q o ;(w A ;Q o )),theretailerwillbuyQ o fromAandq r B (w B ;Q o ) from B; if p B (Q o ;(w A ;Q o )) w B < p B (q m A (w A );0), the retailer will buy q A (w A ;w B )andq B (w A ;w B )fromAandB,respectively;ifp B (q m A (w A );0)w B , the retailer will buy only from manufacturer A. Thus, manufacturer B's prot in thiscaseis B = 8 > > > > > > > < > > > > > > > : m B (w B ) ifw B < b w B r B (w B ;Q o ) if b w B w B <p B (Q o ;(w A ;Q o )) B (w A ;w B ) ifp B (Q o ;(w A ;Q o ))w B < > : b w B <p B (Q o ;0) p A (Q o ;0)w A In (B2-ii), ifw B < b w B , the retailer will buy exclusively from manufacturer B; if b w B w B > > > < > > > > : m B (w B ) ifw B < b w B r B (w B ;Q o ) if b w B w B <p B (Q o ;0) 0 ifp B (Q o ;0)w B : Inthiscase, b w B isdeterminedby T o = max y [R(Q o ;y) b w B y]v(1;b w B ): (B.4) Inanycaseabove,manufacturerBchoosestheoptimalw B tomaximize B . Date1: ManufacturerA'sProblem Correspondingly, by substituting T o using (B.1), (B.2), (B.3) and (B.4), now wecanwritedownmanufacturerA'sprotineachcase. AppendixII.Bcontinued 136 In(B1), A = 8 > < > : 0 ifw B > > > > > > > > > < > > > > > > > > > > : 0 ifw B > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > : 0 ifw B < b w B max y [R(Q o ;y) b w B y] v(1;b w B )cQ o if b w B w B <p B (Q o ;(w A ;Q o )) max y [R(Q o ;y) b w B y] v(1;b w B )cQ o +(w A c)[q A (w A ;w B )Q o ] ifp B (Q o ;(w A ;Q o ))w B < > : 0 ifw B < b w B = max y [R(Q o ;y) b w B y]v(1;b w B )cQ o if b w B w B : As can be seen, although manufacturer A is setting (T o ;Q o ;w A ), it is equiva- lentformanufacturertodesign(b w B ;Q o ;w A )asT o canbesubstitutedby b w B (T o ;Q o ;w A ). As will be shown soon, (B1) and (B2-i-a) will be eliminated fromtheequilibrium,andonly(B2-i-b)and(B2-Both-ii)willemergeastheequi- librium. Lemma18 Ifw B c,thenw e A <c. Proof of Lemma 18. * w B = p B (q m A (w e A );0) by its denition, and c < p B (q m A (c);0) from Assumption II.5. ) w B c () p B (q m A (w e A );0) c r NA (w B ). That is, b w B < w B . Proof of Lemma 19. (By contradiction) Suppose not. That is, in equilibrium w B b w B andhenceitiscase(B1). Step1: Inequilibrium,r AA (w B ;T o ;Q o ;w A )r NA (w B ),thatis, b w B w B . In this case, manufacturer A will denitely earn 0 prot if w B < b w B , so he must ensure manufacturer B's optimal price s.t. b w B w B . However, manufac- turer B will get 0 prot when b w B w B . As a result, the only possibility for AppendixII.Bcontinued 138 (B1) to be the equilibrium is that manufacturer A makes sure b w B c so that w B =c b w B . ByLemma18,wehavew e A <cin(B1)sincew B b w B c. And manufacturerA'sprotin(B1)is A = v(w e A ;1)+(w e A c)q m A (w e A )v(1;b w B ) = R(q m A (w e A );0)cq m A (w e A )v(1;b w B ): Note that R(q m A (w e A );0) c q m A (w e A ) is increasing in w e A since w e A < c. And b w B (T o ;Q o ;w A )canbekeptasconstantbyadjustingT o appropriately. Therefore, we can always increase A by increasingw e A (w A ;Q o ) while keeping b w B as con- stantuntil b w B =w B =p B (q m A (w e A );0). Inotherwords,inequilibrium, r AA (w B ;T o ;Q o ;w A )r NA (w B ): Step2: From b w B = w B ,therealwaysexistsastrictlyprotabledeviation byletting b w B <w B . From b w B = w B = p B (q m A (w e A );0), A = R(q m A (w e A );0) c q m A (w e A ) v(1;p B (q m A (w e A );0)): d A dw e A = [R 1 (q m A (w e A );0)c @v @w B R 21 (q m A (w e A );0)]q m0 A (w e A ): Recallthatw e A <cin(B1),thersttwotermsinthebracketR 1 (q m A (w e A );0)c< 0. Meanwhile, @v @w B R 21 (q m A (w e A );0) < 0 since @v @w B < 0 and R 21 < 0. Thus, d A dw e A > 0. Wecanfurtherincrease A byincreasingw e A until b w B =w B =c: AppendixII.Bcontinued 139 DenotemanufacturerA'sprotwhen b w B =w B =cas b A = R(q m A (b w);0)cq m A (b w)v(1;c) = R( b Q;0)c b Qv(1;c) wherep B (q m A (b w);0) =cand b Q =q m A (b w). Noticethat b w <cfromLemma18: db A db w = [p A (q m A (b w);0)c]q m0 A (b w)> 0 db A d b Q = p A ( b Q;0)c< 0: Thus, b A < R(q m A (w A );0)cq m A (w A )v(1;c);8b w <w A <c (B.5) b A < R(Q o ;0)cQ o v(1;c);8q m A (c)<Q o < b Q: (B.6) Nowwearegoingtoconstructastrictlyprotabledeviationfrom(B1)'sb w B = w B = cusingtheabove(B.5)and(B.6). Notethatwhen b w B = w B = c,wehave w e A = b w: (AA-i):w A b w alittlebits.t. b w < w A <p A (Q o ;0) (B.7) b w < w A <c (B.8) b w B = c (B.9) p B (Q o ;(w A ;Q o )) c (B.10) Here (B.7) and (B.8) are feasible since b w < p A (Q o ;0) and b w < c. So is the case for (B.9) because b w B (T o ;Q o ;w A ) can be kept as constant by appropriately adjusting T o . And (B.10) is for the purpose of making sure p B (Q o ;(w A ;Q o )) c < w B < p A (q m A (w A );0) = w B . And (B.10) is feasible since p A (Q o ;(b w;Q o )) = b w < c = p A (Q o ;(c;Q o )) by the de- nition of (w A ;Q o ) and the fact that p B (Q o ;(w;Q o )) is increasing in w. Meanwhile,p B (Q o ;(w A ;Q o ))c =p B (Q o ;q r B (c;Q o ))()(w A ;Q o ) q B (c;Q o ) () w A = p A (Q o ;(w A ;Q o )) p A (Q o ;q r B (c;Q o )). Thus, (B.10) is equivalent to w A p A (Q o ;q r B (c;Q o )). Hence, (B.7)~(B.10) can besummarizedto b w < w A minfc;p A (Q o ;q B (c;Q o ))g b w B (T o ;Q o ;w A ) = c: Insuchadeviation,wemusthavep B (Q o ;(w A ;Q o ))c<w B AppendixII.Bcontinued 141 <p B (q m A (w A );0) =w B . And the retailer will choose (q A (w A ;w B );q B (w A ;w B )): In addition, b w B = c v(w A ;c)v(1;c)+(w A c)q m A (w A ) ( * q m A (w A ) =q A (w A ;1)>q A (w A ;w B )andw A <c) > v(w A ;1)v(1;c)+(w A c)q m A (w A ) = R(q m A (w A );0)cq m A (w A )v(1;c) > b A (Byinequality(B.5)) Consequently, in AA-i, if we start from (B1)'s b w B = w B = c, then we can alwaysndastrictlyprotabledeviations.t. w B >c = b w B . (AA-ii):p A (Q o ;0)w A :Inthisscenario,w e A =p A (Q o ;0)p A (q m A (w A );0). Fromthefactthatb A =R( b Q;0)c b Qv(1;c),wecanseethatw A does not enter the objective function. So without loss of generality, we can re- strict our attention to the case whenR 1 (Q o ;0) < w A . Starting from (B1)'s b w B =w B =c,thatis,Q o = b Q>q m A (w A ),wecandecreaseQ o q m A (w A ). So is the case for (B.12) because b Q = q m A (b w) > q m A (c) resulting from b w < c. Parallel to the argument in Both-i case, b w B (T o ;Q o ;w A ) can be kept as constant by appropriately adjustingT o . Hence,(B.11)~(B.13)canbesummarizedto maxfq m A (w A );q m A (c)g < Q o < b Q b w B (T o ;Q o ;w A ) = c: In such a deviation, we must have c < w B < p A (Q o ;0) = w B . And the retailer will choose (Q o ;q r B (w B ;Q o )): In addition, b w B = c R(Q o ;0)cQ o v(1;c) ( * Q o p B ( b Q;0) =c) > b A (Byinequality(B.6)) AppendixII.Bcontinued 143 Consequently, in Both-ii, if we start from (B1)'s b w B = w B = c, then we canalwaysndastrictlyprotabledeviations.t. w B >c = b w B . Inshort,wehaveshownthatinequilibrium,wemusthave (i)r AA (w B ;T o ;Q o ;w A ) r NA (w B ) (or equivalently b w B w B ); (ii) For any r AA (w B ;T o ;Q o ;w A ) = r NA (w B ) (or equivalently w B = b w B ), we could nd a strictly protable deviation s.t. w B > c = b w B . As a result, we can conclude that r AA (w B ;T o ;Q o ;w A )>r NA (w B )mustbethecaseinequilibrium. Q.E.D. This lemma shows only (B2) is possible in equilibrium. And the following lemmashowstheorderingofthesetwocutoffswithmarginalcostinequilibrium. Lemma20 Inequilibrium,cc: (By contradiction) Suppose not. That is, w B c. By Lemma 19, we have b w B <w B c. Inthiscase,w B =cw B . A =T o w e A Q o +(w e A c)q m A (w e A ) is increasing in T o , and thus we can increase A by increasing T o until b w B = w B c since b w B will increase withT o . Then the argument in Lemma 19 can be exactlyappliedhereandwecanalwaysndastrictlyprotabledeviationthen. Second,givenw B >c,Ishow b w B >c. (Bycontradiction)Supposenot. Thatis,b w B c. Sincew B >c,manufacturer Bmustset b w B <c<w B <w B . AppendixII.Bcontinued 144 In(B2-i-a), A =v(w A ;b w B )+(w A c)q A (w A ;w B )v(1;b w B ): In(B2-i-b), A = 8 > > > > > > > > > > < > > > > > > > > > > : max y [R(Q o ;y) b w B y] v(1;b w B )cQ o ifw B <p B (Q o ;(w A ;Q o )) max y [R(Q o ;y) b w B y] v(1;b w B )w A Q o +(w A c)q A (w A ;w B ) ifp B (Q o ;(w A ;Q o ))w B csincethiswillnotviolatemanufacturer B's outside option by setting w B b w B and being a monopoly for any b w B c: Therefore,when b w B c,thereisalwaysastrictlyprotabledeviation. Q.E.D. However,aswecanseefromthefollowinglemma,(B2-i-a)isreducedto2PT equilibriumandthuscanbeeliminatedfromour3PTimprovement. Lemma21 (B2-i-a) results in the same prot for manufacturer A as that in 2PT equilibrium. AppendixII.Bcontinued 145 ProofofLemma21. In(B2-i-a),manufacturerA'sprotis A = 8 > > > > > > > > > > < > > > > > > > > > > : 0 ifw B > > > < > > > > : m B (w B ) ifw B < b w B B (w A ;w B ) if b w B w B <p B (q m A (w A );0) 0 ifp B (q m A (w A );0)w B : SinceBwouldneversetp B (q m A (w A );0)w B ,theonlywayforAtoearnstrictly positiveprotistomakesuremax b w B w B <p B (q m A (w A );0) B (w A ;w B ) max w B <b w B m B (w B ). So by the same reasoning as the homogeneous products case,wecanwriteA'sproblemas max w A ;b w B v(w A ;b w B )v(1;b w B )+(w A c)q A (w A ;B(w A )) s:t: B (w A ;B(w A )) = m B (b w B ); whichisexactlythesameas2PTone.Q.E.D. AppendixII.Bcontinued 146 Thenextlemmashowsthat(B2-i-b)atleastmatchestheprotin2PTequilib- rium. Lemma22 (B2-i-b)givesatleastthesameamountofprotin2PTequilibrium. ProofofLemma22. In(B2-i-b),whenp B (Q o ;(w A ;Q o ))w B <p B (q m A (w A );0); A = max y [R(Q o ;y) b w B y]v(1;b w B )w A Q o +(w A c)q A (w A ;w B ): Simply letting Q o = q A (w 2PT A ;b w 2PT B );b w B = b w 2PT B ;w A = w 2PT A results exactly thesameprotin2PTequilibrium,withtheconstraint m B (b w B ) = B (w A ;B(w A )) satised. Q.E.D. Since we are looking for a prot improvement over LP or a 2PT, the two lemmas above tell us that we can focus on (B2-i-b) and (B2-ii) only. It turns out they are equivalent and can be synthesized to one case, which is summarized in Proposition II.5. And from the characterization of the equilibrium, we will see it improvesmanufacturerA'sprotovera2PT. CharacterizationoftheEquilibrium Before proceeding, I make a single-crossing assumption regarding rm B's protfunctions. AssumptionIII.1 (Single-CrossingProts)Assumeboth B (w A ;w)and r B (w;Q o )areconcaveinw;8w A ;8Q o . Moreover, @ B (w A ;w) @w@w A > 0; @ B (w;Qo) @w@Qo < 0: AppendixII.Bcontinued 147 ThisassumptionessentiallyrepresentsthesubstitutabilitybetweenAandB.It says manufacturer B's marginal prot is increasing in A's price while decreasing in A' quantity. And as can be easily checked, both general differentiated linear demandsandCESdemandssatisfythisassumption. We begin with some results relating to manufacturer B's prot curves, which willbeusedsoon. Lemma23 (1) m B (w) B (w A ;w) with "=" at w = c; m B (w) r B (w;Q o ) with"="atw =c. (2) m0 B (w) 0 B (w;w A );8w A : m0 B (w) 0 B (w;Q o );8Q o : Proof of Lemma 23. (1) m B (w) = (w c)q m B (w) = (w c)q B (1;w) > (w c)q B (w A ;w) = B (w A ;w), where the inequality comes from q B (w A ;w) is increasing in w A due to the substitutability between A and B. m B (w) = (w c)q m B (w) = (wc)q r B (w;0)> (wc)q r B (w;Q o ) = B (w;Q o ),wheretheinequal- ity comes fromq r B (w;Q o ) is decreasing inQ o due to the substitutability between AandB. (2) * m B (w) = B (w A = 1;w) = r B (w;Q o = 0) ) This part follows directlyfromAssumptionIII.1. Q.E.D. The next lemma tells us the properties of two prot curves for manufacturer B r B (w B ;Q o )and B (w A ;w B ). Lemma24 (i) If w B c, AppendixII.Bcontinued 148 (ii-a) r B (w B ;Q o ) w B =p B (Qo;(w A ;Qo)) = B (w A ;w B ) w B =p B (Qo;(w A ;Qo)) (ii-b)Ifcw B p B (Q o ;(w A ;Q o )),then r B (w B ;Q o ) B (w A ;w B ); ifp B (Q o ;(w A ;Q o ))<w B ,then r B (w B ;Q o )> B (w A ;w B ). (iii)Atw B =p B (Q o ;(w A ;Q o )), 0 B (w A ;w B ) w B =p B (Qo;(w A ;Qo)) < r0 B (w B ;Q o ) w B =p B (Qo;(w A ;Qo)) . Proof of Lemma 24. Part (i): At w B = p B (Q o ;(w A ;Q o )), q r B (w B ;Q o ) = (w A ;Q o ) =q B (w A ;w B ),q A (w A ;w B ) =Q o . Ifw B <p B (Q o ;(w A ;Q o )),thenq A (w A ;w B )<q A (w A ;p B (Q o ;(w A ;Q o ))) = Q o . Andp B (Q o ;q B (w B ;Q o )) = w B = p B (q A (w A ;w B );q B (w A ;w B )). Hence, q r B (w B ;Q o )<q B (w A ;w B ). Bythesamevein,wecanprovethatifp B (Q o ;(w A ;Q o )) w B ,thenq B (w A ;w B )q r B (w B ;Q o ). Part(ii): Rememberthat r B (w B ;Q o ) = (w B c)q r B (w B ;Q o )and B (w A ;w B ) = (w B c)q B (w A ;w B ),sothispartfollows. AppendixII.Bcontinued 149 Part(iii): Atw B =p B (Q o ;(w A ;Q o )), 0 B (w A ;w B ) w B =p B (Qo;(w A ;Qo)) = q B (w A ;w B ) w B =p B (Qo;(w A ;Qo)) +(w B c) @q B (w A ;w B ) @w B w B =p B (Qo;(w A ;Qo)) = (w A ;Q o ) +(w B c) R 11 R 11 R 22 R 2 12 (Qo;(w A ;Qo)) r0 B (w B ;Q o ) w B =p B (Qo;(w A ;Qo)) = q r B (w B ;Q o ) w B =p B (Qo;(w A ;Qo)) +(w B c) @q r B (w B ;Q o ) @w B w B =p B (Qo;(w A ;Qo)) = (w A ;Q o )+(w B c) 1 R 22 (Qo;(w A ;Qo)) Thus, 0 B (w A ;w B ) w B =p B (Qo;(w A ;Qo)) r0 B (w B ;Q o ) w B =p B (Qo;(w A ;Qo)) = (w B c) R 2 12 (R 11 R 22 R 2 12 )R 22 (Qo;(w A ;Qo)) < 0: Q.E.D. AppendixII.Bcontinued 150 This lemma shows that when p B (Q o ;(w A ;Q o )) c, r B (w B ;Q o ) crosses B (w A ;w B ) from below at w B = p B (Q o ;(w A ;Q o )). They are as shown in FigureII.B-7. FigureII.B-7: r B (w B ;Q 0 )and B (w A ;w B ) (B2-i-b)w A <p A (Q o ;0)and b w B <p B (Q o ;(w A ;Q o )) In this subcase, manufacturer B's prot is depicted as the red line in Figure II.B-8below. FigureII.B-8: ManufacturerB'sProtCurvein(B2-i-b) AppendixII.Bcontinued 151 Withthelemmaabove,manufacturerA'sequilibriumprotismax y [R(Q o ;y) b w B y]v(1;b w B )cQ o . Anditsproblemcanbewrittenas max ToQo;w A fmax y [R(Q o ;y) b w B y]v(1;b w B )cQ o g s:t: r B (w B ;Q o ) max x<b w B m B (x) (B.14) w B = argmax b w B xp B (Qo;(w A ;Qo)) r B (x;Q o ) (B.15) v(1;b w B ) = max y [R(Q o ;y) b w B y]T o (B.16) w A < p A (Q o ;0): (B.17) Notice that the same as the case of homogeneous products, the whole game is reduced to a mechanism design problem from manufacturer A's point of view. Constraint(B.14)isequivalenttoanincentive-compatibilityconstraintinthestan- dard mechanism design problem. Constraint (B.15) is the denition ofw B . And constraint (B.16) is the crossing point condition from retailer's optimal choice. Constraint(B.17)aretherestrictionforcase(B2-i-b). Lemma25 (i) Constraint (B.14) and (B.15) imply that m0 B (b w B ) > 0 and hence max xb w B m B (x) = m B (b w B ): (ii)w B must be an interior solution s.t. b w B < w B < p B (Q o ;(w A ;Q o )) and 0 B (w B ;Q o ) = 0. (iii)Constraint(B.14)mustbebinding. ProofofLemma25. Part(i)(Bycontradiction)Supposenot. Thatis, m0 B (b w B ) 0. From Lemma 23, we know that r0 B (b w B ;Q o ) m0 B (b w B ) 0, which im- AppendixII.Bcontinued 152 plies w B = argmax b w B x<p B (Qo;(w A ;Qo) r B (x;Q o ) = r B (b w B ;Q o ). However, r B (b w B ;Q o ) m B (b w B ) with "=" only when b w B = c from Lemma 23. So for (B.14)tohold,wemusthavew B = b w B =c. However,then m0 B (b w B ) = m0 B (c)> 0 contradicts with our supposition that m0 B (b w B ) 0. Thus, m0 B (b w B ) > 0. And max x<b w B m B (x) = m B (b w B ) follows directly from the concavity of m B (w B ) by AssumptionIII.1. Part (ii): Since r B (b w B ;Q o ) < m B (b w B ) from Lemma 23, in order for (B.14) to hold, we must have r0 B (b w B ;Q o ) > 0 or b w B < w B . Combining with the fact that r0 B (w B ;Q o ) w B =p B (Qo;(w A ;Qo)) 0,part(ii)follows. Part(iii): (Bycontradiction)Supposenot. Noticethat @ A @b w B =q r B (b w B ;Q o )+q m B (b w B )> 0; thenwecanincrease A byincreasing b w B until(B.14)becomesbinding. Q.E.D. Lemma26 w B p B (Q o ;(w A ;Q o )) is equivalent to p A (Q o ;q r B (w B ;Q o )) w A . Proof of Lemma 26. p B (Q o ;q r B (w B ;Q o )) = w B p B (Q o ;(w A ;Q o )) () q B (w B ;Q o )(w A ;Q o )()p A (Q o ;q r B (w B ;Q o ))p A (Q o ;(w A ;Q o )) =w A Q.E.D. AppendixII.Bcontinued 153 Using the lemma above and substitutingT o by equation( B.16), manufacturer A'sproblemcanberewrittenas A = max Qo;b w B fmax y [R(Q o ;y) b w B y]v(1;b w B )cQ o g s:t: r B (w B ;Q o ) = m B (b w B ) (B.18) r0 B (w B ;Q o ) = 0 (B.19) p A (Q o ;q r B (w B ;Q o )) w A <p A (Q o ;0): (B.20) Hence,theequilibriumof(B2-i-b)ischaracterizedbythefollowinglemma. Lemma27 In(B2-i-b),3PTequilibrium(T o ;Q o ;w A ;w B )ischaracterizedby T o = max y [R(Q o ;y) b w B y]v(1;b w B ) p A (Q o ;q r B (w B ;Q o )) w A <p A (Q o ;0) r0 B (w B ;Q o ) = 0; where (Q o ;b w B ) = argmax q;w fmax y [R(q;y)wy]v(1;w)cqg s:t: r B (w ;q) = m B (w) r0 B (w ;q) = 0: Inthisequilibrium,theleaderearnshigherprotthanthatunderLPora2PT. AppendixII.Bcontinued 154 (B2-ii)p A (Q o ;0)w A and b w B > > > < > > > > : m B (w B ) ifw B < b w B r B (w B ;Q o ) if b w B w B < > : 0 ifw B < b w B max y [R(Q o ;y) b w B y]v(1;b w B )cQ o if b w B w B <p B (Q o ;0) : Thus, for possible positive prot, manufacturer A must ensure b w B < w B < p B (Q o ;0). This is equivalent to r B (w B ;Q o ) max xb w B m B (x) 0, where the second inequality comes fromc < b w B . So manufacturer A's problem can be AppendixII.Bcontinued 155 writtenas max ToQo;b w B fmax y [R(Q o ;y) b w B y]v(1;b w B )gcQ o s:t: r B (w B ;Q o ) max xb w B m B (x) (B.21) w B = argmax b w B <x<p B (Qo;0) r B (x;Q o ) (B.22) v(1;b w B ) = max y [R(Q o ;y) b w B y]T o (B.23) p A (Q o ;0) w A : (B.24) Compared with case (B2-i-b), the only difference here is constraint (B.24). Note that w A does not enter either the objective function, nor any of the con- straints. Therefore, two equilibria will result in the same prot for manufacturer A.Accordingly,itsprotinthiscaseishigherthanthatunderLPor2PTequilib- rium. By the same argument, we can characterize the equilibrium of case (B2-ii) as follows. Lemma28 In(B2-ii),3PTequilibrium(T o ;Q o ;w A ;w B )ischaracterizedby T o = max y [R(Q o ;y) b w B y]v(1;b w B ) p A (Q o ;0) w A r0 B (w B ;Q o ) = 0; AppendixII.Bcontinued 156 where (Q o ;b w B ) = argmax q;w 8 > < > : max y [R(q;y)wy]v(1;w)cq s:t: r B (w B ;q) = m B (w) 9 > = > ; : Inthisequilibrium,theleaderearnshigherprotthanthatunderLPora2PT. Combiningbothcases(B2-i-b)and(B2-ii),wehavePropositionII.5. 157 Appendix II.C: Equilibrium for General Differenti- atedLinearDemands Here =c: (();())isdeterminedby 8 > > > > > > > < > > > > > > > : = q 1 (1) 2 1 2 2(1 2 ) [2 2 2(1 2 )] =[2](1) : ((); ())isdeterminedby 8 > > > > < > > > > : = p 1(1) 2 f2[2+(12)+ ]g =(1) : AppendixII.Ccontinued 158 TableII.C-1: LP,2PTand3PTEquilibriumTariffs LP 2PT 3PT Tariffs w LP A =c + (2+)(1) 2(2 2 ) w LP B =c + (4+2 2 )(1) 4(2 2 ) T 2PT = 2 1 8(1 2 ) 2 f[8(2+ 2 ) 2 ] 2 +2[(3 2 2) 2(1 2 )] + 2 [12 2 +2(1 2 )]g w 2PT A =c +(1) w 2PT B =c + 1 2 T 3PT =fc + 2[(1+ )+(1 2 )] 2 g Q 3PT = w 3PT A c + 2(2 2 ) 2 w 3PT B =c + 1 2 AppendixII.Ccontinued 159 TableII.C-2: LP,2PTand3PTEquilibriumProts LP 2PT 3PT Prot LP A = 2 (1)(2+) 2 8(1+)(2 2 ) LP B = 2 (1)(4+2 2 ) 2 16(1+)(2 2 ) 2 LP R = 2 32+3216 2 20 3 + 4 +3 5 32(1+)(2 2 ) 2 2PT A = 2 1 8(1 2 ) 2 f(4+8 2 5 4 ) 2 +2[46 2 + 3 +2 4 2(1 2 )] +2 2 (1 2 ) (44 2 +2 3 )g 2PT B = 2 (1) 2 4(1 2 ) 2PT R = 2 1 8(1 2 ) 2 f(43) 2 +2[1+2(1 2 )] 2(1 2 ) +12 +2 3 g 3PT A = 2 [2(1 2 )(1+ )] 2 3PT B = 2 [1] 2 4 3PT R = 2 1[324 ] 8 AppendixII.Ccontinued 160 TableII.C-3: LP,2PTand3PTEquilibriumTotalSurpluses LP 2PT 3PT TS TS LP = 2 96+3296 2 28 3 +23 4 +5 5 32(1+)(2 2 ) 2 TS 2PT = 2 34+2(42 2 )(43 2 ) 2 8(1 2 ) TS 3PT = 2 3+4[2]+3 2 8 161 AppendixIII.A:TheoreticalProofs Since B (P c ;P e g ;R) = IP (P c ;P e g ;R)+(v g P e g ); ifwecandene F(x;y;z;s) IP (x;y;z)+s(v g y); then B (P c ;P e g ;R) = F(P c ;P e g ;R;) IP (p c ;p g ;r) = F(p c ;p g ;r;0) ProofofPropositionIII.4. F.O.C.sforIP: @F(x;y;z;0) @x = 0 @F(x;y;z;0) @y = 0 F.O.C.sforbundling: @F(x;y;z;) @x = 0() @F(x;y;z;0) @x = 0 @F(x;y;z;) @y = 0() @F(x;y;z;0) @y = AppendixIII.Acontinued 162 Consequently, we can focus on the properties ofF(x;y;z;0) for the compar- ativestatics. DenoteitsHessianmatrixas H =D i D j [F(x;y;z;0)] = [h ij ] = 2 6 4 h 11 h 12 h 21 h 22 3 7 5 : Standardcomparativestaticsgivesthat @p g @ = @y @s = h 11 0 h 21 1 jHj = h 11 jHj < 0 ( * h 11 < 0): And @(p c +p g ) @ = @p c @ + @p g @ = h 11 h 12 jHj < 0: However, @p c @ = @x @s = 0 h 12 1 h 22 jHj = h 12 jHj willdependontheexogenousroyaltyrateR.Q.E.D. AppendixIII.Acontinued 163 ProofofPropositionIII.6. F.O.C.sforIP: @F(x;y;z;0) @x = 0 @F(x;y;z;0) @y = 0 @F(x;y;z;0) @z = 0 F.O.C.sforbundling: @F(x;y;z;) @x = 0() @F(x;y;z;0) @x = 0 @F(x;y;z;) @y = 0() @F(x;y;z;0) @y = @F(x;y;z;) @z = 0() @F(x;y;z;0) @y = 0 Consequently, we can focus on the properties ofF(x;y;z;0) for the compar- ativestatics. DenoteitsHessianmatrixas K =D i D j [F(x;y;z;0)] = [k ij ]: AppendixIII.Acontinued 164 Standardcomparativestaticsgivesthat @p c @ = @x @s = 0 k 12 k 13 1 k 22 k 23 0 k 32 k 33 jKj = k 12 k 13 k 32 k 33 jKj < 0 @p g @ = @y @s = k 11 0 k 13 k 21 1 k 23 k 31 0 k 33 jKj = k 11 k 13 k 31 k 33 jKj < 0 @r @ = @z @s = k 11 k 12 0 k 21 k 22 1 k 31 k 32 0 jKj = k 11 k 12 k 31 k 32 jKj < 0: 30 Q.E.D. 30 The determination of the signs for these terms is followed from direct calculation. And the Mathematicacodesforcomputationareavailableuponrequest. 165 AppendixIII.B:EmpiricalTests TableIII.B-1: Non-NestedModelSelectionTestAnEncompassingTest Linear Log-Linear Log-Log ln(1+#ofBundlesPresent) 0:0730 (0:0436) #ofBundlesPresent 2:5259 0:0220 (1:3992) (0:0173) I(Bundle) 0:4747 0:0044 0:0200 (3:1190) (0:0385) (0:0498) ln(#ofConsolesPresent) 0:2443 (0:0568) #ofConsolesPresent 13:7172 0:1481 (4:0344) (0:0498) ln(#ofSoftwareTitles) 0:0046 (0:0229) #ofSoftwareTitles 0:0017 0:0001 (0:0140) (0:0002) Age 1:1167 0:0145 0:0138 (0:1453) (0:0018) (0:0012) ConsoleFE's No No No AllmodelsincludeMonthxedeffectsandanunreportedconstant. **signicantat95%*signicantat90% AppendixIII.Bcontinued 166 RegressionModelTestsforTheoreticalPrediction1 Box-CoxModelSpecicationTest P () ct = 0 + 1 Age ct + 2 (Numberof Software ct ) () + 3 (1+#of OtherConsoles ct ) () + 4 (1+#of Bundles ct ) () + m=11 P m=1 4+m Month m + ct : TableIII.B-2: Box-CoxModel(Non)LinearSpecicationTest Coef. Std. Err. Lambda :2501 (0:5340) TestH 0 Chi2 Prob>Chi2 Lambda=0 0:21 0:643 Lambda=1 4:82 0:028 TableIII.B-3: Log-LogModelResidualNormalityTest H 0 Chi2(2) Prob>Chi2 ct vN 5:29 0:0710 AppendixIII.Bcontinued 167 RegressionModelTestsforTheoreticalPrediction2 Box-CoxModelSpecicationTest P () g b t = 0 + 1 Age g b t + 2 (Numberof Software g b t ) () + 4 I(Bundles g b t )+ m=11 P m=1 4+m Month m + g b t TableIII.B-4: Box-CoxModel(Non)LinearSpecicationTest Coef. Std. Err. Lambda 1:863 (0:697) TestH 0 Chi2 Prob>Chi2 Lambda=0 3:14 0:076 Lambda=1 1:54 0:214 TableIII.B-5: LinearModelResidualNormalityTest H 0 Chi2(2) Prob>Chi2 ct vN 4:11 0:1281 AppendixIII.Bcontinued 168 RegressionModelTestsforTheoreticalPrediction3 Box-CoxModelSpecicationTest P () gt = 0 + 1 Age gt + 2 (Numberof Software gt ) () + 4 I(Bundles gt )+ m=11 P m=1 4+m Month m + gt TableIII.B-6: Box-CoxModel(Non)LinearSpecicationTest Coef. Std. Err. Lambda 0:556181 (0:0167) TestH 0 Chi2 Prob>Chi2 Lambda=0 1229:36 0:000 Lambda=1 656:02 0:000 TableIII.B-7: NonlinearModelResidualNormalityTest H 0 Chi2(2) Prob>Chi2 ct vN 1:98 0:3724 In this specication test we reject both hypotheses. We, thus, interpret these testresultsasindicatingthatboththelinearandlog-logmodelsareinappropriate. AppendixIII.Bcontinued 169 However, given that lambda in the Box-Cox regression is very near 0:5, a model whichemploysasquare-roottransformationofvariables,weproceedandassume thisspecicationiscorrect. Liketheabovetwotests,wetesttodeterminewhether theresidualsofsuchamodelarenormallydistributed.

Abstract (if available)

Abstract This dissertation studies a variety of bundling and discount strategies adopted by dominant firms, analyze their corresponding competitive effects in the upstream-and-downstream context, as well as in two-sided markets

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Creator Chao, Yong (author)

Core Title Essays on bundling and discounts

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Publication Date 05/21/2010

Defense Date 05/04/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag antitrust,bundling,discounts,OAI-PMH Harvest,oligopoly,price discrimination,three-part tariff,two-sided markets,video game industry

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Tan, Guofu (committee chair), Carrillo, Juan D. (committee member), Cheng, Harrison (committee member), Dukes, Anthony J. (committee member), Wilkie, Simon J. (committee member)

Creator Email chaoyong_cn@yahoo.com,ychao@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3094

Unique identifier UC1366172

Identifier etd-Chao-3748 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-343518 (legacy record id),usctheses-m3094 (legacy record id)

Legacy Identifier etd-Chao-3748.pdf

Dmrecord 343518

Document Type Dissertation

Rights Chao, Yong

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

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