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Three Essays on Bundling and Two-Sided Markets

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PAGE 1

THREE ESSAYS ON BUNDLING AND TWO-SIDED MARKETS By JIN JEON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

PAGE 2

Copyright 2006 by Jin Jeon

PAGE 3

To my parents, wife, and two daughters

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iv ACKNOWLEDGMENTS I must first thank my supervisory committee members. Dr. Jonathan Hamilton, the chair of the committee, always supported me with patience, encouragement, and intellectual guidance. He inspired me to think in new ways and put more emphasis on economic intuition than technical details. Dr Steven Slutsky, a member, generously shared his time to listen to my ideas and gi ve further suggestions. Dr. Roger Blair, a member, also gave me useful comments and provided research ideas. Dr. Joel Demski, the external member, carefully read the ma nuscript and gave helpful comments. I hereby thank them all again. This dissertation would not have been possible without support from my family members. My parents always believed in me a nd kept supporting me. I would also like to give a heartfelt acknowledgment to my wife, Hyo-Jung, and two daughters, Hee-Yeon and Hee-Soo, for their endless loving support.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv TABLE.......................................................................................................................... ....vii LIST OF FIGURES.........................................................................................................viii ABSTRACT.......................................................................................................................ix CHAPTERS 1 INTRODUCTION........................................................................................................1 1.1 Bundling..............................................................................................................2 1.2 Two-Sided Markets.............................................................................................5 2 BUNDLING AND COMMITMENT PR OBLEM IN THE AFTERMARKET.........11 2.1 Introduction.......................................................................................................11 2.2 The Model.........................................................................................................16 2.3 Independent Sale without Commitment............................................................20 2.4 Independent Sale with Commitment.................................................................24 2.5 Bundling: An Alternative Pric ing Strategy without Commitment....................30 2.6 Bundling and Social Welfare............................................................................36 2.7 Bundling and R&D Incentives..........................................................................39 2.8 Conclusion.........................................................................................................41 3 COMPETITION AND WELFARE IN THE TWO-SIDED MARKET: THE CASE OF CREDIT CARD INDUSTRY...................................................................44 3.1 Introduction.......................................................................................................44 3.2 The Model: Nonproprietary Card Scheme........................................................47 3.3 Competition between Identical Ca rd Schemes: Bertrand Competition............49 3.3.1 Single-Homing Consumers...................................................................50 3.3.2 Multi-Homing Consumers....................................................................58 3.4 Competition between Differentiated Ca rd Schemes: Hotelling Competition...66 3.4.1 Single-Homing Consumers...................................................................68 3.4.2 Multi-Homing Consumers....................................................................74 3.5 Proprietary System with Single-Homing Consumers.......................................82 3.5.1 Competition between Identical Card Schemes.....................................83

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vi 3.5.2 Competition between Diffe rentiated Card Schemes.............................88 3.6 Conclusion.........................................................................................................92 4 COMPETITION BETWEEN CARD I SSUERS WITH HETEROGENEOUS EXPENDITURE VOLUMES....................................................................................95 4.1 Introduction.......................................................................................................95 4.2 Equilibrium Cardholder Fee..............................................................................98 4.2.1 The Model.............................................................................................98 4.2.2 Full-Cover Market..............................................................................103 4.2.3 Local Monopoly..................................................................................107 4.2.4 Partial-Cover Market..........................................................................109 4.3 Equilibrium Interchange Fee...........................................................................114 4.3.1 Full-Cover Market..............................................................................114 4.3.2 Local Monopoly..................................................................................116 4.3.3 Partial-Cover Market..........................................................................117 4.4 Extension.........................................................................................................118 4.4.1 Other Comparative Statics..................................................................118 4.4.2 Collusion.............................................................................................120 4.5 Conclusion.......................................................................................................122 5 CONCLUDING REMARKS....................................................................................125 APPENDIX SIMULATION ANALYSIS FOR CHAPTER 4......................................128 REFERENCES................................................................................................................133 BIOGRAPHICAL SKETCH...........................................................................................137

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vii TABLE Table page 4-1 Comparative statics...................................................................................................119

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viii LIST OF FIGURES Figure page 1-1 Credit card schemes......................................................................................................72-1 Consumers surplus in bundling and IS cases when v1 < v2........................................373-1 Welfare and interchange fees of Bertrand competition..............................................663-2 Merchants acceptance decision when a1 > a2 ( m1 > m2)...........................................773-3 Welfare and interchange fees of Hotelling competition when 2()BS BBbbbbc .............................................................................................824-1 Division of consumers in three cases of market coverage........................................1024-2 The effect of a price drop on demand.......................................................................110A-1 The density function................................................................................................129A-2 Effects of an increase in the variance on the cardholder fee ( df / dy )........................130A-3 Change in the interchange fee and the cardholder fee ( df / da ).................................132

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ix Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THREE ESSAYS ON BUNDLING AND TWO-SIDED MARKETS By Jin Jeon December 2006 Chair: Jonathan H. Hamilton Major Department: Economics This work addresses three issues rega rding bundling and two-sided markets. It starts with a brief summary of the theories of bundling and of two-sided markets in Chapter 1. Chapter 2 analyzes various aspects of bundling strategy by the monopolist of a primary good when it faces competition in the complementary good market. The main result is that the monopolist can use a bundli ng strategy in order to avoid commitment problem that arises in optimal pricing. Bundling increases th e monopolists profits without the rival's exit from the market. Bund ling lowers social welfare in most cases, while it may increase consumers surplus. On e of the long-run effects of bundling is that it lowers both firms incentives to invest in R&D. Chapter 3 compares welfare implications of monopoly outcome and competitive outcome. Using a model of the credit card indus try with various settings such as Bertrand and Hotelling competition with single-homing and multi-homing consumers as well as

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x proprietary and nonproprietar y platforms, it is shown that introducing platform competition in two-side markets may lower social welfare compared to the case of monopoly platform. In most cases, monopoly pricing maximizes Marshallian social welfare since the monopolist in a two-sided market can prop erly internalize indirect network externalities by setting unbiased pri ces, while the competing platforms set biased prices in order to attr act the single-homing side. Chapter 4 analyzes the effects of distri bution of consumers expenditure volumes on the market outcomes using a model in which two card issuers compete la Hotelling. The result shows that the effects of distri bution of the expenditure volume are different for various cases of market coverage. For ex ample, as the variance increases, issuers profits decrease when the market is fully covered, while the profits increase when the market is locally monopolized. It is also show n that the neutrality of the interchange fee holds in the full-cover market under the no-surcharge-rule. Simulation results are provided to show other comparative statics th at include the possibility of the positive relationship between the intercha nge fee and the cardholder fee. Finally, Chapter 5 summarizes major findings with some policy implications.

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1 CHAPTER 1 INTRODUCTION This dissertation contains three essays on bundling and two-sided markets. These topics have recently drawn econo mists attention due to the antitrust cases of Kodak and Microsoft, and movements in some countries to regulate the cred it card industry. In the Kodak case, independent service or ganizations (ISOs) alleged that Kodak had unlawfully tied the sale of service for its m achines to the sale of parts, in violation of section 1 of the Sherman Act, and had a ttempted to monopolize the aftermarket in violation of section 2 of the Sherman Act.1 In the Microsoft case, the United States government filed an antitrust lawsuit agai nst Microsoft for illegally bundling Internet Explorer with Windows operating system.2 In the credit card industry, antitrust au thorities around the world have questioned some business practices of the credit card networks, wh ich include the collective determination of the intercha nge fee, the no-surc harge rule, and the honor-all-cards rule. As a result, card schemes in some countries such as Australia, United Kingdom, and South Korea have been required to lower th eir interchange fees or merchant fees. To understand these antitrus t cases, many economic models have been developed. In the following sections, brief summaries of the economic theories of bundling and of two-sided markets will be presented. 1 For more information about the Kodak case, see Klein (1993), Shapiro (1995), Borenstein, MacKie-Mason, and Netz (1995), and Blair and Herndon (1996). 2 See Gilbert & Katz (2001), Whinston ( 2001), and Evans, Nichols and Schmalensee (2001, 2005) for further analysis of the Microsoft case.

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2 1.1 Bundling Economists views regarding bundling or tyi ng have shifted dramatically in recent decades.3 The traditional view of tying can be represented by the leverage theory which postulates that a firm with monopoly power in one market could use the leverage to monopolize another market. The Chicago School criticized the leverage theory, sin ce such leveraging may not increase the profits of th e monopolist. According to th e single monopoly profit theorem supported by the Chicago School, the monopolist earns same profits regardless whether it ties if the tied good market is perfectly competitive. For example, suppose consumers valuation of a combined product of A and B is $10 and marginal cost of producing each good is $1. Good A is supplied only by the monopolist, and good B is available in a competitive market at price equal to the ma rginal cost. Without bundling, the monopolist can charge $9 for Aand $1 for Bto make $8 as unit profit per good A sold. If the monopolist sells A and B as a bundle, it ca n charge $10 for the bundle and earn $8 ($10 $1 $1) per unit bundle. So the monopolist ca nnot increase profits by bundling in this case. Economists led by the Chicago School pr oposed alternative explanations for bundling based on efficiency rationales. Pr obably the most common reason for bundling is it reduces the transaction costs such as consumers searching costs and firms packaging and shipping costs. Examples of th is kind of bundling are ab undant in the real 3 Bundling is the practice of selling two goods together, while tying is the behavior of selling one good conditional on the purchase of another good. There is no difference between tying and bundling if the tied good is valueless wi thout the tying good and two goods are consumed in fixed proportion. See Tirole (2005) and Nalebuff (2003) for the discussions of bundling and tying.

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3 world: shoes are sold in pair s; personal computers (PCs) are sold as bundles of the CPU, a hard drive, a monitor, a keyboard and a mous e; cars are sold with tires and a car audio. In some sense, most products sold in th e real world are bundled goods and services. Another explanation for bundling in line w ith the efficiency rationale is price discrimination. That is, if consumers are he terogeneous in their va luations of products, bundling has a similar effect as price discrimination.4 This advantage of bundling is apparent when consumers valuations ar e negatively correlated. But bundling can be profitable even for nonnegative co rrelation of consumers' valuations (McAfee, McMillan, and Whinston, 1989). In fact, unl ess consumers valuations ar e perfectly correlated, firms can increase profits by bundling.5 Since price discrimination usually increases social welfare as well as firms profits, bundling motivated by price discrimination increases efficiency of the economy. The leverage theory of tying revived w ith the seminal work of Whinston (1990). He showed that the Chicago School arguments regarding tying can break down in certain circumstances which include 1) the monopolized product is not essen tial for all uses of the complementary good, and 2) scale economies are present in the complementary good. If there are uses of the complementary good that do not require the primary good, the monopolist of the primary good cannot capture all profits by selling the primary good only. So the first feature provides an incen tive for the monopolist of the primary good to exclude rival producers of the complement ary good. The second feature provides the monopolist with the abil ity to exclude rivals, since foreclosure of sales in the 4 See Adams and Yellen (1976) and Schmalensee (1984). 5 Bakos and Brynjolfsson (1999) show the bene fit of a very large scale bundling based on the Law of Large Numbers.

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4 complementary market, combined with barri ers to entry through scale economies, can keep rival producers of the complementary good out of the market.6 Bundling can also be used to preserve the monopolists market power in the primary good market by preventing entry into th e complementary market at the first stage (Carlton and Waldman, 2002a). This explains the possibility that Microsoft bundles Internet Explorer with Windows OS in orde r to preserve the monopoly position in the OS market, since Netscapes Navigator combined with Java technology could become a middleware on which other application programs can run regardless of the OS. Choi and Stefanidis (2001) and Choi ( 2004) analyze the effects of tying on R&D incentives. The former shows that tying arra ngement of an incumbent firm that produces two complementary goods and faces possible en tries in both markets reduces entrants R&D incentives since each entrants success is dependent on the others success. The latter analyzes R&D competition between the incumbent and the entrant, and shows that tying increases the incumbents incentives to R&D since it can spre ad out the costs of R&D over a larger number of units, whereas the entrants R&D incentives decrease.7 Chapter 2 presents a model of bundling that follows the basic ideas of the leverage theory. It shows that the monopolist of a primary good that faces competition in the aftermarkets can use the bundling strategy to increase profits to the detriment of the rival firm. Aftermarkets are markets for goods or services used toge ther with durable equipment but purchased after the consumer has invested in the equipment. Examples include maintenance services and parts, app lication programs for operating systems, and 6 Nalebuff (2004) and Carlton and Waldman ( 2005a) also present models that show the entry deterrence effect in the tied good market. 7 In chapter 2, I show that bundling reduces R&D incentives of the m onopolist as well as of the rival.

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5 software upgrades. One of the key elements of the aftermarket is that consumers buy the complementary goods after they have bought the primary good. For the monopolist of the primary good, the best way to maximize its profits is to commit to the second period complementary price. If this commitment is not possible or implementable, bundling can be used. Unlike most of the previous models of th e leverage theory, market foreclosure is not the goal of the bundling in this model. On the contrary, the ex istence of the rival firms is beneficial to the monopolist in so me sense since it can capture some surplus generated by the rival firms product. 1.2 Two-Sided Markets Two-sided markets are defined as markets in which end-users of two distinctive sides obtain benefits from interacting with each other over a common platform.8 These markets are characterized by indi rect network externalities, i.e., benefits of one side depend on the size of the other side.9 According to Rochet and Tirole (2005), a necessary condition for a market to be two-sided is th at the Coase theorem does not apply to the transaction between the two si des. That is, any change in the price structure, holding constant the total level of prices faced by two parties, affects partic ipation levels and the number of interactions on the platform si nce costs on one side cannot be completely passed through to the other side. 8 For general introductions to the two-sided market, see Roson (2005a), and Evans and Schmalensee (2005). 9 In some cases such as media industries, i ndirect network external ities can be negative since the number of advertisers has a negative impact on readers, viewers, or listeners. See Reisinger (2004) for the analysis of two-si ded markets with negativ e externalities.

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6 Examples of the two-sided market are a bundant in the real world. Shopping malls need to attract merchants as well as shoppe rs. Videogame consoles compete for game developers as well as gamers. Credit card sche mes try to attract cardholders as well as merchants who accept the cards. Newspapers ne ed to attract advertisers as well as readers.10 Figure 1-1 shows the structure of the tw o-sided market in case of the credit card industry, both proprietary and nonproprietary schemes. Although some features of tw o-sided markets have been recognized and studied for a long time,11 it is only recently that a general theory of two-sided markets emerged.12 The surge of interest in two-sided markets wa s partly triggered by a series of antitrust cases against the credit card industry in many industrialized countries including the United States, Europe and Australia. The liter ature on the credit card industry has found that the industry has special characteristics; hence conventio nal antitrust policies may not be applicable to the industry.13 Wright (2004b) summarizes fallacies th at can arise from using conventional wisdom from one-sided markets in two-sided markets, which include: an efficient price structure should be set to refl ect relative costs; a high pricecost margin indicates market power; a price below marginal cost indicat es predation; an increase in competition necessarily results in a more efficient struct ure of prices; and an increase in competition necessarily results in a more balanced price structure. 10 See Rochet and Tirole (2003) for more examples of the two-sided market. 11 For example, Baxter (1983) realized the two-sidedness of the credit card industry. 12 The seminal papers include Armstrong (2005) Caillaud and Jullien (2003), and Rochet and Tirole (2003). 13 The literature includes Gans and King (2003) Katz (2001), Rochet and Tirole (2002), Schmalensee (2002), Wright (2003a, 2003b, 2004a).

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7 The theory of two-sided markets is related to the theories of network externalities and of multi-product pricing. While the litera ture on network externalities has found that in some industries there exist externalities that ar e not internalized by end-users, models are developed in the context of one-sided markets.14 Theories of multi-product pricing stress the importance of price structures, but ignore externalities in the consumption of Figure 1-1. Credit card schemes 14 See Katz and Shapiro (1985, 1986), and Farrell and Saloner (1985, 1986). Issuer Acquirer Merchant Cardholder Sells good at price p Pays p + f ( f : cardholder fee) Pays p a ( a : interchange fee) Pays p m ( m : merchant fee) (a) Nonproprietary card scheme Merchant Cardholder Sells good at price p Pays p + f ( f : cardholder fee) Pays p m ( m : merchant fee) (b) Proprietary card scheme Platform ( Card scheme ) Platform ( Card scheme )

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8 different goods since the same consumer buys both goods. That is, the buyer of one product (say, razor) internalizes the benefits that he will derive from buying the other product (blades). The two-sided market theory st arts from the observation that there exist some industries in which consumers on one side do not internalize th e externalities they generate on the other side. The role of platfo rms in two-sided markets is to internalize these indirect externalities by charging appropriate prices to each side. In order to get both sides on board and to balance demands of tw o sides, platforms in two-sided markets must carefully choose pric e structures as well as total price levels.15 So it is possible that one side is charged below marginal cost of serving that side, which would be regarded as predatory pricing in a standard one-sided market. For this reason, many shopping malls offers free parking serv ice to shoppers, and cardholders usually pay no service fees or even negative prices in the form of various rebates. In a standard one-sided market, the price is determined by the marginal cost and the own price elasticity, as is ch aracterized by Lerners formula.16 In two-sided markets, however, there are other factor s that affect the price char ged to each side. These are relative size of cross-group externalities and whether agents on each side single-home or multi-home.17 If one side exerts larger exte rnalities on the other side than vice versa then the platform will set a lower price for this side, ceteris paribus In a media industry, for 15 In the credit card industry, non-proprietary card schemes choose interchange fees which affect the price structure of two sides. 16 The standard Lerners formula is 1 pc p or 1 p c where p is the price, c is the marginal cost, and is the own price elasticity. 17 An end-user is single-homing if she uses one platform, and multi-homing if she uses multiple platforms.

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9 example, viewers pay below the marginal cost of serving while adve rtisers pay above the marginal cost since the externalities from vi ewers to advertisers are larger than those from advertisers to viewers. When two or more platforms compete with each other, end-users may join a single platform or multiple platforms, depending on the benefits and costs of joining platforms. Theoretically, three possible cases emerge: (i ) both sides single-home, (ii) one side single-homes while the other side multi-homes, and (iii) both sides multi-home.18 If interacting with the other side is the main purpose of joining a platform, one can expect case (iii) is not common since end-users of one side need not join multiple platforms if all members of the other side multi-home.19 For example, if every merchant accepts all kinds of credit cards, consumers need to carry onl y one card for transaction purposes. Case (i) is also not common since end-users of one si de can increase interaction with the other side by joining multiple platforms. As long as the increased benefit exceeds the cost of joining additional platform, th e end-users will multi-home. On the contrary, one can find many examples of case (ii) in the real world. Advertisers place ads in several newspapers while readers usually subscribe to only one newspaper. Game developers make the same game for various videogame consoles while gamers each own a single console. Finally merchants accept multiple cards while consumers use a single card.20 18 In most of the models on two-sided mark ets, single-homing and multi-homing of endusers are pre-determined for analytical tr actability. For an analysis of endogenous multihoming, see Roson (2005b). 19 See also Gabszewicz and Wauthy (2004). 20 According to an empirical study by Rysm an (2006), most consumers put a great majority of their payment card purchases on a single network, even when they own multiple cards from different networks.

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10 When end-users of one side single-home wh ile those of the other side multi-home, the single-homing side becomes a bottl eneck (Armstrong, 2005). Platforms compete for the single-homing side, so they will charge lower price to that side. As is shown in Chapter 3, platforms competing for the sing le-homing side may find themselves in a situation of the Prisoners Dilemma. That is, a lower price for the single-homing side combined with a higher price for the multi-hom ing side can decrease total transaction volume and/or total profits compared to the monopoly outcome. Further, competition in two-sided markets may lower social welf are since monopoly platforms can properly internalize the indirect externalities by charging unbiased prices, while competing platforms are likely to distort the price stru cture in favor of the single-homing side. Chapter 3 presents a model of the credit card industry with various settings including single-homing vs. multi-homing car dholders, competition between identical card schemes (Bertrand competition) or diffe rentiated schemes (Hotelling competition), and proprietary vs. non-proprieta ry card schemes. The main finding is that, unlike in a standard one-sided market, competition does not increase social welfare regardless of the model settings. Chapter 4 tackles the assumption made by most models on the credit card industry that cardholders spend the same amounts with credit cards. By allowing heterogeneous expenditures among consumers, it shows the eff ects of a change in the variance of the expenditure on the equilibrium prices and profits. The results show that the effects are different depending on whether the market is fully covered, pa rtially covered, or locally monopolized.

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11 CHAPTER 2 BUNDLING AND COMMITMENT PR OBLEM IN THE AFTERMARKET 2.1 Introduction A monopolist of a primary good that faces co mpetition in the aftermarket of the complementary goods often uses a bundling or tying strategy. Traditionally, bundling was viewed as a practice of transferring th e monopoly power in the tying market to the tied market. This so-called leverage theo ry has been criticized by many economists associated with the Chicago Sc hool in that there exist other motives of bundling such as efficiency-enhancement and price discrimination. Further, they show that there are many circumstances in which firms cannot increa se profits by leveraging monopoly power in one market to the other market, which is known as the single monopoly profit theorem. Since the seminal work of Whinston (1990) the leverage theory revived as many models have been developed to show th at a monopolist can use tying or bundling strategically in order to dete r entry to the complementary ma rket and/or primary market. The research was in part stimulated by the an titrust case against Micr osoft filed in 1998, in which U.S government argued that Microsoft illegally bundles Inte rnet Explorer with Windows operating system.1 Most of the models in this line, however, have a commitment problem since the bundling decisi on or bundling price is not credible when the entrant actually enters or does not exit the market. 1 For further analyses of the Microsoft case, see Gilbert & Katz (2001), Whinston (2001), and Evans, Nichols and Schmalensee (2001).

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12 This paper stands in the tradition of th e leverage theory and shows that the monopolist of a primary good can use a bundling strategy to increase profits as well as the market share in the complementary good market. Unlike the previous models, the monopolists profits increas e with bundling even if the riva l does not exit the market. On the contrary, the existence of a rival firm is beneficial to the monopolist in some sense since the monopolist can capture some su rplus generated by the rival firms complementary good. The model presented here is especially usef ul for the analysis of the Microsoft case. Many new features added toi.e., bundled withthe Windows operating system (OS) had been independent application programs produced by other firms. For example, Netscapes Navigator was a dominant Intern et browser before Microsoft developed Internet Explorer. Therefore, it is Microsof t, not Netscape, that entered the Internet browser market. Since Netscapes software deve lopment cost is already a sunk cost when Microsoft makes a bundling decision, the entr y deterrent effect of bundling cannot be applied. The main result is that the monopolist can use bundling to avoid the commitment problem2 arising in the optimal pricing when consumers purchase the complementary good after they have bought the primary good. If the monopolist cannot commit to its optimal price for the complementary good at the first stage when consumers buy the primary good, then it may have to charge a lower price for the primary good and a higher price for the complementary good compared to its optimal set of prices since consumers 2 This commitment problem is different from the one in the previous literature, in which the commitment problem arises since the bundli ng price is not credib le if the would-be entrant actually enters the market.

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13 rationally expect that the monopolist may raise its comple mentary good price after they have bought the primary goods. A double marginalization problem arises in this case since the monopolist has to char ge the price that maximizes its second stage profits, while it also charges a monopoly price for the primary good at the first stage. Bundling makes it possible for the monopolist to avoid the doubl e marginalization problem by implicitly charging a price equal to zer o for the complementary good. The model also shows that bundling genera lly lowers Marshallian social welfare except for the extreme case wh en the monopolists bundled good is sufficiently superior to the rivals good. Social we lfare decreases with bundling ma inly because it lowers the rivals profits. Consumers surplus generally increases with bundling. However, consumers surplus also decreases when th e rivals complementary good is sufficiently superior to the monopolists. The last result shows the eff ect of bundling on R&D investments. In contrast to the previous result of Choi (2004) that shows tyi ng lowers the rival firms incentive to invest in R&D while it increases the monopolists incentive, I show that bundling lowers both firms incentives to make R&D investments. The literature on bundling or tyi ng is divided into two groups one finds the incentive to bundle from the efficiency-enha ncing motives, and the other finds it from anticompetitive motives.3 In the real world, examples of bundling motivated by efficiency reason are abundant. Shoe make rs sell shoes as a pair, which reduces transaction costs such as consumers searchi ng costs and producers costs of shipping and packaging. The personal computer is another example as it is a bundle of many parts such 3 For a full review of the literature on bundling, see Carlton and Waldman (2005b).

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14 as the CPU, a memory card, a hard drive, a keyboard, a mouse, and a monitor.4 Carlton and Waldman (2002b) explain another effici ency motive for tying by showing that producers of a primary good may use tying in order to induce consumers to make efficient purchase decisions in the af termarket when consumers can buy the complementary goods in variable proporti ons. If the primary good is supplied at a monopoly price while the complementary good is provided competitively, consumers purchase too much of the complementary good and too little of the primary good. Tying can reduce this inefficiency and increase profits. Adams and Yellen (1976) pr ovide a price discriminati on motive for tying. Using some examples, they show that if consumers are heterogeneous in their valuations for the products, bundling has a similar effect as price discrimination. This advantage of bundling is apparent when consumers valua tions are negatively co rrelated. Schmalensee (1984) formalizes this theory assuming consumers valuations follow a normal distribution. McAfee, McMillan, and Wh inston (1989) show that bundling can be profitable even for nonnegativ e correlation of consumers' valuations. Bakos and Brynjolfsson (1999) show the benefit of a ve ry large scale bundling of information goods based on Law of Large Numbers. Since pri ce discrimination usually increases social welfare with an increase in total outpu t, tying or bundling motivated by price discrimination can be welfare improving. The anticompetitive motive of tying is reexamined by Whinston (1990). He recognizes that Chicago Schools criticism of leverage theory only applies when the complementary good market is perfectly comp etitive and characterized by constant 4 See Evans and Salinger (2005) for e fficiency-enhancing motive of tying.

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15 returns to scale, and the pr imary good is essential for use of the complementary good. He shows that in an oligopoly market with increasing returns to scale, tying of two independent goods can deter entry by reducing the entrants profits below the entry cost. As was mentioned earlier, however, his mode l has a credibility problem since bundling is not profitable if entrance actually occurs. Nalebuff (2004) also shows that bundling can be used to deter en try, but without a commitment problem since in his model the incumbent makes higher profits with bundling than independent sale even when the would-be entrant actually enters.5 Carlton and Waldman (2002a) focus on the ability of tying to enhance a monopolists market power in the primary market. Their model shows that by preventing entry into the complementary market at the first stage, tyi ng can also stop the alternative producer from entering the primary market at the second stage. Carlton and Waldman (2005a) shows that if the primary good is a durable good and upgrades for the complementary good are possible, the monopolist may use a tying strategy at the first stage in order to captur e all the upgrade profits at the second stage. Especially when the rivals complementary good is superior to the monopolists, the only way the monopolist sells second-period upgrades is to eliminate the ri vals product in the first period by tying its own complementar y good with its monopolized primary good. By showing that tying can be used strategically even when the primary good is essential for use of the complementary good, it provides another condition under which the Chicago School argument breaks down. 5 However, the optimal bundling price is higher when the entrant enters than the price that is used to threaten the entrant. So ther e exists a credibility pr oblem with the price of the bundled good.

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16 The model presented here also assumes the primary good is essential, but the primary good is not necessarily a durable good and constant retu rns to scale prevail. So it can be added to the conditions under whic h the Chicago School argument breaks down that bundling can be used st rategically when consumers buy the primary good and the complementary good sequentially. The rest of Chapter 2 is organized in th e following way. Section 2.2 describes the basic setting of the model. Sections 2.3 to 2.5 show and compare the cases of independent sale, pricing with commitment, and bundling, respectively. Section 2.6 analyzes the welfare effect of bundling. Section 2.7 is devoted to the effect of bundling on R&D investments. The last section summarizes the results. 2.2 The Model Suppose there are two goods and two firm s in an industry. A primary good is produced solely by a monopolist, firm 1. The other good is a complementary good that is produced by both the monopolist and a rival, firm 2. The purchases of the primary good and the complementary good are made sequentially, i.e., consumers buy the complementary good after they have bought the primary good. Consumers buy at most one unit of each good,6 and are divided into two gr oups. Both groups have same reservation value v0 for the primary good. For the complementary good, however, one group has zero reservation va lue and the other group has positive reservation value vi, where i = 1, 2 indicates the producer.7 For modeling convenience, it is assumed that the 6 So there is no vari able proportion issue. 7 Consumption of the complementary good may increase the reserv ation value of the primary good. It is assumed that vi also includes this additional value.

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17 marginal cost of producing each good is zero and there is no fixed cost for producing any good.8 The PC software industry fits in this m odel, in which Microsoft Windows OS is the monopolized primary good and other application programs are complementary goods. Microsoft also produces application progr ams that compete with others in the complementary good market. Sometimes Micros oft bundles application programs such as an Internet browser and a media player that could be sold separate ly into Windows OS. Consumers usually buy the Windows OS at the time they buy a PC, then buy application software later. Let the total number of consumers be normalized to one, and be the portion of the consumers, group S who have positive valuations for the complementary good. It is assumed that the consumers in S are distributed uniformly on the unit interval, in which the monopolist and firm 2 are located at 0 and 1, respectively. The two complimentary goods are differentiated in a Hotelling fashion. A consumer located at x incurs an additional transportation cost tx when she buys the monopolist's complementary good, and t (1 x ) when she buys firm 2s. So the gross utility of the complementary good for the consumer is v1 tx when she buys from the monopolist, and v2 t (1 x ) when she buys from firm 2. v1 and v2 are assumed to be greater than t in order to make sure that consumers in S cannot have a negative gross utility for any complementary good regardless of their positions. Further, in order to 8 Unlike the models that explain tying as an entry deterrence device, the model in this paper assumes constant returns to scale.

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18 make sure that all the consumers in S buy the complementary goods at equilibrium, it is assumed that9 v1 + v2 > 3 t (2-1) The model presented here allows a difference between v1 and v2 in order to analyze bundling decision when the monopolist produces inferioror superiorcomplementary good and the effect of bundling on R&D investment s. But the difference is assumed to be less than t i.e., | v1 v2 | t (2-2) since otherwise all consumers find one of th e complementary goods superior to the other good.10 In the software industry, the primar y good is the operating system (OS), and application programs like an In ternet browser or a word processor are examples of complementary goods. The OS itself can be seen a collection of many functions and commands. Bakos and Brynjolfsson (1999) s how that the reserv ation values among consumers of a large scale bundle converge to a single number, which justifies the assumption that consumers have the same valuation for the primary good. A single application program, however, is not as broadly used as an OS, so the valuation for the 9 The prices chosen by two firms could be t oo high so that some of the consumers in S may not want to buy the complementary good. The assumption v1+ v2 > 3 t guarantees that every consumer in S buy a complementary good at equilibrium. 10 This is also a sufficient condition for the existence of the various equilibria.

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19 complementary good may vary among consumers. Furthermore, not all the application programs are produced for all consumers. So me of them are developed for a certain group of consumers such as business customers. The game consists of two stages.11 At the first stage consumers buy the primary good or bundled good at the price that the monopolist sets. The m onopolist can set the price of its own complementary good with or without commitment, or sell both goods as a bundle. At the second stage, consumers buy one of the complementary goods, the prices of which are determined by the competition between the two firms. Let p0, p1, and p2 be the prices of the monopolists primary good, the monopolists complementary good, and firm 2s complementary good, respectively. Then the net utilities of the consumer located at x if she consumes the primary good only, the primary good with the monopolists complementary good, and the primary good with firm 2s complementary good are, respectively, u0 = v0 p0 u1 = v0 + v1 tx p0 p1 u2 = v0 + v2 t (1 x ) p0 p2 The consumer will buy only the primary good if u0 > u1, u0 > u2, and u0 0 She will buy the primary good and the monopolists complementary good if 11 In section 2.7, an earlier stage will be added at which two firms make investment decisions that determine vis.

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20 u1 u2, u1 u0, and u1 0 She will buy the primary good and firm 2s complementary good if u2 u1, u2 u0, and u2 0 Lastly, she will buy nothing if u0 < 0, u1 < 0, and u2 < 0 2.3 Independent Sale without Commitment In this section, it is assumed that the monopolist cannot commit to p1 at the first stage. Without commitment, p1 must be chosen to be optimal at the second stage. That is, in game-theoretic terms, the equilib rium price must be subgame perfect. As in a standard sequential game, the equi librium set of prices can be obtained by backward induction. Let x* be the critical consumer w ho is indifferent between the monopolists complementary good and firm 2s good. One can find this critical consumer by solving v1 tx* p1 = v2 t (1 x*) p2, which gives 12211 22 vvpp x t (2-3) There are two cases to be considered: when the monopolist sells the primary good to all consumers, and when it sells its products to group S only. Consider first the case that the monopolist sells the primary good to all consumers. At th e second stage, the monopolist will set p1 to maximize p1x*, while firm 2 will set p2 to maximize p2(1 x*).

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21 By solving each firms maximization pr oblem, one can obtain the following best response functions: 2ijj ivvpt p i j = 1, 2 and i j (2-4) from which one can obtain the following equilibrium prices for the case of the independent sale without commitment (IA case): 3ij IA ivv p t i j = 1, 2 and i j Plugging these into (2-3) gives the lo cation of the critical consumer: 121 26IAvv x t At the first stage, the monopolist will se t the price of the primary good equal to v0 since consumers outside of group S will not buy the good for the price higher than v0: 0IA p = v0 One needs to check whether consumers actual ly buy the goods for this set of prices. This can be done by plugging the prices into th e net utility of the cr itical consumer, i.e., 12 101013 ()0 2IAIAIAIAvvt uxvvtxpp

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22 where the last inequality holds because of th e assumption given in (2-1). As was noted in footnote 10, this assumption guara ntees that all consumers in S buy both goods at equilibrium. The profits of the firms at equilibrium are 2 12 1010(3) 18IAIAIAIAvvt ppxv t 2 21 22(3) (1) 18IAIAIAvvt px t The monopolist may find it profitable to sell the primary goods exclusively to group S by charging the price higher than v0. If a consumer located at x have bought the primary good at the first stage, the maximu m prices she is willing to pay for the monopolists and firm 2s complement ary goods at the second stage are v1 tx and v2 t (1 x ), respectively, regardless how much she paid for the primary good at the first stage. Since the payment at stage one is a sunk cost to the consumer, she will buy a complementary good as long as the net ut ility from the complementary good is nonnegative. This implies that when th e monopolist sells the primary good to group S only without commitment to p1 (IS case), the equilibrium prices and the location of the critical consumer at the second stage are ex actly the same as in the IA case.12 That is, 3ij IS ivv p t i j = 1, 2 and i j 12 There may exist multiple equilibria b ecause of the coordination problem among consumers. For example, suppose consumers around at xIS did not buy the primary good at stage 1. Then at stage 2, the tw o firms will charge higher prices than pi IS. At this price set, consumers who did not buy the base good will be satisfied with their decision.

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23 121 26ISvv x t When consumers buy the primary good at stage 1, they rationally predict that the second period prices of the complementary goods are IS i p So the monopolist will set the primary good price to make the critical consumer indifferent between buying the complementary good and not buying, which yi elds the following equilibrium price: 12 003 2ISvvt pv Note that the primary good price is higher than v0 as is expected. By excluding the consumers who buy only the primary good, the monopolist can charge a higher price in order to capture some surplus that would ot herwise be enjoyed by the consumers of the complementary goods. The monopolists profits may increase or decrease depending on the size of while firm 2s profits remain the same as in the IA case since the price and the quantity demanded in IS case are exactly the same as in the IA case: 2 1212 1010()56 () 186ISISISISvvvvt ppxv t 2 21 22(3) (1) 18ISISISvvt px t By comparing 1 IS and 1 IA one can derive the condition in which the monopolist prefers the IS outcome to the IA outcome:

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24 0 0122 2(3)ISv vvvt Note that IS lies between 0 and 1 since 1230 vvt is assumed in (2-1). 2.4 Independent Sale with Commitment The results of the previous section may not be optimal for the monopolist if it can choose both p0 and p1 simultaneously at the first stage and commit to p1. To see this, suppose the monopolist can set both prices at the first stage with commitment. As in the previous section, one can distinguish two cas es depending on the coverage of the primary good market. When the monopolist sells its primary good to all consumers with commitment to p1 (CA case), the model shrinks to a simple game in which the monopolist set p1 at the first stage and firm 2 set p2 at the second stage since the primary good price should be set equal to v0, i.e., 00CA p v The equilibrium prices of the complementary goods can be derived using a st andard Stackelberg leader-follower model. The equilibrium can be found using backwa rd induction. At the second stage, the critical consumer who is indifferent be tween the monopolists complementary good and firm 2s good is determined by (2-3) with p2 replaced by firm 2s best response function given by (2-4), i.e., 1213 44 vvp x t (2-5) The monopolist will set p1 to maximize p1 x*, which gives the following optimal price:

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25 12 13 2CAvvt p The remaining equilibrium values can be obtained by plugging this into (2-4) and (2-5): 21 25 4CAvvt p 123 88CAvv x t 2 12 10(3) 16CAvvt v t 2 21 2(5) 32CAvvt t The differences between the equilibrium prices of CA case and IA case are 12 113 6CAIAvvt pp > 0 12 223 12CAIAvvt pp > 0 The price differences are positive since the difference between v1 and v2 is assumed to be less than t Since the monopolists complementary good is a substitute for firm 2s good, p1 and p2 are strategic complements. If one firm can set its price first, it will set a higher price so that the rival also raises its own price compared to the simultaneous move game. With the increase in the prices, both firms enjoy higher profits as the following calculation shows:

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26 2 12 11(3) 144CAIAvvt t > 0 1212 22(3)[277()] 288CAIAvvttvv t > 0 The profit of the monopolist must increase sin ce it chooses a different price even if it could commit to 1 IA p at the first stage. Firm 2s prof it also increases as both firms prices of complementary goods increase whil e the price of the primary good remains the same. When the monopolist covers only the consumers in group S with commitment to p1 (CS case), the equilibrium can be found in a si milar way as in the CA case. At the second stage, firm 2's best response function is the sa me as (2-4) and the critical consumer is also determined by (2-5). Since the monopolist will make the critical consumer indifferent between buying and not buying the complementary good, p0 will be set to satisfy the following condition: p0 = v0 + v1 p1 tx* (2-6) Using (2-5) and (2-6), the monopolist's prof its can be rewritten as a function of p1 in the following way: 121121 101033() () 44 vvtpvvp ppxv t Maximizing this profit function w.r.t. p1 yields the optimal price for the monopolist's complementary good, which is

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27 12 12CSvv p Plugging this back to (2-4), (2-5) and (2-6), one can derive the remaining equilibrium values: 12 00356 8CSvvt pv 21 22 4CSvvt p 123 48CSvv x t 2 1212 10()33 164CSvvvvt v t 2 21 2(2) 32CSvvt t The differences between the equilibrium prices of CS case and IS case are as follows: 21 006 8CSISvvt pp > 0 12 116 8CSISvvt pp < 0 12 226 12CSISvvt pp < 0 When the monopolist can commit to its complementary good price, it charges a higher price for the primary good and a lower price for the complementary good. And the

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28 rival firm also charges a lower pri ce for its own complementary good. Since p1 and p2 are strategic complements, the monopolis t can induce firm 2 to decrease p2 by lowering p1, which makes it possible for the monopolist to raise p0 for higher profits. This would not be possible if the monopolist cannot commit to p1 at the first stag e since the monopolist has an incentive to raise the complementary good price at the second stage after consumers have bought the primary good. The difference between the profits of CS case and IS case are as follows: 2 21 11(6) 144CSISvvt t > 0 1212 22[187()][6()] 288CSIStvvtvv t < 0 The monopolist's profits increase when it ca n commit as in CA case. However, firm 2s profits decrease since th e monopolist can capture some of the consumers surplus generated by firm 2s complementary good by charging a higher price for the primary good. Comparing 1CS and 1CA one can derive the fo llowing condition for the monopolist to prefer the CS outcome to the CA outcome: 0 01216 16(61021)CSv vvvt CS lies between 0 and 1 since 121221610218(3)2()30 vvtvvtvvt from the assumptions given in (2-1 ) and (2-2). The difference between IS and CS is

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29 021 0120122(223) 0 (23)(1661021)ISCSvvvt vvvtvvvt The critical level of with commitment is lower than with independent sale since the profit gain from commitment is highe r in the CS case than in the CA case.13 That is, the monopolist is willing to sell both goods to a smaller group of consumers when it can commit to the price of its own comple mentary good sold in the second period. The problem that the monopolist earns lowe r profits when it cannot commit to the second period price of the complementar y good is common in cases of durable goods with aftermarkets.14 That is, rational consumers expect that the monopolist will set its second period price to maximize its second pe riod profit regardless of its choice in the first period. The monopolist has an incentive to ch arge a higher p1 after consumers in S have bought the primary good at the first stage, since the price consumers have paid for the primary goods is sunk cost at stage 2.15 If the monopolist cannot commit to CSp1, therefore, some consumers in S would not buy the primary good at the first stage. So the monopolist would have to set a lower p0 (0IS p ) and a higher p1 (1IS p ) because of the holdup problem. One of the problems in relation to the pric ing with commitment is that the optimal prices may not be implemented since CSp1 is negative when v1 < v2.16 The bundling 13 Note that 21 1111[2()3] ()()0 16CSISCAIAvvt 14 See Blair and Herndon (1996) 15 After consumers have bought the primary goods at stage 1, the monopolist has an incentive to charge 1 IS p which is higher than CSp1. 16 If the marginal cost of producing the co mplementary good is positive, the optimal price

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30 strategy that will be presented in the following section can resolve this problem as well as the commitment problem. 2.5 Bundling: An Alternative Pric ing Strategy without Commitment An alternative strategy for the monopolis t when it cannot co mmit to the second period price or implement a negative price is bundling. That is, it sells both the primary good and its own complementary good for a single price. Note first th at it is not optimal for the monopolist to sell th e bundled good to all consumers si nce the bundled price must be equal to v0 in that case. So the monopo list will sell the bundled good to group S only if it chooses the bundling strategy. It is assumed that tying is reversible i.e., a consumer who buys a bundled good may also buy another complementary good and consume it with the primary good.17 Further, suppose consumers use only one complementary good, so the monopolists bundled complementary good is valueless to the consumers who use firm 2s complementary good.18 At the second stage, a consumer wh o has bought the bundled good earlier may buy firm 2's good or not, depending on her location x If she buys firm 2's complementary good, her net gain at stage 2 is v2 t (1 x ) p2. If she does not buy, she can use the monopolist's complementary good included in the bundle without extra cost, and get net gain of v1 t x So the critical consumer who is indifferent between buying firm 2's complementary good and using the bundled complementary good is can be positive even if v1 < v2. 17 In the software industry, a consumer who uses Windows OS bundled with Internet Explorer may install anot her Internet browser. 18 As long as there is no compatibility problem, consumers will use only one complementary good they prefer.

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31 1221 22 vvp x t (2-7) Since the price paid for the bundled good is a sunk cost at the second stage, the critical consumer is determined by p2 only. Firm 2 will choose p2 to maximize p2(1 x*), which yields the following optimal price for firm 2: 21 22BSvvt p Plugging this into (2-7) give s the location of the critic al consumer as follows: 123 44BSvv x t For this critical consumer to exist between 0 and 1, it is required that 3 t v1 v2 t So the assumption of | v1 v2 | t given in (2-2) is also a sufficient condition for the existence of a bundling equilibrium wit hout the exit of the rival firm. If v1 v2 t then all consumers buy the bundled good only so the rival firm will exit the market. If v1 v2 3 t on the other hand, all consumers buy both the bundled good and firm 2s complementary good. At stage 1, the monopolist will set the bundled good price, pb, that makes the critical consumer indifferent between buying and not buying:19 19 At the second stage, the monopolist may ha ve an incentive to unbundle the product and sell the primary good to the consumers outside of group S as long as the consumers second stage valuation for the good is positive, i.e., higher than the marginal cost. Knowing this, some consumers in group S may want to wait until the second period,

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32 12 033 4BS bvvt pv When consumers choose the monopolists co mplementary good, the total price for the primary good and the complementary good d ecreases compared to the IS case since 21 013 ()0 12BSISIS bvvt ppp (2-8) If consumers buy firm 2s complementary good as well as the monopolists bundled good, the total price increases compared to the IS case since 12 2023 ()()0 12BSBSISIS bvvt pppp (2-9) Comparing (2-8) and (2-9) one can find that the total price decrease for the consumers of monopolists complementary good is exactly the same as the total price increase for the consumers of firm 2s good. With the decrease of the total price, the number of consumers who choose to use the monopolists complementary good increases compared to the IS case as the following shows: 123 0 12BSISvvt xx t (2-10) The profits of the firms are which will lower the monopolists profits. To avoid this, the m onopolist will try to commit to not unbundling. One way to comm it is to make unbundling technologically difficult or impossible, as Microsoft combined Internet Explorer with Windows OS.

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33 12 1033 4BSBS bvvt pv 2 21 22() (1) 8BSBSBSvvt px t The following proposition shows that bundling increases monopolist's profits compared to the IS case. Proposition 2-1 Suppose the monopolist sells its goods to consumers in S only. Then the monopolist's profit in the bundling equilibrium is strictly higher than under IS, but not higher than under CS. Proof. The difference between profit s with bundling and IS case is 1112123 (3)()0 182BSISvvtvvt t The inequality holds since | v1 v2 | t On the other hand, the difference betw een profits with bundling and CS is 2 12 11() 0 16BSCSvv t where the inequality holds steadily when v1 v2. Q.E.D.

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34 In most of the previous analysis of bund ling based on the leverage therory, one of the main purposes of the bundling strategy is to foreclose the complementary good market. By lowering expected profits of the would-be entrants, bundling can be used to deter entry. The difference between the prev ious models and the current one is that bundling increases the profits of the monopolist even though the rival firm does not exit the market. On the contrary, the existence of the rival firm helps the monopolist in some sense since it creates demand for the monopolists bundled good. When compared to the CS case, bundling st rategy generates the same profits for the monopolist if v1 = v2. Technically, bundling strate gy is equivalent to setting p1 = pb and p2 = 0. When v1 = v2, the equilibrium commitmen t price for the monopolists complementary good, 1CS p is zero, hence the monopolists profits of bundling and CS cases are equal.20 Since the optimal commitment price is either positive or negative if v1 v2, the monopolists bundling profits is less than the CS case. By comparing bundling case with IA case, one can find the critical level of above which the monopolist finds bundling is more pr ofitable if commitment is not possible. The difference between the monopolists profits is 2 1212 1100()5715 1812BSIAvvvvt vv t 20 If the marginal cost (MC) of producing the complementary good is positive ( c ), the optimal commitment price for the good is c when v1 = v2 since the monopolist can avoid double marginalization problem by MC prici ng for the downstream good. In this case, bundling cannot generate same profits as the CS case even when v1 = v2 since it implicitly charges zero price inst ead of the one equal to MC.

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35 And the critical level of at which the monopolist is indifferent between bundling and independent sale is 0 2 0121236 363(5715)2()BSvt vttvvtvv If is higher than BS the monopolist can make hi gher profit by bundling both goods together and selling it to group S only than by separately selling the primary good to all consumers. That is, bundling is profita ble if the complementary good is widely used by the consumers of the primary good. In the software industry, Microsoft bundles Internet Explorer into Windows OS, while it sells MS Office as an independent product since Internet browser is a widely used product whereas the Office products are used by relatively small group of consumers. Note that BS lies between 0 and 1 since 22 121212 1223(5715)2()3(5715)2 15(3)2(3)0 tvvtvvtvvtt tvvttvt where the first and second inequalities hold b ecause of the assumptions given in (2-2) and (2-1), respectively. The difference between IS and BS is 01212 2 012120122(3)[32()] 0 [3(125715)2()](23)ISBSvvvttvv tvvvtvvvvvt The inequality holds because of the assu mptions (2-1) and (2-2). Since the

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36 monopolist can make much higher profits by bundli ng than IS case, it is willing to sell its goods to a smaller group of consumers th an IS case if bundling is possible. The difference between CS and BS is 01212 2 012120124[4()3][32()] [3(125715)2()](1661021)CSBSvvvttvv tvvvtvvvvvt Using assumptions (2-1) and (2-2), one can find that CS is higher than BS except when 213 4 tvvt Even though CS is not smaller than BS the profit gain from selling group S only is higher in bundling case than CS case except 213 4 tvvt as the following shows: 1212[4()3][32()] ()() 144BSIACSCAvvttvv t This explains why the monopo list is willing to sell the goods to a smaller group of consumers than the commitment case. 2.6 Bundling and Social Welfare Most previous analyses on bundling have ambiguous conc lusions about the welfare effect of bundling. It has been said that bundling could increa se or decrease welfare. In the model presented here, bundling decreases Marshallian social welfare except for an extreme case. Marshallian social welfare consists of th e monopolist profits, firm 2s profits, and consumers surplus. When the monopolist bun dles, its profits always increase compared

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37 to the IS case. Firm 2's profits, on the othe r hand, decreases in bundling equilibrium since 1212 22(3)[5()9] 0 72BSISvvtvvt t Consumers' surpluses wi th bundling and IS are 1 01022 0 22 1212(1) [()2()5] 16BS BSx BSBSBSBS bb xCSvvtxpdxvvtxppdx vvtvvt t 1 01010202 0 22 12(1) [()9] 36IS ISx ISISISISIS xCSvvtxppdxvvtxppdx vvt t The shaded area of Figure 2-1 shows c onsumers surplus of each case when v1 < v2. The difference between consumers surplus with bundling and IS is (a) Bundling (b) Independent sale (IS) Figure 2-1. Consumers surplus in bundling and IS cases when v1 < v2 v0 + v2 v0 + v1 2 B SBS b p p B S bp 01 I SISpp v0 + v1 x BS 0 1 0 1 x I S 02 I SISpp

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38 1212(3)[5()3] 144BSISvvtvvt CSCS t which shows that consumers' surplus incr eases by the monopolists decision to bundle unless v2 v1 > (3/5) t That is, unless firm 2s product is much superior to the monopolists complementary good, consumers surplus increases as the monopolist bundles. The consumers surplus increases mainly because consumers who pay less in bundling case than in IS case outnumber consumers who pay more in bundling equilibrium. Unlike consumers surplus, however, social welfare is more likely to decrease with bundling strategy by the monopo list, as the following proposition shows. Proposition 2-2 Suppose the monopolist sells its goods to consumers in S only. Then Marshallian social welfare decreases with the monopolists decision to bundle unless 123 7 tvvt Proof. Marshallian social welfare is define d as the sum of consumers surplus and profits of all firms. So so cial welfare w ith bundling is 2 1212 1203()1065 1616BSBSBSBSvvvvt WCSv t And social welfare with IS is

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39 2 1212 1205()22 364ISISISISvvvvt WCSv t The difference between them is 1212(3)[7()3] 144BSISWWvvtvvt t which is negative if t < v1 v2 < (3/7) t and positive otherwise. Since | v1 v2 | < t the social welfare decreases except (3/7) t < v1 v2 < t Q.E.D. The above proposition shows that unless the monopolists complementary good is superior enough, the monopolists bundling strate gy lowers the social welfare. Especially, the social welfare always decreases when the monopolist bundles an inferior good or a good with the same quality as the rivals, i.e., 12vv 2.7 Bundling and R&D Incentives One of the concerns about the bundling strategy by the monopolist of a primary good is that it may reduce R&D incentives in the complementary good industry. This section is devoted to the analysis of the effect of bundling on R&D incentives. To analyze this, one needs to introduce an earlier stage at which two firms make decisions on the level of R&D investment s to develop complementary goods. The whole game consists of three stages now. Let R ( v ) be the minimum required investment level to develop a complementary good of value v A simple form of the investment function is R ( v ) = ev2, e > 0

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40 Using this, the firms profit functions can be rewritten as follows:21 2 2 1212 1 01()56 186ISvvvvt vev t 2 2 21 2 2(3) 18ISvvt ev t 2 12 1 0133 4BSvvt vev 2 2 21 2 2() 8BSvvt ev t The following proposition shows that the monopolists bundling strategy reduces not only the R&D incentive of the riva l firm, but also its own incentive. Proposition 2-3 Suppose the investment cost satisfies 3 8 e t Then the equilibrium values of vi ( i = 1, 2) are higher in the IS equili brium than in the bundling equilibrium, i.e., ISv2 > 2BSv and ISv1 > 1BSv Further, firm 2's incentiv e decreases more than the monopolist's by bundling. Proof. The first order conditions for profit maxi mization problems yield each firms best response functions from which one can obt ain the following equilibrium levels of vis for each equilibrium: 21 In previous sections, it is assumed that the complementary goods already have been developed before the start of the game. The exclusion of the investment costs in profit function does not affect equilibr ium since they are sunk costs.

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41 1(907) 24(9)ISet v eet 2(367) 24(9)ISet v eet 13 8BSv e 2(83) 8(8)BSet v eet Since 0 1, the assumption 3 8 e t guarantees non-negative equilibrium values. Now the following comparisons prove the main argument: 11(92) 0 24(9)ISBSet vv eet 2 22(10) 0 248(9)(8)ISBSet vv eeetet 2 2211 ()()0 4(9)(8)ISBSISBSt vvvv etet Q.E.D. Firm 2 has a lower incentive to invest in R&D because part of the rents from the investment will be transferred to the m onopolist by bundling. The monopolist also has a lower incentive to invest because the bundli ng strategy reduces competitive pressure in the complementary good market. 2.8 Conclusion It has been shown that the monopolist of a primary good has an incentive to bundle its own complementary good with the primary good if it cannot commit to the optimal set of prices when consumers buy the primary good and the complementary good

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42 sequentially. Since the monopolist can increase its profits and the market share of its own complementary good by bundling, the model provi des another case in which the Chicago Schools single monopoly price theorem does not hold. While bundling lowers the rival firm's profits and Marshallian social welfare in general, it increases consumers surplus except when the monopolist's complementary good is sufficiently inferior to the rival's good. Bundling also has a negative eff ect on R&D incentives of both firms. Since bundling may increase consumers surplu s while it lowers social welfare, the implication for the antitrust policy is ambiguous If antitrust authori ties care more about consumers surplus than rival firms profits this kind of bundling may be allowed. Even if total consumers surplus increases, however, consumers who prefer the rivals complementary good can be worse off since th ey have to pay higher price for both the bundled good and the alternative compleme ntary good. So bundling transfers surplus from one group to another group of consumers. In addition to the problem of a redistribut ion of consumers surplus, bundling also has a negative long-term effect on welfare since it reduces both firms R&D incentives. This long-term effect of bundling on R&D i nvestment may be more important than immediate effects on competitor's profit or cons umers' surplus, especially for so-called high-tech industries that are characterized by high levels of R&D investments. For example, if a software company anticipates that development of a software program will induce the monopolist of the operating system to develop a competing product and bundle it with the OS, then the firm may have less incentive to invest or give up developing the software. This could be a new version of market foreclosure.

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43 A related issue is that if the risk of R&D investments includes the possibility of the monopolists developing and bundling of an a lternative product, it can be said that bundling increases social costs of R&D inve stments. Furthermore, since the monopolist is more likely bundle a complementary good th at has a broad customer base, bundling may induce R&D investments to be biased to the complementary goods that are for special group of consumers. A possible extens ion of the model lies in this direction. Another extension could be to introduce competition in the primary good market, which is suitable for the Kodak case.22 It has been pointed out that when the primary good market is competitive, the anticompetitive effect of bundling is limited. In the model presented here, firm 1 (the monopolist) could not set the bundling price so high if it faced competition in the primary good market However, if the primary goods are also differentiated so that the producers of th em have some (limited) monopoly powers, bundling may have anticompetitive effects. The result can be more complicatedbut more realisticif it is combined with the possibility of upgrade which is common in the software industry. 22 See Klein (1993), Shapiro (1995), Borenste in, MacKie-Mason, and Netz (1995), and Blair and Herndon (1996).

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44 CHAPTER 3 COMPETITION AND WELFARE IN THE TWO-SIDED MARKET: THE CASE OF CREDIT CARD INDUSTRY 3.1 Introduction It is well known that a two-sided marketor more generally a multi-sided marketworks differently from a conventiona l one-sided market. In order to get both sides on board and to balance the demands of both sides, a platform with two sides may have to subsidize one side (i.e ., set the price of one side lower than the marginal cost of serving the side). In the credit card industr y, cardholders usually pay no service fee or even a negative fee in various forms of rebate. In terms of the traditional one-sided market logic, this can be seen as a practice of predatory pricing. Several models of twosided markets, however, show th at the pricing rule of the tw o-sided market is different from the rule of the one-sided market, and a price below marginal cost may not be anticompetitive.1 Another feature of the two-sided market is that competition may not necessarily lower the price charged to the customers. In the credit card industry, competition between nonproprietary card schemes may raise the interchange fee, which in turn forces the acquirers to raise the merchant fee. The in terchange fee is a fee that is paid by the acquirer to the issuer for each transaction made by the credit card. If the interchange fee decreases as a result of competition, the cardholder fee is forced to increase. For the 1 Published papers include Ba xter (1983), Rochet and Tiro le (2002), Schmalensee (2002), and Wright (2003a, 2003b, 2004a).

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45 proprietary card schemes that set the cardholder fees and th e merchant fees directly, competition may lower one of the fees but not both fees. The distinctive relationship between compe tition and prices raises a question about the welfare effect of competition in the two-si ded market. Even if competition lowers the overall level of prices, it doe s not necessarily lead to a mo re efficient price structure. Previous models about competition in the two-sided markets focus mainly on the effect of competition on the price structure and deri ve ambiguous results on the welfare effects of competition. I present a model of the credit card industry in order to show the effects of competition on social welfare as well as on the price structure and level. The main result is that while the effects of comp etition on the price structure are different depending on the assumptions about whether consumers single-home or multi-home2 and whether card schemes are identical (Bertrand competition) or differentiated (Hotelling competition), the effects of competition on social welfare do not vary regardless of different model settings. That is, competition does not improve the social welfare in the various models presented here. The main reason for this result is that competition forces the platforms to set the price(s) in favor of one side that is a bottleneck part, wh ile a monopoly platform can fully internalize the indirect network externali ties that arise in the two-sided market.3 In order to maximize the transaction volume (for nonproprietary schemes) or profits (for proprietary schemes), the monopolist first need s to make the total size of the network 2 If a cardholder (or merchant) chooses to use (or accept) only one card, she is said to single-home. If she uses multiple cards, she is said to multi-home. 3 In a two-sided market, the benefit of one si de depends on the size of the other side. This indirect network externality cannot be internalized by the end-users of the two-sided market. See Rochet and Tirole (2005).

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46 externalities as large as possible. Competing card schemes, on the contrary, set biased prices since they share the market and tr y to attract single-homing consumers or merchants. Since the first formal model by Baxter ( 1983), various models of two-sided markets have been developed. Many of them focus on the price structure of a monopolistic twosided market.4 It is in recent years that consid erable attention has been paid to competition in two-sided markets. Rochet and Tirole (2003) st udy competition between differentiated platforms and show that if both buyer (consumer) and seller (merchant) demands are linear, then the price stru ctures of a monopoly platform, competing proprietary platforms and competing (non-prop rietary) associations are the same and Ramsey optimal. They measure the price struct ure and Ramsey optimality in terms of the price-elasticity ratio, so price levels and re lative prices are not the same for different competitive environments. While they assume that consumers always hold both cards, the model presented here distinguishes case s with single-homing consumers and multihoming consumers and uses Marshallian we lfare measure which includes platforms profits as well as consumers and merchants surpluses. Guthrie and Wright (2005) present a mode l of competition between identical card schemes. They introduce the business steali ng effect by allowing competing merchants and show that competition may or may not improve social welfare. I extend their model to the case of the competition between differentiated card schemes as well as the cases of proprietary card schemes, while removing the business stealing effect for simpler results. 4 The interchange fee is the main topic in thes e analyses of the credit card industry. See Rochet and Tirole (2002), Schmalens ee (2002) and Wright (2003a, 2003b, 2004a) for the analyses of the credit card i ndustry with monopoly card scheme.

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47 Chakravorti and Roson (2004) also provide a model of competing card schemes and show that competition is always welfare enhancing for both consumers and merchants since the cardholder fee and the merchant fee in duopoly are always lower than in monopoly. To derive the results, they assume that consumers pay an annual fee while merchants pay a per-transaction fee and cardholder benefits are platform specific and independent of each other. In contrast to their model, this paper assumes both consumers and merchants pay per-transaction fees5 and cardholder benefits are either identical or differentiated according to th e Hotelling model, and concludes that competition does not improve Marshallian social welfare. Further, it shows competition may not always lower both the cardholder a nd merchant fees even for the proprietary scheme as well as non-proprietary scheme. The rest of Chapter 3 proceeds as follows. Section 3.2 sets up the basic model of the non-proprietary card scheme. Section 3.3 and 3.4 show the effects of competition on the price structure and welfare for the cases of single-homing consumers and multihoming consumers. Section 3.5 extends the mo del to the case of the proprietary card scheme and compares the results with those of the non-proprietary card scheme. The last section concludes with a di scussion of some extensions and policy implications. 3.2 The Model: Nonproprietary Card Scheme Suppose there are two payment card schemes, i = 1, 2, both of which are not-forprofit organizations of many member banks A cardholder or consumer receives a pertransaction benefit bBi from using card i which is assumed to be uniformly distributed between ( Bb, Bb). A merchant receives a per-transaction benefit, bS, which is also 5 The per-transaction fee paid by consumers can be negative in the various forms of rebates.

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48 uniformly distributed between ( Sb Sb). It is assumed that merchants find no difference between two card schemes. There are two types of member banks. Issuers provide service to consumers, while acquirers provide service to merchants. Following Guthrie and Wright (2006), both the issuer market and the acquirer market are a ssumed to be perfectly competitive. Card schemes set the interchange fees in order to maximize total transaction volumes.6 For modeling convenience, it is assumed that there is no fixed cost or fixed fee. Let Ic and Ac be per-transaction costs of a issuer and a acquirer, respectively. Then card scheme i 's per-transaction cardholder fee and merchant fee are, respectively, iIi f ca iAimca where ai is scheme i s interchange fee. Note that the sum of the cardholder fee and the merchant fee is independent of the interchange fee since iiIA f mccc In order to rule out the possibility th at no merchant accepts the card and all merchants accept the card, it is assumed that 6 Rochet and Tirole (2003) assume consta nt profit margins for the issuers and the acquirers. Under this assumption, maximi zing member banks profits is same as maximizing total transaction volume, and the sum of the cardholder fee and the merchant fee is also independe nt of the interchange fee.

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49 BBS Sbbcbb (3-1) Both the numbers of consumers and merc hants are normalized to one. Consumers have a unit demand for each good sold by a monopolistic merchant.7 Merchants charge the same price to cash-paying consumer s and card-paying consumers, i.e., the nosurcharge-rule applies. The timing of the game proceeds as follows: i) at stage 1, the card schemes set the interchange fees, and the issuers and acquirers set the cardholder fees and merchant fees, respectively; ii) at stage 2, consumers choose which card to hold and use, and merchants choose which card to accept. 3.3 Competition between Identical Card Schemes: Bertrand Competition In this section, two card schemes are assumed to be identical, i.e., bB 1 = bB 2 ( bB). Consumers can hold one or both cards depe nding on the assumption of single-homing or multi-homing, while merchants are assumed to freely choose whether to accept one card, both cards, or none. One of the key features of the two-sided ma rket is that there exist indirect network externalities. As the number of members or acti vities increase on one side, the benefits to the members of the other side also increase. In the cred it card industry, cardholders benefits increase as the number of mercha nts that accept the card increases, while the merchants benefits increase as the number of cardholders who use the card increases. Some of the previous analyses of the cred it card industry did not fully incorporate this network effect in their models by a ssuming homogeneous merchants, in which case 7 Since merchants do not compete with each ot her, the business stea ling effect does not exist in this model.

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50 either all merchants or none accept the card.8 So at any equilibrium where transactions occur, all merchants accept card and consumer s do not need to worry about the size of the other side of the network. The model presen ted here takes into account this indirect network effect by assuming merchants are heterogeneous and the net utility of a consumer with bB takes the following form: ()()BiBiSiBIiSiUbfQbcaQ i = 1, 2 where QSi is the number of merchants that accept card i For modeling convenience, it is assumed throughout this section that the issuer market is not fully covered at equilibrium, which requires 2()BS BBbbbbc 3.3.1 Single-Homing Consumers If consumers are restricted to hold only one card, they will choose to hold card i if UBi > UBj and UBi 0. Note that the cardholding decisi on depends on the size of the other side as well as the price char ged to the consumers. Even if fi > fj, a consumer may choose card i as long as the number of merchants that accept card i ( QSi) is large enough compared to the number of merchants accepting card j ( QSj). Merchants will accept card i as long as bS mi since accepting both cards is always a dominant strategy for an individual mercha nt when consumers single-home. So the 8 See Rochet and Tirole (2002) and Guthrie and Wright (2006).

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51 number of merchants that accept card i (quasi-demand function for acquiring service) is9 SS iAi Si SS SSbmbca Q bbbb (3-2) Using (3-2), the consumers ne t utility can be rewritten as ()()S BiIiAi Bi S Sbcabca U bb Let bB be the benefit of the cri tical consumer who is indi fferent between card 1 and 2. One can obtain bB by solving UB 1 = UB 2, which is 12S BIAbbccaa A consumer with low bB is more sensitive to the transaction fee, so she prefers the card with lower cardholder fee (i.e., higher interchange fee). On the other hand, a consumer with high bB gets a larger surplus for each car d transaction, so she prefers the card that is accepted by more mercha nts. Therefore, a consumer whose bB is higher than bB will choose a card with lower ai, and a consumer whose bB is lower than bB will choose a card with higher ai. If ai = aj, then consumers are indifferent between two cards, so they are assumed to randomize between car d 1 and 2. This can be summarized by the following quasi-demand function of consumers: 9 Schmalensee (2002) calls QSi and QBi partial demands, and Rochet and Tirole (2003) call them quasi-demands since the actual dema nd is determined by the decisions of both sides in a two-sided market.

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52 *= if = if if 2()S Aj Bi ij BB BB BS B IAij B Biij BB BB B Ii ij B Bbca bf aa bbbb bbccaa bb Qaa bbbb bca aa bb (3-3) At stage 1, the card schemes choose th e interchange fees to maximize the transaction volume whic h is the product of QBi and QSi. The following proposition shows the equilibrium interchange fee of the single-homing case of Bertrand competition. Proposition 3-1 If two identical card schemes comp ete with each other and consumers single-home, (i) the equilibrium interchange fee is 1 2()() 3bs SB AIabcbc (ii) bsa maximizes total consumers surplus Proof. (i) Without loss of generality, suppose a1 > a2. Then scheme 2 will maximize the following objective function: 212 22122()() (;) ()()SBS AIA BS BS BSbacbbccaa TaaQQ bbbb

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53 from which scheme 2s best response function can be obtained as follows: 2111 ()22 2SB IA R abbacc Scheme 1s objective function is 12 11211()() (;) ()()SS A A BS BS BSbacbac TaaQQ bbbb Since the function is a linear function of a1 with negative coefficient, scheme 1 will set a1 as low as possible, i.e., as close to a2 as possible. So the be st response function of scheme 1 is R1(a2) = a2 Solving R1(a2) and R2(a1) together, one can obtain the following Nash equilibrium: ** 121 [2()()] 3bs SB AIaabcbca The equilibrium transaction volume of scheme i when a1 = a2 = abs is 2() (;) 9()()BS bsbsbs i BS BSbbc TaaT bbbb

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54 Since scheme 1s best response function seems to contradict the premise that 12aa it is necessary to show that card sc hemes do not have an incentive to deviate from the equilibrium. To see this, suppose scheme 1 changes a1 by a Then the transaction volume of scheme 1 becomes 1()(3) if 0 9()() (;) (3)(3) if 0 9()()BSBS BS BS bsbs BSBS BS BSbbcbbca a bbbb Taaa bbcabbca a bbbb Both of them are less than bsT so there is no incentive for scheme 1 to deviate from abs. (ii) At symmetric equilibrium with common a the consumers demands for the card services are given by (3-3). So the total consumers surplus is 2 2 1 2(())(()) (()) 2()() ()() 2()()Bb BS bs BSiBi f BS i BS BS IA BS BSbfabma TUfaQQdf bbbb bcabca bbbb The optimal a that maximizes bs BTU is *1 [2()()] 3SB AIabcbc which is same as bsa Q.E.D.

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55 When consumers single-home, each card scheme has monopoly power over the merchants that want to sell their products to the consumers. This makes the card schemes try to attract as many consum ers as possible by setting the interchange fee favorable to consumers. The resulting in terchange fee chosen by the card schemes is one that maximizes total consumers surplus. An interchange fee higher than bsa may attract more consumers due to the lower cardholder fee, but fewer merchants will accept the card due to the higher merchant fee. Therefore, a card scheme can increase the tr ansaction volume by lowering its interchange fee, which attracts higher types of consum ers who care more about the number of merchants that accept the card. On the othe r hand, an interchange fee lower than bsa may attract more merchants, but fewer consumers will use the card. In this case, a card scheme can increase the transaction volume by raising its interchange fee. In order to see how competi tion in the two-sided market affects the price structure, it is necessary to analyze the case in whic h the two card schemes are jointly owned by one entity. As the following proposition shows, it turns out that joint ownership or monopoly generates a lowe r interchange fee, which implie s a higher cardholder fee and a lower merchant fee. In other words, competition between card schemes when consumers single-home raises the interchange fee. Proposition 3-2 If two identical card schemes ar e jointly owned and consumers singlehome, (i) the symmetric equilibrium interchange fee is

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56 1 ()() 2bjbs SB AIabcbca (ii) the joint entity may engage in pri ce discrimination in which one scheme sets the interchange fee equal to bja and the other scheme sets th e interchange fee at any level above bja, but the total transaction volume cannot increase by the price discrimination, (iii) bja maximizes the social welfare, which is defined as the sum of the total consumers surplus and the to tal merchants surplus. Proof. (i) Since the card schemes are identi cal, there is no difference between operating only one scheme and operating both schemes w ith same interchange fees. So suppose the joint entity operates only one scheme Then the quasi-demand functions are B B I B BB BBbca bf Q bbbb S S A S SS SSbca bm Q bbbb The joint entity will choose the optimal a in order to maximize the transaction volume QBQS. The optimal interchange fee obtaine d from the first-order condition is *1 ()() 2bj SB AIabcbca which is less than bsa since

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57 1 0 6bsbj SBaabbc (ii) Without loss of generality, suppose a1 > a2. Then scheme 1 will attract low-type consumers and scheme 2 will attract high-type consumers. The quasi-demand functions are determined by (3-2) and (3-3). And the total transaction volume is 22 1122()() ()()BS IA BSBS BS BSbcabca QQQQ bbbb (3-4) Note that (3-4) is independent of a1, which implies a1 can be set at any level above a2. The optimal a2 can be obtained from the first-order condition for maximizing (3-4): 21 ()() 2bj SB AIabcbca It is not difficult to check that the total tr ansaction volume at equilibrium is also the same as in the symmetric equilibrium. (iii) The sum of the total consumers surplus and the total merchants surplus is ()()() 2()()BSbb bsjbsjbsj BSSBBS fm BSBS IA BS BSTUTUTUQQdfQQdm bcabcabbc bbbb The optimal a that maximizes bsjTU is

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58 1 ()() 2bw SB AIabcbc which is same as bja. So bja maximizes social welfare. Q.E.D. The most interesting result of the propositi on is that the joint entity, which acts like a monopolist, chooses the socially optimal inte rchange fee. This is possible because both the issuing and acquiring sides are competitive even though the platform is monopolized, and the joint entity can internalize the indi rect network external ities of both sides. Comparing propositions 3-1 and 3-2, one can find that competition between card schemes lowers social welfare as well as decreases to tal transaction volume. In a typical example of prisoners dilemma in game theory, co mpeting firms choose higher quantity and/or lower price, which is detrimental to themselv es but beneficial to the society. But this example of the two-sided market shows that competitive outcome can be detrimental to the society as well as to themselves. 3.3.2 Multi-Homing Consumers In this subsection, consumers are allowed to multi-home. Since there is no fixed fee or cost, individual consumer is always be tter off by holding both cards as long as bBi > fi. So the number of consumers who hold card i is BB iIi Bi BB BBbfbca Q bbbb (3-5) On the other hand, since merchants have monopoly power over the products they sell, they may strategical ly refuse to accept card i even if bS > mi.

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59 If a merchant accepts card i only, it receives a surplus equal to ()() ()B SAiIi SiSiBi B Bbcabca UbmQ bb (3-6) If the merchant accept bo th cards, the surplus is USb = (bS m1)Qb 1 + (bS m2)Qb 2 = (bS cA a1)Qb 1 + (bS cA a2)Qb 2 (3-7) where Qbi is the number of consumers who will use card i if the merchant accepts both cards.10 When a consumer holding both cards buys from a merchant that accept both cards, the consumer will choose to use the card that gives a higher net benefit, i.e., she will use card i if BiiBjjbfbf And the consumer w ill randomize between card i and j if BiiBjjbfbf. If the two card schemes are identical (bB 1 = bB 2), consumers will use the card that has a lower consumer fee if merchant accepts both cards, i.e., if () 0 if () (1/2) if ()Biijij biijij BiijijQaaff Qaaff Qaaff (3-8) A merchant with bS will accept card i only if USi > USj and USi > USb. It will accept both cards if USb USi, i = 1, 2. To see the acceptance decision by a merchant, suppose 10 Consumers card-holding decision and card-us ing decision can be different since they can hold both cards but use onl y one card for each merchant.

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60 a1 > a2 without loss of generality. Then the net surplus to the merchant if it accepts both cards is USb = (bS cA a1)QB 1 + (bS cA a2)0 = US 1 Merchants are indifferent between accep ting card 1 only and accepting both cards since consumers will only use card 1 if merc hants accept both cards. In other words, there is no gain from accepting both cards if consumers multi-home. So merchants decision can be simplified to the c hoice between two cards. Let bS be the critical merchant that is indifferent between accepting card 1 only a nd card 2 only, which can be obtained by setting US 1 = US 2: 12 B SIAbbccaa Merchants with low bS will be sensitive to the mercha nt fee and prefer a card with low merchant fee (low interchange fee), while merchants with high bS will prefer a card with low consumer fee (high interchange fee) since they care more about the number of consumers who use the card. Therefore, if m1 > m2 (a1 > a2), merchants with bS smaller than bS (and greater than m2) will accept card 2 only and merchant with bS higher than bS will accept card 1. If a1 = a2, all cardholders have both cards a nd it is indifferent for merchants whether they accept card 1, card 2 or both. Fo r modeling simplicity, it is assumed that merchants will accept both cards if a1 = a2. The following summarizes the number of merchants that accept card i:

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61 *= if = if = if SB S IAij S ij SS SS B Ij Si Siij SS SS SS iAi ij SS SSbbccaa bb aa bbbb bca bm Qaa bbbb bmbca aa bbbb (3-9) Proposition 3-3 If two identical card schemes co mpete with each other and consumers multi-home, (i) the equilibrium interchange fee is 1 ()2() 3bm SB AIabcbc (ii) bma maximizes total merchants surplus. Proof. (i) Without loss of generality, suppose a1 > a2 (m1 > m2). Then scheme 1s best response function can be obtai ned by solving the optimizatio n problem of the scheme, which is 1221 ()22 2SB IA R abbcca Scheme 2s objective function is 12 22122()() (;) ()()BB II BS BS BSbcabca TaaQQ bbbb

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62 Since the function is linear in a2 with positive coefficient, scheme 2 will set a2 as high as possible, i.e., as close to a1 as possible. So the best response function of scheme 2 is R2(a1) = a1 Solving R1(a2) and R2(a1) together, one can obtain the following Nash equilibrium: ** 121 ()2() 3bm SB AIaabcbca The equilibrium transaction volume of scheme i when a1 = a2 = bma is 2() (;) 9()()BS bmbmbm i BS BSbbc TaaT bbbb As in Proposition 3-1, it is necessary to s how that the card schemes do not have an incentive to deviate from bma in order to justify the e quilibrium. To see this, suppose scheme 1 changes a1 by a Then the transaction volume of scheme 1 becomes 1(3)(3) if 0 9()() (;) ()(3) if 0 9()()BSBS BS BS bmbm BSBS BS BSbbcabbca a bbbb Taaa bbcbbca a bbbb

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63 Both of them are less than bmT so there is no incentive for the scheme to deviate from bma (ii) At symmetric e quilibrium with common a the total merchants surplus is 2 2 1 2(())(()) (()) 2()() ()() 2()()Sb SB bm SBiSi m BS i BS SB AI BS BSbmabfa TUmaQQdm bbbb bcabca bbbb The optimal interchange fee that maximizes bm STU is *1 ()2() 3SB AIabcbc which is equal to bma So bma maximizes total merchants surplus. Q.E.D. When consumers multi-home, the card schemes care more about merchants since they can strategically refuse to accept one ca rd. By setting the interchange fee so as to maximize the merchants surplus, the card schemes can attract as many merchants as possible. As in the single-homing case, an interchange fee hi gher or lower than bma is suboptimal and a card scheme can increase its transaction volume by changing the interchange fee closer to bma The interchange fee in the multi-homing case is lower than in the single-homing case since the fee is set in favor of the merchants. The following proposition shows that

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64 the interchange fee is higher if the card schemes are jointly owned, which implies the interchange fee decreases as a result of competition between card schemes when consumers multi-home. It also shows that comp etition lowers social welfare as in the single-homing case. Proposition 3-4 If two identical card schemes are jointly owned and consumers multihome, (i) the symmetric equilibrium interchange fee is 1 ()() 2bjbm SB AIabcbca (ii) the joint entity may engage in pri ce discrimination in which one scheme sets the interchange fee equal to bja and the other scheme sets th e interchange fee at any level below bja, but the total transaction volume cannot increase by the price discrimination, (iii) bja maximizes social welfare. Proof. (i) Regardless whether consumers single-home or multi-home, there is no difference for the joint entity between operating two card schemes with same interchange fee and operating only one scheme since the card schemes are identical. So the proof is the same as the first part of Proposition 3-2. And for multi-homing consumers, the monopolistic interchange fee is higher th an the competitive interchange fee since 1 0 6bjbm SBaabbc

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65 (ii) Without loss of generality, suppose a1 > a2. Then scheme 1 will attract lowtype merchants and scheme 2 will attract high-type ones. Then the total transaction volume is 11 1122()() ()()BS IA BSBS BS BSbcabca QQQQ bbbb (3-10) Note that (3-10) is independent of a2, which implies a2 can be set at any level below a1. The optimal a1 obtained from the first-order condition is 11 ()() 2SB AIabcbc which is equal to abj. The total transaction volume at equilibrium is 2() 4()()BS BS BSbbc bbbb which is the same as in the symmetric equilibrium. (iii) The proof is the same as in part (iii) of Proposition 3-2. Q.E.D. The optimal interchange fee for the joint en tity is the same as in the single-homing case since the card schemes do not compete for consumers or merchants. Unlike the single-homing case, however, the interchange fee decreases as a result of competition between the card schemes when consumers multi-home. Social welfare deteriorates since

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66 competing card schemes set the interchange fee too low in order to attract more merchants. Figure 3-1 shows the results of this section. As is clear in the figure, competitive equilibrium interchange fees maximize either consumers surplus or merchants surplus. Since monopoly interchange fee maximizes total surplus, competitive outcome is suboptimal in terms of social welfare. 3.4 Competition between Differentiated Card Schemes: Hotelling Competition In this section, card schemes are assumed to be differentiated and compete la Hotelling. As in a standard Hotelli ng model, suppose cons umers are uniformly distributed between 0 and 1, and the card scheme 1 is located at 0 and scheme 2 is at 1. A consumer located at x receives a net benefit of Bbtx ( bB 1) if she uses card 1, and (1)Bbtx ( bB 2) if she uses card 2. In order to comply with the assumption that consumers benefits from card usage is uniformly distributed between ( Bb, Bb), the transportation cost t is assumed to be equal to B Bbb Figure 3-1. Welfare and interchange fees of Bertrand competition TUS TUS TU = TUB + TUS B Icb S A bc bmabjabsa

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67 The net utilities of a consumer located at x when she uses card 1 and 2 are 11111 22222(1) (1)(1)BB B BSIS BB B BSISUbtxfQbxbxcaQ UbtxfQbxbxcaQ The critical consumer, x*, who is indifferent between card 1 and 2 can be obtained by solving UB 1 = UB 2: 1122 12()() ()()B B ISIS B B SSbcaQbcaQ x bbQQ (3-11) If the issuer market is not fully cove red, each card scheme has a full monopoly power over the consumers and the resulting equilibrium will be the same as in the monopoly case of the previous section. In order to obtain competitive outcomes, the issuer market is assumed to be fully covere d at equilibrium. This requires the following assumption:11 S Bbbc Depending on whether consumers single-hom e or multi-home, and whether card schemes compete or collude, various equilibria can be derived. There may exist multiple equilibria including asymmetric ones. For expositional simplicity, however, only symmetric equilibria will be considered unless otherwise noted. 11 For the issuer market to be fully covered, the net utility of the consumer located at x = must be non-negative for th e monopolistic interchange fee abj.

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68 3.4.1 Single-Homing Consumers When consumers are restricted to single-home, merchants will accept card i as long as bS mi as in the previous section. So QSi is determined by (3-2). Since the issuer market is fully covered, the number of consumers who choose card i is 1 BQx and 21BQx where x* is defined in (3-11). The following proposition shows the symm etric equilibrium of the Hotelling competition when consumers single-home. Proposition 3-5 If two differentiated card scheme s compete la Hotelling and consumers single-home, (i) the symmetric equilibrium interchange fee is 1 () if 2() 2 = 1 2()2()() if 2() 4BBS BBB I hs SBBBS BBB AIcbbbbbbc a bcbcbbbbbbc (3-12) (ii) ahs maximizes the weighted sum of to tal consumers surplus and total merchants surplus, (1)hh BSwTUwTU where the weight for consumers surplus is 13()2() 6()2()BS BB BS BBbbbbc w bbbbc if 2()BS BBbbbbc

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69 22()4() 3()8()BS BB BS BBbbbbc w bbbbc if 2()BS BBbbbbc Proof. (i) For a given a2, card scheme 1 will set a1 to maximize its transaction volume. The symmetric equilibrium interchange fee can be obtained from th e first-order condition in which a1 and a2 are set to be equal to each other for symmetry: *1 2()2()() 4SBB B AIabcbcbb For this fee to be an equilibrium net benefit of the consumer at x = must be nonnegative since the issuer market is assu med to be fully covered, which requires *1 ()2()()0 24BSB BBt bfabbcbb That is, a* is an equilibrium interchange fee if 2()BS BBbbbbc If 2()BS BBbbbbc the equilibrium interchange fee can be obtained by setting consumers net benefit at x = equal to zero: **1 () 2B B Iacbb For a** to be an equilibrium, it needs to be shown that the card schemes have no incentive to deviate from a**. The transaction volume of card scheme 1 at a** is

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70 **** 122 (,) 4()BS B S Sbbbc Taa bb The right and left derivative s of scheme 1s profit at a1 = a** are, respectively, ******** 11 0(,)(,)2()() lim0 4()()SB BB a BS BSTaaaTaabbcbb a bbbb ******** 11 0(,)(,)() lim0 ()()S B a BS BSTaaaTaabbc a bbbb So a** is an equilibrium when 2()BS BBbbbbc Note that a* = a** when 2()BS BBbbbbc (ii) First, note that QBi = at symmetric equilibrium since the market is fully covered. The weighted sum of total consum ers surplus and merchants surplus for scheme 1 is 2 12 0 11 (1) 2 ()(344)2(1)() 4()Sb hh BSBBSi m i SBS B AIA S Sw wTUwTUwUdxUdxQdm bcawbbcawbca bb (3-13) The optimal interchange fee that maximizes this weighted surplus is *4(21)()(34) 4(31)SB B A I wwbcwbbc a w (3-14)

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71 The size of the weight can be obtained by setting hs waa which is 13()2() 6()2()BS BB BS BBbbbbc w bbbbc if 2()BS BBbbbbc 22()4() 3()8()BS BB BS BBbbbbc w bbbbc if 2()BS BBbbbbc Note that w1 = w2 = 4/7 if 2()BS BBbbbbc Q.E.D. When the card schemes compete la Ho telling, they have some monopoly power over the consumers. So unlike the Bertrand comp etition case, they do not need to set the interchange fee so high as to maximize total consumers surplus. While the weight for consumers surplus ( w ) in Bertrand competition is equal to 1, the weight in Hotelling competition ranges between 4/7 and 1. If ()2()BS BBbbbbc the weight is equal to 4/7. It becomes close to one as B Bbb approaches zero. Note that B Bbb is equal to the transportation cost t As in a standard Hotelling m odel, the monopoly power of a card scheme weakens as t becomes smaller. Therefore, the card scheme will set the interchange fee so as to maximize total cons umers surplus when the transportation cost becomes zero. The following proposition shows the monopol y interchange fee in the Hotelling model also maximizes the social we lfare as in the Bertrand model. Proposition 3-6 If the two differentiated card sc hemes are jointly owned and consumers single-home,

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72 (i) the joint entity will set the interchange fee equal to 1 () 2hj B B Iacbb (ii) hja maximizes the sum of the total consumers surplus and the total merchants surplus. Proof. (i) I will prove this proposition in two cases: (a) when the joint entity sets the same interchange fees for scheme 1 and 2, and (b) when it sets two different fees (price discrimination). (a) When the joint entity sets the same interchange fees for both schemes, the joint transaction volume is 1122(,)S A MBSBS S Sbca TaaQQQQ bb where QB 1 = QB 2 = since the issuer market is assumed to be fully covered. Note that TM is decreasing in a which implies that the optimal a is the minimum possible level that keeps the issuer market covered. This fee can be obtained by setting the consumers net benefit at x = equal to zero, which is hja (b) Now suppose the joint entity trie s a price discrimination by setting a1 = hja+ a and a2 = hja a a > 0. The joint transaction volume when it charges same fee, hja is

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73 2()() (,) 2()SB hjhj BB M S Sbbcbb Taa bb while the joint transaction volume of the price discrimination is 2()()2 (,) 2()SB hjhj BB M S Sbbcbba Taaaa bb It is not beneficial to engage in price discrimination since (,)(,)0hjhjhjhj MM S Sa TaaaaTaa bb (ii) Since QB 1 = QB 2 = at full-cover market equilibrium, the sum of total consumers surplus and tota l merchants surplus is 2 12 0 11 2 ()(34222) 4()Sb hh BSBBSi m i SBS B AIA S STUTUUdxUdxQdm bcabbcbca bb The optimal a that maximizes social welfare is12 *1 3 4B B UIacbb 12 The fee is equivalent to *wa in (3-14) when w =

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74 Note that the market is not fully covered at *Ua since *Ua < hja In other words, *Ua is not feasible. Therefore, hja maximizes the sum of the total consumers surplus and the total merchants surplus when the market is fully covered. Q.E.D. Note that hshjaa if 2()BS BBbbbbc and hshjaa if 2()BS BBbbbbc. As in the Bertrand competition case, competition does not lower the equilibrium interchange fee nor increase social welfare when the card schemes compete la Hotelling and consumers single-home. 3.4.2 Multi-Homing Consumers If consumers are allowed to multi-home, they will hold card i as long as bBi > fi. So the number of consumers who hold card i is the same as (3-5). If the issuer market is fully covered and the merchants accept both cards, th e critical consumer who is indifferent between card 1 and 2 is obtained by solving 12(1)BBbtxfbtxf which is 211211 222 2()B B f faa x t bb The number of consumers who use card i if merchants accept both cards is Qb 1 = x*, and Qb 2 = 1 x* Lemma 3-1 If ai > aj, merchants accept either card j only or both cards, i.e., no merchant will accept card i only.

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75 Proof. Without loss of generality, suppose a1 > a2. The critical merchant that is indifferent between accepting card 1 only a nd accepting card 2 only can be obtained by setting US 1 = US 2, where USi is defined in (3.6): 12B SIAbbccaa Merchants with low bS will be more sensitive to the merchant fee, while merchants with high bS will care more about the number of consumers who use the card. So if *SSbb, the merchant prefers card 1 to card 2 and vice versa. The critical merchant that is indifferent between accepting card i only and accepting both cards can be obtained by setting USb = USi, where 1122()()SbbbUbsmQbsmQ Let *Sib be the critical merchant. That is, *()()2()2() 2()B BBB jiijijAIAI Si B Iiaaaabbaabccccb b cab If *SSibb, accepting both cards is more pr ofitable than accepting only card i since merchants with high Sb care more about the transacti on volume. The difference between *Sb and *Sib is

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76 22 ** 1212()(2)2()() 2()BB BB III SSi B Iiaaaabbcbcbc bb cab Note that the numerator is independent of i and the denominator is positive.13 Since a1 > a2, *** 21SSSbbb if the numerator is positive, and *** 21SSSbbb if the numerator is negative. Note also that *Sb is larger than mi since *B Sijbmbf > 0, i j and 1Sb is smaller than m1 since 1212 11 1()()() 0 2()B B S Baabbaa bm fb which implies *** 21 SSSbbb. Note that the difference between two interchange fees, which is same as the difference between two cardholder fees, cannot exceed the difference between Bb and Bb since B B ibfb As is shown in Figure 3-2, merchants will accept card 2 only if bS [ m2, 2 Sb ), and accept both cards if bS [* 2Sb Sb].14 Q.E.D. 13 0B Iicab since it is equal to B i f b and the cardholder fee must be higher than Bb. 14 Since 12 *()()() 2()B B ji Sij B iaabbff bm fb 12 Sbm (* 21 Sbm) if and only if 12 B Bbbff

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77 Figure 3-2. Merchants acceptance decision when a1 > a2 ( m1 > m2) Based on Lemma 3-1, the number of merchants that accept card i is if () if ()S Sj ijij S S Si S i ijij S Sbb aamm bb Q bm aamm bb Let Qai be the number of merchants that accept card i only, and QSb be the number of merchants that accept both cards. That is, if () 0 if () where ()SiSjijij ai ijij SbSiijijQQaamm Q aamm QQaamm The following proposition summarizes the equilibrium interchange fee of the Hotelling competition with multi-homing consumers. Proposition 3-7 If consumers can multi-home and card schemes compete la Hotelling, accepting card 2accepting both cards accepting card 2accepting both cards Sb Sb Sb Sb* 1 Sb2m1m* 2Sb*Sb2m* 1Sb* 2Sb1m*Sb

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78 (i) the symmetric equilibrium interchange fee is 1 if 2() 2 1 () if 2() 2SBS BBB AI hm BBS BBB IbccbAbbbbc a cbbbbbbc where 222()()BS BBAbbbbc (ii) hmhsaa, where the equality holds when 2()BS BBbbbbc (iii) hma maximizes the weighted sum of total consumers surplus and total merchants surplus, (1)hshs BSwTUwTU, where the weight for the consumers surplus is 12()2 2()3()6S B SB BBbbcA w bbcbbA if 2()BS BBbbbbc 22()4() 3()8()BS BB BS BBbbbbc w bbbbc if 2()BS BBbbbbc Proof. (i) Without loss of generality, suppose a1 a2 ( m1 m2). Then scheme 1 and 2s transaction volumes are, respectively, 1121(;)bSbTaaQQ 221222(;)BabSbTaaQQQQ The symmetric equilibrium can be obtained by taking derivative of Ti w.r.t. ai at ai = aj, which yields

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79 *221 2()() 2SBS BBB AIabccbbbbbc At the symmetric equilibrium, all merchants accept both cards (i.e., Qai = 0) and Qb 1 = Qb 2 = So the transaction volume of each card scheme is 22 **2()() (;) 4()SBS BBB i S Sbbcbbbbc Taa bb To see the card schemes do not have an incentive to deviate from a*, suppose scheme 1 changes a1 by a Then the transaction volume of the scheme becomes 1 ** 1 222 if 0 (;) if 0bSb BabSbQQa Taaa QQQQa The transaction volume doe s not increase by changing a since **** 11 2 2(;)(;) (2()) 0 if 0 2()()(()) (32()()3) 0 if 0 2()()(()2)B B BSS BSB BS BB BSS BSBTaaaTaa abba a bbbbAbbc aAbbbbca a bbbbAbbca So the card schemes do not have an incentive to deviate from a*. For a* to be an equilibrium, the issuer market must be fully covered at equilibrium. The net benefit of the consumer located at x = is

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80 *2211 ()()2()() 22BBSBS BB Ibtcabbcbbbbc This is nonnegativ e if and only if 2()BS BBbbbbc since 222()2()()()2()()BSBSBSB BBBBBbbcbbbbcbbbbcbb If 2()BS BBbbbbc, as in the single-homing case, the equilibrium interchange fee can be obtained by setting consumers net benefit at x = equal to zero, which is **1 () 2B B Iacbb Note that, as in the single-homing case, a* = a** when 2()BS BBbbbbc (ii) If 2()BS BBbbbbc, [2()2()()]hs SBB B AIabcbcbb and the difference between the two equilibrium fees is 22 221 22()()3() 4 1 22()[(1/2)()]3()0 4hshm BSB BBB BBB BBBaabbbblbb bbbbbb If 2()BS BBbbbbc, both hsa and hma are equal to ()B B Icbb (iii) The weighted sum of total consumers surplus and total merchants surplus is the same as (3-13), hence the optimal interc hange fee maximizing the weighted surplus is

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81 also the same as (3-14). The level of the weight can be obtained by setting hm waa, which is 12()2 2()3()6S B SB BBbbcA w bbcbbA if 2()BS BBbbbbc 22()4() 3()8()BS BB BS BBbbbbc w bbbbc if 2()BS BBbbbbc Note that, as in the single-homing case, w1 = w2 = 4/7 if 2()BS BBbbbbc Q.E.D. When consumers multi-home, the equilibrium interchange fee is lower than that of the single-homing case. But unlike the Bert rand competition case in which card schemes set the interchange fee so as to maximize the merchants surplus, the card schemes do not lower the fee enough. In the Bertrand competition with multi-homing consumers, merchants accept only one card if the merchant fees set by two card schemes are different. Therefore, a card scheme can maximize its transaction volume by attracting as many merchants as possible. In Hotelling compe tition, however, each card scheme has its own patronizing consumers since it provides differentiated serv ice. This weakens merchant resistance, which forces many merchants to accept both cards.15 Therefore, card schemes do not need to provide maximu m surplus to the merchants. If the card schemes are jointly owned, the result will be the same as in the singlehoming case since the joint entity will split the is suer market so that each consumer holds 15 See Rochet and Tirole (2002) for a discussion of merchant resistance.

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82 only one card at equilibrium. Figure 3-3 shows the relationship of vari ous equilibrium interchange fees and welfare, which is drawn for the case of 2()BS BBbbbbc .16 The left side of hja is not feasible since the market cannot be fully covered. As is clear from the figure, competition not only increases the equilibrium interchange fee but also lowers social welfare. It also shows that allowing consum ers to multi-home increases social welfare in the Hotelling competition case, although it lowers total consumers surplus. 3.5 Proprietary System with Single-Homing Consumers The analysis of the previous sections has been restricted to the competition between non-proprietary card schemes that set interc hange fees and let the cardholder fees and Figure 3-3 Welfare and interchange f ees of Hotelling competition when 2()BS BBbbbbc 16 When 2()BS BBbbbbc, hjhmhsaaa. hja* Uahsa S A bc hma BSTUTUTU BTU STU

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83 merchant fees be determined by issuers and acquirers, respectivel y. Another type of credit card scheme, a proprietary scheme, serv es as both issuer and acquirer. It sets the cardholder fee and merchant fee directly, so there is no need for an interchange fee.17 3.5.1 Competition between Identical Card Schemes One of the features of the proprietary card scheme is that competition may not only alter the price structure but may also change th e price level. In the previous sections, the sum of the cardholder fee and merchant fee does not change even afte r the introduction of competition between card schemes.18 When a card scheme sets both the cardholder fee and the merchant fee, it may change one of th e fees more than the other since the effects of competition on two sides are not equivalent. To see how competition affects the equilibrium fees of the proprietary card scheme, the equilibrium of the monopoly case will be presented first. For the sake of simplicity, only the case of single-homing consumers will be considered. When the monopoly proprietary card scheme sets f and m the quasi-demand functions of consumers and merchants are B B B Bbf Q bb and S S S Sbm Q bb 17 In the United States, Discover and American Express are examples of this type of card scheme. 18 This feature of the non-proprietary sc heme requires an assumption of perfect competition among issuers and acquirers. If the perfect competition assumption is removed, competition may alter the price level as well as the price structure in the nonproprietary card scheme model.

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84 and the profit of the scheme is19 ()BS f mcQQ From the first order condition for the pr ofit maximization problem, one can obtain the following equilibrium cardhol der fee and merchant fee: 1 2 3 1 2 3M BS M SB f bbc mbbc (3-15) The following lemma shows that there does not exist a pure strategy equilibrium when two identical proprietary card schemes compete with each other. Lemma 3-2 If two identical proprietary card sche mes compete in a Bertrand fashion, no pure strategy equilibrium exists. Proof. Note first that any set of prices that generates positive profit cannot be a symmetric equilibrium. If an equilibrium set of prices is ( f m ) such that f + m > c a card scheme can increase profit by lowering the ca rdholder fee marginally while keeping the merchant fee since the scheme can attract all consumers instead of sharing them with the other scheme. Second, a set of prices which satisfies f + m = c cannot be an equilibrium, either. To see this, let the equilibrium set of prices is ( f m ) such that f + m = c Without loss of 19 The proprietary card scheme maximizes pr ofits instead of card transaction volume.

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85 generality, suppose scheme 2 lower the cardholder fee by d and raise the merchant fee by e where e > d > 0. As in the Bertrand competition case of the previous section, consumers whose bB is higher than *Bb will choose card 1 while consumers with bB lower than Bb will choose card 2, in which Bb is defined as *()S Bbmd bfd e The quasi-demands of consumers and merc hants for scheme 2s card service are 2() () ()S B B BB BBbfd dbm Q bbebb 2()S S S Sbme Q bb The profit of the scheme 2 is 2()()() 0 ()()SS BS BSdbmbmefmced ebbbb Since the scheme 2 can make positive profits by deviating from ( f m ), it cannot be an equilibrium set of prices. Q.E.D. The above lemma does not exclude the possi bility of a mixed strategy equilibrium or asymmetric equilibrium. As the following proposition shows, however, competition

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86 cannot improve social welfare since the monopo listic equilibrium set of prices maximizes social welfare. Proposition 3-8 The equilibrium prices set by the monopolistic proprietary card scheme in the Bertrand model maximize Marshallian so cial welfare which is defined as the sum of cardholders surplus, merchants surplus and card schemes profits. Proof. Marshallian social welfare is defined as follows: () ()()(2) 2()()BSbb BSBSBSBS fm BSBS BS BSWTUTUQQdfQQdmfmcQQ bfbmbbfmc bbbb (3-16) The optimal prices that maximize welfare are 1 2 3W BS f bbc 1 2 3W SBmbbc These are same as M f and M m respectively. Q.E.D. For comparison with other models, one may derive a set of Ramsey-optimal prices which is the solution of the following problem:

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87 .. BS fm M axTUTUstfmc From the first-order condition of this maximization problem, the following Ramsey-optimal prices can be obtained: 1 2R BS f bbc 1 2R SBmbbc The differences between two different op timal prices are same for both cardholder and merchant fees. That is, 1 0 6WRWR BSffmmbbc Ramsey-optimal prices are lower than the prices that maximize Marshallian welfare since the former does not allow profits of the firms while the latter puts the same weight on profits as on customers surplus. If social welfare is measured by the Ramsey standard, competition may increase the social welfar e as long as competition lowers both cardholder and merchant fees. It is also worth noting that the Ramsey-opt imal fees of the proprietary scheme is equal to the consumer and merchant f ees that are determined by the monopoly interchange fee of the nonpr oprietary scheme, i.e., Rbj I f ca and Rbj Amca which confirms that bja maximizes both Marshallian and Ramsey social welfares.

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88 3.5.2 Competition between Differentiated Card Schemes When two proprietary card schemes are di fferentiated and compete la Hotelling, the critical consumer, x*, who is indifferent between card 1 and 2 is determined in the same way as (3-11) except that the card schemes set fi and mi instead of ai: 1122 12()()()() ()(2)BSS B BS Bbfbmfbbm x bbbmm If the issuer market is not fully cove red, each card scheme has a monopoly power over its own consumers, so the equilibrium set of prices will be same as M f and M m in (3-15).20 In order to obtain a non-tr ivial result, suppos e the issuer market is fully covered at equilibrium as in the previous secti on. This requires the following assumption:21 2()BS BBbbbbc Using the first-order conditions, one can de rive the best response functions of card schemes from which the following equilibrium prices can be obtained: 1 5322 4phs BS B f bbbc 1 22 4phs SB Bmbbbc 20 Since merchants accept card i as long as bS mi, the existence of competing card schemes does not affect the equilibrium merchant fee. 21 For the issuer market to be fully covered, the net utility of the consumer located at x = must be non-negative for the monopoly prices, M f and M m

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89 If two schemes collude and act like a monopolist, the joint entity will set the cardholder fee such that the critic al consumer who is located at x = is indifferent between using card and cash as well as betw een card 1 and 2. Since the transportation cost is assumed to be equal to B Bbb the cardholder fee that will be set by the joint entity is 1 2phj B B f bb (3-17) Given this cardholder fee, the joint profit can be rewritten as follows: 1122()(22) ()() 2()SB phj B BSBS S Sbmbbcm fmcQQQQ bb The optimal merchant fee that maximizes this profit function is 1 22 4phj SB Bmbbbc Note that the merchant fee set by the join t entity is the same as the competitive merchant fee, i.e., phsphjmm This is because the issuer market is fully covered in both cases and the multi-homing merchants will accept a ny card as long as the merchant fee is less than Sb

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90 Proposition 3-9 When the two proprietary card sc hemes are differentiated in a Hotelling fashion, competition does not improve Marshallian social welfare. Proof. If two card schemes charge same prices and the issuer market is fully covered, Marshallian social welfare is 12 22 12 011 21 () 2 (34)()()()() 4()2()() ()[32(2)] 4()SBS b BBSiBiSi m ii BSSS B SSS SSS SBS B S SWTUTU UdxUdxQdmfmcQQ bbfbmbmfmcbm bbbbbb bmbbbcm bb (3-18) Note that social welf are is independent of f That is, the cardholder fee has no effect on the welfare as long as the f ee is low enough for the issuer market to be fully covered. An increase in the cardholder fee just tran sfers surplus from consumers to the card schemes. Since the social welfare is only affected by the merchant fee and the equilibrium merchant fees of the competitive case and the monopoly case are equal to each other, competition does not improve the social welfare. Q.E.D. The cardholder fee cannot affect social welf are since the issuer market is fully covered, i.e., the consumers quasi-demand is fixed regardless of the cardholder fee. When the cardholder fee changes, it does not affect the demand of the issuer market, but

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91 does affect consumers surplus and card scheme s profits. Since an increase (decrease) in consumers surplus is exactly offset by a decr ease (increase) in the profits of the card schemes, the social welfare remains the same even though total customers surplus, which is the sum of the consumers surplus and the merchants surplus, may increase due to the competition between the card schemes. Note that the merchant fee that maximizes the social welfare represented by (3-18) is 1 3 4phw B Bmcbb (3-19) This merchant fee is not feasible since the card schemes profits are negative at this fee. For the issuer market to be fu lly covered, the cardholder fee cannot exceed phj f in (3-17). So the maximum possible profit ma rgin when the card scheme charges phj f and phwm is 1 ()0 4phjphw B Bfmcbb When the first-best price is not feasible, one can think of the second-best price, or Ramsey price, which is the optimal price am ong the feasible prices. The Ramsey prices can be obtained by setting th e cardholder fee equal to phj f and the merchant fee equal to phjcf i.e.,22 22 These are the fees implied by the monopoly interchange fee ahj in the nonproprietary Hotelling model, i.e., phrhj I f ca and phrhj Amca

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92 1 2phr B B f bb 1 2phr B Bmcbb The merchant fee determined by the mark etregardless whether it is monopoly or duopolyis too high from the so cial point of view since the difference between the equilibrium merchant fee and Ramsey-optimal fee is 1 ()2()0 4phsphr BS BBmmbbbbc 3.6 Conclusion This chapter shows the effects of compe tition in a two-sided market on the price structure and welfare using a formal model w ith various settings including single-homing vs. multi-homing consumers, Bertrand vs. Hote lling competition, and proprietary vs. nonproprietary card schemes. The effect of competition on the price structure depends on whether consumers single-home or multi-home since competing card schemes set lower prices for the single-homing side. The most surprising result is that competition never improves social welfare regardless whether consumers single-home or multi-home, whether card schemes are identical or differentiated, or whether card schemes are propr ietary or nonproprietary. In most cases, monopoly pricing maximizes Marsha llian social welfare since the monopolist in a two-sided market can internalize indirect network externalities without bias to one side. The only exception is the case of the Hotelling model of the pr oprietary card scheme,

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93 in which monopoly pricing dose not maximize the social welfare. But even in this case, competition does not improve Mashallian welfare. The welfare effect of competition in the tw o-sided market may be different if the business stealing effect is introduced. Compe ting merchants may accept credit cards even if the merchant fees are higher than the direct benefit from the card service since accepting credit cards can attract card-using c onsumers. As is pointed by Rochet and Tirole (2002) and Guthrie and Wright (2005), th e equilibrium interchange fees tend to be higher when there is a business stealing effect. Therefore, if the business stealing effect exists, monopoly pricing may not maximize th e social welfare. And competition may improve social welfare if consumers mu lti-home and card schemes are nonproprietary and identical. If consumers si ngle-home or card schemes comp ete la Hotelling, however, competition may deteriorate social welfar e since both competition and the business stealing effect tend to force the interchange fee upward. Policy makers in many countries have inve stigated interchange fees and the rules set by the members of payment card systems, then moved to regulate card associations. Collective determination of the interchange fee and a lack of competition between card schemes are treated as main cause of the hi gh interchange fee. Card schemes in some countries such as Australia, United Kingdo m and South Korea have been required to lower their interchange fees or merchant fees This chapter shows th at high interchange fees or merchant fees may be a result of competition, not a result of the lack of competition. On the one hand, this implies that the interests of the regulatory authority and the card schemes can be aligned. That is if lowering interchange fee or merchant fee increases merchant acceptance, both the social welfare and the card schemes profits or

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94 transaction volume can increase at the same time. On the other hand, it implies that twosided markets should be regulated with disc retion since, even though they may not be desirable, the outcomes of the market cannot be categorized as collusive or predatory actions i.e., anticompetitive actions. A possible extension of the model in this chapter, in addition to introducing the business stealing effect, lies in endogenizi ng the mechanism that determines singlehoming or multi-homing of each side. One way to do it is, as most of other models do, introducing a fixed fee or a fixed cost of holding or accepting a card. Although the model deals with the credit card industry, it can be easily extended to the other two-sided markets such as videogame consoles, shopping malls, telecommunications, and media industries.

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95 CHAPTER 4 COMPETITION BETWEEN CARD I SSUERS WITH HETEROGENEOUS EXPENDITURE VOLUMES 4.1 Introduction As credit and debit cards become an incr easingly important part of the payment system, the card industry has drawn economists attention. Theoretica lly, the credit card industry is analyzed as one of the typical ex amples of two-sided market. A two-sided (or multi-sided) market is a market in which two (or more) parties interact on a platform. The end-users enjoy indirect networ k externalities which increase as the size of the other side increases and cannot be in ternalized by themselves.1 The platform enables the interaction by appropriately charging each side. The two largest credit card networksMasterCard and Visause interchange fees to balance the dema nds of two sides. The interchange fee is a payment between the merchants bank, known as the acquirer, and the consumers bank, known as the issuer. Another reason for the recent surge of inte rest in the credit ca rd industry lies in government policies. Antitr ust authorities around the wo rld have questioned some business practices of the credit card networ ks. These include the collective determination of the interchange fee, the no-surcha rge rule, and the honor-all-card rule. 1 Rochet and Tirole (2004) point out that a necessary condition for a market to be twosided is that the Coase theorem does not appl y to the transaction between the two sides. For general introductions to the two-si ded market, see Armstrong (2004), Roson (2005a), and Evans and Schmalensee (2005).

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96 Because of its importance in theory and pr actice, the interchange fee has been the main topic in most of the literature on th e credit card industry. To mitigate the complexity caused by two-sidedness of the market, however, these models make relatively simple assumptions on each side of the market. One of these simplifying assumptions is that cardholders have unit demand for all goods. Th e unit demand is a reasonable assumption in the analysis of ma ny traditional marketsespecially markets for durable goods such as houses and automobiles. But for the cred it card industry, unit demand assumption is implausible since it implies cardholders pref erences are identical except for the credit card service. The model presented here adopts a more plausible assumption that cardholders are heterogeneous in terms of the expenditure volume. Not only is this a more plausible assumption, it also makes possible a richer anal ysis on the competition on the issuer side of the market. The main finding is that the effects of a change in the variance of the expenditure volume on the equilibrium cardhol der fees and profits are different for various cases of market coverage. As the va riance of the expenditure volume increases, issuers profits as well as the equilibrium cardholder fee d ecrease when the market is fully covered. When the market is locally monopolized, the profits increase while the cardholder fee remains the same as the varian ce increases. In case of the partial-cover market, the effect of an increase in the va riance is mixed (i.e., the cardholder fee may increase or decrease as the variance increases). The model also contains some new findings about the interchange fee. One of them is the neutrality of the interchange fee holds in the full-cover market even under the nosurcharge-rule. When the market is not fully covered, the neutrality does not hold since

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97 there exist potential consumers that card i ssuers can attract. A simulation result also shows the possibility of the positive relationship between the interchange fee and the cardholder fee. The first formal analysis of credit card i ndustry in the context of two-sided market was provided by Baxter (1983). Although it is normativ e rather than positive, his model clearly shows that interchange f ee is necessary to balance the demands of the two sides. It is only recently that more rigorous models were developed by economists as they started to pay attention to the two-sided market. Sc hmalensee (2002), Rochet and Tirole (2002), and Wright (2003a, 2003b, 2004) deve lop Baxters idea in rigorous models with a single platform. Rochet and Tirole (2003) and Guthrie and Wright (2006) extend the models by allowing competition between platforms. The main focuses of these papers are how interchange fees are determin ed and how they are different from ordinary cartel pricefixing behavior. Although their models are more sophisticated than Baxters, their treatment of each sideespecially the issuer sideis relatively simple. For example, Schmalensee (2002) allows imperfect compe tition on both issuer and acquirer sides but the demands of each side are given, not derived. Rochet and Tirole (2002) derive th e demand for card se rvice by endogenizing consumer behavior, but there is no differen ce between card-holding and card-using since they assume all consumers purchase the same amount of goods from each merchant. Further, by assuming identical merchants, their model cannot capture the trade-off between consumer demand and merchant dema nd caused by a change in the interchange fee. Wright (2002) allows he terogeneity among merchants, but also assumes unit demand by consumers.

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98 Chapter 4 is organized as follows. The fo llowing section provides a basic model of the issuer market in the credit card i ndustry and shows how th e cardholder fee is determined in the various cases of market c overage. It also shows that the effects of a change in the variance of the expenditure volume are quite di fferent for various cases of market coverage. Section 4.3 provides the determ ination of the interc hange fee, also in different cases of market coverage. Section 4.4 provides other comparative statics and the results of the collusion between issuers. Th e last section summarizes the results and provides concluding remarks. 4.2 Equilibrium Cardholder Fee 4.2.1 The Model Suppose there are two card issuers, i = 1, 2, associated with a single card scheme.2 The issuers set cardholder fees, fi, which can be negative, and the card scheme sets the interchange fee, a which is a payment to card issuer s from card acquirers. Merchants are not allowed to impose surcharges on cons umers who pay with a card (i.e., the nosurcharge rule prevails). Consumers or cardholders have the same valuation, b for the card service but have different expenditure volumes, v which are drawn with a positive density g ( v ) over the interval ] [ v v It is assumed that each consumer spends the same amount at every merchant. That is, v can be interpreted as the purchas ing amount from each merchant. So a consumers total charging volume with the credit card is vQm, where Qm is the number 2 Visa and MasterCard are examples of this type of the credit card scheme. Competition between the card schemes is not an issue in this paper. Since ma ny issuers issue both cards and most merchants accept both, they can be treated as one monopoly platform.

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99 of merchants that accept the card. Consumer s are also assumed to be distributed uniformly over the interval [0,1], where the issuers are locate d at two extremes. The issuers compete la Hote lling. A consumer located at x on the unit interval incurs a transpor tation cost of tx when she uses card 1 and t (1 x ) when using card 2.3 The net utilities a consumer with v located at x receives from using card 1 and 2 are 11 22() ()(1)m muvbfQtx uvbfQtx (4-1) A consumer will choose to use card i if the following two conditions are satisfied: ui uj (IR1), i j = 1,2, i j ui 0 (IR2) (IR1) requires the net utility from card i be at least as good as from card j while (IR2) requires the net utility from card i be at least as good as from the other payment method, say cash. Note that the benefit of the card service, b is measured relative to the benefit of using cash. 3 There are two types of transportation co sts. One is the shipping cost which is proportional to the purchasing amount, and th e other is the shopping cost which is a one-time cost and independent of the expendi ture volume. In the credit card industry, both types of transaction costs exist. For ex ample, average percentage rate (APR) for purchases is a shipping cost while APR for balance transfer is a shopping cost. For modeling simplicity, this pape r assumes shopping costs only and no shipping costs. If one assumes shipping costs only, the expenditure volume plays no role in the model. To see this, suppose u1 = v ( b f1 tx ) Qm and u2 = v [ b f2 t (1 x )] Qm. Then the critical consumer who is indiffere nt between two cards is 211 22 ff x t which is independent of v

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100 The critical consumers, x*, who are indifferent betw een card 1 and 2 can be obtained by setting u1 = u2:4 21 12() 1 (,;) 22mvffQ xffv t (4-2) Note that the locations of critical consumers varies as v changes if f1 f2. Consumers whose (IR2) condition is binding, *i x are determined by setting ui = 0. These consumers are indifferent between using card i and using cash: 1 11 2 22() (;) () (;)1m mvbfQ xfv t vbfQ xfv t (4-3) There may exist a consumer who is indiffere nt between using card 1 and 2, or not using any of the cards, i.e., both (IR1) and (IR 2) conditions are bindi ng. This consumers expenditure volume, v*, is determined by setting 2 1x x : 12 12(,) (2)mt vff bffQ (4-4) Depending on the size of *v compared to v and v, one can distinguish three regimes: full-cover market when *vv local monopoly when *vv, and partial-cover market 4 Consumers are assumed to use a single card or none at all, hen ce the possibility of multi-homing is excluded. In fact, there is no extra gain from multi-homing since the two issuers are in the same network.

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101 when *vvv Figure 4-1 shows these three cases with the division of consumers in three parts. Consumers in area I and II use card 1 and 2, respectively. Consumers in area III choose not to use any of the cards.5 The consumers demand for card i s service, qi, is the sum of all consumers expenditure volumes in area I or II. The demand functions of full-cover market are 112 221(;)() (;)(1)()v v v vqffvxgvdv qffvxgvdv (4-5) In case of the local monopol y, the demand functions are 111 222()() ()(1)()v v v vqfvxgvdv qfvxgvdv (4-6) Last, the demand functions of partial-cover market are * *** 1121 ** 2212(;)()() (;)(1)()(1)()vv vv vv vvqffvxgvdvvxgvdv qffvxgvdvvxgvdv (4-7) Issuer i s profit is as follows: 5 The split lines, *()i x v never cross the vertical axes since, at around x = 0 and x = 1, there always exist some consumers who will use one of the credit cards regardless of the transportation costs as long as b > fi.

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102 Figure 4-1. Division of consumers in three cases of market coverage v v v0 1 x v I III II x ( v ) ) (* 1v x ) (* 2v x v v1 x v I II x ( v ) v v0 1 x v I III II ) (* 1v x ) (* 2v x (a) Full-cover market (b) Local monopoly (c) Partial-cover market 0

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103 ()iimi f acQq (4-8) where c is the marginal cost of the issuer, whic h is assumed to be same for both issuers. The game proceeds as follows: at stage 1, the card scheme sets a ; at stage 2, the competing card issuers set fi; at stage 3, each consumer chooses whether to use card 1, 2 or not. The model can be solved by usi ng backward induction. Since consumers behavior at stage 3 has already been analyzed above, the next subsections will focus on the analysis of stage 2. 4.2.2 Full-Cover Market In this subsection, the issuer market is a ssumed to be fully covered. This is possible if *v v or *2()mtvbfQ where f is the equilibrium cardholder fee of the symmetric issuers. When the market is fully covered, th e total demand for each issuer can be simplified as follows: 22 2121 112()() 11 (;)()()[][] 2222vv mm vvffQffQ qffvgvdvvgvdvEvEv tt 2 12 221() 1 (;)[][] 22mffQ qffEvEv t (4-5) where E [ ] is the mean of the variable in the bracket, i.e., []()v vEvvgvdv and 22[]()v vEvvgvdv Issuer i 's profit can be rewritten by pluggi ng (4-5) into (4-8), which is

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104 2 2()() 1 ()[][] 22ijim iimfacffQ f acQEvEv t Proposition 4-1 When the market is fully covere d, the equilibrium cardholder fee and profits decrease as the variance of the e xpenditure volume increases while the mean of the volume remains constant. Proof. At stage 2, each card issuer will maximize its profits by choosing optimal fi. The first order condition for the pr ofit maximization problem is 2 2(2) [][]0 22jim im iffacQ dQ EvEv dft (4-9) Using (4-9), one can derive the issue rs best response function as follows: 2[] () 22[]j ij mfac tEv ff QEv The symmetric Nash equilibrium of the model is 2[] []FC mtEv fca QEv (4-10) Note that, as in a standard Hotelling mode l, the equilibrium fee is the sum of the net marginal cost ( c a ) and the profit margin 2[] []mtEv QEv

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105 The equilibrium profit is 2 2[] 2[]FCtEv Ev (4-11) When the variance increases while the mean remains constant, E [ v2] must increase since the variance of v is equal to E [ v2] E [ v ]2. So the equilibrium fee and profits decrease as the variance of v increases, holding the mean constant. Q.E.D. When a card issuer lowers its cardholder fee, the demand increases faster for the higher variance of v since 2[] 2im idqQ Ev dft But the other issuer will match the price decrease so the quantity sold will always be equal to 1 [] 2 Ev at equilibrium, which implies that the demand becomes mo re elastic as the variance of v increases. As in a standard economic model, the equilibriu m price decreases due to the increasing competition when the elasticity of demand increases. The profits decrease as a result of decreasing price without an in crease in quantity sold. Note that the equilibrium fee can also be expressed in terms of elasticity (Lerners formula): () 1FC FC FCfca (4-12) where ( c a ) is the net marginal cost, FC i FC idq f dfq and 1 (;)[] 2FC iqqffEv

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106 As is clear in (4-10) and (4-11), the e quilibrium fee and profits increase when the mean of the expenditure volume increases wh ile the variance remains the same. So it is not clear whether the fee and profits increase or decrease when both the mean and the variance of v increase. However, the following propos ition shows that the equilibrium fee decreases when every cardholder increases expe nditure volume at the same rate, so that both the mean and the variance increase. Proposition 4-2 If every cardholder increases her expenditure volume at the same rate, the equilibrium cardholder f ee decreases while the pr ofits remain the same. Proof. Suppose each cardholders increased expenditure volume is w = v > 1. Then the density of w h ( w ), is equal to () gv since w is distributed more widely. The means of w and w2 are, respectively, []()()[] EwwhwdwvgvdvEv 222222[]()()[] EwwhwdwvgvdvEv The equilibrium cardholder fee can be obtained using (4-10): 2[][] [][]FC w mmtEwtEv fcaca QEwQEv Since > 1, the equilibrium fee decreases when every cardholder increases expenditure volume at the same rate.

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107 Issuers profits remain the same since 2222 2222[][][] 2[]2[]2[]FCFC wtEwtEvtEv EwEvEv Q.E.D. This is a quite surprising result since the combined effects of increases in both mean and variance of the expenditure volume is to decrease the equilibrium cardholder fee. As the credit card industry grows, cardholders use more cards than other payment methods such as cash and checks. This increas es the variance of the charging volume as well as the mean of the volume. So the in creasing variance of the expenditure volume combined with an increase in the mean may be one of the reasons for the decrease in the cardholder fees over the history. 4.2.3 Local Monopoly The local monopoly case arises when *vv, or *2()mtvbfQ. In this case, the demand for each issuers service can be simplified as 2 2()() ()()[]v imim ii vvbfQbfQ qfgvdvEv tt (4-6) Using (4-6) and (4-8), issuer i 's profit can be rewritten as 2 2()() []iim ifacbfQ Ev t

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108 From the first-order condition for the profit maximization problem, one can derive the equilibrium cardholder fee as follows: 2LMbac f (4-13) As in the full-cover market case, demand increases faster for a given drop of the price as the variance of v increases since 2[]im idqQ Ev dft Unlike in the full-cover market, however, the quantity demanded also increases when the variance of v increases since equilibrium quantity for each issuer is 2() [] 2LM m ibacQ qEv t As a result, the elasticity of demand is independent of the variance of v Since the cardholder fee follows the Lerners formula,6 it is also independent of the variance of v The equilibrium profit of the local monopoly is 22 2() [] 4LM mbacQ Ev t (4-14) Contrary to the full-cover market cas e, it is an increasing function of E [ v2]. This is because, as the variance of v increases, the quantity demanded also increases while the 6 () 1LM LM LMfca where LM i LM iidq f dfq.

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109 price (cardholder fee) remains the same. Th e following proposition summarizes the above analysis. Proposition 4-3 In case of a local monopoly, equilibrium profits increase while the equilibrium cardholder fee does not change when the variance of v increases, holding the mean constant. 4.2.4 Partial-Cover Market The market is partially covered if *vvv or **2()2()mmvbfQtvbfQ. When the market is partially covered, the de mand for the card service is represented by (4-7), which can be rewritten as **** 1121 *** 2212(;)()()() (;)(1)()()()vv vv vv vvqffvxgvdvvxxgvdv qffvxgvdvvxxgvdv (4-7) Plugging (4-2) and (4-3) into (4-7), one can obtain *22 *() 11 (;)()[]() 222vv jim iij vvffQ qffvgvdvEvvgvdv tv (4-7) The derivative of (4-7) with respect to fi is *22[]() 2v im v idqQ Evvgvdv dft (4-15)

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110 It is not clear from (4-15) that whether, as the variance of v increases, the demand changes faster for a given change in the price. Since *2()v vvgvdv also changesit could increase or decreaseas the variance of v increases, *22[]()v v E vvgvdv may be increasing or decreasing in the variance of v Since *222[]()()vv vv E vvgvdvvgvdv it is clear that ** *2222[]()2()()vvv vvvEvvgvdvvgvdvvgvdv Figure 4-2 shows why the expenditure volumes below v* have a higher weight. When issuer 1 lowers f1, consumers whose expenditure volumes are below v* respond more sensitively than those with higher v This is because issuer 1 competes with issuer 2 for higher type consumers, but not for lower type consumers. For notational convenience, define Figure 4-2. The effect of a price drop on demand v* v v 1 x v x *( v ) ) (* 1v x ) (* 2v x

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111 *2 22[]()v vEvvgvdv Then the derivative of the profit func tion given in (4-8) with respect to fi is 2() 2iim mi idfacQ Qq dft (4-16) Lemma 4-1 The equilibrium cardholder fee is un ique in the partial-cover market. Proof. Using (4-16) and symmetry of the firms, the equilibrium fee, P C f is implicitly determined by 2() (;) 2PC PCPC m ifacQ qff t (4-17) When f = c a the LHS of (4-17) is positive while the RHS of it is equal to zero. As f increases, the LHS of (4-17) decreases monotonically, while the RHS increases monotonically. So the equilibrium fee is uniquely determined by (4-17). Q.E.D. Using (4-7) and (4-17), one can obtai n the equilibrium cardholder fee as a function of v*:

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112 * **2 * 22()() () []()vv vv PC v m vvvgvdvvgvdv t fvca Qv Evvgvdv (4-18) Since v* itself is a function of f the equilibrium fee is still implicitly determined by (4-18). One of the benefits of expres sing the equilibrium fee in terms of v* is that it can be used to check the continuity of th e equilibrium fee. To see this, suppose *vv Then (4-18) becomes equal to (4-10) so that P CFC f f at *vv When *vv, (4-18) shrinks to () 2PC mt fvca Qv (4-19) Using the definition of v*, one can show that (4-19) is equal to (4-13) so that P CLM f f at *vv. Propositions 4-1 and 4-3 have shown that as the variance of the expenditure volume increases, the equilibrium fee decreases in the full-cover market or remains constant in the local monopoly case. From thes e results one may conjecture that in case of partial-cover market, the effect of a cha nge in the variance of the expenditure volume should lie between the results of full-cover market and loca l monopoly cases. That is, the equilibrium fee may decrease or remain consta nt but never increase as the variance of the expenditure volume increases. But it turns out that the equilibrium fee may increase as well as decrease when the variance of v increases.7 7 A simulation model is presented in the a ppendix that shows this result graphically.

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113 Lemma 4-2 A sufficient condition fo r the equilibrium cardholder fee of the partial-cover market to increase when the variance of v increases, holding the mean constant, is ** 2() 1v vdvvgvdv d where 222[][] EvEv Proof. A change in the variance of v affects the equilibrium fee in two ways. First, it affects the equilibrium fee dir ectly through the changes in each term in the bracket of the right-hand side of (4-18). There also exists a second-order effect due to a change in v*. However, the second-order effect is minor a nd cannot offset the first-order effect. So I will focus on the first-order effect only. Note first that the bracket in (4-18) is less than 1 since *****222()()()()[]vvvv vvvvvvgvdvvvgvdvvgvdvvgvdvEv So the equilibrium cardholder fee will in crease if the numerator increases more than the denominator when the variance increases. Since 2 2[] 1 dEv d the above condition is indeed a sufficient condition. Q.E.D. Note that the equilibrium fee can be expre ssed in terms of the Lerners formula as in the other two cases. That is, using (415) and (4-17), the equilibrium fee can be

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114 rewritten as () 1PC PC PCfca where PC i P C iidq f dfq 4.3 Equilibrium Interchange Fee The analysis of the interchange fee is one of the main topics of the two-sided market literature focused on the credit card i ndustry. Previous models on the credit card industry emphasize the balancing role of the interchange f ee. A card scheme tries to optimize card transactions by achieving the right balance of cardholder demand and merchant acceptance. The interchange fee cannot be optimal if the demand of one side is too high while the demand of the other side is too low. By balancing the demands of both sides, the card scheme maximizes the aggreg ate profits of the member banks or total transaction volumes made by the card. The model presented here follows the previ ous literature in that the card scheme sets the interchange fee in order to maximize combined profits. But the following analysis of the interchange fee is not complete due to the lack of in-depth analysis of the acquirer side. Most of the results here ar e obtained assuming the acquirer market is perfectly competitive, which simplifies the card schemes objective as to maximize issuers profits. 4.3.1 Full-Cover Market As can be seen in (4-11), the e quilibrium profit is independent of a and Qm when the market is fully covered. This is because any gain from an increase in a or Qm will be

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115 competed away among the issuers. If the acqui rer market is perfectly competitive as is often assumed in the credit card literature,8 the card scheme will simply maximize the issuers profits that are independent of the interchange fee. So the choice of the interchange fee is irrelevant to the overall prof its as long as Qm( a ) > 0 and f*( a ) < b Even if the acquirer market is not perfec tly competitive, the interchange fee may not affect the profits when the acquirer market is also fully covered. Suppose the acquirer market structure is similar to the issuer market. Then the resulting merchant fee and acquirers profits will have similar structure as those of issuers so that the profits will be independent of the interchange fee. When a change in the interchange fee doe s not alter any real variable in the economy, the interchange fee is neutral. Previous work finds that the neutrality of the interchange fee holds when both issuer and acquirer markets are perfectly competitive (Carlton and Frankel, 1995) or surcharg e is possible (Roche t and Tirole, 2002).9 As will be clear in the following subsectio ns, the neutrality of the interchange fee can hold in case of the full-cover market even if surcharge is not possible. When the market is not fully covered, lowering interc hange fee can attract more consumers, which will affect issuers profits as well as the quantity demanded. This can be summarized as follows. 8 Unlike the issuer market, there are few ways to differentiate in the acquiring market. Rochet and Tirole (2002) and Wright (2003) also assume perfect competition in the acquirer side. If acquirers are assumed to have market power, one would need to consider relative strength of issuers and acqui rers to determine the interchange fee. See Schmalensee (2002). 9 See Gans and King (2003) for a discussion of the neutrality of the interchange fee.

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116 Proposition 4-4 Neutrality of the interchange fe e holds under the no-surcharge-rule if both the issuer market and the acquirer market are fully covered. 4.3.2 Local Monopoly When the market is locally monopolized, the card scheme will choose a to maximize industry profits represented by (414) if the acquirer side is perfectly competitive. Then the optimal interchange fee is implicitly determined by the following equation derived from th e first-order condition: /LM m mQ acb dQda (4-20) or using the elas ticity formula, () 1LM m macb (4-21) where m m mdQ a daQ In a standard two-sided market model, the importance of an interchange fee lies in the role of balancing the de mands of both sides. So it may look unconventional that (421) implies that the optimal interchange fee is related to the elasticity of the acquirer side only. But the fact is the elasticity of the issu er side is zero at the optimal level of the interchange fee. To see this, note first that the elasticity of the issuer side is

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117 LM LMLM LM i am LMLM idq aa daqbac (4-22) A simple calculation using (4-20) shows that (4-22) is equal to zero. This implies that the optimal interchange fee is set so as to maximize the quantity demanded in the issuer side. Rearranging (4-22), one can obtain the following expression: () 1LM LM ma LM maacb So the optimal interchange fee is indeed related to both sides. 4.3.3 Partial-Cover Market In the partial-cover market, the equilibri um profits can be obtained by plugging (417) into the profit func tion (4-8), which is **22 *22() []() 2v m i vfacQ Evvgvdv t (4-23) If the acquirer side is perfectly competi tive, the card scheme will choose optimal a to maximize i A change in a affects profits in various ways. The derivative of the profit function w.r.t. the intercha nge fee can be decomposed as **** iimii mdQ f daaaQaf (4-24)

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118 Since the issuer chooses the optimal f* to maximize profit, the last term in (4-24) is zero at equilibrium. Using the envelope theorem, the first order condition can be simplified as 22 *3*() ()[2()()]20 2PC PC imm mdfacQdQ facvgvQ datda (4-25) It follows from (4-25) that the optimal interchange fee is implicitly determined by 2 2 *3*2 / 2()()PCPC m mQ acf dQda vgv Unfortunately, no further analysis is po ssible unless one has more information about the distribution of v and the merchants demand f unction for the card service. 4.4 Extension 4.4.1 Other Comparative Statics The effects of a change in the variance of the expenditure volume on the equilibrium cardholder fees and profits have been analyzed in section 4.2. This subsection is devoted to the analysis of the other comparative statics. Table 4-1 shows the main results of the co mparative statics. One of the interesting results is the effect of t on cardholder fees and profits. Just like the variance of v the effects of a change in t are opposite between full-cover ma rket and local monopoly cases. In the full-cover market, transportation cost t works as in a standard Hotelling model. That is, a customer incurs a higher cost to switch to the other issuer as t increases. So the

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119 Table 4-1 Comparative Statics 2df d 2d d *df dt *d dt mdf dQ md dQ *df da Full-cover market + + 0 Local monopoly 0 + 0 0 + Partial-cover market + issuers can charge higher pr ice and make higher profits.10 In the local monopoly case, however, customers e ither use a least-cost card or stop using the card. So when t increases, customers cost of us ing the card increases while the cost of the alternative payment methodcashrema ins the same. So the marginal customers will stop using the card for the given price, which causes a decrease in profits. The cardholder fee remains the same since, as t increases, the demand decreases proportionally for each level of v so that the elasticity of demand is independent of t In the partial-cover market, th e effect of an increase in t is mixed. For customers above v*, issuers can charge higher price when t increases since their switching cost increases just as in full-cover market. But for customers below v*, issuers lose marginal customers due to the increase in the car d-usage cost. Unlike local monopoly case, however, the demand does not decrease pr oportionally so it may be optimal for the issuers to lower the cardholder fee. The overall effects of a change in t on the cardholder fee and profits depend on the relati ve size of the two opposing effects. 10 When t increases, the cost of using the card also increases. But the switching cost exceeds the usage cost.

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120 Another interesting result is the effect of a change in the interchange fee on the equilibrium cardholder fee. Unlike conventiona l wisdom that cardholder fee decreases when the interchange fee increases, it turns out that the cardholder fee may increase as the interchange fee increases. The logic behind this is as follows. When the interchange fee increases, it raises the merchant fee so that the number of merchant that accept the card ( Qm) decreases, which in turn can cause an increase in the cardholder fee due to a decrease in the demand elasticity.11 The derivative of the full-cover mark et equilibrium cardholder fee w.r.t. a is 2[] 1 []FC m mdftEv daQaEv The sign of this derivative depends on th e relative size of each parameter. Although it is likely to be negati ve since the value of t should be small for the market to be fully covered, the sign could be positive especially when the elasticity of the acquirer market (m) is extremely high. 4.4.2 Collusion Since the equilibrium profits decrease in the full-cover market when the variance of v increases, the issuers have an incentive to exclude some low-volume consumers in order to reduce the variance. This is po ssible when they collude, and the following proposition shows that it is profitab le to exclude some consumers. 11 The negative relationship between fFC and Qm can be verified in (10).

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121 Proposition 4-5 If two card issuers collude, the market cannot be fully covered at equilibrium. Proof. It is enough to show that the issuers have an incentive to rais e cooperatively the cardholder fee at *vv When the two issuers collude, each firm s (common) demand function is *2 *11 (;)()() 22vv C vvqffvgvdvvgvdv v The first-order condition for the jo int profit maximization problem is *2() ()0C v C m m vfacQ d Qqvgvdv dft (4-26) The equilibrium cardholder fee, C f is implicitly determined by the following condition: *2() (;)()C v CCC m vfacQ qffvgvdv t (4-27) When 2mt fb vQ *vv by the definition of v*. At this level of cardholder fee, 2[]0 2mC m t fb vQQ d Ev df

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122 since the second term inside the bracket in (4-26) is zero and [] 2CEv q This implies that the optimal fee must be higher than 2mt b vQ so that *v is greater than v at equilibrium. Q.E.D. Comparing (4-27) with (4-17), it is clear that the collusive equilibrium fee is higher than the one without collusion in the partial-cover market, i.e., CPC f f since *22[]()v v E vvgvdv When the issuers collusively exclude low spending consumers, they can decrease the variance and increase the mean of the expenditure volume. Ev en though the total demand may decrease by this measur e, the resulting profits increase. Regulatory authorities in some countries are moving to regulate the credit card industry because of the allegedly too high in terchange fees. If a policy measure lowering the interchange fee is accompanied by a highe r cardholder fee, it also helps to reduce the variance of the charging volume among the credit card users. 4.5 Conclusion Chapter 4 has proposed a framework for studying competition between card issuers when cardholders have heterogeneous expendi ture volumes. What has been found is the effects of a change in variables on the competition vary depending on whether the market is fully covered, partially covered, or local ly monopolized. When the market is locally monopolized, card issuers compete with other pa yment methods but not with each other. So any change that strengthen (weak en) the monopoly power will have a positive (negative) effect on profits. In case of the full-cover market, however, the only

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123 competition issuers face is the one with each other. So any change that affects the competition between them has influence over issuers profits. For example, suppose the transportation cost ( t ) increases. Then it decreases issuers profits in case of local monopoly since it weakens the competitive power over the alternative payment methods. But in the full-cover market case, it increases the profits since it strengthens competitive power over the other issuer. The results are mixed if the market is partially covered. The effects of a change in variables on the equilib rium price and profits are not constrained to the range between those of full-cover market and local monopoly. As the simulation results in the appendix show, however, the effects tend to become clos er to those of full-cover market, the more the market is covered. As the credit card industry gr ows, the market will become closer and closer to the full-cover market. If this happe ns with increasing variance of the expenditure volume, the overall profits of the industry may decrease even without competition in the card scheme level. One of the policy implications is that re gulating the interchange fee may have the same effect as reduced competition on the issu er side of the credit card industry if it induces a higher cardholder fee. Since the higher cardholder fee will exclude consumers with low expenditure volume from using the cr edit card even if the benefit from using the card ( b ) is the same as the high volume consum ers, the policy may have an undesirable effect in terms of equality. Although the model sheds new light on issuer side of the credit card industry by introducing heterogeneous expenditure volumes, a comparable analysis of the acquirer

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124 side is missing. Since the card industry is ca tegorized as a two-sided market, modeling of both sides is necessary in order to fully unde rstand the working of the industry. Another possible extension of the model is to introduce shipping cost fo r the transportation cost of the model since many differentiated card benefits such as rebates in the form of frequent flyer miles are proportional to the charging volume. Last but not least, allowing platform competition will help understanding the difference between competition on the member bank levelissuers and acquirersand platform competition.

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125 CHAPTER 5 CONCLUDING REMARKS This dissertation analyzes pricing strategi es of multiproduct firms when they face competition in the complementary aftermarket (bundling), or there exist indirect network externalities that cannot be internalized by consumers of each good (two-sided markets). Chapter 2 deals with a multiproduct firm that produces a monopolistic primary good and a competitive complementary good. If consumers buy the complementary good after they have bought the primary good, i.e ., the complementary goods are sold in the aftermarket, the monopolist can make th e highest profits by committing to the aftermarket price. But if credibility of comm itment is an issue or the committed price is not feasible, the monopolist can sell them as a bundle and make higher profits than when it sells them independently. It is also shown that bundling lowers social welfare in most cases while it increases consumers surplus. So whether this kind of bundling should be allowed depends on the objective of policy makers. That is, bundling may be allowed if policy makers maximize consumers surplus, whereas it should be regulated as an anticompetitive practice if Marshallian social welfare is the main concer n. In the long-term poi nt of view, however, bundling should be viewed with concern since it decreases both firms incentives to invest in R&D. Chapter 3 presents a model of a multiproduct firm (platform) that sells two products to two different types of end-user s who interact with each other through the platform. Since the end-users cannot internalize indirect network externalities in this two-

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126 sided market, the platform must choose the appropr iate set of prices in order to get both sides on board and to maximize profits (for a proprietary platform) or output (for a nonproprietary platform). Using a case of the credit card industry, it is shown that competition may not improve social welfare or even have a negative effect on welfare since competing platforms set unbalanced pri ces in favor of the single-homing side. A lower price to the single-homing side is accompanied by a higher price to the multihoming side. That is, in two-sided markets, higher prices may be a direct result of competition, not a sign of lack of competition. Besides, competing platforms choose a price structure that maximizes consumers surplus if consumers single-home and merchants multi-home. So antitrust policy on two-sided market should be implemented with discretion. Chapter 4 delves into the issuer side of the credit card industry by allowing heterogeneous expenditure volumes among cons umers. The effects of a change in the variance of the expenditure volumes are mainly analyzed. The main finding is that the effects of a change in the variance on the e quilibrium price and profits are different for various cases of market covera ge. Especially, when all cons umers expenditures increase at the same rate so that both the mean a nd the variance increase, the equilibrium price decreases when the market is fully covered. This gives the issuers an incentive to cooperatively exclude consumers with low expe nditure. This implies that any policy that increases cardholder fees may have a negative effect on consum ers welfare since it helps card issuers to reduce the variance of cardholders charging volumes by excluding consumers with low expenditure.

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127 Theories of two-sided markets are fair ly new and are still in the middle of development, which includes the analysis of va rious strategies in two-sided markets, such as tying (Rochet and Tirole, 2006) and excl usive dealing (Armstr ong and Wright, 2005). Despite rapid development in theory, empirical studies on two-sided markets are rare. As in the other areas of economic theory, more ba lanced empirical research is required to deepen our understanding of two-sided markets and to derive unbiased policy implications.

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128 APPENDIX SIMULATION ANALYSIS FOR CHAPTER 4 In this appendix, simulation results that show the behavior of the partial-cover market will be presented. Since it is difficu lt to solve the general model mathematically, a simulation model can be used to help understanding the partial-cover market. In order to incorporate effects of a cha nge in the variance of the expenditure volume, suppose the density function of v take the following form. 22()12()() () () yvvvvvvv gvy vv (A-1) Using this density function, the mean and the variance of v can be derived as [] 2 vv Ev 2321 []()7()24 24 Evyvvvvvv 321 []()() 24 Varvyvvvv As can also be seen in Figure A-1, the variance of v increases when y increases, while the mean is independent of y The derivative of the variance w.r.t y is 3[]() 24 dVarvvv dy > 0

PAGE 139

129 Figure A-1. The density function The model can be simplified without loss of generality by setting v= 0. To solve the model, note first that v* increases as t increases in (4-4). As is shown in Table 4-1, fi may increase or decrease when t increases. But even if fi decreases, the resulting decrease in v* is a second-order effect which cannot offset the first-order effect. This implies that there exists only one t for each v* at equilibrium. Define ()dmtdbfQv 0 d 2 (A-2) where d is the parameter that correspond to v*. Then td also corresponds to each value of v*. Setting t = td yields *2 d vv So *0 () vv if d = 0, and *vv if d = 2. Since the density function is symmetric, v* = E [ v ] if d = 1. Now the equilibrium cardholder fee can be obtained as a function d (or v*) using the first-order condition (4-16). Let *() f d be the equilibrium cardholder fee obtained from this calculation. This equilibrium fee cannot be used directly to do comparative v v y > 0 E [ v ] y = 0

PAGE 140

130 statics because t cannot be held constant. To resolve this problem, one needs to apply implicit function theorem using (4-18). That is, define * **2 22()() []()vv vv v m vvvgvdvvgvdv t Hcaf Qv Evvgvdv Then the derivative of f w.r.t. a parameter x is df dHdH dxdf dx (A-3) where x can be any parameter such as y a t and Qm. The final result is obtained by plugging td and f*( d ) into (A-3). The effect of an increase in the variance of v on the cardholder fee can be captured by df / dy Figure A-2 shows that this derivative can be both negative and positive Figure A-2. Effects of an increase in the variance on the cardholder fee ( df / dy ) 0.5 1 1.5 2 -0.06 -0.04 -0.02 0.02 y = 0 2/ y v 1/ yv d

PAGE 141

131 depending on d (or v*).76 The three graphs are drawn for the cases of three different distributions of the expenditure volume. As is clear in the figure, the distribution of v does not affect the result qualitatively. Another interesting result of the comparative statics is the effect of a change in the interchange fee on th e cardholder fee ( df / da ). In order to derive df / da an assumption regarding the acquirer market is necessary since Qm also changes when a changes. To make the model tractable, assume a linear demand function in th e acquirer market. ()mQrka This linear demand function assumes that the acquirers transfer the whole interchange fee to the merchant fee, and the merchants accept the cr edit card as long as the merchant fee is lower than the per-transaction benefit they receive from the card service. Figure A-3 shows the effects of the interc hange fee on the cardholder fee. As can be seen in the figure, the cardholder fee may increase or decrease when the interchange fee increases depending on the parameter value. To obtain numerical results, arbitrary numbers are assigned to the parameters The thick graph is drawn assuming y = 0, b = 6, c = 2, k = 4, and a = 2. And the thin graph is drawn assuming the same parameter values except a = 1. The higher the interchange fee is, the bigger is the elasticity of the acquirer market. And the issuers may raise the cardholder fee due to a lower Qm despite of a higher a 76 When drawing the graphs, it is assumed that ()1 abcv to get a numerical result.

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132 Figure A-3. Change in the interc hange fee and the cardholder fee ( df / da ) a = 2 a = 1 0.5 1 1.5 2 -1 -0.8 -0.6 -0.4 -0.2 0.2 a = aH a = aL (< aH)

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133 REFERENCES Adams, William J. and Janet L. Yellen (1976) Commodity Bundling and the Burden of Monopoly, Quarterly Journal of Economics 90, pp. 475. Armstrong, Mark (2005) Competition in Two-Sided Markets, Rand Journal of Economics forthcoming. Armstrong, Mark and Julian Wright (2005) Two-Sided Markets, Competitive Bottlenecks and Exclusive Contracts, mimeo. Bakos, Yannis and Erik Brynj olfsson (1999) Bundling Info rmation Goods: Pricing, Profits and Efficiency, Management Science 45, pp. 16131630. Baxter, William F. (1983) Bank Intercha nge of Transactional Paper: Legal and Economic Perspectives, Journal of Law & Economics 26, pp. 541. Blair, Roger D. and Jill B. Herndon (1996) Restraints of Trade by Durable Good Producers, Review of Industrial Organization 11, pp. 339353. Borenstein, Severin, Jeffrey K. MacKie-Mas on, and Janet S. Netz (1995) Antitrust Policy in Aftermarkets, Antitrust Law Journal 63, pp. 455. Caillaud, Bernard and Bruno Jullien (2003) Chicken & Egg: Competition among Intermediation Service Providers, Rand Journal of Economics 24, pp.309. Carlton, Dennis W. and Michael Waldman (2002a) The Strategic Use of Tying to Preserve and Create Market Po wer in Evolving Industries, Rand Journal of Economics 33, pp. 194220. Carlton, Dennis W. and Michael Wald man (2002b) Competition, Monopoly, and Aftermarket, mimeo. Carlton, Dennis W. and Michael Waldma n (2005a) Tying, Upgrades, and Switching Costs in Durable-Goods Market, NBER Working Paper 11407. Carlton, Dennis W. and Michael Waldman ( 2005b) Theories of Tying and Implications for Antitrust, mimeo. Chakravorti, Sujit and Roberto Roson (2006) Platform Competition in Two-Sided Markets: The Case of Payment Networks, Review of Network Economics 5, pp. 118142.

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134 Choi, Jay P. (2004) Tying and Innova tion: A Dynamic Analysis of Tying Arrangements, The Economic Journal 114, pp. 83101. Choi, Jay P. and Christodoulos Stefanadis (2001) Tying, Investment, and the Dynamic Leverage Theory, Rand Journal of Economics 32, pp. 5271. Evans, David (2003) The Antitrust Econom ics of Multi-Sided Platform Markets, Yale Journal on Regulation 20, pp. 325381. Evans, David and Richard Schmalensee (1999) Paying with Plastic: The Digital Revolution in Buying and Borrowing MIT Press: Cambridge, MA. Evans, David and Richard Schmalensee (2005) The Industrial Orga nization of Markets with Two-Sided Platforms, NBER Working Paper, 11603. Evans, David S. and Michael Salinger (2005) Why Do Firms Bundle or Tie? Evidence from Competitive Markets and Implications for Tying Law, Yale Journal of Regulation 22, pp. 3789. Evans, David S., Albert L. Nichols and Richard Schmalensee (2001) An Analysis of the Governments Economic Case in U.S. v. Microsoft, The Antitrust Bulletin 46, pp. 163251. Evans, David S., Albert L. Nichols and Ri chard Schmalensee (2005) U.S. v. Microsoft: Did Consumers Win?, NBER Working Paper 11727. Farrell, Joseph and Garth Saloner (1985) Standardization, Compatibility, and Innovation, Rand Journal of Economics 16, pp. 70. Farrell, Joseph and Garth Saloner (1986) Ins talled Base and Compatibility: Innovation, Product Preannouncements, and Predation, American Economic Review 76, pp. 940. Gabszewicz Jean J. and Xavier Y. Waut hy (2004) Two-Sided Markets and Price Competition with Multi-homing, mimeo. Gans, Joshua S. and Stephan P. King (2003) The Neutrality of Interchange Fees in Payment Systems, Topics in Economic Analysis & Policy 3, pp. 1. Gilbert, Richard J. and Michael L. Katz (2001) An Economists Guide to U.S. v. Microsoft, Journal of Economic Perspectives 15, pp. 25. Guthrie, Graeme and Julian Wright (2006) Competing Payment Schemes, Journal of Industrial Economics forthcoming. Katz, Michael L. (2001) Reform of Credit Card Schemes in Australia II Reserve Bank of Australia.

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135 Katz, Michael L. and Carl Shapiro (1985) Network Externalities, Competition, and Compatibility, American Economic Review 75, pp. 424. Katz, Michael L. and Carl Shapiro (1986) Technology Adoption in the Presence of Network Externalities, Journal of Political Economy 94, pp. 822. Klein, Benjamin (1993) Market Power in Antitrust: Economic Analysis after Kodak, Supreme Court Economic Review 3, pp. 43. McAfee, R. Preston, John McMillan, a nd Michael D. Whinston (1989) Multiproduct Monopoly, Commodity Bundling, and Correlati on of Values, Quarterly Journal of Economics, 104, pp. 371. Nalebuff, Barry (2003) Bundling, Tying, and Portfolio Effects DTI Economics Paper No. 1. Nalebuff, Barry (2004) Bundling as an Entry Barrier, Quarterly Journal of Economics 119, pp. 159187. Reisinger, Markus (2004) Two-Sided Mark ets with Negative Externalities, mimeo. Rochet, Jean-Charles and Jean Tirole (2002) Cooperation among Competitors: Some Economics of Payment Card Associations, Rand Journal of Economics 33, pp. 549. Rochet, Jean-Charles and Jean Tirole ( 2003) Platform Competition in Two-Sided Markets, Journal of European Economic Association 1, pp. 9901029. Rochet, Jean-Charles and Jean Tirole (2005) Two-Sided Markets: A Progress Report, Rand Journal of Economics forthcoming. Rochet, Jean-Charles and Jean Tirole ( 2006) Tying in Two-Sided Markets and the Honor All Cards Rule, mimeo. Roson, Roberto (2005a) Two-Sided Markets: A Tentative Survey, Review of Network Economics 4, pp. 142. Roson, Roberto (2005b) Platform Competiti on with Endogenous Multihoming, mimeo. Rysman, Marc (2006) An Empirical Analys is of Payment Card Usage, mimeo. Schmalensee, Richard (1984) Gaussia n Demand and Commodity Bundling, Journal of Business 57, pp. S211S230. Schmalensee, Richard (2002) Payment Systems and Interchange Fees, Journal of Industrial Economics 50, pp. 103.

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136 Shapiro, Carl (1995) Aftermarkets and C onsumer Welfare: Making Sense of Kodak, Antitrust Law Journal 63, pp. 483511. Shapiro, Carl and David J. Teece (1994) Systems Competition and Aftermarkets: an Economic Analysis of Kodak, Antitrust Bulletin 39, pp. 135162. Tirole, Jean (2005) The Analysis of Tying Cases: A Primer, Competition Policy International 1, pp. 125. Whinston, Michael D. (1990) Tying, Foreclosure, and Exclusion, American Economic Review 80, pp. 837859. Whinston, Michael D. (2001) Exclusivity a nd Tying in U.S. v. Microsoft: What We Know, and Dont know, Journal of Economic Perspectives 15, pp. 6380. Wright, Julian (2003a) Optimal Card Payment Systems, European Economic Review 47, pp. 587. Wright, Julian (2003b) Pricing in Debit and Credit Card Schemes, Economics Letters 80, pp. 305. Wright, Julian (2004a). "The Determinants of Optimal Interchange Fees in Payment Systems." Journal of Industrial Economics 52, pp. 1. Wright, Julian (2004b) One-Sided Logic in Two-Sided Markets, Review of Network Economics 3, pp. 44.

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137 BIOGRAPHICAL SKETCH Jin Jeon was born in Kunsan, Korea, in 1967. He received his B.A. and M.A. in economics from Seoul National University in Seoul, Korea. He joined the doctoral program at economics department of the University of Florida in 1999. He will receive his Ph.D. in economics in December 2006.


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THREE ESSAYS ON BUNDLING AND TWO-SIDED MARKETS


By

JIN JEON

















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Jin Jeon

































To my parents, wife, and two daughters















ACKNOWLEDGMENTS

I must first thank my supervisory committee members. Dr. Jonathan Hamilton, the

chair of the committee, always supported me with patience, encouragement, and

intellectual guidance. He inspired me to think in new ways and put more emphasis on

economic intuition than technical details. Dr. Steven Slutsky, a member, generously

shared his time to listen to my ideas and give further suggestions. Dr. Roger Blair, a

member, also gave me useful comments and provided research ideas. Dr. Joel Demski,

the external member, carefully read the manuscript and gave helpful comments. I hereby

thank them all again.

This dissertation would not have been possible without support from my family

members. My parents always believed in me and kept supporting me. I would also like to

give a heartfelt acknowledgment to my wife, Hyo-Jung, and two daughters, Hee-Yeon

and Hee-Soo, for their endless loving support.















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

T A B L E .............................................................................................................. ..... v ii

L IST O F FIG U R E S .............. ............................ ............. ........... ... ........ viii

ABSTRACT ........ .............. ............. ...... ...................... ix

CHAPTERS

1 IN TR OD U CTION ............................................... .. ......................... ..

1 .1 B u n d lin g ..............................................................................................................2
1.2 Tw o-Sided M markets ....................................... ........ ........ .. ........ ..

2 BUNDLING AND COMMITMENT PROBLEM IN THE AFTERMARKET .........11

2 .1 In tro d u ctio n ...................... .. .. ............. ..................... ................ 1 1
2.2 The M odel ............................ ..... .............. ................. ........... 16
2.3 Independent Sale without Commitment .................... ..............................20
2.4 Independent Sale with Commitment...................... .... .. ................24
2.5 Bundling: An Alternative Pricing Strategy without Commitment....................30
2.6 Bundling and Social W welfare ........................................ ........................ 36
2.7 Bundling and R& D Incentives ........................................ ....... ............... 39
2 .8 C o n clu sio n ................................................. ................ 4 1

3 COMPETITION AND WELFARE IN THE TWO-SIDED MARKET: THE
CASE OF CREDIT CARD INDUSTRY ..........................................................44

3.1 Introduction ................................. ............................... ........44
3.2 The Model: Nonproprietary Card Scheme......................................................47
3.3 Competition between Identical Card Schemes: Bertrand Competition ............49
3.3.1 Single-H om ing Consum ers........................................ ............... 50
3.3.2 M ulti-H om ing Consum ers ................................... ..................58
3.4 Competition between Differentiated Card Schemes: Hotelling Competition...66
3.4.1 Single-H om ing Consum ers........................................ ............... 68
3.4.2 M ulti-H om ing Consum ers ....................................... ............... 74
3.5 Proprietary System with Single-Homing Consumers ......................................82
3.5.1 Competition between Identical Card Schemes ................................... 83









3.5.2 Competition between Differentiated Card Schemes.............................88
3 .6 C o n clu sio n ................................................. ................ 9 2

4 COMPETITION BETWEEN CARD ISSUERS WITH HETEROGENEOUS
EXPENDITURE VOLUMES ........................................................95

4.1 Introduction ................. ................. ............................... ........95
4.2 Equilibrium C ardholder Fee......................................... ......................... 98
4 .2 .1 T he M odel ................................................................ ............... 98
4.2.2 Full-C over M market ................................... ................ .................... 103
4.2.3 L ocal M monopoly ....................................................... ............... 107
4.2.4 Partial-Cover M market ........................................................................ 109
4.3 Equilibrium Interchange Fee.................................... ..................................... 114
4 .3 .1 F ull-C ov er M market ..................................................................... ...... 114
4 .3 .2 L ocal M on op oly .................................................................... .. ...... 116
4.3.3 P artial-C over M market .................................. ...................................... 117
4 .4 E xten sion .................................................................................... ........ 118
4.4.1 Other Comparative Statics............... ...................................118
4 .4 .2 C ollu sion ........................................... ................ 12 0
4 .5 C o n clu sio n ................................................ ................ 12 2

5 C O N CLU D IN G R EM A R K S ......................................................... .....................125

APPENDIX SIMULATION ANALYSIS FOR CHAPTER 4..................................... 128

R E F E R E N C E S ........................................ .......................................... ............... .... 13 3

BIOGRAPHICAL SKETCH ............................................................. ...............137
















TABLE

Table page

4 -1 C om p arativ e statics........................................................................... .................. 119
















LIST OF FIGURES


Figure page

1-1 Credit card scheme es .............. .... ........ .. ..... .................. ........7

2-1 Consumers' surplus in bundling and IS cases when vl
3-1 Welfare and interchange fees of Bertrand competition............................................66

3-2 Merchants' acceptance decision when al > a2 (ml > m2) ..................................77

3-3 Welfare and interchange fees of Hotelling competition when
bB b < 2(bB + bs c) ........................................ ....................................... 82

4-1 Division of consumers in three cases of market coverage................. ...............102

4-2 The effect of a price drop on demand ............................ ..... ... ............... 110

A The density function ......... ................................ ....................................... 129

A-2 Effects of an increase in the variance on the cardholder fee (dfldy) ........................130

A-3 Change in the interchange fee and the cardholder fee (df/da)..............................132















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THREE ESSAYS ON BUNDLING AND TWO-SIDED MARKETS

By

Jin Jeon

December 2006


Chair: Jonathan H. Hamilton
Major Department: Economics

This work addresses three issues regarding bundling and two-sided markets. It

starts with a brief summary of the theories of bundling and of two-sided markets in

Chapter 1.

Chapter 2 analyzes various aspects of bundling strategy by the monopolist of a

primary good when it faces competition in the complementary good market. The main

result is that the monopolist can use a bundling strategy in order to avoid commitment

problem that arises in optimal pricing. Bundling increases the monopolist's profits

without the rival's exit from the market. Bundling lowers social welfare in most cases,

while it may increase consumers' surplus. One of the long-run effects of bundling is that

it lowers both firms' incentives to invest in R&D.

Chapter 3 compares welfare implications of monopoly outcome and competitive

outcome. Using a model of the credit card industry with various settings such as Bertrand

and Hotelling competition with single-homing and multi-homing consumers as well as









proprietary and nonproprietary platforms, it is shown that introducing platform

competition in two-side markets may lower social welfare compared to the case of

monopoly platform. In most cases, monopoly pricing maximizes Marshallian social

welfare since the monopolist in a two-sided market can properly internalize indirect

network externalities by setting unbiased prices, while the competing platforms set biased

prices in order to attract the single-homing side.

Chapter 4 analyzes the effects of distribution of consumers' expenditure volumes

on the market outcomes using a model in which two card issuers compete a la Hotelling.

The result shows that the effects of distribution of the expenditure volume are different

for various cases of market coverage. For example, as the variance increases, issuers'

profits decrease when the market is fully covered, while the profits increase when the

market is locally monopolized. It is also shown that the neutrality of the interchange fee

holds in the full-cover market under the no-surcharge-rule. Simulation results are

provided to show other comparative statics that include the possibility of the positive

relationship between the interchange fee and the cardholder fee.

Finally, Chapter 5 summarizes major findings with some policy implications.














CHAPTER 1
INTRODUCTION

This dissertation contains three essays on bundling and two-sided markets. These

topics have recently drawn economists' attention due to the antitrust cases of Kodak and

Microsoft, and movements in some countries to regulate the credit card industry.

In the Kodak case, independent service organizations (ISOs) alleged that Kodak

had unlawfully tied the sale of service for its machines to the sale of parts, in violation of

section 1 of the Sherman Act, and had attempted to monopolize the aftermarket in

violation of section 2 of the Sherman Act. 1 In the Microsoft case, the United States

government filed an antitrust lawsuit against Microsoft for illegally bundling Internet

Explorer with Windows operating system.2

In the credit card industry, antitrust authorities around the world have questioned

some business practices of the credit card networks, which include the collective

determination of the interchange fee, the no-surcharge rule, and the honor-all-cards rule.

As a result, card schemes in some countries such as Australia, United Kingdom, and

South Korea have been required to lower their interchange fees or merchant fees.

To understand these antitrust cases, many economic models have been developed.

In the following sections, brief summaries of the economic theories of bundling and of

two-sided markets will be presented.

1 For more information about the Kodak case, see Klein (1993), Shapiro (1995),
Borenstein, MacKie-Mason, and Netz (1995), and Blair and Herndon (1996).
2 See Gilbert & Katz (2001), Whinston (2001), and Evans, Nichols and Schmalensee
(2001, 2005) for further analysis of the Microsoft case.









1.1 Bundling

Economists' views regarding bundling or tying have shifted dramatically in recent

decades.3 The traditional view of tying can be represented by the leverage theory which

postulates that a firm with monopoly power in one market could use the leverage to

monopolize another market.

The Chicago School criticized the leverage theory, since such leveraging may not

increase the profits of the monopolist. According to the single monopoly profit theorem

supported by the Chicago School, the monopolist earns same profits regardless whether it

ties if the tied good market is perfectly competitive. For example, suppose consumers'

valuation of a combined product of A and B is $10 and marginal cost of producing each

good is $1. Good A is supplied only by the monopolist, and good B is available in a

competitive market at price equal to the marginal cost. Without bundling, the monopolist

can charge $9 for A-and $1 for B-to make $8 as unit profit per good A sold. If the

monopolist sells A and B as a bundle, it can charge $10 for the bundle and earn $8 ($10 -

$1 $1) per unit bundle. So the monopolist cannot increase profits by bundling in this

case.

Economists led by the Chicago School proposed alternative explanations for

bundling based on efficiency rationales. Probably the most common reason for bundling

is it reduces the transaction costs such as consumers' searching costs and firms'

packaging and shipping costs. Examples of this kind of bundling are abundant in the real



3 Bundling is the practice of selling two goods together, while tying is the behavior of
selling one good conditional on the purchase of another good. There is no difference
between tying and bundling if the tied good is valueless without the tying good and two
goods are consumed in fixed proportion. See Tirole (2005) and Nalebuff (2003) for the
discussions of bundling and tying.









world: shoes are sold in pairs; personal computers (PCs) are sold as bundles of the CPU,

a hard drive, a monitor, a keyboard and a mouse; cars are sold with tires and a car audio.

In some sense, most products sold in the real world are bundled goods and services.

Another explanation for bundling in line with the efficiency rationale is price

discrimination. That is, if consumers are heterogeneous in their valuations of products,

bundling has a similar effect as price discrimination.4 This advantage of bundling is

apparent when consumers' valuations are negatively correlated. But bundling can be

profitable even for nonnegative correlation of consumers' valuations (McAfee, McMillan,

and Whinston, 1989). In fact, unless consumers' valuations are perfectly correlated, firms

can increase profits by bundling.5 Since price discrimination usually increases social

welfare as well as firm's profits, bundling motivated by price discrimination increases

efficiency of the economy.

The leverage theory of tying revived with the seminal work of Whinston (1990).

He showed that the Chicago School arguments regarding tying can break down in certain

circumstances which include 1) the monopolized product is not essential for all uses of

the complementary good, and 2) scale economies are present in the complementary good.

If there are uses of the complementary good that do not require the primary good, the

monopolist of the primary good cannot capture all profits by selling the primary good

only. So the first feature provides an incentive for the monopolist of the primary good to

exclude rival producers of the complementary good. The second feature provides the

monopolist with the ability to exclude rivals, since foreclosure of sales in the


4 See Adams and Yellen (1976) and Schmalensee (1984).
5 Bakos and Brynjolfsson (1999) show the benefit of a very large scale bundling based on
the Law of Large Numbers.









complementary market, combined with barriers to entry through scale economies, can

keep rival producers of the complementary good out of the market.6

Bundling can also be used to preserve the monopolist's market power in the

primary good market by preventing entry into the complementary market at the first stage

(Carlton and Waldman, 2002a). This explains the possibility that Microsoft bundles

Internet Explorer with Windows OS in order to preserve the monopoly position in the OS

market, since Netscape's Navigator combined with Java technology could become a

middleware on which other application programs can run regardless of the OS.

Choi and Stefanidis (2001) and Choi (2004) analyze the effects of tying on R&D

incentives. The former shows that tying arrangement of an incumbent firm that produces

two complementary goods and faces possible entries in both markets reduces entrants'

R&D incentives since each entrant's success is dependent on the other's success. The

latter analyzes R&D competition between the incumbent and the entrant, and shows that

tying increases the incumbent's incentives to R&D since it can spread out the costs of

R&D over a larger number of units, whereas the entrant's R&D incentives decrease.7

Chapter 2 presents a model of bundling that follows the basic ideas of the leverage

theory. It shows that the monopolist of a primary good that faces competition in the

aftermarkets can use the bundling strategy to increase profits to the detriment of the rival

firm. Aftermarkets are markets for goods or services used together with durable

equipment but purchased after the consumer has invested in the equipment. Examples

include maintenance services and parts, application programs for operating systems, and

6 Nalebuff (2004) and Carlton and Waldman (2005a) also present models that show the
entry deterrence effect in the tied good market.
7 In chapter 2, I show that bundling reduces R&D incentives of the monopolist as well as
of the rival.









software upgrades. One of the key elements of the aftermarket is that consumers buy the

complementary goods after they have bought the primary good. For the monopolist of the

primary good, the best way to maximize its profits is to commit to the second period

complementary price. If this commitment is not possible or implementable, bundling can

be used.

Unlike most of the previous models of the leverage theory, market foreclosure is

not the goal of the bundling in this model. On the contrary, the existence of the rival

firms is beneficial to the monopolist in some sense since it can capture some surplus

generated by the rival firm's product.

1.2 Two-Sided Markets

Two-sided markets are defined as markets in which end-users of two distinctive

sides obtain benefits from interacting with each other over a common platform. 8 These

markets are characterized by indirect network externalities, i.e., benefits of one side

depend on the size of the other side.9 According to Rochet and Tirole (2005), a necessary

condition for a market to be two-sided is that the Coase theorem does not apply to the

transaction between the two sides. That is, any change in the price structure, holding

constant the total level of prices faced by two parties, affects participation levels and the

number of interactions on the platform since costs on one side cannot be completely

passed through to the other side.




SFor general introductions to the two-sided market, see Roson (2005a), and Evans and
Schmalensee (2005).
9 In some cases such as media industries, indirect network externalities can be negative
since the number of advertisers has a negative impact on readers, viewers, or listeners.
See Reisinger (2004) for the analysis of two-sided markets with negative externalities.









Examples of the two-sided market are abundant in the real world. Shopping malls

need to attract merchants as well as shoppers. Videogame consoles compete for game

developers as well as gamers. Credit card schemes try to attract cardholders as well as

merchants who accept the cards. Newspapers need to attract advertisers as well as

readers. 10 Figure 1-1 shows the structure of the two-sided market in case of the credit

card industry, both proprietary and nonproprietary schemes.

Although some features of two-sided markets have been recognized and studied for

a long time, 11 it is only recently that a general theory of two-sided markets emerged. 12

The surge of interest in two-sided markets was partly triggered by a series of antitrust

cases against the credit card industry in many industrialized countries including the

United States, Europe and Australia. The literature on the credit card industry has found

that the industry has special characteristics; hence conventional antitrust policies may not

be applicable to the industry. 13

Wright (2004b) summarizes fallacies that can arise from using conventional

wisdom from one-sided markets in two-sided markets, which include: an efficient price

structure should be set to reflect relative costs; a high price-cost margin indicates market

power; a price below marginal cost indicates predation; an increase in competition

necessarily results in a more efficient structure of prices; and an increase in competition

necessarily results in a more balanced price structure.


10 See Rochet and Tirole (2003) for more examples of the two-sided market.
11 For example, Baxter (1983) realized the two-sidedness of the credit card industry.
12 The seminal papers include Armstrong (2005), Caillaud and Jullien (2003), and Rochet
and Tirole (2003).
13 The literature includes Gans and King (2003), Katz (2001), Rochet and Tirole (2002),
Schmalensee (2002), Wright (2003a, 2003b, 2004a).









The theory of two-sided markets is related to the theories of network externalities

and of multi-product pricing. While the literature on network externalities has found that

in some industries there exist externalities that are not internalized by end-users, models

are developed in the context of one-sided markets. 14 Theories of multi-product pricing

stress the importance of price structures, but ignore externalities in the consumption of





Issuer ^ (a : interchange fee) Aq r
....................................... t .................................... P l a t fo r m ........................................... .........................................
Platform
(Card scheme)
Pays p +f (CPaysp m
(f: cardholder fee) (m : merchant fee)


Cardholder 4---------------------- Merchant
Sells good at price

(a) Nonproprietary card scheme



Platform
(Card scheme)

Paysp +f Paysp-m
(f: cardholder fee) (m : merchant fee)



(Cardholder ----------------------- Merchant
Sells good at price

(b) Proprietary card scheme

Figure 1-1. Credit card schemes


14 See Katz and Shapiro (1985, 1986), and Farrell and Saloner (1985, 1986).









different goods since the same consumer buys both goods. That is, the buyer of one

product (say, razor) internalizes the benefits that he will derive from buying the other

product (blades). The two-sided market theory starts from the observation that there exist

some industries in which consumers on one side do not internalize the externalities they

generate on the other side. The role of platforms in two-sided markets is to internalize

these indirect externalities by charging appropriate prices to each side.

In order to get both sides on board and to balance demands of two sides, platforms

in two-sided markets must carefully choose price structures as well as total price levels.15

So it is possible that one side is charged below marginal cost of serving that side, which

would be regarded as predatory pricing in a standard one-sided market. For this reason,

many shopping malls offers free parking service to shoppers, and cardholders usually pay

no service fees or even negative prices in the form of various rebates.

In a standard one-sided market, the price is determined by the marginal cost and the

own price elasticity, as is characterized by Lerner's formula.16 In two-sided markets,

however, there are other factors that affect the price charged to each side. These are

relative size of cross-group externalities and whether agents on each side single-home or

multi-home. 17

If one side exerts larger externalities on the other side than vice versa, then the

platform will set a lower price for this side, ceterisparibus. In a media industry, for

15 In the credit card industry, non-proprietary card schemes choose interchange fees
which affect the price structure of two sides.
16 The standard Lerner'-s formula is -C 1
SThe standard Lerner's formula is or p = c, where is the price, c is the
P E E-I
p e s-1
marginal cost, and e is the own price elasticity.
17 An end-user is "single-homing" if she uses one platform, and "multi-homing" if she
uses multiple platforms.









example, viewers pay below the marginal cost of serving while advertisers pay above the

marginal cost since the externalities from viewers to advertisers are larger than those

from advertisers to viewers.

When two or more platforms compete with each other, end-users may join a single

platform or multiple platforms, depending on the benefits and costs of joining platforms.

Theoretically, three possible cases emerge: (i) both sides single-home, (ii) one side

single-homes while the other side multi-homes, and (iii) both sides multi-home. 18 If

interacting with the other side is the main purpose of joining a platform, one can expect

case (iii) is not common since end-users of one side need not join multiple platforms if all

members of the other side multi-home.19 For example, if every merchant accepts all kinds

of credit cards, consumers need to carry only one card for transaction purposes. Case (i)

is also not common since end-users of one side can increase interaction with the other

side by joining multiple platforms. As long as the increased benefit exceeds the cost of

joining additional platform, the end-users will multi-home.

On the contrary, one can find many examples of case (ii) in the real world.

Advertisers place ads in several newspapers while readers usually subscribe to only one

newspaper. Game developers make the same game for various videogame consoles while

gamers each own a single console. Finally, merchants accept multiple cards while

consumers use a single card.20


18 In most of the models on two-sided markets, single-homing and multi-homing of end-
users are pre-determined for analytical tractability. For an analysis of endogenous multi-
homing, see Roson (2005b).
19 See also Gabszewicz and Wauthy (2004).
20 According to an empirical study by Rysman (2006), most consumers put a great
majority of their payment card purchases on a single network, even when they own
multiple cards from different networks.









When end-users of one side single-home while those of the other side multi-home,

the single-homing side becomes a "bottleneck" (Armstrong, 2005). Platforms compete

for the single-homing side, so they will charge lower price to that side. As is shown in

Chapter 3, platforms competing for the single-homing side may find themselves in a

situation of the "Prisoner's Dilemma". That is, a lower price for the single-homing side

combined with a higher price for the multi-homing side can decrease total transaction

volume and/or total profits compared to the monopoly outcome. Further, competition in

two-sided markets may lower social welfare since monopoly platforms can properly

internalize the indirect externalities by charging unbiased prices, while competing

platforms are likely to distort the price structure in favor of the single-homing side.

Chapter 3 presents a model of the credit card industry with various settings

including single-homing vs. multi-homing cardholders, competition between identical

card schemes (Bertrand competition) or differentiated schemes (Hotelling competition),

and proprietary vs. non-proprietary card schemes. The main finding is that, unlike in a

standard one-sided market, competition does not increase social welfare regardless of the

model settings.

Chapter 4 tackles the assumption made by most models on the credit card industry

that cardholders spend the same amounts with credit cards. By allowing heterogeneous

expenditures among consumers, it shows the effects of a change in the variance of the

expenditure on the equilibrium prices and profits. The results show that the effects are

different depending on whether the market is fully covered, partially covered, or locally

monopolized.














CHAPTER 2
BUNDLING AND COMMITMENT PROBLEM IN THE AFTERMARKET

2.1 Introduction

A monopolist of a primary good that faces competition in the aftermarket of the

complementary goods often uses a bundling or tying strategy. Traditionally, bundling

was viewed as a practice of transferring the monopoly power in the tying market to the

tied market. This so-called "leverage theory" has been criticized by many economists

associated with the Chicago School in that there exist other motives of bundling such as

efficiency-enhancement and price discrimination. Further, they show that there are many

circumstances in which firms cannot increase profits by leveraging monopoly power in

one market to the other market, which is known as the single monopoly profit theorem.

Since the seminal work of Whinston (1990), the leverage theory revived as many

models have been developed to show that a monopolist can use tying or bundling

strategically in order to deter entry to the complementary market and/or primary market.

The research was in part stimulated by the antitrust case against Microsoft filed in 1998,

in which U.S government argued that Microsoft illegally bundles Internet Explorer with

Windows operating system.1 Most of the models in this line, however, have a

commitment problem since the bundling decision or bundling price is not credible when

the entrant actually enters or does not exit the market.




1 For further analyses of the Microsoft case, see Gilbert & Katz (2001), Whinston (2001),
and Evans, Nichols and Schmalensee (2001).









This paper stands in the tradition of the leverage theory and shows that the

monopolist of a primary good can use a bundling strategy to increase profits as well as

the market share in the complementary good market. Unlike the previous models, the

monopolist's profits increase with bundling even if the rival does not exit the market. On

the contrary, the existence of a rival firm is beneficial to the monopolist in some sense

since the monopolist can capture some surplus generated by the rival firm's

complementary good.

The model presented here is especially useful for the analysis of the Microsoft case.

Many new features added to-i.e., bundled with-the Windows operating system (OS)

had been independent application programs produced by other firms. For example,

Netscape's Navigator was a dominant Internet browser before Microsoft developed

Internet Explorer. Therefore, it is Microsoft, not Netscape, that entered the Internet

browser market. Since Netscape's software development cost is already a sunk cost when

Microsoft makes a bundling decision, the entry deterrent effect of bundling cannot be

applied.

The main result is that the monopolist can use bundling to avoid the commitment

problem2 arising in the optimal pricing when consumers purchase the complementary

good after they have bought the primary good. If the monopolist cannot commit to its

optimal price for the complementary good at the first stage when consumers buy the

primary good, then it may have to charge a lower price for the primary good and a higher

price for the complementary good compared to its optimal set of prices since consumers


2 This commitment problem is different from the one in the previous literature, in which
the commitment problem arises since the bundling price is not credible if the would-be
entrant actually enters the market.









rationally expect that the monopolist may raise its complementary good price after they

have bought the primary goods. A double marginalization problem arises in this case

since the monopolist has to charge the price that maximizes its second stage profits, while

it also charges a monopoly price for the primary good at the first stage. Bundling makes it

possible for the monopolist to avoid the double marginalization problem by implicitly

charging a price equal to zero for the complementary good.

The model also shows that bundling generally lowers Marshallian social welfare

except for the extreme case when the monopolist's bundled good is sufficiently superior

to the rival's good. Social welfare decreases with bundling mainly because it lowers the

rival's profits. Consumers' surplus generally increases with bundling. However,

consumers' surplus also decreases when the rival's complementary good is sufficiently

superior to the monopolist's.

The last result shows the effect of bundling on R&D investments. In contrast to the

previous result of Choi (2004) that shows tying lowers the rival firm's incentive to invest

in R&D while it increases the monopolist's incentive, I show that bundling lowers both

firms' incentives to make R&D investments.

The literature on bundling or tying is divided into two groups one finds the

incentive to bundle from the efficiency-enhancing motives, and the other finds it from

anticompetitive motives.3 In the real world, examples of bundling motivated by

efficiency reason are abundant. Shoe makers sell shoes as a pair, which reduces

transaction costs such as consumers' searching costs and producers' costs of shipping and

packaging. The personal computer is another example as it is a bundle of many parts such



3 For a full review of the literature on bundling, see Carlton and Waldman (2005b).









as the CPU, a memory card, a hard drive, a keyboard, a mouse, and a monitor.4 Carlton

and Waldman (2002b) explain another efficiency motive for tying by showing that

producers of a primary good may use tying in order to induce consumers to make

efficient purchase decisions in the aftermarket when consumers can buy the

complementary goods in variable proportions. If the primary good is supplied at a

monopoly price while the complementary good is provided competitively, consumers

purchase too much of the complementary good and too little of the primary good. Tying

can reduce this inefficiency and increase profits.

Adams and Yellen (1976) provide a price discrimination motive for tying. Using

some examples, they show that if consumers are heterogeneous in their valuations for the

products, bundling has a similar effect as price discrimination. This advantage of

bundling is apparent when consumers' valuations are negatively correlated. Schmalensee

(1984) formalizes this theory assuming consumers' valuations follow a normal

distribution. McAfee, McMillan, and Whinston (1989) show that bundling can be

profitable even for nonnegative correlation of consumers' valuations. Bakos and

Brynjolfsson (1999) show the benefit of a very large scale bundling of information goods

based on Law of Large Numbers. Since price discrimination usually increases social

welfare with an increase in total output, tying or bundling motivated by price

discrimination can be welfare improving.

The anticompetitive motive of tying is reexamined by Whinston (1990). He

recognizes that Chicago School's criticism of leverage theory only applies when the

complementary good market is perfectly competitive and characterized by constant



4 See Evans and Salinger (2005) for efficiency-enhancing motive of tying.









returns to scale, and the primary good is essential for use of the complementary good. He

shows that in an oligopoly market with increasing returns to scale, tying of two

independent goods can deter entry by reducing the entrant's profits below the entry cost.

As was mentioned earlier, however, his model has a credibility problem since bundling is

not profitable if entrance actually occurs.

Nalebuff (2004) also shows that bundling can be used to deter entry, but without a

commitment problem since in his model the incumbent makes higher profits with

bundling than independent sale even when the would-be entrant actually enters.5 Carlton

and Waldman (2002a) focus on the ability of tying to enhance a monopolist's market

power in the primary market. Their model shows that by preventing entry into the

complementary market at the first stage, tying can also stop the alternative producer from

entering the primary market at the second stage.

Carlton and Waldman (2005a) shows that if the primary good is a durable good and

upgrades for the complementary good are possible, the monopolist may use a tying

strategy at the first stage in order to capture all the upgrade profits at the second stage.

Especially when the rival's complementary good is superior to the monopolist's, the only

way the monopolist sells second-period upgrades is to eliminate the rival's product in the

first period by tying its own complementary good with its monopolized primary good. By

showing that tying can be used strategically even when the primary good is essential for

use of the complementary good, it provides another condition under which the Chicago

School argument breaks down.



5 However, the optimal bundling price is higher when the entrant enters than the price
that is used to threaten the entrant. So there exists a credibility problem with the price of
the bundled good.









The model presented here also assumes the primary good is essential, but the

primary good is not necessarily a durable good and constant returns to scale prevail. So it

can be added to the conditions under which the Chicago School argument breaks down

that bundling can be used strategically when consumers buy the primary good and the

complementary good sequentially.

The rest of Chapter 2 is organized in the following way. Section 2.2 describes the

basic setting of the model. Sections 2.3 to 2.5 show and compare the cases of independent

sale, pricing with commitment, and bundling, respectively. Section 2.6 analyzes the

welfare effect of bundling. Section 2.7 is devoted to the effect of bundling on R&D

investments. The last section summarizes the results.

2.2 The Model

Suppose there are two goods and two firms in an industry. A primary good is

produced solely by a monopolist, firm 1. The other good is a complementary good that is

produced by both the monopolist and a rival, firm 2. The purchases of the primary good

and the complementary good are made sequentially, i.e., consumers buy the

complementary good after they have bought the primary good. Consumers buy at most

one unit of each good,6 and are divided into two groups. Both groups have same

reservation value vo for the primary good. For the complementary good, however, one

group has zero reservation value and the other group has positive reservation value v,,

where i= 1,2 indicates the producer.7 For modeling convenience, it is assumed that the




6 So there is no variable proportion issue.

7 Consumption of the complementary good may increase the reservation value of the
primary good. It is assumed that v, also includes this additional value.









marginal cost of producing each good is zero and there is no fixed cost for producing any

good.8

The PC software industry fits in this model, in which Microsoft Windows OS is the

monopolized primary good and other application programs are complementary goods.

Microsoft also produces application programs that compete with others in the

complementary good market. Sometimes Microsoft bundles application programs such as

an Internet browser and a media player that could be sold separately into Windows OS.

Consumers usually buy the Windows OS at the time they buy a PC, then buy application

software later.

Let the total number of consumers be normalized to one, and ac be the portion of the

consumers, group S, who have positive valuations for the complementary good. It is

assumed that the consumers in S are distributed uniformly on the unit interval, in which

the monopolist and firm 2 are located at 0 and 1, respectively.

The two complimentary goods are differentiated in a Hotelling fashion. A

consumer located at x incurs an additional transportation cost tx when she buys the

monopolist's complementary good, and t(1 x) when she buys firm 2's. So the gross

utility of the complementary good for the consumer is vl tx when she buys from the

monopolist, and v2 t(1 x) when she buys from firm 2. vl and v2 are assumed to be

greater than t in order to make sure that consumers in S cannot have a negative gross

utility for any complementary good regardless of their positions. Further, in order to





8 Unlike the models that explain tying as an entry deterrence device, the model in this
paper assumes constant returns to scale.









make sure that all the consumers in S buy the complementary goods at equilibrium, it is

assumed that9



v + V2 > 3t (2-1)



The model presented here allows a difference between vi and v2 in order to analyze

bundling decision when the monopolist produces inferior-or superior-complementary

good and the effect of bundling on R&D investments. But the difference is assumed to be

less than t, i.e.,



I v v2 < t (2-2)



since otherwise all consumers find one of the complementary goods superior to the other

good. 10

In the software industry, the primary good is the operating system (OS), and

application programs like an Internet browser or a word processor are examples of

complementary goods. The OS itself can be seen a collection of many functions and

commands. Bakos and Brynjolfsson (1999) show that the reservation values among

consumers of a large scale bundle converge to a single number, which justifies the

assumption that consumers have the same valuation for the primary good. A single

application program, however, is not as broadly used as an OS, so the valuation for the



9 The prices chosen by two firms could be too high so that some of the consumers in S
may not want to buy the complementary good. The assumption vl+v2 > 3t guarantees
that every consumer in S buy a complementary good at equilibrium.
10 This is also a sufficient condition for the existence of the various equilibria.









complementary good may vary among consumers. Furthermore, not all the application

programs are produced for all consumers. Some of them are developed for a certain

group of consumers such as business customers.

The game consists of two stages.11 At the first stage consumers buy the primary

good or bundled good at the price that the monopolist sets. The monopolist can set the

price of its own complementary good with or without commitment, or sell both goods as

a bundle. At the second stage, consumers buy one of the complementary goods, the prices

of which are determined by the competition between the two firms.

Letpo,pi, and p2 be the prices of the monopolist's primary good, the monopolist's

complementary good, and firm 2's complementary good, respectively. Then the net

utilities of the consumer located at x if she consumes the primary good only, the primary

good with the monopolist's complementary good, and the primary good with firm 2's

complementary good are, respectively,



Uo= vo -po

u = Vo+ vi- tx-po-pl

u2 = V+ V2- t(1 -x) -Po-P2



The consumer will buy only the primary good if



Uo > U1, Uo > u2, and uo > 0



She will buy the primary good and the monopolist's complementary good if

11 In section 2.7, an earlier stage will be added at which two firms make investment
decisions that determine v,'s.









ul > u2, ul > uo, and ul > 0



She will buy the primary good and firm 2's complementary good if



u2 u> u2 > uo, and u2 > 0



Lastly, she will buy nothing if



uo < 0, ul < 0, and u2 < 0



2.3 Independent Sale without Commitment

In this section, it is assumed that the monopolist cannot commit topl at the first

stage. Without commitment, pi must be chosen to be optimal at the second stage. That is,

in game-theoretic terms, the equilibrium price must be subgame perfect.

As in a standard sequential game, the equilibrium set of prices can be obtained by

backward induction. Let x* be the critical consumer who is indifferent between the

monopolist's complementary good and firm 2's good. One can find this critical consumer

by solving vi tx*-pl = v2 t(1 x*) -p2, which gives



1 v, v 2 2 1-
x --+ (2-3)
2 2t



There are two cases to be considered: when the monopolist sells the primary good

to all consumers, and when it sells its products to group S only. Consider first the case

that the monopolist sells the primary good to all consumers. At the second stage, the

monopolist will setpl to maximize oapx*, while firm 2 will setp2 to maximize ap2(l -x*).









By solving each firm's maximization problem, one can obtain the following best

response functions:


vl V + p +t
2


i,j= 1,2andi j


from which one can obtain the following equilibrium prices for the case of the

independent sale without commitment (IA case):



t V V
pA=v +t, i,j= ,2andi j
3



Plugging these into (2-3) gives the location of the critical consumer:



IA 1 V1 -V2
2 6t



At the first stage, the monopolist will set the price of the primary good equal to vo

since consumers outside of group S will not buy the good for the price higher than vo:


One needs to check whether consumers actually buy the goods for this set of prices.

This can be done by plugging the prices into the net utility of the critical consumer, i.e.,


A A +V2 3t
Po -p1 >0
2


(2-4)


U, (XIA) = Vg + V1 tXI









where the last inequality holds because of the assumption given in (2-1). As was noted in

footnote 10, this assumption guarantees that all consumers in S buy both goods at

equilibrium.

The profits of the firms at equilibrium are



A p t px(vI v2 + 3t)2
7Tj = P, + apDX, =VO +-a
18t

IA IA 1 a(v2- v + 3t)2
18t



The monopolist may find it profitable to sell the primary goods exclusively to

group S by charging the price higher than vo. If a consumer located at x have bought the

primary good at the first stage, the maximum prices she is willing to pay for the

monopolist's and firm 2's complementary goods at the second stage are v tx and v2

t(1 x), respectively, regardless how much she paid for the primary good at the first stage.

Since the payment at stage one is a sunk cost to the consumer, she will buy a

complementary good as long as the net utility from the complementary good is non-

negative. This implies that when the monopolist sells the primary good to group S only

without commitment to p (IS case), the equilibrium prices and the location of the critical

consumer at the second stage are exactly the same as in the IA case. 12 That is,



V -V
s= v- +t, i,j=1,2andi j
3

12 There may exist multiple equilibria because of the coordination problem among
consumers. For example, suppose consumers around at xs did not buy the primary good
at stage 1. Then at stage 2, the two firms will charge higher prices than p,s. At this
price set, consumers who did not buy the base good will be satisfied with their decision.










XIS 1 + 2
2 6t



When consumers buy the primary good at stage 1, they rationally predict that the

second period prices of the complementary goods are pf So the monopolist will set the

primary good price to make the critical consumer indifferent between buying the

complementary good and not buying, which yields the following equilibrium price:



IS v +v2- 3t
Po = Vo +
2



Note that the primary good price is higher than vo as is expected. By excluding the

consumers who buy only the primary good, the monopolist can charge a higher price in

order to capture some surplus that would otherwise be enjoyed by the consumers of the

complementary goods.

The monopolist's profits may increase or decrease depending on the size of a,

while firm 2's profits remain the same as in the IA case since the price and the quantity

demanded in IS case are exactly the same as in the IA case:



IJs = a(pis is) = a ( -v1 2)2 5v, +v2 -6t
01 1 1 18t 6


s = (1 I) a(v2 v + 3t)2
18t



By comparing tis and A, one can derive the condition in which the monopolist

prefers the IS outcome to the IA outcome:









2v ^Is
a> --- a
2v + (v + v, 3t)



^IS
Note that a lies between 0 and 1 since v +v2 3t > 0 is assumed in (2-1).

2.4 Independent Sale with Commitment

The results of the previous section may not be optimal for the monopolist if it can

choose bothpo and pi simultaneously at the first stage and commit topi. To see this,

suppose the monopolist can set both prices at the first stage with commitment. As in the

previous section, one can distinguish two cases depending on the coverage of the primary

good market. When the monopolist sells its primary good to all consumers with

commitment topl (CA case), the model shrinks to a simple game in which the

monopolist setpl at the first stage and firm 2 setp2 at the second stage since the primary

good price should be set equal to vo, i.e., pf = 0 The equilibrium prices of the

complementary goods can be derived using a standard Stackelberg leader-follower model.

The equilibrium can be found using backward induction. At the second stage, the

critical consumer who is indifferent between the monopolist's complementary good and

firm 2's good is determined by (2-3) with p2 replaced by firm 2's best response function

given by (2-4), i.e.,



3 Vl -v v -p p
x -+ (2-5)
4 4t



The monopolist will set p to maximize ~ap x*, which gives the following optimal


price:










CA V1 V2 +3t
2



The remaining equilibrium values can be obtained by plugging this into (2-4) and

(2-5):



CA 2 v + 5t
P2

CA 3 V1 V2
8 8t

CA (v1 vi + 3t)2
16t

CA a(v2 v1 + 5t)2
2 32t



The differences between the equilibrium prices of CA case and IA case are



CPA _pA v -v+ 3t > 0
6
CA IA v v2 3t>
P2 -p = > 0
12



The price differences are positive since the difference between vl and v2 is assumed

to be less than t. Since the monopolist's complementary good is a substitute for firm 2's

good, pi andp2 are strategic complements. If one firm can set its price first, it will set a

higher price so that the rival also raises its own price compared to the simultaneous move

game. With the increase in the prices, both firms enjoy higher profits as the following

calculation shows:










c -- a(vl v + 3t)2 > 0
144t

CA (vl v2 + 3t)[27t -7(v, -v)] > 0
288t



The profit of the monopolist must increase since it chooses a different price even if

it could commit to p1A at the first stage. Firm 2's profit also increases as both firms'

prices of complementary goods increase while the price of the primary good remains the

same.

When the monopolist covers only the consumers in group S with commitment topl

(CS case), the equilibrium can be found in a similar way as in the CA case. At the second

stage, firm 2's best response function is the same as (2-4) and the critical consumer is also

determined by (2-5). Since the monopolist will make the critical consumer indifferent

between buying and not buying the complementary good, po will be set to satisfy the

following condition:



po= vo+ vl-pi- tx' (2-6)


Using (2-5) and (2-6), the monopolist's profits can be rewritten as a function of p

in the following way:



S 3v, + v, 3t P (v, v2 P1)
Tr, = a(po + px) = a + +3t (V



Maximizing this profit function w.r.t. pi yields the optimal price for the

monopolist's complementary good, which is










CS V_ V2
2



Plugging this back to (2-4), (2-5) and (2-6), one can derive the remaining

equilibrium values:



s 3v, +5v2 6t
PoCS = V +-
8

cs v2 -v + 2t
P2 4
4

cs 3 v, -v2
x^ =-+
4 8t

cs (I -v )2 3v, + V -3t
S16t 4

2cs a(v2 +2t)2
32t



The differences between the equilibrium prices of CS case and IS case are as

follows:






Is Is V v2 + 6t
pCS <0

8
C-S 1 <- 6



cs is vI 6t
P2 p2 12
12



When the monopolist can commit to its complementary good price, it charges a

higher price for the primary good and a lower price for the complementary good. And the









rival firm also charges a lower price for its own complementary good. Sincepl andp2 are

strategic complements, the monopolist can induce firm 2 to decrease p2 by lowering pi,

which makes it possible for the monopolist to raise po for higher profits. This would not

be possible if the monopolist cannot commit topl at the first stage since the monopolist

has an incentive to raise the complementary good price at the second stage after

consumers have bought the primary good.

The difference between the profits of CS case and IS case are as follows:



s_ Is a(v,2 + 6t)2>
144t

cs I_ s a[18t-7(v v2)][6t -(v v2 ) 0
288t



The monopolist's profits increase when it can commit as in CA case. However, firm

2's profits decrease since the monopolist can capture some of the consumers' surplus

generated by firm 2's complementary good by charging a higher price for the primary

good.

Comparing TCS and z c, one can derive the following condition for the

monopolist to prefer the CS outcome to the CA outcome:



16vo ^cs
a > -a
16vo + (6v, +10v2 -21t)



CS
a lies between 0 and 1 since 6v, +10v2 21t = 8(v, +v2 -3t)+ 2(v2 -v)+3t > 0
^IS ^CS
from the assumptions given in (2-1) and (2-2). The difference between a and a is









^Is ^cs 2vo(2v 2v, +3t)
a =->0
(2vo + v, + v2 3t)(16v0 + 6v, +10v2 21t)



The critical level of a with commitment is lower than with independent sale since

the profit gain from commitment is higher in the CS case than in the CA case.13 That is,

the monopolist is willing to sell both goods to a smaller group of consumers when it can

commit to the price of its own complementary good sold in the second period.

The problem that the monopolist earns lower profits when it cannot commit to the

second period price of the complementary good is common in cases of durable goods

with aftermarkets.14 That is, rational consumers expect that the monopolist will set its

second period price to maximize its second period profit regardless of its choice in the

first period. The monopolist has an incentive to charge a higherpl after consumers in S

have bought the primary good at the first stage, since the price consumers have paid for

the primary goods is sunk cost at stage 2.15 If the monopolist cannot commit to p1cs

therefore, some consumers in S would not buy the primary good at the first stage. So the

monopolist would have to set a lowerpo (p- ) and a higher p (pf ) because of the hold-

up problem.

One of the problems in relation to the pricing with commitment is that the optimal

prices may not be implemented since ps is negative when vl < v2.16 The bundling



13 Note that (cs s)-( RCA A a[2(v2 v)+ 3t]
=t, -)T1 16 >0
16
14 See Blair and Herndon (1996)
15 After consumers have bought the primary goods at stage 1, the monopolist has an
incentive to charge p1s which is higher than pcs.
16 If the marginal cost of producing the complementary good is positive, the optimal price









strategy that will be presented in the following section can resolve this problem as well as

the commitment problem.

2.5 Bundling: An Alternative Pricing Strategy without Commitment

An alternative strategy for the monopolist when it cannot commit to the second

period price or implement a negative price is bundling. That is, it sells both the primary

good and its own complementary good for a single price. Note first that it is not optimal

for the monopolist to sell the bundled good to all consumers since the bundled price must

be equal to vo in that case. So the monopolist will sell the bundled good to group S only if

it chooses the bundling strategy.

It is assumed that tying is reversible, i.e., a consumer who buys a bundled good

may also buy another complementary good and consume it with the primary good.17

Further, suppose consumers use only one complementary good, so the monopolist's

bundled complementary good is valueless to the consumers who use firm 2's

complementary good.18

At the second stage, a consumer who has bought the bundled good earlier may buy

firm 2's good or not, depending on her location x. If she buys firm 2's complementary

good, her net gain at stage 2 is v2 t(1 x) -p2. If she does not buy, she can use the

monopolist's complementary good included in the bundle without extra cost, and get net

gain of v tx. So the critical consumer who is indifferent between buying firm 2's

complementary good and using the bundled complementary good is


can be positive even if v < v2.
17 In the software industry, a consumer who uses Windows OS bundled with Internet
Explorer may install another Internet browser.
18 As long as there is no compatibility problem, consumers will use only one
complementary good they prefer.









1 v, v, + p
x- +-- (2-7)
2 2t



Since the price paid for the bundled good is a sunk cost at the second stage, the

critical consumer is determined by p2 only. Firm 2 will choosep2 to maximize p2( -x*),

which yields the following optimal price for firm 2:



BS V 2- V, + t
P2 2



Plugging this into (2-7) gives the location of the critical consumer as follows:



xs 3 v, v2
x = -+
4 4t



For this critical consumer to exist between 0 and 1, it is required that -3tvi- v2
So the assumption of I vi 2 | < t given in (2-2) is also a sufficient condition for the

existence of a bundling equilibrium without the exit of the rival firm. If vi -v2 > t, then all

consumers buy the bundled good only so the rival firm will exit the market. If vi v2 <

-3t, on the other hand, all consumers buy both the bundled good and firm 2's

complementary good.

At stage 1, the monopolist will set the bundled good price, pb, that makes the

critical consumer indifferent between buying and not buying: 19



19 At the second stage, the monopolist may have an incentive to unbundle the product and
sell the primary good to the consumers outside of group S as long as the consumer's
second stage valuation for the good is positive, i.e., higher than the marginal cost.
Knowing this, some consumers in group S may want to wait until the second period,









S 3v, +v2 -3t
PBS = 2
Pb Vo+-



When consumers choose the monopolist's complementary good, the total price for

the primary good and the complementary good decreases compared to the IS case since



BPS _s r V j v -3t
PBS Sp 2 +P ) 1 < 0 (2-8)
12



If consumers buy firm 2's complementary good as well as the monopolist's

bundled good, the total price increases compared to the IS case since


BS + p) __ (pS + pI) =V -- 2 +- 3t
(p +BS + )= 1+ > 0 (2-9)
12



Comparing (2-8) and (2-9) one can find that the total price decrease for the

consumers of monopolist's complementary good is exactly the same as the total price

increase for the consumers of firm 2's good. With the decrease of the total price, the

number of consumers who choose to use the monopolist's complementary good increases

compared to the IS case as the following shows:



xBS _- 3t >0 (2-10)
12t



The profits of the firms are

which will lower the monopolist's profits. To avoid this, the monopolist will try to
commit to not unbundling. One way to commit is to make unbundling technologically
difficult or impossible, as Microsoft combined Internet Explorer with Windows OS.










S aBS 3v1 +v2 -3t]


BS BS (_BS) a(v-2_ + t)2
2 = ap2 (8-XBS)=t
8t



The following proposition shows that bundling increases monopolist's profits

compared to the IS case.



Proposition 2-1 Suppose the monopolist sells its goods to consumers in S only. Then the

monopolist's profit in the bundling equilibrium is strictly higher than under IS, but not

higher than under CS.



Proof. The difference between profits with bundling and IS case is



BS IS 3
S- (VI 2+ 3t)(v v2 t) >
18t 2



The inequality holds since | vl v2 < t.

On the other hand, the difference between profits with bundling and CS is



-BS C=s a(v 2 <0
S16t
16t


where the inequality holds steadily when vi # v2.


Q.E.D.









In most of the previous analysis of bundling based on the leverage theory, one of

the main purposes of the bundling strategy is to foreclose the complementary good

market. By lowering expected profits of the would-be entrants, bundling can be used to

deter entry. The difference between the previous models and the current one is that

bundling increases the profits of the monopolist even though the rival firm does not exit

the market. On the contrary, the existence of the rival firm helps the monopolist in some

sense since it creates demand for the monopolist's bundled good.

When compared to the CS case, bundling strategy generates the same profits for the

monopolist if vl = v2. Technically, bundling strategy is equivalent to settingpl =pb andp2

= 0. When vl = v2, the equilibrium commitment price for the monopolist's

complementary good, pS is zero, hence the monopolist's profits of bundling and CS

cases are equal.20 Since the optimal commitment price is either positive or negative if vl

/ v2, the monopolist's bundling profits is less than the CS case.

By comparing bundling case with IA case, one can find the critical level of a above

which the monopolist finds bundling is more profitable if commitment is not possible.

The difference between the monopolist's profits is



BS IA V (v, V + 5v, +72 -15t1
I 18t 12





20 If the marginal cost (MC) of producing the complementary good is positive (c), the
optimal commitment price for the good is c when vl = v2 since the monopolist can avoid
double marginalization problem by MC pricing for the downstream good. In this case,
bundling cannot generate same profits as the CS case even when vl = v2 since it
implicitly charges zero price instead of the one equal to MC.










And the critical level of a at which the monopolist is indifferent between bundling

and independent sale is



^BS 36v t
36vot + 3t(5v, +7v2 -15t)- 2(v, v2)2



^BS
If a is higher than a the monopolist can make higher profit by bundling both

goods together and selling it to group S only than by separately selling the primary good

to all consumers. That is, bundling is profitable if the complementary good is widely used

by the consumers of the primary good. In the software industry, Microsoft bundles

Internet Explorer into Windows OS, while it sells MS Office as an independent product

since Internet browser is a widely used product whereas the Office products are used by

relatively small group of consumers.

^BS
Note that a lies between 0 and 1 since



3t(5v, + 7v2 -15t)- 2(v v2 )2 > 3t(5v, + 7v2 -15t)- 2t2

= 15t(v, + v2 3t) + 2t(3v2 t) > 0



where the first and second inequalities hold because of the assumptions given in (2-2) and

SIS ^-BS
(2-1), respectively. The difference between a and a is



^s -BS 2vo (v v + 3t)[3t 2(v, v2)
[3t(12v + 5v, +7v2 -15t)- 2(v v2)2 ](2v + v, + 3t)


The inequality holds because of the assumptions (2-1) and (2-2). Since the









monopolist can make much higher profits by bundling than IS case, it is willing to sell its

goods to a smaller group of consumers than IS case if bundling is possible.

^CS ^BS
The difference between a and a is



^cs -Bs 4vo [4(v, v,) + 3t] [3t 2(v, v,)]




^CS ^BS
Using assumptions (2-1) and (2-2), one can find that a is higher than a except

3
when -t < v2- v1 < t. Even though sC' is not smaller than sTBS, the profit gain from
4

3
selling group S only is higher in bundling case than CS case except -t < v2- v < t as the
4

following shows:



(ZBS ZA CS ZCA) a[4(v v2) + 3t][3t 2(v -v2)]
144t



This explains why the monopolist is willing to sell the goods to a smaller group of

consumers than the commitment case.

2.6 Bundling and Social Welfare

Most previous analyses on bundling have ambiguous conclusions about the welfare

effect of bundling. It has been said that bundling could increase or decrease welfare. In

the model presented here, bundling decreases Marshallian social welfare except for an

extreme case.

Marshallian social welfare consists of the monopolist profits, firm 2's profits, and

consumers' surplus. When the monopolist bundles, its profits always increase compared









to the IS case. Firm 2's profits, on the other hand, decreases in bundling equilibrium since



BS Is a(v, -v2+ 3t)[5(v v2)-9t]
z <0
2 2 72t



Consumers' surpluses with bundling and IS are


BS p
CSs =a (v, +, tx-pb)dxBS +a( +v bt(l x) BS pBS)dx

a[(v- v2 2 + 2t(v1 v2)+5t2]
16t
IsS IS
CSs =aJ (vo + v tx -S ps)d x ++a j(v +v (1 x) S -

a[(v v2)2 + 9t2]
36t



The shaded area of Figure 2-1 shows consumers' surplus of each case when v1 < v2.

The difference between consumers' surplus with bundling and IS is



Sv + V2


ip Po + Pi


(a) Bundling (b) Independent sale (IS)


Figure 2-1. Consumers' surplus in bundling and IS cases when vi < V2










CSBS CSS = a(v, -v2 + 3t)[5(v v2) + 3t]
144t



which shows that consumers' surplus increases by the monopolist's decision to bundle

unless v2- v > (3/5)t. That is, unless firm 2's product is much superior to the

monopolist's complementary good, consumers' surplus increases as the monopolist

bundles. The consumers' surplus increases mainly because consumers who pay less in

bundling case than in IS case outnumber consumers who pay more in bundling

equilibrium. Unlike consumers' surplus, however, social welfare is more likely to

decrease with bundling strategy by the monopolist, as the following proposition shows.



Proposition 2-2 Suppose the monopolist sells its goods to consumers in S only. Then

Marshallian social welfare decreases with the monopolist's decision to bundle unless

3
-t 7



Proof. Marshallian social welfare is defined as the sum of consumers' surplus and

profits of all firms. So social welfare with bundling is



Bs = CSBS +BS BS =a vo 3(vl V2)2 10v +6v2 5t
;12 6t 16


And social welfare with IS is










W = CS + + = v o +
S36t 4



The difference between them is



WBS WIs 1= a (v -v2 +3t)[7(vl -v2)-3t]
144t



which is negative if -3t < v v2 < (3/7)t, and positive otherwise. Since I vl v2 < t, the

social welfare decreases except (3/7)t < v v2 < t. Q.E.D.



The above proposition shows that unless the monopolist's complementary good is

superior enough, the monopolist's bundling strategy lowers the social welfare. Especially,

the social welfare always decreases when the monopolist bundles an inferior good or a

good with the same quality as the rival's, i.e., v, < v2.

2.7 Bundling and R&D Incentives

One of the concerns about the bundling strategy by the monopolist of a primary

good is that it may reduce R&D incentives in the complementary good industry. This

section is devoted to the analysis of the effect of bundling on R&D incentives.

To analyze this, one needs to introduce an earlier stage at which two firms make

decisions on the level of R&D investments to develop complementary goods. The whole

game consists of three stages now. Let R(v) be the minimum required investment level to

develop a complementary good of value v. A simple form of the investment function is


R(v) = ev2, e > 0









Using this, the firms' profit functions can be rewritten as follows:21



is V (v V )2 5v, + v 6 2
I1 =a vo+ 18/ + 6 ev,
18t 6

is a(v2 -1 + 3t)2 2
22 = -ev2
18t

aBS v 3v, + V -3t 2
"1 =a 0 -ev,
4

-B a(v2 +t)2 2
"2 -- ev2
8t



The following proposition shows that the monopolist's bundling strategy reduces

not only the R&D incentive of the rival firm, but also its own incentive.



3
Proposition 2-3 Suppose the investment cost satisfies e > -. Then the equilibrium
8t

values of v, (i = 1, 2) are higher in the IS equilibrium than in the bundling equilibrium,

i.e., v2 > ys, and ys > BS Further, firm 2's incentive decreases more than the

monopolist's by bundling.



Proof. The first order conditions for profit maximization problems yield each firm's best

response functions from which one can obtain the following equilibrium levels of v,'s for

each equilibrium:




21 In previous sections, it is assumed that the complementary goods already have been
developed before the start of the game. The exclusion of the investment costs in profit
function does not affect equilibrium since they are sunk costs.










s a(90et 7a)
1 24e(9et a)

Is a(36et- 7a)
2 24e(9et a)

"BS 3a
S 8e

"BS a(8et- 3a)
8e(8et a)



3
Since 0 < c < 1, the assumption e > guarantees non-negative equilibrium values.
8t

Now the following comparisons prove the main argument:



s _BS a(9et + 2a)
V > 0
24e(9et a)

_is _Bs a a2(1et-a) >
v2 -v2 =-+ >0
24e 8e(9et- a)(8et-a)

2
(- BS _=S > 0 Q.E.D.
2 2 4(9et a)(8et a)




Firm 2 has a lower incentive to invest in R&D because part of the rents from the

investment will be transferred to the monopolist by bundling. The monopolist also has a

lower incentive to invest because the bundling strategy reduces competitive pressure in

the complementary good market.

2.8 Conclusion

It has been shown that the monopolist of a primary good has an incentive to bundle

its own complementary good with the primary good if it cannot commit to the optimal set

of prices when consumers buy the primary good and the complementary good









sequentially. Since the monopolist can increase its profits and the market share of its own

complementary good by bundling, the model provides another case in which the Chicago

School's single monopoly price theorem does not hold. While bundling lowers the rival

firm's profits and Marshallian social welfare in general, it increases consumers' surplus

except when the monopolist's complementary good is sufficiently inferior to the rival's

good. Bundling also has a negative effect on R&D incentives of both firms.

Since bundling may increase consumers' surplus while it lowers social welfare, the

implication for the antitrust policy is ambiguous. If antitrust authorities care more about

consumers' surplus than rival firm's profits, this kind of bundling may be allowed. Even

if total consumers' surplus increases, however, consumers who prefer the rival's

complementary good can be worse off since they have to pay higher price for both the

bundled good and the alternative complementary good. So bundling transfers surplus

from one group to another group of consumers.

In addition to the problem of a redistribution of consumers' surplus, bundling also

has a negative long-term effect on welfare since it reduces both firms' R&D incentives.

This long-term effect of bundling on R&D investment may be more important than

immediate effects on competitor's profit or consumers' surplus, especially for so-called

high-tech industries that are characterized by high levels of R&D investments. For

example, if a software company anticipates that development of a software program will

induce the monopolist of the operating system to develop a competing product and

bundle it with the OS, then the firm may have less incentive to invest or give up

developing the software. This could be a new version of market foreclosure.









A related issue is that if the risk of R&D investments includes the possibility of the

monopolist's developing and bundling of an alternative product, it can be said that

bundling increases social costs of R&D investments. Furthermore, since the monopolist

is more likely bundle a complementary good that has a broad customer base, bundling

may induce R&D investments to be biased to the complementary goods that are for

special group of consumers. A possible extension of the model lies in this direction.

Another extension could be to introduce competition in the primary good market,

which is suitable for the Kodak case.22 It has been pointed out that when the primary

good market is competitive, the anticompetitive effect of bundling is limited. In the

model presented here, firm 1 (the monopolist) could not set the bundling price so high if

it faced competition in the primary good market. However, if the primary goods are also

differentiated so that the producers of them have some (limited) monopoly powers,

bundling may have anticompetitive effects. The result can be more complicated-but

more realistic-if it is combined with the possibility of upgrade which is common in the

software industry.

















22 See Klein (1993), Shapiro (1995), Borenstein, MacKie-Mason, and Netz (1995), and
Blair and Herndon (1996).














CHAPTER 3
COMPETITION AND WELFARE IN THE TWO-SIDED MARKET:
THE CASE OF CREDIT CARD INDUSTRY

3.1 Introduction

It is well known that a two-sided market-or more generally a multi-sided

market-works differently from a conventional one-sided market. In order to get both

sides on board and to balance the demands of both sides, a platform with two sides may

have to subsidize one side (i.e., set the price of one side lower than the marginal cost of

serving the side). In the credit card industry, cardholders usually pay no service fee or

even a negative fee in various forms of rebate. In terms of the traditional one-sided

market logic, this can be seen as a practice of predatory pricing. Several models of two-

sided markets, however, show that the pricing rule of the two-sided market is different

from the rule of the one-sided market, and a price below marginal cost may not be anti-

competitive. 1

Another feature of the two-sided market is that competition may not necessarily

lower the price charged to the customers. In the credit card industry, competition between

nonproprietary card schemes may raise the interchange fee, which in turn forces the

acquirers to raise the merchant fee. The interchange fee is a fee that is paid by the

acquirer to the issuer for each transaction made by the credit card. If the interchange fee

decreases as a result of competition, the cardholder fee is forced to increase. For the



1 Published papers include Baxter (1983), Rochet and Tirole (2002), Schmalensee (2002),
and Wright (2003a, 2003b, 2004a).









proprietary card schemes that set the cardholder fees and the merchant fees directly,

competition may lower one of the fees but not both fees.

The distinctive relationship between competition and prices raises a question about

the welfare effect of competition in the two-sided market. Even if competition lowers the

overall level of prices, it does not necessarily lead to a more efficient price structure.

Previous models about competition in the two-sided markets focus mainly on the effect

of competition on the price structure and derive ambiguous results on the welfare effects

of competition. I present a model of the credit card industry in order to show the effects

of competition on social welfare as well as on the price structure and level. The main

result is that while the effects of competition on the price structure are different

depending on the assumptions about whether consumers single-home or multi-home2 and

whether card schemes are identical (Bertrand competition) or differentiated (Hotelling

competition), the effects of competition on social welfare do not vary regardless of

different model settings. That is, competition does not improve the social welfare in the

various models presented here.

The main reason for this result is that competition forces the platforms to set the

prices) in favor of one side that is a bottleneck part, while a monopoly platform can fully

internalize the indirect network externalities that arise in the two-sided market.3 In order

to maximize the transaction volume (for nonproprietary schemes) or profits (for

proprietary schemes), the monopolist first needs to make the total size of the network


2 If a cardholder (or merchant) chooses to use (or accept) only one card, she is said to
single-home. If she uses multiple cards, she is said to multi-home.
3 In a two-sided market, the benefit of one side depends on the size of the other side. This
indirect network externality cannot be internalized by the end-users of the two-sided
market. See Rochet and Tirole (2005).









externalities as large as possible. Competing card schemes, on the contrary, set biased

prices since they share the market and try to attract single-homing consumers or

merchants.

Since the first formal model by Baxter (1983), various models of two-sided markets

have been developed. Many of them focus on the price structure of a monopolistic two-

sided market.4 It is in recent years that considerable attention has been paid to

competition in two-sided markets. Rochet and Tirole (2003) study competition between

differentiated platforms and show that if both buyer (consumer) and seller (merchant)

demands are linear, then the price structures of a monopoly platform, competing

proprietary platforms and competing (non-proprietary) associations are the same and

Ramsey optimal. They measure the price structure and Ramsey optimality in terms of the

price-elasticity ratio, so price levels and relative prices are not the same for different

competitive environments. While they assume that consumers always hold both cards, the

model presented here distinguishes cases with single-homing consumers and multi-

homing consumers and uses Marshallian welfare measure which includes platforms'

profits as well as consumers' and merchants' surpluses.

Guthrie and Wright (2005) present a model of competition between identical card

schemes. They introduce the business stealing effect by allowing competing merchants

and show that competition may or may not improve social welfare. I extend their model

to the case of the competition between differentiated card schemes as well as the cases of

proprietary card schemes, while removing the business stealing effect for simpler results.



4 The interchange fee is the main topic in these analyses of the credit card industry. See
Rochet and Tirole (2002), Schmalensee (2002) and Wright (2003a, 2003b, 2004a) for
the analyses of the credit card industry with monopoly card scheme.









Chakravorti and Roson (2004) also provide a model of competing card schemes

and show that competition is always welfare enhancing for both consumers and

merchants since the cardholder fee and the merchant fee in duopoly are always lower

than in monopoly. To derive the results, they assume that consumers pay an annual fee

while merchants pay a per-transaction fee and cardholder benefits are platform specific

and independent of each other. In contrast to their model, this paper assumes both

consumers and merchants pay per-transaction fees5 and cardholder benefits are either

identical or differentiated according to the Hotelling model, and concludes that

competition does not improve Marshallian social welfare. Further, it shows competition

may not always lower both the cardholder and merchant fees even for the proprietary

scheme as well as non-proprietary scheme.

The rest of Chapter 3 proceeds as follows. Section 3.2 sets up the basic model of

the non-proprietary card scheme. Section 3.3 and 3.4 show the effects of competition on

the price structure and welfare for the cases of single-homing consumers and multi-

homing consumers. Section 3.5 extends the model to the case of the proprietary card

scheme and compares the results with those of the non-proprietary card scheme. The last

section concludes with a discussion of some extensions and policy implications.

3.2 The Model: Nonproprietary Card Scheme

Suppose there are two payment card schemes, i = 1, 2, both of which are not-for-

profit organizations of many member banks. A cardholder or consumer receives a per-

transaction benefit bBk from using card i, which is assumed to be uniformly distributed

between (bB, bB). A merchant receives a per-transaction benefit, bs, which is also

5 The per-transaction fee paid by consumers can be negative in the various forms of
rebates.









uniformly distributed between (bs, bs). It is assumed that merchants find no difference

between two card schemes.

There are two types of member banks. Issuers provide service to consumers, while

acquirers provide service to merchants. Following Guthrie and Wright (2006), both the

issuer market and the acquirer market are assumed to be perfectly competitive. Card

schemes set the interchange fees in order to maximize total transaction volumes.6

For modeling convenience, it is assumed that there is no fixed cost or fixed fee. Let

c, and cA be per-transaction costs of a issuer and a acquirer, respectively. Then card

scheme i's per-transaction cardholder fee and merchant fee are, respectively,



Jf = c1 a

m, = CA +c



where a, is scheme i's interchange fee. Note that the sum of the cardholder fee and the

merchant fee is independent of the interchange fee since



f + m, =c +cA



In order to rule out the possibility that no merchant accepts the card and all

merchants accept the card, it is assumed that




6 Rochet and Tirole (2003) assume constant profit margins for the issuers and the
acquirers. Under this assumption, maximizing member banks' profits is same as
maximizing total transaction volume, and the sum of the cardholder fee and the
merchant fee is also independent of the interchange fee.









bB + b


Both the numbers of consumers and merchants are normalized to one. Consumers

have a unit demand for each good sold by a monopolistic merchant.7 Merchants charge

the same price to cash-paying consumers and card-paying consumers, i.e., the no-

surcharge-rule applies.

The timing of the game proceeds as follows: i) at stage 1, the card schemes set the

interchange fees, and the issuers and acquirers set the cardholder fees and merchant fees,

respectively; ii) at stage 2, consumers choose which card to hold and use, and merchants

choose which card to accept.

3.3 Competition between Identical Card Schemes: Bertrand Competition

In this section, two card schemes are assumed to be identical, i.e., bB, = bB2 (- bB).

Consumers can hold one or both cards depending on the assumption of single-homing or

multi-homing, while merchants are assumed to freely choose whether to accept one card,

both cards, or none.

One of the key features of the two-sided market is that there exist indirect network

externalities. As the number of members or activities increase on one side, the benefits to

the members of the other side also increase. In the credit card industry, cardholders'

benefits increase as the number of merchants that accept the card increases, while the

merchants' benefits increase as the number of cardholders who use the card increases.

Some of the previous analyses of the credit card industry did not fully incorporate

this network effect in their models by assuming homogeneous merchants, in which case


7 Since merchants do not compete with each other, the business stealing effect does not
exist in this model.









either all merchants or none accept the card.8 So at any equilibrium where transactions

occur, all merchants accept card and consumers do not need to worry about the size of the

other side of the network. The model presented here takes into account this indirect

network effect by assuming merchants are heterogeneous and the net utility of a

consumer with bB takes the following form:



UB, = (bB- f)Qs, = (b c + a )s, i= 1,2



where Qsi is the number of merchants that accept card i.

For modeling convenience, it is assumed throughout this section that the issuer

market is not fully covered at equilibrium, which requires



bB b > 2(b + bs -c)



3.3.1 Single-Homing Consumers

If consumers are restricted to hold only one card, they will choose to hold card i if

UB, > UB, and UB, > 0. Note that the cardholding decision depends on the size of the other

side as well as the price charged to the consumers. Even iff >f, a consumer may choose

card i as long as the number of merchants that accept card i (Qsi) is large enough

compared to the number of merchants accepting card (Qsj).

Merchants will accept card i as long as bs > m, since accepting both cards is always

a dominant strategy for an individual merchant when consumers single-home. So the


8 See Rochet and Tirole (2002) and Guthrie and Wright (2006).









number of merchants that accept card i (quasi-demand function for acquiring service) is9



bs -m, bs cA a-2)
Qs, = (3-2)
bs- b bs b



Using (3-2), the consumer's net utility can be rewritten as



(b, -c, + -a c )(bs-c-a)
bs -bS



Let bB* be the benefit of the critical consumer who is indifferent between card 1 and

2. One can obtain bB* by solving UB1 = UB2, which is



b = bs +c -cA -a1 -a



A consumer with low bB is more sensitive to the transaction fee, so she prefers the

card with lower cardholder fee (i.e., higher interchange fee). On the other hand, a

consumer with high bB gets a larger surplus for each card transaction, so she prefers the

card that is accepted by more merchants. Therefore, a consumer whose bB is higher than

bB* will choose a card with lower a,, and a consumer whose bB is lower than bB* will

choose a card with higher a,. If a, = a,, then consumers are indifferent between two cards,

so they are assumed to randomize between card 1 and 2. This can be summarized by the

following quasi-demand function of consumers:


9 Schmalensee (2002) calls Qs, and QB, partial demands, and Rochet and Tirole (2003)
call them quasi-demands since the actual demand is determined by the decisions of both
sides in a two-sided market.










b _bs -c, -a,
bB ifa,>a
be -bB be -b B

be -b* be -bs -c, +c, +a, +a
QB, = b if a < aJ (3-3)
be -b, be -b
bB c, +a .
2bB -C + if a =a
2(bs-bB)



At stage 1, the card schemes choose the interchange fees to maximize the

transaction volume which is the product of QB, and Qsi. The following proposition shows

the equilibrium interchange fee of the single-homing case of Bertrand competition.



Proposition 3-1 If two identical card schemes compete with each other and consumers

single-home,

(i) the equilibrium interchange fee is



bs 3 [2(bs c,)- (b- c,)]




(ii) ab maximizes total consumers' surplus



Proof. (i) Without loss of generality, suppose al > a2. Then scheme 2 will maximize the

following objective function:



((bs -a2 -c )(bB -bs -c, + c + a + a)
T (a; al) = Q(b Qs) =b b
(bB- b B)(s -bs)









from which scheme 2's best response function can be obtained as follows:



I1- -
R2(a) = (2bs -bB -a, +c1 -2c



Scheme l's objective function is


T,(a,; a,) =Q, Ql


(bs a, cA)(bs a c)
(b- bb,)(bs -b,)


Since the function is a linear function of al with negative coefficient, scheme 1 will

set al as low as possible, i.e., as close to a2 as possible. So the best response function of

scheme 1 is



Rl(a2)= a2



Solving Ri(a2) and R2(al) together, one can obtain the following Nash equilibrium:



a=a = [2(bs c)-(b c)]- ab"
3



The equilibrium transaction volume of scheme i when a, = a2 = as is



S(ab;a bs) (bB +bs c) 2 bT
9(bB -b,)(bs -bs)









Since scheme l's best response function seems to contradict the premise that

a1 > a2, it is necessary to show that card schemes do not have an incentive to deviate

from the equilibrium. To see this, suppose scheme 1 changes al by Aa. Then the

transaction volume of scheme 1 becomes



(bB + bs- c)(bB + bs -c 3Aa)i
if Aa > 0
9(b bB)(bs bs)
7(abs + Aa; a ") =
(bB + bs -c + 3Aa)(bB +bs -c- 3Aa)i
if Aa < 0
9(bB -b,)(bs -bs)



Both of them are less than Tb, so there is no incentive for scheme 1 to deviate

from as.

(ii) At symmetric equilibrium with common a, the consumers' demands for the

card services are given by (3-3). So the total consumers' surplus is



bB (b f (a))2 (bs -m(a))
TU (f (a)) = Qs, QB, df =
l 2f 2(bB -b,)(bs -bs)
_(bB c + a)2(bs -c a)
2(b bb)(bs -b,)



The optimal a that maximizes TUB is



1 -
a = [2(bs- c,)- (bB c)

which is same as a. .E.D.

which is same as abs. Q.E.D.









When consumers single-home, each card scheme has monopoly power over the

merchants that want to sell their products to the consumers. This makes the card schemes

try to attract as many consumers as possible by setting the interchange fee favorable to

consumers. The resulting interchange fee chosen by the card schemes is one that

maximizes total consumers' surplus.

An interchange fee higher than ab" may attract more consumers due to the lower

cardholder fee, but fewer merchants will accept the card due to the higher merchant fee.

Therefore, a card scheme can increase the transaction volume by lowering its interchange

fee, which attracts higher types of consumers who care more about the number of

merchants that accept the card. On the other hand, an interchange fee lower than ab may

attract more merchants, but fewer consumers will use the card. In this case, a card scheme

can increase the transaction volume by raising its interchange fee.

In order to see how competition in the two-sided market affects the price structure,

it is necessary to analyze the case in which the two card schemes are jointly owned by

one entity. As the following proposition shows, it turns out that joint ownership or

monopoly generates a lower interchange fee, which implies a higher cardholder fee and a

lower merchant fee. In other words, competition between card schemes when consumers

single-home raises the interchange fee.



Proposition 3-2 If two identical card schemes are jointly owned and consumers single-

home,


(i) the symmetric equilibrium interchange fee is









ab 2 [(bs- C)-(b -c,)]


(ii) the joint entity may engage in price discrimination in which one scheme sets

the interchange fee equal to abi and the other scheme sets the interchange fee at any level

above abJ, but the total transaction volume cannot increase by the price discrimination,

(iii) abJ maximizes the social welfare, which is defined as the sum of the total

consumers' surplus and the total merchants' surplus.



Proof. (i) Since the card schemes are identical, there is no difference between operating

only one scheme and operating both schemes with same interchange fees. So suppose the

joint entity operates only one scheme. Then the quasi-demand functions are



be -f be -c, +a
QB =-
bB bB bB b

bs -m bs -cA -a
bs b bs -bs



The joint entity will choose the optimal a in order to maximize the transaction

volume QBQs. The optimal interchange fee obtained from the first-order condition is



a* c)( c)] abj
--
a=2 [(bs c) -(b cr) ab


which is less than abs since










a-a =a bs +bB -c)>O



(ii) Without loss of generality, suppose al > a2. Then scheme 1 will attract low-type

consumers and scheme 2 will attract high-type consumers. The quasi-demand functions

are determined by (3-2) and (3-3). And the total transaction volume is



(bs c, + a2)(bs c, a,)
QBS +QB2QS2 = (3 -4)
(b bB)(bs -bs)



Note that (3-4) is independent of al, which implies al can be set at any level above

a2. The optimal a2 can be obtained from the first-order condition for maximizing (3-4):



1-j
a = [(bs c,) -(bB -c)] -a



It is not difficult to check that the total transaction volume at equilibrium is also the

same as in the symmetric equilibrium.

(iii) The sum of the total consumers' surplus and the total merchants' surplus is



TUs = TU + TU* = Qs bdf +Q dm

(bB -c, +a)(bs c -a)(bB +bs -c)
2(bB -b)(bs -b,)


The optimal a that maximizes TUb, is










ab= [(bs ) (b c,)]




which is same as ab~. So abi maximizes social welfare. Q.E.D.



The most interesting result of the proposition is that the joint entity, which acts like

a monopolist, chooses the socially optimal interchange fee. This is possible because both

the issuing and acquiring sides are competitive even though the platform is monopolized,

and the joint entity can internalize the indirect network externalities of both sides.

Comparing propositions 3-1 and 3-2, one can find that competition between card schemes

lowers social welfare as well as decreases total transaction volume. In a typical example

of prisoner's dilemma in game theory, competing firms choose higher quantity and/or

lower price, which is detrimental to themselves but beneficial to the society. But this

example of the two-sided market shows that competitive outcome can be detrimental to

the society as well as to themselves.

3.3.2 Multi-Homing Consumers

In this subsection, consumers are allowed to multi-home. Since there is no fixed fee

or cost, individual consumer is always better off by holding both cards as long as bB, >f.

So the number of consumers who hold card i is



be B bB -c, +a (35)
O=_ (3-5)
bB b bB b



On the other hand, since merchants have monopoly power over the products they

sell, they may strategically refuse to accept card i even if bs > m,.








If a merchant accepts card i only, it receives a surplus equal to


(b -CA- a)(bB -c +a,)
be -bB


(3-6)


If the merchant accept both cards, the surplus is


Usb = (bs ml)Qbl + (bs m2)Qb2 = (bs cA al)Qbl + (bs CA a2)Qb2


(3-7)


where Qb, is the number of consumers who will use card i if the merchant accepts both

cards. 10 When a consumer holding both cards buys from a merchant that accept both

cards, the consumer will choose to use the card that gives a higher net benefit, i.e., she

will use card i if b, f > bJ f,. And the consumer will randomize between card i andj

if b, / = b, fj .

If the two card schemes are identical (bBl = bB2), consumers will use the card that

has a lower consumer fee if merchant accepts both cards, i.e.,


QB,
Qb, 0
tI (/2)QB,


if a > a (f < f)
if a < a (f > f )
if a = :a (f = f )


A merchant with bs will accept card i only if Us, > Us, and Us, > Usb. It will accept

both cards if Usb > Usi, i = 1, 2. To see the acceptance decision by a merchant, suppose


10 Consumers' card-holding decision and card-using decision can be different since they
can hold both cards but use only one card for each merchant.


(3-8)


Us, = (bs M, )QB,









al > a2 without loss of generality. Then the net surplus to the merchant if it accepts both

cards is



Usb = (bs cA al)QB1 + (bs cA a2)O = Us'



Merchants are indifferent between accepting card 1 only and accepting both cards

since consumers will only use card 1 if merchants accept both cards. In other words, there

is no gain from accepting both cards if consumers multi-home. So merchants' decision

can be simplified to the choice between two cards. Let bs* be the critical merchant that is

indifferent between accepting card 1 only and card 2 only, which can be obtained by

setting Us, = Us2:



bs* =bB C + +al+ a2



Merchants with low bs will be sensitive to the merchant fee and prefer a card with

low merchant fee (low interchange fee), while merchants with high bs will prefer a card

with low consumer fee (high interchange fee) since they care more about the number of

consumers who use the card. Therefore, if ml > m2 (al > a2), merchants with bs smaller

than bs* (and greater than m2) will accept card 2 only and merchant with bs higher than

bs* will accept card 1.

If al = a2, all cardholders have both cards and it is indifferent for merchants

whether they accept card 1, card 2 or both. For modeling simplicity, it is assumed that

merchants will accept both cards if aa = a2. The following summarizes the number of

merchants that accept card i:










bs- bB +c -c, a
bs -bS

bB c +aJ
bs b,

bs -cA -a
bs -b,


if a >a


if a < a


if a, = a


(3-9)


Proposition 3-3 If two identical card schemes compete with each other and consumers

multi-home,

(i) the equilibrium interchange fee is


abm [(bs -c,) 2(b- c)
3


(ii) abm maximizes total merchants' surplus.



Proof. (i) Without loss of generality, suppose al > a2 (ml > m2). Then scheme l's best

response function can be obtained by solving the optimization problem of the scheme,

which is


R,(a) bs
2


2bB +2c, -c,


Scheme 2's objective function is



(bB -c, +a)(bB -c, +a2)
T2 (;2a1) QB2QS2 -
(bB b )(bs -bs)


bs
bs

bs

_bs-
bs









Since the function is linear in a2 with positive coefficient, scheme 2 will set a2 as

high as possible, i.e., as close to al as possible. So the best response function of scheme 2

is


R2(ai) = a


Solving Ri(a2) and R2(al) together, one can obtain the following Nash equilibrium:



a = a2 = bs A)- 2( -c)] ab



The equilibrium transaction volume of scheme i when a, = a2 abm is



T ab) (b+bs c)2 b
T(abm;a 9(bB b
9(b b_)(bs bs)


As in Proposition 3-1, it is necessary to show that the card schemes do not have an

incentive to deviate from abm in order to justify the equilibrium. To see this, suppose

scheme 1 changes al by Aa. Then the transaction volume of scheme 1 becomes



(bB+ bs c 3Aa)(bB +bs-c+ 3Aa)
if Aa > 0
9(bB -b)(bs -b,)
T,(abm +Aa;abm) -B -s
(bB +bs c)(bB +bs -c+ 3Aa)
if(b Aab < 0)(bs
9(bB -bB)(bs bs)










Both of them are less than Tb, so there is no incentive for the scheme to deviate

from abm.

(ii) At symmetric equilibrium with common a, the total merchants' surplus is


2 bs
-^ -
TUb, (M(a)) Q Qm

(bs -c, )2(bB
2(bB b,)(bs


(bs m(a)) (bs f(a))
2(bB -bB)(bs -b,)
-ct +a)
-bs)


The optimal interchange fee that maximizes TU)" is


a 3 = [(bs -c,)-2(bB -c,)
3=


which is equal to ab". So ab" maximizes total merchants' surplus.

Q.E.D.



When consumers multi-home, the card schemes care more about merchants since

they can strategically refuse to accept one card. By setting the interchange fee so as to

maximize the merchants' surplus, the card schemes can attract as many merchants as

possible. As in the single-homing case, an interchange fee higher or lower than ab" is

suboptimal and a card scheme can increase its transaction volume by changing the

interchange fee closer to ab".

The interchange fee in the multi-homing case is lower than in the single-homing

case since the fee is set in favor of the merchants. The following proposition shows that









the interchange fee is higher if the card schemes are jointly owned, which implies the

interchange fee decreases as a result of competition between card schemes when

consumers multi-home. It also shows that competition lowers social welfare as in the

single-homing case.



Proposition 3-4 If two identical card schemes are jointly owned and consumers multi-

home,

(i) the symmetric equilibrium interchange fee is



a =2 [(bs -A) (bB -c)]>abm



(ii) the joint entity may engage in price discrimination in which one scheme sets

the interchange fee equal to abj and the other scheme sets the interchange fee at any level

below ab but the total transaction volume cannot increase by the price discrimination,

(iii) abJ maximizes social welfare.



Proof. (i) Regardless whether consumers single-home or multi-home, there is no

difference for the joint entity between operating two card schemes with same interchange

fee and operating only one scheme since the card schemes are identical. So the proof is

the same as the first part of Proposition 3-2. And for multi-homing consumers, the

monopolistic interchange fee is higher than the competitive interchange fee since



'(b- -B









(ii) Without loss of generality, suppose al > a2. Then scheme 1 will attract low-

type merchants and scheme 2 will attract high-type ones. Then the total transaction

volume is


QBlQSl + QB2QS2


(bB -c1 +a)(bs -c -a,)
(b- -b,)(bs -b)


Note that (3-10) is independent of a2, which implies a2 can be set at any level

below al. The optimal al obtained from the first-order condition is


1 [( c (b c,)
az= (s cA -b -z


which is equal to ab.

The total transaction volume at equilibrium is


(b +bs -c)2
4(bB -b)(bs -bs)


which is the


same as in the symmetric equilibrium.


(iii) The proof is the same as in part (iii) of Proposition 3-2.

Q.E.D.



The optimal interchange fee for the joint entity is the same as in the single-homing

case since the card schemes do not compete for consumers or merchants. Unlike the

single-homing case, however, the interchange fee decreases as a result of competition

between the card schemes when consumers multi-home. Social welfare deteriorates since


(3-10)









competing card schemes set the interchange fee too low in order to attract more

merchants.

Figure 3-1 shows the results of this section. As is clear in the figure, competitive

equilibrium interchange fees maximize either consumers' surplus or merchants' surplus.

Since monopoly interchange fee maximizes total surplus, competitive outcome is

suboptimal in terms of social welfare.

3.4 Competition between Differentiated Card Schemes: Hotelling Competition

In this section, card schemes are assumed to be differentiated and compete a la

Hotelling. As in a standard Hotelling model, suppose consumers are uniformly

distributed between 0 and 1, and the card scheme 1 is located at 0 and scheme 2 is at 1. A

consumer located at x receives a net benefit of be tx (- bB,) if she uses card 1, and

b t(1 x) (- bB2) if she uses card 2. In order to comply with the assumption that


consumers' benefits from card usage is uniformly distributed between (bB bB ), the

transportation cost t is assumed to be equal to bB -bB.


TU = TUB + TUs


cI -bB abm ab ab


Figure 3-1. Welfare and interchange fees of Bertrand competition









The net utilities of a consumer located at x when she uses card 1 and 2 are



UB =(bB -tx- f)Q= (bB(1-x)+bBx-cI+a)Qs

UB2 (bB t(1-x)-f2)Q2= (bBx+bB(1-x) -c+a Q



The critical consumer, x*, who is indifferent between card 1 and 2 can be obtained

by solving UBs = UB2:



bB (b -c, + a)Qsl -(b -c +a2)Qs2 (3-11)
(b bB)(QsI Q2)



If the issuer market is not fully covered, each card scheme has a full monopoly

power over the consumers and the resulting equilibrium will be the same as in the

monopoly case of the previous section. In order to obtain competitive outcomes, the

issuer market is assumed to be fully covered at equilibrium. This requires the following

assumption:1



bB +bs > c



Depending on whether consumers single-home or multi-home, and whether card

schemes compete or collude, various equilibria can be derived. There may exist multiple

equilibria including asymmetric ones. For expositional simplicity, however, only

symmetric equilibria will be considered unless otherwise noted.


11 For the issuer market to be fully covered, the net utility of the consumer located at x =
12 must be non-negative for the monopolistic interchange fee abJ.









3.4.1 Single-Homing Consumers

When consumers are restricted to single-home, merchants will accept card i as long

as bs > m, as in the previous section. So Qs, is determined by (3-2). Since the issuer

market is fully covered, the number of consumers who choose card i is



QB =x* and Q = 1- x*


where x* is defined in (3-11).

The following proposition shows the symmetric equilibrium of the Hotelling

competition when consumers single-home.



Proposition 3-5 If two differentiated card schemes compete a la Hotelling and

consumers single-home,

(i) the symmetric equilibrium interchange fee is



c --(bBs+bB) if b -b, > 2(b +bs -c)
a0 = (3-12)
4 [2(bs-cA)-2(bB-c,)-(bB-bB)] if bB -bB <2(bB +bs -c)



(ii) ahs maximizes the weighted sum of total consumers' surplus and total

merchants' surplus, w TU1 + (1 w)TUs where the weight for consumers' surplus is



3(bB-bB,)+ 2(b, +bs-c)
w= B if bB-bB <2(bB +bs-c)
6(bB b)+2(b + bs c)









2(bs- bB)+ 4(b + bs -c)
w2 = if be-b B > 2(bB + bs- c)
3(bB-bB)+ 8(b +bs -c)



Proof. (i) For a given a2, card scheme 1 will set al to maximize its transaction volume.

The symmetric equilibrium interchange fee can be obtained from the first-order condition

in which al and a2 are set to be equal to each other for symmetry:


1-'
a=4 [2(bs-cA)-2(bB -c)- (b -bB)]



For this fee to be an equilibrium, net benefit of the consumer at x = 2 must be

nonnegative since the issuer market is assumed to be fully covered, which requires


t [B
bB 2 f(a)= 2(b,+bs-c)-(bB-bB)]>0



That is, a* is an equilibrium interchange fee if bB b < 2(bB + bs c).

If bB -bB > 2(b + bs c), the equilibrium interchange fee can be obtained by

setting consumer's net benefit at x = 12 equal to zero:


1-
a= c -(bB +bB)



For a** to be an equilibrium, it needs to be shown that the card schemes have no

incentive to deviate from a*. The transaction volume of card scheme 1 at a** is










**, bB+bB+2bs -2c
7T(a ,a )=- 4-
4(bs -bs)



The right and left derivatives of scheme 1's profit at al = a* are, respectively,


lim T(a- + Aa,a")- T7(a",a')
Aa-o+ Aa

lim T(a** +Aa,a**)- T(a**, a**)
Aa-^o Aa


2(bs + bB c) -(b b)
< 0
4(b bb)(bs -b,)

(b + bs -c)
(b B)( < 0)
(bs b,)(bs bs)


So a is an equilibrium when bB bB > 2(b, + b c). Note that a* a when

bB-bB =2(b +bs c).

(ii) First, note that QB = 12 at symmetric equilibrium since the market is fully

covered. The weighted sum of total consumers' surplus and merchants' surplus for

scheme 1 is


wTU +(1 w)TU = w f 2 UBdx+ I B2 1 b s Q, dm
; 22 ,2 i

(bs -cA -a)[w(3bB +b -4c +4a)+ 2(1 w)(bs c a)
4(bs -bs)



The optimal interchange fee that maximizes this weighted surplus is



4(2w 1)(bs cA)- w(3bB+bB- 4c,)
4(3w 1)


(3-13)


(3-14)









The size of the weight can be obtained by setting ah = a*, which is



3(bB-b b)+ 2(b, +bs-c)
w, = if bB-b <2(b, +bs -c)
6(bB b)+ 2(bB +bs c)

2(bB bB) + 4(b, + bs c)
w2 = -- if bB-bB >2(b + bs- c)
3(b -b)+ 8(b +bs- c)



Note that wl = w2 = 4/7 if bB b = 2(b +bs -c). Q.E.D.



When the card schemes compete a la Hotelling, they have some monopoly power

over the consumers. So unlike the Bertrand competition case, they do not need to set the

interchange fee so high as to maximize total consumers' surplus. While the weight for

consumers' surplus (w) in Bertrand competition is equal to 1, the weight in Hotelling

competition ranges between 4/7 and 1. If (bB bB) = 2(bB + bs -c), the weight is equal to

4/7. It becomes close to one as bB bB approaches zero. Note that bB bB is equal to the

transportation cost t. As in a standard Hotelling model, the monopoly power of a card

scheme weakens as t becomes smaller. Therefore, the card scheme will set the

interchange fee so as to maximize total consumers' surplus when the transportation cost

becomes zero.

The following proposition shows the monopoly interchange fee in the Hotelling

model also maximizes the social welfare as in the Bertrand model.



Proposition 3-6 If the two differentiated card schemes are jointly owned and consumers

single-home,









(i) the joint entity will set the interchange fee equal to



1-
a = -2 (b +b,)




(ii) ah maximizes the sum of the total consumers' surplus and the total merchants'

surplus.



Proof. (i) I will prove this proposition in two cases: (a) when the joint entity sets the

same interchange fees for scheme 1 and 2, and (b) when it sets two different fees (price

discrimination).

(a) When the joint entity sets the same interchange fees for both schemes, the joint

transaction volume is



bs cA -a
TM (aa) = QBQs + 0QB2S2 CA
bs -bs



where QB1 = QB2 = 12 since the issuer market is assumed to be fully covered.

Note that TM is decreasing in a, which implies that the optimal a is the minimum

possible level that keeps the issuer market covered. This fee can be obtained by setting

the consumer's net benefit at x = 12 equal to zero, which is ah.

(b) Now suppose the joint entity tries a price discrimination by setting al = aj + Aa

and a2 = a Aa, Aa > 0. The joint transaction volume when it charges same fee, ah, is









S( 2(bs +bB -c) +(bB b)
SM(a ",a 1)= -
2(bs -bs)



while the joint transaction volume of the price discrimination is



T +,2(bs b -c)(b -b) -2Aa
TM(ahj + Aa,ahj Aa) (bB
2(bs -bs)



It is not beneficial to engage in price discrimination since


T, (ah + Aa, ah Aa) T, (a", ah )


Aa
bs -bs


(ii) Since QB1 = QB2 = 12 at full-cover market equilibrium, the sum of total

consumers' surplus and total merchants' surplus is


TUh + TU~h


JU2ldx + UBdx+ 2lQsdm

(bs c a)(3bB + b 4c + 2bs 2cA + 2a)
4(bs -bs)


The optimal a that maximizes social welfare is12



a= c 3bB +bB
4L=7 ^^5^


12 The fee is equivalent to a* in (3-14) when w = /2.









Note that the market is not fully covered at a since a* < ah In other words, a

is not feasible. Therefore, ah maximizes the sum of the total consumers' surplus and the

total merchants' surplus when the market is fully covered. Q.E.D.



Note that ah = ahj if bB b > 2(bB + bs -c) and ah > ahj if

bB bB < 2(bB + bs -c). As in the Bertrand competition case, competition does not lower

the equilibrium interchange fee nor increase social welfare when the card schemes

compete a la Hotelling and consumers single-home.

3.4.2 Multi-Homing Consumers

If consumers are allowed to multi-home, they will hold card i as long as bBZ >f. So

the number of consumers who hold card i is the same as (3-5). If the issuer market is fully

covered and the merchants accept both cards, the critical consumer who is indifferent

between card 1 and 2 is obtained by solving b tx f = bB t(1 x) f2, which is



x f-+ -+J
X I + 1+ a1- a2
2 2t 2 2(b b,)



The number of consumers who use card i if merchants accept both cards is



Qbl = x*, and Qb2= 1 x*



Lemma 3-1 If a, > a,, merchants accept either card only or both cards, i.e., no

merchant will accept card i only.









Proof. Without loss of generality, suppose al > a2. The critical merchant that is

indifferent between accepting card 1 only and accepting card 2 only can be obtained by

setting Usi = Us2, where Us, is defined in (3.6):



bs = bB c + c+al+ a2



Merchants with low bs will be more sensitive to the merchant fee, while merchants

with high bs will care more about the number of consumers who use the card. So if

bs > bs, the merchant prefers card 1 to card 2 and vice versa.

The critical merchant that is indifferent between accepting card i only and

accepting both cards can be obtained by setting Usb = Us,, where



Usb = (bs m,)Qbl + (bs m )Qb2



Let bs, be the critical merchant. That is,



(a, a,)(a, + a + b b,)- 2a,(a + b + c )+ 2c (c bs)
2(c, -a -b,)



If bs > bs, accepting both cards is more profitable than accepting only card i since

merchants with high bs care more about the transaction volume. The difference between


bs and bs is










bs* _b a2 + ( + a2 )(bB + b -2c,)+ 2(bB c)(bB -cI)
2(c -a -bB)



Note that the numerator is independent of i and the denominator is positive.13 Since

al > a2, bs > bS2 > bs if the numerator is positive, and bs < bS2 < bs, if the numerator is

negative. Note also that bs is larger than m, since



bs m, = bB- f >0, i j,



and bs, is smaller than m\ since



(a, -a2,) (bs bB) (a -a2)]
bs- --m--- < 0
2(f -bB)



which implies bs > bS2 > bsl. Note that the difference between two interchange fees,

which is same as the difference between two cardholder fees, cannot exceed the

difference between bB and bB since _B< < / < bB.

As is shown in Figure 3-2, merchants will accept card 2 only if bs e [m2, b2 ), and

accept both cards if bs e [ bS2, bs ].14 Q.E.D.


13 C -a, -bB > 0 since it is equal to f -b and the cardholder fee must be higher than
b,.

14 S (a -0) [(b+ b)- (f +f2)]
Since b- m = b, < m2 (bbs2 > m ) if and only if
S2(f, b, _)
be+ b >f+ f2









accepting card 2 accepting both cards

I I I'I I I I
bs bs m2 m b* b bs

accepting card 2 accepting both cards

I I I I I I s
b, m, bb b2 m1 b


Figure 3-2. Merchants' acceptance decision when al > a2 (ml > m2)



Based on Lemma 3-1, the number of merchants that accept card i is


bs -b
bs b

bs -m
bs -b,


if a, >a, (m, >m)


if a, < a (m,

Let Qa, be the number of merchants that accept card i only, and QSb be the number

of merchants that accept both cards. That is,


Q s, -Qs if a 0 if a, > a (m, > m )

Qsb = Qs, where a, > a (m, > m )


The following proposition summarizes the equilibrium interchange fee of the

Hotelling competition with multi-homing consumers.


Proposition 3-7 If consumers can multi-home and card schemes compete a la Hotelling,









(i) the symmetric equilibrium interchange fee is


ahm { I2b


CA +I -b,

1-(
2 (bs + bB)
2


if bB

if bB


bB < 2(b +bs -c)

b > 2(b +bs -c)


where A = 2(bB -


bB)2 + (b+bs -c)2


(ii) ahm < ah where the equality holds when bB b > 2(bB + bs -c)

(iii) ahm maximizes the weighted sum of total consumers' surplus and total

merchants' surplus, w TUB + (1 -w)TU s, where the weight for the consumers' surplus is


2(b+bs -c)+2A
w, =----
2(b +bs c)- 3(b bs)+6A

2(bB bB) + 4(bB + bs c)
3(b- b )+ 8(bB +bs -c)


if bB -bB <2(bB +bs -c)


if b-bB >2(b+bs -c)


Proof. (i) Without loss of generality, suppose al > a2 (ml > m2). Then scheme 1 and 2's

transaction volumes are, respectively,



TI (a; a2) QblQSb

T2 (a2; a) = B2,2 + b2Sb



The symmetric equilibrium can be obtained by taking derivative of T, w.r.t. a, at

a, = a,, which yields









a bs-c +c -b -2(b -bB +(b+bs c)



At the symmetric equilibrium, all merchants accept both cards (i.e., Qae = 0) and

Qb = Qb2 = /2. So the transaction volume of each card scheme is



S bB + bs-c+ 2(bB-bB)2 +(b+bs )2
T(a ;a *)=
4(bs bs )



To see the card schemes do not have an incentive to deviate from a*, suppose

scheme 1 changes al by Aa. Then the transaction volume of the scheme becomes


T7(a + Aa; a) = Qb
QBO, + Qb2Qsb


if Aa > 0
if Aa <0


The transaction volume does not increase by changing a since


7T(a + Aa; a)- T7(a ;a )

Aa (2(bB b) + Aa)
--- <0
2(bB b)(bs b)(A (bs + b c))

Aa2 (3A 2(bB- b, )-(bs + bB -c)- 3Aa)
2(bB bB)(bs bs)(A (bs + b -c)- 2Aa)


if Aa > 0


<0 ifAa<0


So the card schemes do not have an incentive to deviate from a .

For a* to be an equilibrium, the issuer market must be fully covered at equilibrium.

The net benefit of the consumer located at x = /2 is









bB--t-(c-a)= (b+bs c) -2(bB b)2 ( bs c)2
2 2L



This is nonnegative if and only if b b < 2(bB + bs c) since

(bB +bs-c)2 -[2(bB b,) +(b+bs -c)2]= (b -BB)[2(B +bs -c)-(bB -b)].

If bB b > 2(bB + bs c), as in the single-homing case, the equilibrium

interchange fee can be obtained by setting consumer's net benefit at x = /2 equal to zero,

which is


S 1 1
a c -(bs +bB)



Note that, as in the single-homing case, a* a** when be bB = 2(bB + bs c).

(ii) If bB bB < 2(bB + bs -c), ah = 1[2(bs cA)- 2(bB c) -(bB bB)] and the

difference between the two equilibrium fees is



a a _hm ( 22(bB -bB)2 +(bB +bs -)2 3(bs -bB))


> 22(bB b)2 +[(1/2)( b)]2 -3(bB bB)) = 0



If bB -bB > 2(b +bs -c), both ah" and ahm are equal to c, -12(bB +b,).

(iii) The weighted sum of total consumers' surplus and total merchants' surplus is

the same as (3-13), hence the optimal interchange fee maximizing the weighted surplus is









also the same as (3-14). The level of the weight can be obtained by setting ah' = aW,

which is



2(b_ + bs c)+ 2A
w,= -- if bB -bB <2(b, +bs -c)
2(bB +bs -c)- 3(b bB)+6A

2(b- b, )+ 4(b +bs -c)
w = if be -b > 2(bB +bs -c)
3(b- bB )+ 8(b +bs -c)



Note that, as in the single-homing case, wl = w2 = 4/7 if bB bB = 2(b + bs c).

Q.E.D.



When consumers multi-home, the equilibrium interchange fee is lower than that of

the single-homing case. But unlike the Bertrand competition case in which card schemes

set the interchange fee so as to maximize the merchants' surplus, the card schemes do not

lower the fee enough. In the Bertrand competition with multi-homing consumers,

merchants accept only one card if the merchant fees set by two card schemes are different.

Therefore, a card scheme can maximize its transaction volume by attracting as many

merchants as possible. In Hotelling competition, however, each card scheme has its own

patronizing consumers since it provides differentiated service. This weakens merchant

resistance, which forces many merchants to accept both cards. 15 Therefore, card schemes

do not need to provide maximum surplus to the merchants.

If the card schemes are jointly owned, the result will be the same as in the single-

homing case since the joint entity will split the issuer market so that each consumer holds


15 See Rochet and Tirole (2002) for a discussion of merchant resistance.









only one card at equilibrium.

Figure 3-3 shows the relationship of various equilibrium interchange fees and

welfare, which is drawn for the case of be bB < 2(bB + bs c).16 The left side of ahi is

not feasible since the market cannot be fully covered. As is clear from the figure,

competition not only increases the equilibrium interchange fee but also lowers social

welfare. It also shows that allowing consumers to multi-home increases social welfare in

the Hotelling competition case, although it lowers total consumers' surplus.

3.5 Proprietary System with Single-Homing Consumers

The analysis of the previous sections has been restricted to the competition between

non-proprietary card schemes that set interchange fees and let the cardholder fees and


TU = TUB +TUs


TUB


bs LA


Figure 3-3 Welfare and interchange fees of Hotelling competition when
bB b, <2(bB +bs -c)


16 When bB -bB > 2(b + bs c), ahj = ahm = a h









merchant fees be determined by issuers and acquirers, respectively. Another type of

credit card scheme, a proprietary scheme, serves as both issuer and acquirer. It sets the

cardholder fee and merchant fee directly, so there is no need for an interchange fee. 17

3.5.1 Competition between Identical Card Schemes

One of the features of the proprietary card scheme is that competition may not only

alter the price structure but may also change the price level. In the previous sections, the

sum of the cardholder fee and merchant fee does not change even after the introduction of

competition between card schemes. When a card scheme sets both the cardholder fee

and the merchant fee, it may change one of the fees more than the other since the effects

of competition on two sides are not equivalent.

To see how competition affects the equilibrium fees of the proprietary card scheme,

the equilibrium of the monopoly case will be presented first. For the sake of simplicity,

only the case of single-homing consumers will be considered.

When the monopoly proprietary card scheme setsfand m, the quasi-demand

functions of consumers and merchants are



b f bs -m
QB = and Q = --
be b bs -bs






17 In the United States, Discover and American Express are examples of this type of card
scheme.
18 This feature of the non-proprietary scheme requires an assumption of perfect
competition among issuers and acquirers. If the perfect competition assumption is
removed, competition may alter the price level as well as the price structure in the non-
proprietary card scheme model.









and the profit of the scheme is19



Tr = (f +m-c)QBQ



From the first order condition for the profit maximization problem, one can obtain

the following equilibrium cardholder fee and merchant fee:


1---
f = 3 (2b -bs+c)
(3-15)
mnM =(2bs -b +c)



The following lemma shows that there does not exist a pure strategy equilibrium

when two identical proprietary card schemes compete with each other.



Lemma 3-2 If two identical proprietary card schemes compete in a Bertrand fashion, no

pure strategy equilibrium exists.



Proof. Note first that any set of prices that generates positive profit cannot be a

symmetric equilibrium. If an equilibrium set of prices is (f m) such that+ m > c, a card

scheme can increase profit by lowering the cardholder fee marginally while keeping the

merchant fee since the scheme can attract all consumers instead of sharing them with the

other scheme.

Second, a set of prices which satisfiesf+ m = c cannot be an equilibrium, either. To

see this, let the equilibrium set of prices is (f m) such that+ m = c. Without loss of

19 The proprietary card scheme maximizes profits instead of card transaction volume.









generality, suppose scheme 2 lower the cardholder fee by d and raise the merchant fee by

e, where e > d > 0. As in the Bertrand competition case of the previous section,

consumers whose bB is higher than b* will choose card 1 while consumers with bB lower

than b* will choose card 2, in which b* is defined as



f (bs m)d
b =f -d+
e



The quasi-demands of consumers and merchants for scheme 2's card service are



b2 (f -d) d(bs m)
be -bB e(bB -bB)

bs -(m+e)
9S2 =
bs -bs



The profit of the scheme 2 is



d(bs m)(bs -m -e)(f + m c + e -d)
)T2 = -- -- > 0
e(bB -bs)(bs -bs)



Since the scheme 2 can make positive profits by deviating from (f m), it cannot be

an equilibrium set of prices. Q.E.D.



The above lemma does not exclude the possibility of a mixed strategy equilibrium

or asymmetric equilibrium. As the following proposition shows, however, competition









cannot improve social welfare since the monopolistic equilibrium set of prices maximizes

social welfare.



Proposition 3-8 The equilibrium prices set by the monopolistic proprietary card scheme

in the Bertrand model maximize Marshallian social welfare which is defined as the sum

of cardholders' surplus, merchants' surplus and card schemes' profits.



Proof. Marshallian social welfare is defined as follows:



W =TU, +TUs r = QQsdf + Q,Qsdm + (f +m c)QQs
(3-16)
(bB f)(bs m)(bB + bs + f + m- 2c)
2(bB -b,)(bs -bs)



The optimal prices that maximize welfare are



f =3 (2b -bs+c)

mW = Ibs -b +c
3W (2bs b



These are same as fM and mM, respectively. Q.E.D.



For comparison with other models, one may derive a set of Ramsey-optimal prices

which is the solution of the following problem:










Max TU + TUs s.t. f+m = c
f,m



From the first-order condition of this maximization problem, the following

Ramsey-optimal prices can be obtained:



fR=1 -
fR (b -bs +c)

R=1 -
mR =(bs -bB +c




The differences between two different optimal prices are same for both cardholder

and merchant fees. That is,



61/- -b c z
fw fR =mW _mR = b +bs-c>0
6(



Ramsey-optimal prices are lower than the prices that maximize Marshallian welfare

since the former does not allow profits of the firms while the latter puts the same weight

on profits as on customers' surplus. If social welfare is measured by the Ramsey standard,

competition may increase the social welfare as long as competition lowers both

cardholder and merchant fees.

It is also worth noting that the Ramsey-optimal fees of the proprietary scheme is

equal to the consumer and merchant fees that are determined by the monopoly

interchange fee of the nonproprietary scheme, i.e., fR =c, -ab' and mR = cA + ab, which

confirms that abj maximizes both Marshallian and Ramsey social welfare.









3.5.2 Competition between Differentiated Card Schemes

When two proprietary card schemes are differentiated and compete a la Hotelling,

the critical consumer, x*, who is indifferent between card 1 and 2 is determined in the

same way as (3-11) except that the card schemes setf and m, instead of a,:



X*= (b f)(bs -m,)+(f2 -bB)(bs -m2)
(bB b)(2bs m -m2)



If the issuer market is not fully covered, each card scheme has a monopoly power

over its own consumers, so the equilibrium set of prices will be same as fM and mM in

(3-15).20 In order to obtain a non-trivial result, suppose the issuer market is fully covered

at equilibrium as in the previous section. This requires the following assumption:21



bB-bb


Using the first-order conditions, one can derive the best response functions of card

schemes from which the following equilibrium prices can be obtained:



fP = l5bB-3_bB 2bs +2c)


mp 4 =(2bs -bB-bB +2c)




20 Since merchants accept card i as long as bs > m,, the existence of competing card
schemes does not affect the equilibrium merchant fee.
21 For the issuer market to be fully covered, the net utility of the consumer located at x =
12 must be non-negative for the monopoly prices, fM and mM.









If two schemes collude and act like a monopolist, the joint entity will set the

cardholder fee such that the critical consumer who is located at x = 12 is indifferent

between using card and cash as well as between card 1 and 2. Since the transportation

cost is assumed to be equal to bB -bB, the cardholder fee that will be set by the joint

entity is



f (bB +bB) (3-17)



Given this cardholder fee, the joint profit can be rewritten as follows:



; Ti p( = (f bs m)(bB+ bB 2c + 2m)
r = ( f ^ + m-c)(QQ,,Qs + QzQs2) =
2(bs -bs)



The optimal merchant fee that maximizes this profit function is



m ph =(2bs -bB-b +2c)



Note that the merchant fee set by the joint entity is the same as the competitive

merchant fee, i.e., mph' = mp This is because the issuer market is fully covered in both

cases and the multi-homing merchants will accept any card as long as the merchant fee is

less than bs.









Proposition 3-9 When the two proprietary card schemes are differentiated in a Hotelling

fashion, competition does not improve Marshallian social welfare.



Proof. If two card schemes charge same prices and the issuer market is fully covered,

Marshallian social welfare is



W = TUB +TUs + T1 + 2

= UBl+dxf UB2d++ 1Jf rQsdm+ (f +m-c)QBQs,
2 m 1=i
(3bB+bB -4f)(bs -m) (bs -m)2 (f +m-c)(bs -m) (3-18)
+ +
4(bs -b ) 2(bs -bs) (bs -bs)

(bs -m)[3bB + bB + 2(bs 2c + m)]
4(bs -bs)



Note that social welfare is independent off That is, the cardholder fee has no effect

on the welfare as long as the fee is low enough for the issuer market to be fully covered.

An increase in the cardholder fee just transfers surplus from consumers to the card

schemes.

Since the social welfare is only affected by the merchant fee and the equilibrium

merchant fees of the competitive case and the monopoly case are equal to each other,

competition does not improve the social welfare. Q.E.D.



The cardholder fee cannot affect social welfare since the issuer market is fully

covered, i.e., the consumers' quasi-demand is fixed regardless of the cardholder fee.

When the cardholder fee changes, it does not affect the demand of the issuer market, but