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 Three Essays on Bundling and TwoSided Markets
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Complementary goods ( jstor ) Consumer equilibrium ( jstor ) Consumer goods ( jstor ) Consumer prices ( jstor ) Consumer surplus ( jstor ) Credit cards ( jstor ) Fees ( jstor ) Merchants ( jstor ) Social welfare ( jstor )
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THREE ESSAYS ON BUNDLING AND TWOSIDED 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, HyoJung, and two daughters, HeeYeon
and HeeSoo, 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 oSided 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 TWOSIDED 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 SingleH om ing Consum ers........................................ ............... 50
3.3.2 M ultiH om ing Consum ers ................................... ..................58
3.4 Competition between Differentiated Card Schemes: Hotelling Competition...66
3.4.1 SingleH om ing Consum ers........................................ ............... 68
3.4.2 M ultiH om ing Consum ers ....................................... ............... 74
3.5 Proprietary System with SingleHoming 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 FullC over M market ................................... ................ .................... 103
4.2.3 L ocal M monopoly ....................................................... ............... 107
4.2.4 PartialCover M market ........................................................................ 109
4.3 Equilibrium Interchange Fee.................................... ..................................... 114
4 .3 .1 F ullC ov er M market ..................................................................... ...... 114
4 .3 .2 L ocal M on op oly .................................................................... .. ...... 116
4.3.3 P artialC 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
11 Credit card scheme es .............. .... ........ .. ..... .................. ........7
21 Consumers' surplus in bundling and IS cases when vl
31 Welfare and interchange fees of Bertrand competition............................................66
32 Merchants' acceptance decision when al > a2 (ml > m2) ..................................77
33 Welfare and interchange fees of Hotelling competition when
bB b < 2(bB + bs c) ........................................ ....................................... 82
41 Division of consumers in three cases of market coverage................. ...............102
42 The effect of a price drop on demand ............................ ..... ... ............... 110
A The density function ......... ................................ ....................................... 129
A2 Effects of an increase in the variance on the cardholder fee (dfldy) ........................130
A3 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 TWOSIDED MARKETS
By
Jin Jeon
December 2006
Chair: Jonathan H. Hamilton
Major Department: Economics
This work addresses three issues regarding bundling and twosided markets. It
starts with a brief summary of the theories of bundling and of twosided 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 longrun 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 singlehoming and multihoming consumers as well as
proprietary and nonproprietary platforms, it is shown that introducing platform
competition in twoside 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 twosided market can properly internalize indirect
network externalities by setting unbiased prices, while the competing platforms set biased
prices in order to attract the singlehoming 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 fullcover market under the nosurchargerule. 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 twosided 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 nosurcharge rule, and the honorallcards 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
twosided markets will be presented.
1 For more information about the Kodak case, see Klein (1993), Shapiro (1995),
Borenstein, MacKieMason, 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 Aand $1 for Bto 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 TwoSided Markets
Twosided markets are defined as markets in which endusers 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 twosided 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 twosided 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 twosided markets with negative externalities.
Examples of the twosided 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 11 shows the structure of the twosided market in case of the credit
card industry, both proprietary and nonproprietary schemes.
Although some features of twosided markets have been recognized and studied for
a long time, 11 it is only recently that a general theory of twosided markets emerged. 12
The surge of interest in twosided 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 onesided markets in twosided markets, which include: an efficient price
structure should be set to reflect relative costs; a high pricecost 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 twosided market.
11 For example, Baxter (1983) realized the twosidedness 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 twosided markets is related to the theories of network externalities
and of multiproduct pricing. While the literature on network externalities has found that
in some industries there exist externalities that are not internalized by endusers, models
are developed in the context of onesided markets. 14 Theories of multiproduct 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 Payspm
(f: cardholder fee) (m : merchant fee)
(Cardholder  Merchant
Sells good at price
(b) Proprietary card scheme
Figure 11. 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 twosided 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 twosided 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 twosided 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 onesided 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 onesided market, the price is determined by the marginal cost and the
own price elasticity, as is characterized by Lerner's formula.16 In twosided markets,
however, there are other factors that affect the price charged to each side. These are
relative size of crossgroup externalities and whether agents on each side singlehome or
multihome. 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, nonproprietary 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 EI
p e s1
marginal cost, and e is the own price elasticity.
17 An enduser is "singlehoming" if she uses one platform, and "multihoming" 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, endusers 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 singlehome, (ii) one side
singlehomes while the other side multihomes, and (iii) both sides multihome. 18 If
interacting with the other side is the main purpose of joining a platform, one can expect
case (iii) is not common since endusers of one side need not join multiple platforms if all
members of the other side multihome.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 endusers 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 endusers will multihome.
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 twosided markets, singlehoming and multihoming of end
users are predetermined 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 endusers of one side singlehome while those of the other side multihome,
the singlehoming side becomes a "bottleneck" (Armstrong, 2005). Platforms compete
for the singlehoming side, so they will charge lower price to that side. As is shown in
Chapter 3, platforms competing for the singlehoming side may find themselves in a
situation of the "Prisoner's Dilemma". That is, a lower price for the singlehoming side
combined with a higher price for the multihoming side can decrease total transaction
volume and/or total profits compared to the monopoly outcome. Further, competition in
twosided 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 singlehoming side.
Chapter 3 presents a model of the credit card industry with various settings
including singlehoming vs. multihoming cardholders, competition between identical
card schemes (Bertrand competition) or differentiated schemes (Hotelling competition),
and proprietary vs. nonproprietary card schemes. The main finding is that, unlike in a
standard onesided 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 socalled "leverage theory" has been criticized by many economists
associated with the Chicago School in that there exist other motives of bundling such as
efficiencyenhancement 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 toi.e., bundled withthe 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 wouldbe
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 efficiencyenhancing 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 efficiencyenhancing 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 wouldbe 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 secondperiod 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 (21)
The model presented here allows a difference between vi and v2 in order to analyze
bundling decision when the monopolist produces inferioror superiorcomplementary
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 (22)
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 txpopl
u2 = V+ V2 t(1 x) PoP2
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 gametheoretic 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 + (23)
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 (23) 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
(24)
U, (XIA) = Vg + V1 tXI
where the last inequality holds because of the assumption given in (21). 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 (21).
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 leaderfollower 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 (23) with p2 replaced by firm 2's best response function
given by (24), i.e.,
3 Vl v v p p
x + (25)
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 (24) and
(25):
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 (24) and the critical consumer is also
determined by (25). 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+ vlpi tx' (26)
Using (25) and (26), 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 (24), (25) and (26), 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
CS 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[18t7(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 (21) and (22). 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 + (27)
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 (27) 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 (22) 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 (28)
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 (29)
12
Comparing (28) and (29) 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 (210)
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(v2_ + t)2
2 = ap2 (8XBS)=t
8t
The following proposition shows that bundling increases monopolist's profits
compared to the IS case.
Proposition 21 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 wouldbe 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 (22) and
SIS ^BS
(21), 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 (21) and (22). 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 (21) and (22), 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, +, txpb)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 21 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 21. 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 22 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 23 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 nonnegative 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(1eta) >
v2 v2 =+ >0
24e 8e(9et a)(8eta)
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 longterm effect on welfare since it reduces both firms' R&D incentives.
This longterm effect of bundling on R&D investment may be more important than
immediate effects on competitor's profit or consumers' surplus, especially for socalled
hightech 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 complicatedbut
more realisticif it is combined with the possibility of upgrade which is common in the
software industry.
22 See Klein (1993), Shapiro (1995), Borenstein, MacKieMason, and Netz (1995), and
Blair and Herndon (1996).
CHAPTER 3
COMPETITION AND WELFARE IN THE TWOSIDED MARKET:
THE CASE OF CREDIT CARD INDUSTRY
3.1 Introduction
It is well known that a twosided marketor more generally a multisided
marketworks differently from a conventional onesided 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 onesided
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 twosided market is different
from the rule of the onesided market, and a price below marginal cost may not be anti
competitive. 1
Another feature of the twosided 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 twosided 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 twosided 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 singlehome or multihome2 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 twosided 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
singlehome. If she uses multiple cards, she is said to multihome.
3 In a twosided market, the benefit of one side depends on the size of the other side. This
indirect network externality cannot be internalized by the endusers of the twosided
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 singlehoming consumers or
merchants.
Since the first formal model by Baxter (1983), various models of twosided 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 twosided 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 (nonproprietary) associations are the same and
Ramsey optimal. They measure the price structure and Ramsey optimality in terms of the
priceelasticity 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 singlehoming 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 pertransaction 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 pertransaction 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 nonproprietary scheme.
The rest of Chapter 3 proceeds as follows. Section 3.2 sets up the basic model of
the nonproprietary card scheme. Section 3.3 and 3.4 show the effects of competition on
the price structure and welfare for the cases of singlehoming 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 nonproprietary 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 notfor
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 pertransaction benefit, bs, which is also
5 The pertransaction 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 pertransaction costs of a issuer and a acquirer, respectively. Then card
scheme i's pertransaction 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 cashpaying consumers and cardpaying consumers, i.e., the no
surchargerule 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 singlehoming or
multihoming, while merchants are assumed to freely choose whether to accept one card,
both cards, or none.
One of the key features of the twosided 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 SingleHoming 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 singlehome. So the
8 See Rochet and Tirole (2002) and Guthrie and Wright (2006).
number of merchants that accept card i (quasidemand function for acquiring service) is9
bs m, bs cA a2)
Qs, = (32)
bs b bs b
Using (32), the consumer's net utility can be rewritten as
(b, c, + a c )(bsca)
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 quasidemand function of consumers:
9 Schmalensee (2002) calls Qs, and QB, partial demands, and Rochet and Tirole (2003)
call them quasidemands since the actual demand is determined by the decisions of both
sides in a twosided 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 (33)
be b, be b
bB c, +a .
2bB C + if a =a
2(bsbB)
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 singlehoming case of Bertrand competition.
Proposition 31 If two identical card schemes compete with each other and consumers
singlehome,
(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 (33). 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 singlehome, 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 twosided 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
singlehome raises the interchange fee.
Proposition 32 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 quasidemand 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 firstorder condition is
a* c)( c)] abj

a=2 [(bs c) (b cr) ab
which is less than abs since
aa =a bs +bB c)>O
(ii) Without loss of generality, suppose al > a2. Then scheme 1 will attract lowtype
consumers and scheme 2 will attract hightype consumers. The quasidemand functions
are determined by (32) and (33). And the total transaction volume is
(bs c, + a2)(bs c, a,)
QBS +QB2QS2 = (3 4)
(b bB)(bs bs)
Note that (34) is independent of al, which implies al can be set at any level above
a2. The optimal a2 can be obtained from the firstorder condition for maximizing (34):
1j
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 31 and 32, 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 twosided market shows that competitive outcome can be detrimental to
the society as well as to themselves.
3.3.2 MultiHoming Consumers
In this subsection, consumers are allowed to multihome. 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=_ (35)
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
(36)
If the merchant accept both cards, the surplus is
Usb = (bs ml)Qbl + (bs m2)Qb2 = (bs cA al)Qbl + (bs CA a2)Qb2
(37)
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' cardholding decision and cardusing decision can be different since they
can hold both cards but use only one card for each merchant.
(38)
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 multihome. 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
(39)
Proposition 33 If two identical card schemes compete with each other and consumers
multihome,
(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 31, 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 +bsc+ 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 multihome, 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 singlehoming 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 multihoming case is lower than in the singlehoming
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 multihome. It also shows that competition lowers social welfare as in the
singlehoming case.
Proposition 34 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 singlehome or multihome, 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 32. And for multihoming 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 hightype ones. Then the total transaction
volume is
QBlQSl + QB2QS2
(bB c1 +a)(bs c a,)
(b b,)(bs b)
Note that (310) is independent of a2, which implies a2 can be set at any level
below al. The optimal al obtained from the firstorder 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 32.
Q.E.D.
The optimal interchange fee for the joint entity is the same as in the singlehoming
case since the card schemes do not compete for consumers or merchants. Unlike the
singlehoming case, however, the interchange fee decreases as a result of competition
between the card schemes when consumers multihome. Social welfare deteriorates since
(310)
competing card schemes set the interchange fee too low in order to attract more
merchants.
Figure 31 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 31. 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(1x)+bBxcI+a)Qs
UB2 (bB t(1x)f2)Q2= (bBx+bB(1x) 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 (311)
(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 singlehome or multihome, 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 nonnegative for the monopolistic interchange fee abJ.
3.4.1 SingleHoming Consumers
When consumers are restricted to singlehome, merchants will accept card i as long
as bs > m, as in the previous section. So Qs, is determined by (32). 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 (311).
The following proposition shows the symmetric equilibrium of the Hotelling
competition when consumers singlehome.
Proposition 35 If two differentiated card schemes compete a la Hotelling and
consumers singlehome,
(i) the symmetric equilibrium interchange fee is
c (bBs+bB) if b b, > 2(b +bs c)
a0 = (312)
4 [2(bscA)2(bBc,)(bBbB)] 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(bBbB,)+ 2(b, +bsc)
w= B if bBbB <2(bB +bsc)
6(bB b)+2(b + bs c)
2(bs bB)+ 4(b + bs c)
w2 = if beb B > 2(bB + bs c)
3(bBbB)+ 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 firstorder condition
in which al and a2 are set to be equal to each other for symmetry:
1'
a=4 [2(bscA)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,+bsc)(bBbB)]>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')
Aao+ 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
bBbB =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)
(313)
(314)
The size of the weight can be obtained by setting ah = a*, which is
3(bBb b)+ 2(b, +bsc)
w, = if bBb <2(b, +bs c)
6(bB b)+ 2(bB +bs c)
2(bB bB) + 4(b, + bs c)
w2 =  if bBbB >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 36 If the two differentiated card schemes are jointly owned and consumers
singlehome,
(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 fullcover 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 (314) 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 singlehome.
3.4.2 MultiHoming Consumers
If consumers are allowed to multihome, they will hold card i as long as bBZ >f. So
the number of consumers who hold card i is the same as (35). 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 31 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 32, 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 32. Merchants' acceptance decision when al > a2 (ml > m2)
Based on Lemma 31, 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 multihoming consumers.
Proposition 37 If consumers can multihome 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 bbB >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 bsc +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 + bsc+ 2(bBbB)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
bBt(ca)= (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 +bsc)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 singlehoming 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 singlehoming 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 (313), hence the optimal interchange fee maximizing the weighted surplus is
also the same as (314). 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 singlehoming case, wl = w2 = 4/7 if bB bB = 2(b + bs c).
Q.E.D.
When consumers multihome, the equilibrium interchange fee is lower than that of
the singlehoming 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 multihoming 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 33 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 multihome increases social welfare in
the Hotelling competition case, although it lowers total consumers' surplus.
3.5 Proprietary System with SingleHoming Consumers
The analysis of the previous sections has been restricted to the competition between
nonproprietary card schemes that set interchange fees and let the cardholder fees and
TU = TUB +TUs
TUB
bs LA
Figure 33 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 singlehoming consumers will be considered.
When the monopoly proprietary card scheme setsfand m, the quasidemand
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 nonproprietary 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 +mc)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)
(315)
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 32 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 quasidemands 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 38 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
(316)
(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 Ramseyoptimal prices
which is the solution of the following problem:
Max TU + TUs s.t. f+m = c
f,m
From the firstorder condition of this maximization problem, the following
Ramseyoptimal 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 +bsc>0
6(
Ramseyoptimal 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 Ramseyoptimal 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 (311) 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
(315).20 In order to obtain a nontrivial result, suppose the issuer market is fully covered
at equilibrium as in the previous section. This requires the following assumption:21
bBbb
Using the firstorder conditions, one can derive the best response functions of card
schemes from which the following equilibrium prices can be obtained:
fP = l5bB3_bB 2bs +2c)
mp 4 =(2bs bBbB +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 nonnegative 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) (317)
Given this cardholder fee, the joint profit can be rewritten as follows:
; Ti p( = (f bs m)(bB+ bB 2c + 2m)
r = ( f ^ + mc)(QQ,,Qs + QzQs2) =
2(bs bs)
The optimal merchant fee that maximizes this profit function is
m ph =(2bs bBb +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 multihoming merchants will accept any card as long as the merchant fee is
less than bs.
Proposition 39 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 +mc)QBQs,
2 m 1=i
(3bB+bB 4f)(bs m) (bs m)2 (f +mc)(bs m) (318)
+ +
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' quasidemand 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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THREE ESSAYS ON BUNDLING AND TWOSIDED 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
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Copyright 2006 by Jin Jeon
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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, HyoJung, and two daughters, HeeYeon and HeeSoo, 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 TwoSided 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 TWOSIDED 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 SingleHoming Consumers...................................................................50 3.3.2 MultiHoming Consumers....................................................................58 3.4 Competition between Differentiated Ca rd Schemes: Hotelling Competition...66 3.4.1 SingleHoming Consumers...................................................................68 3.4.2 MultiHoming Consumers....................................................................74 3.5 Proprietary System with SingleHoming 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 FullCover Market..............................................................................103 4.2.3 Local Monopoly..................................................................................107 4.2.4 PartialCover Market..........................................................................109 4.3 Equilibrium Interchange Fee...........................................................................114 4.3.1 FullCover Market..............................................................................114 4.3.2 Local Monopoly..................................................................................116 4.3.3 PartialCover 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 41 Comparative statics...................................................................................................119
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viii LIST OF FIGURES Figure page 11 Credit card schemes......................................................................................................721 Consumers surplus in bundling and IS cases when v1 < v2........................................3731 Welfare and interchange fees of Bertrand competition..............................................6632 Merchants acceptance decision when a1 > a2 ( m1 > m2)...........................................7733 Welfare and interchange fees of Hotelling competition when 2()BS BBbbbbc .............................................................................................8241 Division of consumers in three cases of market coverage........................................10242 The effect of a price drop on demand.......................................................................110A1 The density function................................................................................................129A2 Effects of an increase in the variance on the cardholder fee ( df / dy )........................130A3 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 TWOSIDED MARKETS By Jin Jeon December 2006 Chair: Jonathan H. Hamilton Major Department: Economics This work addresses three issues rega rding bundling and twosided markets. It starts with a brief summary of the theories of bundling and of twosided 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 longrun 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 singlehoming and multihoming consumers as well as
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x proprietary and nonproprietar y platforms, it is shown that introducing platform competition in twoside 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 twosided 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 singlehoming 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 fullcover market under the nosurchargerule. 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 twosided 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 nosurc harge rule, and the honorallcards 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 twosided markets will be presented. 1 For more information about the Kodak case, see Klein (1993), Shapiro (1995), Borenstein, MacKieMason, 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 TwoSided Markets Twosided markets are defined as markets in which endusers 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 twosided 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 twosided 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 twosi ded markets with negativ e externalities.
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6 Examples of the twosided 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 11 shows the structure of the tw osided market in case of the credit card industry, both proprietary and nonproprietary schemes. Although some features of tw osided markets have been recognized and studied for a long time,11 it is only recently that a general theory of twosided markets emerged.12 The surge of interest in twosided 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 onesided markets in twosided 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 twosided market. 11 For example, Baxter (1983) realized the twosidedness 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 twosided markets is related to the theories of network externalities and of multiproduct pricing. While the litera ture on network externalities has found that in some industries there exist externalities that ar e not internalized by endusers, models are developed in the context of onesided markets.14 Theories of multiproduct pricing stress the importance of price structures, but ignore externalities in the consumption of Figure 11. 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 twosided 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 twosided 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 twosided 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 onesided 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 onesided market, the price is determined by the marginal cost and the own price elasticity, as is ch aracterized by Lerners formula.16 In twosided markets, however, there are other factor s that affect the price char ged to each side. These are relative size of crossgroup externalities and whether agents on each side singlehome or multihome.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, nonproprietary 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 enduser is singlehoming if she uses one platform, and multihoming 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, endusers 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 singlehome, (ii) one side singlehomes while the other side multihomes, and (iii) both sides multihome.18 If interacting with the other side is the main purpose of joining a platform, one can expect case (iii) is not common since endusers of one side need not join multiple platforms if all members of the other side multihome.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 endusers 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 endusers will multihome. 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 twosided mark ets, singlehoming and multihoming of endusers are predetermined 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 endusers of one side singlehome wh ile those of the other side multihome, the singlehoming side becomes a bottl eneck (Armstrong, 2005). Platforms compete for the singlehoming side, so they will charge lower price to that side. As is shown in Chapter 3, platforms competing for the sing lehoming side may find themselves in a situation of the Prisoners Dilemma. That is, a lower price for the singlehoming side combined with a higher price for the multihom ing side can decrease total transaction volume and/or total profits compared to the monopoly outcome. Further, competition in twosided 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 singlehoming side. Chapter 3 presents a model of the credit card industry with various settings including singlehoming vs. multihoming car dholders, competition between identical card schemes (Bertrand competition) or diffe rentiated schemes (Hotelling competition), and proprietary vs. nonproprieta ry card schemes. The main finding is that, unlike in a standard onesided 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 socalled 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 efficiencyenhancement 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 wouldbe 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 efficiencyenha 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 fficiencyenhancing 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 wouldbe 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 secondperiod 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 (21) 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 (22) 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 gametheoretic 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 (23) 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 (24) 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 (23) 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 (21). 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 (21). 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 leaderfollower 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 (23) with p2 replaced by firm 2s best response function given by (24), i.e., 1213 44 vvp x t (25) 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 (24) and (25): 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 (24) and the critical consumer is also determined by (25). 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* (26) Using (25) and (26), 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 (24), (25) and (26), 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 (21 ) and (22). 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 (27) 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 (27) 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 (22) 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 (28) 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 (29) Comparing (28) and (29) 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 (210) 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 21 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 wouldbe 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 (22) and (21), 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 (21) and (22). 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 (21) and (22), 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 21 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 21. 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 22 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 23 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 nonnegative 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 longterm effect on welfare since it reduces both firms R&D incentives. This longterm 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 socalled hightech 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, MacKieMason, and Netz (1995), and Blair and Herndon (1996).
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44 CHAPTER 3 COMPETITION AND WELFARE IN THE TWOSIDED MARKET: THE CASE OF CREDIT CARD INDUSTRY 3.1 Introduction It is well known that a twosided marketor more generally a multisided marketworks differently from a conventiona l onesided 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 onesided 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 osided market is different from the rule of the onesided market, and a price below marginal cost may not be anticompetitive.1 Another feature of the twosided 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 twosi 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 twosided 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 singlehome or multihome2 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 twosided 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 singlehome. If she uses multiple cards, she is said to multihome. 3 In a twosided market, the benefit of one si de depends on the size of the other side. This indirect network externality cannot be internalized by the endusers of the twosided 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 singlehoming consumers or merchants. Since the first formal model by Baxter ( 1983), various models of twosided 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 twosided 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 (nonprop rietary) associations are the same and Ramsey optimal. They measure the price struct ure and Ramsey optimality in terms of the priceelasticity 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 singlehoming 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 pertransaction 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 pertransaction 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 nonproprietary scheme. The rest of Chapter 3 proceeds as follows. Section 3.2 sets up the basic model of the nonproprietary card scheme. Section 3.3 and 3.4 show the effects of competition on the price structure and welfare for the cases of singlehoming 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 nonproprietary 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 notforprofit 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 pertransaction benefit, bS, which is also 5 The pertransaction 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 pertransaction costs of a issuer and a acquirer, respectively. Then card scheme i 's pertransaction 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 (31) 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 cashpaying consumer s and cardpaying consumers, i.e., the nosurchargerule 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 singlehoming or multihoming, while merchants are assumed to freely choose whether to accept one card, both cards, or none. One of the key features of the twosided 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 SingleHoming 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 singlehome. 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 (quasidemand function for acquiring service) is9 SS iAi Si SS SSbmbca Q bbbb (32) Using (32), 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 quasidemand function of consumers: 9 Schmalensee (2002) calls QSi and QBi partial demands, and Rochet and Tirole (2003) call them quasidemands since the actual dema nd is determined by the decisions of both sides in a twosided 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 (33) 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 singlehoming case of Bertrand competition. Proposition 31 If two identical card schemes comp ete with each other and consumers singlehome, (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 (33). 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 singlehome, 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 twosided 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 singlehome raises the interchange fee. Proposition 32 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 quasidemand 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 firstorder 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 lowtype consumers and scheme 2 will attract hightype consumers. The quasidemand functions are determined by (32) and (33). And the total transaction volume is 22 1122()() ()()BS IA BSBS BS BSbcabca QQQQ bbbb (34) Note that (34) is independent of a1, which implies a1 can be set at any level above a2. The optimal a2 can be obtained from the firstorder condition for maximizing (34): 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 31 and 32, 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 twosided market shows that competitive outcome can be detrimental to the society as well as to themselves. 3.3.2 MultiHoming Consumers In this subsection, consumers are allowed to multihome. 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 (35) 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 (36) 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 (37) 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 (38) 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 cardholding decision and cardus 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 multihome. 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 (39) Proposition 33 If two identical card schemes co mpete with each other and consumers multihome, (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 31, 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 multihome, 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 singlehoming 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 multihoming case is lower than in the singlehoming 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 multihome. It also shows that comp etition lowers social welfare as in the singlehoming case. Proposition 34 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 singlehome or multihome, 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 32. And for multihoming 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 hightype ones. Then the total transaction volume is 11 1122()() ()()BS IA BSBS BS BSbcabca QQQQ bbbb (310) Note that (310) is independent of a2, which implies a2 can be set at any level below a1. The optimal a1 obtained from the firstorder 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 32. Q.E.D. The optimal interchange fee for the joint en tity is the same as in the singlehoming case since the card schemes do not compete for consumers or merchants. Unlike the singlehoming case, however, the interchange fee decreases as a result of competition between the card schemes when consumers multihome. 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 31 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 31. 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 (311) 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 singlehom e or multihome, 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 nonnegative for th e monopolistic interchange fee abj.
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68 3.4.1 SingleHoming Consumers When consumers are restricted to singlehome, merchants will accept card i as long as bS mi as in the previous section. So QSi is determined by (32). 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 (311). The following proposition shows the symm etric equilibrium of the Hotelling competition when consumers singlehome. Proposition 35 If two differentiated card scheme s compete la Hotelling and consumers singlehome, (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 (312) (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 firstorder 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 (313) The optimal interchange fee that maximizes this weighted surplus is *4(21)()(34) 4(31)SB B A I wwbcwbbc a w (314)
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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 36 If the two differentiated card sc hemes are jointly owned and consumers singlehome,
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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 fullcover 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 (314) 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 singlehome. 3.4.2 MultiHoming Consumers If consumers are allowed to multihome, they will hold card i as long as bBi > fi. So the number of consumers who hold card i is the same as (35). 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 31 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 32, 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 32. Merchants acceptance decision when a1 > a2 ( m1 > m2) Based on Lemma 31, 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 multihoming consumers. Proposition 37 If consumers can multihome 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 singlehoming 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 singlehoming 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 (313), hence the optimal interc hange fee maximizing the weighted surplus is
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81 also the same as (314). 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 singlehoming case, w1 = w2 = 4/7 if 2()BS BBbbbbc Q.E.D. When consumers multihome, the equilibrium interchange fee is lower than that of the singlehoming 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 multihoming 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 33 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 multihome increases social welfare in the Hotelling competition case, although it lowers total consumers surplus. 3.5 Proprietary System with SingleHoming Consumers The analysis of the previous sections has been restricted to the competition between nonproprietary card schemes that set interc hange fees and let the cardholder fees and Figure 33 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 singlehoming consumers will be considered. When the monopoly proprietary card scheme sets f and m the quasidemand 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 nonproprietary 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 (315) The following lemma shows that there does not exist a pure strategy equilibrium when two identical proprietary card schemes compete with each other. Lemma 32 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 quasidemands 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 38 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 (316) 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 Ramseyoptimal prices which is the solution of the following problem:
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87 .. BS fm M axTUTUstfmc From the firstorder condition of this maximization problem, the following Ramseyoptimal 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 Ramseyoptimal 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 Ramseyopt 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 (311) 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 (315).20 In order to obtain a nontr 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 firstorder 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 nonnegative 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 (317) 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 multihoming merchants will accept a ny card as long as the merchant fee is less than Sb
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90 Proposition 39 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 (318) 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 quasidemand 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 (318) is 1 3 4phw B Bmcbb (319) 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 (317). So the maximum possible profit ma rgin when the card scheme charges phj f and phwm is 1 ()0 4phjphw B Bfmcbb When the firstbest price is not feasible, one can think of the secondbest 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 Ramseyoptimal fee is 1 ()2()0 4phsphr BS BBmmbbbbc 3.6 Conclusion This chapter shows the effects of compe tition in a twosided market on the price structure and welfare using a formal model w ith various settings including singlehoming vs. multihoming consumers, Bertrand vs. Hote lling competition, and proprietary vs. nonproprietary card schemes. The effect of competition on the price structure depends on whether consumers singlehome or multihome since competing card schemes set lower prices for the singlehoming side. The most surprising result is that competition never improves social welfare regardless whether consumers singlehome or multihome, 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 twosided 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 osided 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 cardusing 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 ltihome and card schemes are nonproprietary and identical. If consumers si nglehome 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 multihoming 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 twosided 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 twosided market. A twosided (or multisided) market is a market in which two (or more) parties interact on a platform. The endusers 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 nosurcha rge rule, and the honorallcard 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 twosi 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 twosidedness 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 partialcover 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 fullcover market even under the nosurchargerule. 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 twosided 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 twosided 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 cardholding and cardusing since they assume all consumers purchase the same amount of goods from each merchant. Further, by assuming identical merchants, their model cannot capture the tradeoff 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 (41) 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 onetime 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 (42) 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 (43) 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 (44) Depending on the size of *v compared to v and v, one can distinguish three regimes: fullcover market when *vv local monopoly when *vv, and partialcover market 4 Consumers are assumed to use a single card or none at all, hen ce the possibility of multihoming is excluded. In fact, there is no extra gain from multihoming since the two issuers are in the same network.
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101 when *vvv Figure 41 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 fullcover market are 112 221(;)() (;)(1)()v v v vqffvxgvdv qffvxgvdv (45) In case of the local monopol y, the demand functions are 111 222()() ()(1)()v v v vqfvxgvdv qfvxgvdv (46) Last, the demand functions of partialcover market are * *** 1121 ** 2212(;)()() (;)(1)()(1)()vv vv vv vvqffvxgvdvvxgvdv qffvxgvdvvxgvdv (47) 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 41. 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) Fullcover market (b) Local monopoly (c) Partialcover market 0
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103 ()iimi f acQq (48) 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 FullCover 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 (45) 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 (45) into (48), which is
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104 2 2()() 1 ()[][] 22ijim iimfacffQ f acQEvEv t Proposition 41 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 (49) Using (49), 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 (410) 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 (411) 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 (412) 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 (410) and (411), 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 42 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 (410): 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 (46) Using (46) and (48), issuer i 's profit can be rewritten as 2 2()() []iim ifacbfQ Ev t
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108 From the firstorder condition for the profit maximization problem, one can derive the equilibrium cardholder fee as follows: 2LMbac f (413) As in the fullcover 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 fullcover 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 (414) Contrary to the fullcover 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 43 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 PartialCover 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 (47), which can be rewritten as **** 1121 *** 2212(;)()()() (;)(1)()()()vv vv vv vvqffvxgvdvvxxgvdv qffvxgvdvvxxgvdv (47) Plugging (42) and (43) into (47), one can obtain *22 *() 11 (;)()[]() 222vv jim iij vvffQ qffvgvdvEvvgvdv tv (47) The derivative of (47) with respect to fi is *22[]() 2v im v idqQ Evvgvdv dft (415)
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110 It is not clear from (415) 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 42 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 42. 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 (48) with respect to fi is 2() 2iim mi idfacQ Qq dft (416) Lemma 41 The equilibrium cardholder fee is un ique in the partialcover market. Proof. Using (416) and symmetry of the firms, the equilibrium fee, P C f is implicitly determined by 2() (;) 2PC PCPC m ifacQ qff t (417) When f = c a the LHS of (417) is positive while the RHS of it is equal to zero. As f increases, the LHS of (417) decreases monotonically, while the RHS increases monotonically. So the equilibrium fee is uniquely determined by (417). Q.E.D. Using (47) and (417), 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 (418) Since v* itself is a function of f the equilibrium fee is still implicitly determined by (418). 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 (418) becomes equal to (410) so that P CFC f f at *vv When *vv, (418) shrinks to () 2PC mt fvca Qv (419) Using the definition of v*, one can show that (419) is equal to (413) so that P CLM f f at *vv. Propositions 41 and 43 have shown that as the variance of the expenditure volume increases, the equilibrium fee decreases in the fullcover market or remains constant in the local monopoly case. From thes e results one may conjecture that in case of partialcover market, the effect of a cha nge in the variance of the expenditure volume should lie between the results of fullcover 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 42 A sufficient condition fo r the equilibrium cardholder fee of the partialcover 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 righthand side of (418). There also exists a secondorder effect due to a change in v*. However, the secondorder effect is minor a nd cannot offset the firstorder effect. So I will focus on the firstorder effect only. Note first that the bracket in (418) 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 (417), 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 twosided 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 indepth 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 FullCover Market As can be seen in (411), 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 fullcover 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 44 Neutrality of the interchange fe e holds under the nosurchargerule 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 firstorder condition: /LM m mQ acb dQda (420) or using the elas ticity formula, () 1LM m macb (421) where m m mdQ a daQ In a standard twosided 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 (422) A simple calculation using (420) shows that (422) 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 (422), 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 PartialCover Market In the partialcover market, the equilibri um profits can be obtained by plugging (417) into the profit func tion (48), which is **22 *22() []() 2v m i vfacQ Evvgvdv t (423) 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 (424)
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118 Since the issuer chooses the optimal f* to maximize profit, the last term in (424) 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 (425) It follows from (425) 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 41 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 fullcover ma rket and local monopoly cases. In the fullcover 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 41 Comparative Statics 2df d 2d d *df dt *d dt mdf dQ md dQ *df da Fullcover market + + 0 Local monopoly 0 + 0 0 + Partialcover market + issuers can charge higher pr ice and make higher profits.10 In the local monopoly case, however, customers e ither use a leastcost 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 partialcover 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 fullcover market. But for customers below v*, issuers lose marginal customers due to the increase in the car dusage 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 fullcover 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 fullcover market when the variance of v increases, the issuers have an incentive to exclude some lowvolume 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 45 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 firstorder condition for the jo int profit maximization problem is *2() ()0C v C m m vfacQ d Qqvgvdv dft (426) The equilibrium cardholder fee, C f is implicitly determined by the following condition: *2() (;)()C v CCC m vfacQ qffvgvdv t (427) 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 (426) 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 (427) with (417), it is clear that the collusive equilibrium fee is higher than the one without collusion in the partialcover 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 fullcover 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 fullcover 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 fullcover market and local monopoly. As the simulation results in the appendix show, however, the effects tend to become clos er to those of fullcover market, the more the market is covered. As the credit card industry gr ows, the market will become closer and closer to the fullcover 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 twosided 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 (twosided 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 longterm 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 enduser s who interact with each other through the platform. Since the endusers 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 singlehoming side. A lower price to the singlehoming side is accompanied by a higher price to the multihoming side. That is, in twosided 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 singlehome and merchants multihome. So antitrust policy on twosided 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 twosided markets are fair ly new and are still in the middle of development, which includes the analysis of va rious strategies in twosided 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 twosided markets are rare. As in the other areas of economic theory, more ba lanced empirical research is required to deepen our understanding of twosided 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 partialcover 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 partialcover 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 (A1) 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 A1, 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
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129 Figure A1. 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 (44). As is shown in Table 41, fi may increase or decrease when t increases. But even if fi decreases, the resulting decrease in v* is a secondorder effect which cannot offset the firstorder effect. This implies that there exists only one t for each v* at equilibrium. Define ()dmtdbfQv 0 d 2 (A2) 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 firstorder condition (416). 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
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130 statics because t cannot be held constant. To resolve this problem, one needs to apply implicit function theorem using (418). 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 (A3) 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 (A3). The effect of an increase in the variance of v on the cardholder fee can be captured by df / dy Figure A2 shows that this derivative can be both negative and positive Figure A2. 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
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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 pertransaction benefit they receive from the card service. Figure A3 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 A3. 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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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.

