THREE ESSAYS ON BUNDLING AND TWO-SIDED MARKETS
By
JIN JEON
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2006
Copyright 2006
by
Jin Jeon
To my parents, wife, and two daughters
ACKNOWLEDGMENTS
I must first thank my supervisory committee members. Dr. Jonathan Hamilton, the
chair of the committee, always supported me with patience, encouragement, and
intellectual guidance. He inspired me to think in new ways and put more emphasis on
economic intuition than technical details. Dr. Steven Slutsky, a member, generously
shared his time to listen to my ideas and give further suggestions. Dr. Roger Blair, a
member, also gave me useful comments and provided research ideas. Dr. Joel Demski,
the external member, carefully read the manuscript and gave helpful comments. I hereby
thank them all again.
This dissertation would not have been possible without support from my family
members. My parents always believed in me and kept supporting me. I would also like to
give a heartfelt acknowledgment to my wife, Hyo-Jung, and two daughters, Hee-Yeon
and Hee-Soo, for their endless loving support.
TABLE OF CONTENTS
page
A C K N O W L E D G M E N T S ................................................................................................. iv
T A B L E .............................................................................................................. ..... v ii
L IST O F FIG U R E S .............. ............................ ............. ........... ... ........ viii
ABSTRACT ........ .............. ............. ...... ...................... ix
CHAPTERS
1 IN TR OD U CTION ............................................... .. ......................... ..
1 .1 B u n d lin g ..............................................................................................................2
1.2 Tw o-Sided M markets ....................................... ........ ........ .. ........ ..
2 BUNDLING AND COMMITMENT PROBLEM IN THE AFTERMARKET .........11
2 .1 In tro d u ctio n ...................... .. .. ............. ..................... ................ 1 1
2.2 The M odel ............................ ..... .............. ................. ........... 16
2.3 Independent Sale without Commitment .................... ..............................20
2.4 Independent Sale with Commitment...................... .... .. ................24
2.5 Bundling: An Alternative Pricing Strategy without Commitment....................30
2.6 Bundling and Social W welfare ........................................ ........................ 36
2.7 Bundling and R& D Incentives ........................................ ....... ............... 39
2 .8 C o n clu sio n ................................................. ................ 4 1
3 COMPETITION AND WELFARE IN THE TWO-SIDED MARKET: THE
CASE OF CREDIT CARD INDUSTRY ..........................................................44
3.1 Introduction ................................. ............................... ........44
3.2 The Model: Nonproprietary Card Scheme......................................................47
3.3 Competition between Identical Card Schemes: Bertrand Competition ............49
3.3.1 Single-H om ing Consum ers........................................ ............... 50
3.3.2 M ulti-H om ing Consum ers ................................... ..................58
3.4 Competition between Differentiated Card Schemes: Hotelling Competition...66
3.4.1 Single-H om ing Consum ers........................................ ............... 68
3.4.2 M ulti-H om ing Consum ers ....................................... ............... 74
3.5 Proprietary System with Single-Homing Consumers ......................................82
3.5.1 Competition between Identical Card Schemes ................................... 83
3.5.2 Competition between Differentiated Card Schemes.............................88
3 .6 C o n clu sio n ................................................. ................ 9 2
4 COMPETITION BETWEEN CARD ISSUERS WITH HETEROGENEOUS
EXPENDITURE VOLUMES ........................................................95
4.1 Introduction ................. ................. ............................... ........95
4.2 Equilibrium C ardholder Fee......................................... ......................... 98
4 .2 .1 T he M odel ................................................................ ............... 98
4.2.2 Full-C over M market ................................... ................ .................... 103
4.2.3 L ocal M monopoly ....................................................... ............... 107
4.2.4 Partial-Cover M market ........................................................................ 109
4.3 Equilibrium Interchange Fee.................................... ..................................... 114
4 .3 .1 F ull-C ov er M market ..................................................................... ...... 114
4 .3 .2 L ocal M on op oly .................................................................... .. ...... 116
4.3.3 P artial-C over M market .................................. ...................................... 117
4 .4 E xten sion .................................................................................... ........ 118
4.4.1 Other Comparative Statics............... ...................................118
4 .4 .2 C ollu sion ........................................... ................ 12 0
4 .5 C o n clu sio n ................................................ ................ 12 2
5 C O N CLU D IN G R EM A R K S ......................................................... .....................125
APPENDIX SIMULATION ANALYSIS FOR CHAPTER 4..................................... 128
R E F E R E N C E S ........................................ .......................................... ............... .... 13 3
BIOGRAPHICAL SKETCH ............................................................. ...............137
TABLE
Table page
4 -1 C om p arativ e statics........................................................................... .................. 119
LIST OF FIGURES
Figure page
1-1 Credit card scheme es .............. .... ........ .. ..... .................. ........7
2-1 Consumers' surplus in bundling and IS cases when vl
3-1 Welfare and interchange fees of Bertrand competition............................................66
3-2 Merchants' acceptance decision when al > a2 (ml > m2) ..................................77
3-3 Welfare and interchange fees of Hotelling competition when
bB b < 2(bB + bs c) ........................................ ....................................... 82
4-1 Division of consumers in three cases of market coverage................. ...............102
4-2 The effect of a price drop on demand ............................ ..... ... ............... 110
A The density function ......... ................................ ....................................... 129
A-2 Effects of an increase in the variance on the cardholder fee (dfldy) ........................130
A-3 Change in the interchange fee and the cardholder fee (df/da)..............................132
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THREE ESSAYS ON BUNDLING AND TWO-SIDED MARKETS
By
Jin Jeon
December 2006
Chair: Jonathan H. Hamilton
Major Department: Economics
This work addresses three issues regarding bundling and two-sided markets. It
starts with a brief summary of the theories of bundling and of two-sided markets in
Chapter 1.
Chapter 2 analyzes various aspects of bundling strategy by the monopolist of a
primary good when it faces competition in the complementary good market. The main
result is that the monopolist can use a bundling strategy in order to avoid commitment
problem that arises in optimal pricing. Bundling increases the monopolist's profits
without the rival's exit from the market. Bundling lowers social welfare in most cases,
while it may increase consumers' surplus. One of the long-run effects of bundling is that
it lowers both firms' incentives to invest in R&D.
Chapter 3 compares welfare implications of monopoly outcome and competitive
outcome. Using a model of the credit card industry with various settings such as Bertrand
and Hotelling competition with single-homing and multi-homing consumers as well as
proprietary and nonproprietary platforms, it is shown that introducing platform
competition in two-side markets may lower social welfare compared to the case of
monopoly platform. In most cases, monopoly pricing maximizes Marshallian social
welfare since the monopolist in a two-sided market can properly internalize indirect
network externalities by setting unbiased prices, while the competing platforms set biased
prices in order to attract the single-homing side.
Chapter 4 analyzes the effects of distribution of consumers' expenditure volumes
on the market outcomes using a model in which two card issuers compete a la Hotelling.
The result shows that the effects of distribution of the expenditure volume are different
for various cases of market coverage. For example, as the variance increases, issuers'
profits decrease when the market is fully covered, while the profits increase when the
market is locally monopolized. It is also shown that the neutrality of the interchange fee
holds in the full-cover market under the no-surcharge-rule. Simulation results are
provided to show other comparative statics that include the possibility of the positive
relationship between the interchange fee and the cardholder fee.
Finally, Chapter 5 summarizes major findings with some policy implications.
CHAPTER 1
INTRODUCTION
This dissertation contains three essays on bundling and two-sided markets. These
topics have recently drawn economists' attention due to the antitrust cases of Kodak and
Microsoft, and movements in some countries to regulate the credit card industry.
In the Kodak case, independent service organizations (ISOs) alleged that Kodak
had unlawfully tied the sale of service for its machines to the sale of parts, in violation of
section 1 of the Sherman Act, and had attempted to monopolize the aftermarket in
violation of section 2 of the Sherman Act. 1 In the Microsoft case, the United States
government filed an antitrust lawsuit against Microsoft for illegally bundling Internet
Explorer with Windows operating system.2
In the credit card industry, antitrust authorities around the world have questioned
some business practices of the credit card networks, which include the collective
determination of the interchange fee, the no-surcharge rule, and the honor-all-cards rule.
As a result, card schemes in some countries such as Australia, United Kingdom, and
South Korea have been required to lower their interchange fees or merchant fees.
To understand these antitrust cases, many economic models have been developed.
In the following sections, brief summaries of the economic theories of bundling and of
two-sided markets will be presented.
1 For more information about the Kodak case, see Klein (1993), Shapiro (1995),
Borenstein, MacKie-Mason, and Netz (1995), and Blair and Herndon (1996).
2 See Gilbert & Katz (2001), Whinston (2001), and Evans, Nichols and Schmalensee
(2001, 2005) for further analysis of the Microsoft case.
1.1 Bundling
Economists' views regarding bundling or tying have shifted dramatically in recent
decades.3 The traditional view of tying can be represented by the leverage theory which
postulates that a firm with monopoly power in one market could use the leverage to
monopolize another market.
The Chicago School criticized the leverage theory, since such leveraging may not
increase the profits of the monopolist. According to the single monopoly profit theorem
supported by the Chicago School, the monopolist earns same profits regardless whether it
ties if the tied good market is perfectly competitive. For example, suppose consumers'
valuation of a combined product of A and B is $10 and marginal cost of producing each
good is $1. Good A is supplied only by the monopolist, and good B is available in a
competitive market at price equal to the marginal cost. Without bundling, the monopolist
can charge $9 for A-and $1 for B-to make $8 as unit profit per good A sold. If the
monopolist sells A and B as a bundle, it can charge $10 for the bundle and earn $8 ($10 -
$1 $1) per unit bundle. So the monopolist cannot increase profits by bundling in this
case.
Economists led by the Chicago School proposed alternative explanations for
bundling based on efficiency rationales. Probably the most common reason for bundling
is it reduces the transaction costs such as consumers' searching costs and firms'
packaging and shipping costs. Examples of this kind of bundling are abundant in the real
3 Bundling is the practice of selling two goods together, while tying is the behavior of
selling one good conditional on the purchase of another good. There is no difference
between tying and bundling if the tied good is valueless without the tying good and two
goods are consumed in fixed proportion. See Tirole (2005) and Nalebuff (2003) for the
discussions of bundling and tying.
world: shoes are sold in pairs; personal computers (PCs) are sold as bundles of the CPU,
a hard drive, a monitor, a keyboard and a mouse; cars are sold with tires and a car audio.
In some sense, most products sold in the real world are bundled goods and services.
Another explanation for bundling in line with the efficiency rationale is price
discrimination. That is, if consumers are heterogeneous in their valuations of products,
bundling has a similar effect as price discrimination.4 This advantage of bundling is
apparent when consumers' valuations are negatively correlated. But bundling can be
profitable even for nonnegative correlation of consumers' valuations (McAfee, McMillan,
and Whinston, 1989). In fact, unless consumers' valuations are perfectly correlated, firms
can increase profits by bundling.5 Since price discrimination usually increases social
welfare as well as firm's profits, bundling motivated by price discrimination increases
efficiency of the economy.
The leverage theory of tying revived with the seminal work of Whinston (1990).
He showed that the Chicago School arguments regarding tying can break down in certain
circumstances which include 1) the monopolized product is not essential for all uses of
the complementary good, and 2) scale economies are present in the complementary good.
If there are uses of the complementary good that do not require the primary good, the
monopolist of the primary good cannot capture all profits by selling the primary good
only. So the first feature provides an incentive for the monopolist of the primary good to
exclude rival producers of the complementary good. The second feature provides the
monopolist with the ability to exclude rivals, since foreclosure of sales in the
4 See Adams and Yellen (1976) and Schmalensee (1984).
5 Bakos and Brynjolfsson (1999) show the benefit of a very large scale bundling based on
the Law of Large Numbers.
complementary market, combined with barriers to entry through scale economies, can
keep rival producers of the complementary good out of the market.6
Bundling can also be used to preserve the monopolist's market power in the
primary good market by preventing entry into the complementary market at the first stage
(Carlton and Waldman, 2002a). This explains the possibility that Microsoft bundles
Internet Explorer with Windows OS in order to preserve the monopoly position in the OS
market, since Netscape's Navigator combined with Java technology could become a
middleware on which other application programs can run regardless of the OS.
Choi and Stefanidis (2001) and Choi (2004) analyze the effects of tying on R&D
incentives. The former shows that tying arrangement of an incumbent firm that produces
two complementary goods and faces possible entries in both markets reduces entrants'
R&D incentives since each entrant's success is dependent on the other's success. The
latter analyzes R&D competition between the incumbent and the entrant, and shows that
tying increases the incumbent's incentives to R&D since it can spread out the costs of
R&D over a larger number of units, whereas the entrant's R&D incentives decrease.7
Chapter 2 presents a model of bundling that follows the basic ideas of the leverage
theory. It shows that the monopolist of a primary good that faces competition in the
aftermarkets can use the bundling strategy to increase profits to the detriment of the rival
firm. Aftermarkets are markets for goods or services used together with durable
equipment but purchased after the consumer has invested in the equipment. Examples
include maintenance services and parts, application programs for operating systems, and
6 Nalebuff (2004) and Carlton and Waldman (2005a) also present models that show the
entry deterrence effect in the tied good market.
7 In chapter 2, I show that bundling reduces R&D incentives of the monopolist as well as
of the rival.
software upgrades. One of the key elements of the aftermarket is that consumers buy the
complementary goods after they have bought the primary good. For the monopolist of the
primary good, the best way to maximize its profits is to commit to the second period
complementary price. If this commitment is not possible or implementable, bundling can
be used.
Unlike most of the previous models of the leverage theory, market foreclosure is
not the goal of the bundling in this model. On the contrary, the existence of the rival
firms is beneficial to the monopolist in some sense since it can capture some surplus
generated by the rival firm's product.
1.2 Two-Sided Markets
Two-sided markets are defined as markets in which end-users of two distinctive
sides obtain benefits from interacting with each other over a common platform. 8 These
markets are characterized by indirect network externalities, i.e., benefits of one side
depend on the size of the other side.9 According to Rochet and Tirole (2005), a necessary
condition for a market to be two-sided is that the Coase theorem does not apply to the
transaction between the two sides. That is, any change in the price structure, holding
constant the total level of prices faced by two parties, affects participation levels and the
number of interactions on the platform since costs on one side cannot be completely
passed through to the other side.
SFor general introductions to the two-sided market, see Roson (2005a), and Evans and
Schmalensee (2005).
9 In some cases such as media industries, indirect network externalities can be negative
since the number of advertisers has a negative impact on readers, viewers, or listeners.
See Reisinger (2004) for the analysis of two-sided markets with negative externalities.
Examples of the two-sided market are abundant in the real world. Shopping malls
need to attract merchants as well as shoppers. Videogame consoles compete for game
developers as well as gamers. Credit card schemes try to attract cardholders as well as
merchants who accept the cards. Newspapers need to attract advertisers as well as
readers. 10 Figure 1-1 shows the structure of the two-sided market in case of the credit
card industry, both proprietary and nonproprietary schemes.
Although some features of two-sided markets have been recognized and studied for
a long time, 11 it is only recently that a general theory of two-sided markets emerged. 12
The surge of interest in two-sided markets was partly triggered by a series of antitrust
cases against the credit card industry in many industrialized countries including the
United States, Europe and Australia. The literature on the credit card industry has found
that the industry has special characteristics; hence conventional antitrust policies may not
be applicable to the industry. 13
Wright (2004b) summarizes fallacies that can arise from using conventional
wisdom from one-sided markets in two-sided markets, which include: an efficient price
structure should be set to reflect relative costs; a high price-cost margin indicates market
power; a price below marginal cost indicates predation; an increase in competition
necessarily results in a more efficient structure of prices; and an increase in competition
necessarily results in a more balanced price structure.
10 See Rochet and Tirole (2003) for more examples of the two-sided market.
11 For example, Baxter (1983) realized the two-sidedness of the credit card industry.
12 The seminal papers include Armstrong (2005), Caillaud and Jullien (2003), and Rochet
and Tirole (2003).
13 The literature includes Gans and King (2003), Katz (2001), Rochet and Tirole (2002),
Schmalensee (2002), Wright (2003a, 2003b, 2004a).
The theory of two-sided markets is related to the theories of network externalities
and of multi-product pricing. While the literature on network externalities has found that
in some industries there exist externalities that are not internalized by end-users, models
are developed in the context of one-sided markets. 14 Theories of multi-product pricing
stress the importance of price structures, but ignore externalities in the consumption of
Issuer ^ (a : interchange fee) Aq r
....................................... t .................................... P l a t fo r m ........................................... .........................................
Platform
(Card scheme)
Pays p +f (CPaysp m
(f: cardholder fee) (m : merchant fee)
Cardholder 4---------------------- Merchant
Sells good at price
(a) Nonproprietary card scheme
Platform
(Card scheme)
Paysp +f Paysp-m
(f: cardholder fee) (m : merchant fee)
(Cardholder ----------------------- Merchant
Sells good at price
(b) Proprietary card scheme
Figure 1-1. Credit card schemes
14 See Katz and Shapiro (1985, 1986), and Farrell and Saloner (1985, 1986).
different goods since the same consumer buys both goods. That is, the buyer of one
product (say, razor) internalizes the benefits that he will derive from buying the other
product (blades). The two-sided market theory starts from the observation that there exist
some industries in which consumers on one side do not internalize the externalities they
generate on the other side. The role of platforms in two-sided markets is to internalize
these indirect externalities by charging appropriate prices to each side.
In order to get both sides on board and to balance demands of two sides, platforms
in two-sided markets must carefully choose price structures as well as total price levels.15
So it is possible that one side is charged below marginal cost of serving that side, which
would be regarded as predatory pricing in a standard one-sided market. For this reason,
many shopping malls offers free parking service to shoppers, and cardholders usually pay
no service fees or even negative prices in the form of various rebates.
In a standard one-sided market, the price is determined by the marginal cost and the
own price elasticity, as is characterized by Lerner's formula.16 In two-sided markets,
however, there are other factors that affect the price charged to each side. These are
relative size of cross-group externalities and whether agents on each side single-home or
multi-home. 17
If one side exerts larger externalities on the other side than vice versa, then the
platform will set a lower price for this side, ceterisparibus. In a media industry, for
15 In the credit card industry, non-proprietary card schemes choose interchange fees
which affect the price structure of two sides.
16 The standard Lerner'-s formula is -C 1
SThe standard Lerner's formula is or p = c, where is the price, c is the
P E E-I
p e s-1
marginal cost, and e is the own price elasticity.
17 An end-user is "single-homing" if she uses one platform, and "multi-homing" if she
uses multiple platforms.
example, viewers pay below the marginal cost of serving while advertisers pay above the
marginal cost since the externalities from viewers to advertisers are larger than those
from advertisers to viewers.
When two or more platforms compete with each other, end-users may join a single
platform or multiple platforms, depending on the benefits and costs of joining platforms.
Theoretically, three possible cases emerge: (i) both sides single-home, (ii) one side
single-homes while the other side multi-homes, and (iii) both sides multi-home. 18 If
interacting with the other side is the main purpose of joining a platform, one can expect
case (iii) is not common since end-users of one side need not join multiple platforms if all
members of the other side multi-home.19 For example, if every merchant accepts all kinds
of credit cards, consumers need to carry only one card for transaction purposes. Case (i)
is also not common since end-users of one side can increase interaction with the other
side by joining multiple platforms. As long as the increased benefit exceeds the cost of
joining additional platform, the end-users will multi-home.
On the contrary, one can find many examples of case (ii) in the real world.
Advertisers place ads in several newspapers while readers usually subscribe to only one
newspaper. Game developers make the same game for various videogame consoles while
gamers each own a single console. Finally, merchants accept multiple cards while
consumers use a single card.20
18 In most of the models on two-sided markets, single-homing and multi-homing of end-
users are pre-determined for analytical tractability. For an analysis of endogenous multi-
homing, see Roson (2005b).
19 See also Gabszewicz and Wauthy (2004).
20 According to an empirical study by Rysman (2006), most consumers put a great
majority of their payment card purchases on a single network, even when they own
multiple cards from different networks.
When end-users of one side single-home while those of the other side multi-home,
the single-homing side becomes a "bottleneck" (Armstrong, 2005). Platforms compete
for the single-homing side, so they will charge lower price to that side. As is shown in
Chapter 3, platforms competing for the single-homing side may find themselves in a
situation of the "Prisoner's Dilemma". That is, a lower price for the single-homing side
combined with a higher price for the multi-homing side can decrease total transaction
volume and/or total profits compared to the monopoly outcome. Further, competition in
two-sided markets may lower social welfare since monopoly platforms can properly
internalize the indirect externalities by charging unbiased prices, while competing
platforms are likely to distort the price structure in favor of the single-homing side.
Chapter 3 presents a model of the credit card industry with various settings
including single-homing vs. multi-homing cardholders, competition between identical
card schemes (Bertrand competition) or differentiated schemes (Hotelling competition),
and proprietary vs. non-proprietary card schemes. The main finding is that, unlike in a
standard one-sided market, competition does not increase social welfare regardless of the
model settings.
Chapter 4 tackles the assumption made by most models on the credit card industry
that cardholders spend the same amounts with credit cards. By allowing heterogeneous
expenditures among consumers, it shows the effects of a change in the variance of the
expenditure on the equilibrium prices and profits. The results show that the effects are
different depending on whether the market is fully covered, partially covered, or locally
monopolized.
CHAPTER 2
BUNDLING AND COMMITMENT PROBLEM IN THE AFTERMARKET
2.1 Introduction
A monopolist of a primary good that faces competition in the aftermarket of the
complementary goods often uses a bundling or tying strategy. Traditionally, bundling
was viewed as a practice of transferring the monopoly power in the tying market to the
tied market. This so-called "leverage theory" has been criticized by many economists
associated with the Chicago School in that there exist other motives of bundling such as
efficiency-enhancement and price discrimination. Further, they show that there are many
circumstances in which firms cannot increase profits by leveraging monopoly power in
one market to the other market, which is known as the single monopoly profit theorem.
Since the seminal work of Whinston (1990), the leverage theory revived as many
models have been developed to show that a monopolist can use tying or bundling
strategically in order to deter entry to the complementary market and/or primary market.
The research was in part stimulated by the antitrust case against Microsoft filed in 1998,
in which U.S government argued that Microsoft illegally bundles Internet Explorer with
Windows operating system.1 Most of the models in this line, however, have a
commitment problem since the bundling decision or bundling price is not credible when
the entrant actually enters or does not exit the market.
1 For further analyses of the Microsoft case, see Gilbert & Katz (2001), Whinston (2001),
and Evans, Nichols and Schmalensee (2001).
This paper stands in the tradition of the leverage theory and shows that the
monopolist of a primary good can use a bundling strategy to increase profits as well as
the market share in the complementary good market. Unlike the previous models, the
monopolist's profits increase with bundling even if the rival does not exit the market. On
the contrary, the existence of a rival firm is beneficial to the monopolist in some sense
since the monopolist can capture some surplus generated by the rival firm's
complementary good.
The model presented here is especially useful for the analysis of the Microsoft case.
Many new features added to-i.e., bundled with-the Windows operating system (OS)
had been independent application programs produced by other firms. For example,
Netscape's Navigator was a dominant Internet browser before Microsoft developed
Internet Explorer. Therefore, it is Microsoft, not Netscape, that entered the Internet
browser market. Since Netscape's software development cost is already a sunk cost when
Microsoft makes a bundling decision, the entry deterrent effect of bundling cannot be
applied.
The main result is that the monopolist can use bundling to avoid the commitment
problem2 arising in the optimal pricing when consumers purchase the complementary
good after they have bought the primary good. If the monopolist cannot commit to its
optimal price for the complementary good at the first stage when consumers buy the
primary good, then it may have to charge a lower price for the primary good and a higher
price for the complementary good compared to its optimal set of prices since consumers
2 This commitment problem is different from the one in the previous literature, in which
the commitment problem arises since the bundling price is not credible if the would-be
entrant actually enters the market.
rationally expect that the monopolist may raise its complementary good price after they
have bought the primary goods. A double marginalization problem arises in this case
since the monopolist has to charge the price that maximizes its second stage profits, while
it also charges a monopoly price for the primary good at the first stage. Bundling makes it
possible for the monopolist to avoid the double marginalization problem by implicitly
charging a price equal to zero for the complementary good.
The model also shows that bundling generally lowers Marshallian social welfare
except for the extreme case when the monopolist's bundled good is sufficiently superior
to the rival's good. Social welfare decreases with bundling mainly because it lowers the
rival's profits. Consumers' surplus generally increases with bundling. However,
consumers' surplus also decreases when the rival's complementary good is sufficiently
superior to the monopolist's.
The last result shows the effect of bundling on R&D investments. In contrast to the
previous result of Choi (2004) that shows tying lowers the rival firm's incentive to invest
in R&D while it increases the monopolist's incentive, I show that bundling lowers both
firms' incentives to make R&D investments.
The literature on bundling or tying is divided into two groups one finds the
incentive to bundle from the efficiency-enhancing motives, and the other finds it from
anticompetitive motives.3 In the real world, examples of bundling motivated by
efficiency reason are abundant. Shoe makers sell shoes as a pair, which reduces
transaction costs such as consumers' searching costs and producers' costs of shipping and
packaging. The personal computer is another example as it is a bundle of many parts such
3 For a full review of the literature on bundling, see Carlton and Waldman (2005b).
as the CPU, a memory card, a hard drive, a keyboard, a mouse, and a monitor.4 Carlton
and Waldman (2002b) explain another efficiency motive for tying by showing that
producers of a primary good may use tying in order to induce consumers to make
efficient purchase decisions in the aftermarket when consumers can buy the
complementary goods in variable proportions. If the primary good is supplied at a
monopoly price while the complementary good is provided competitively, consumers
purchase too much of the complementary good and too little of the primary good. Tying
can reduce this inefficiency and increase profits.
Adams and Yellen (1976) provide a price discrimination motive for tying. Using
some examples, they show that if consumers are heterogeneous in their valuations for the
products, bundling has a similar effect as price discrimination. This advantage of
bundling is apparent when consumers' valuations are negatively correlated. Schmalensee
(1984) formalizes this theory assuming consumers' valuations follow a normal
distribution. McAfee, McMillan, and Whinston (1989) show that bundling can be
profitable even for nonnegative correlation of consumers' valuations. Bakos and
Brynjolfsson (1999) show the benefit of a very large scale bundling of information goods
based on Law of Large Numbers. Since price discrimination usually increases social
welfare with an increase in total output, tying or bundling motivated by price
discrimination can be welfare improving.
The anticompetitive motive of tying is reexamined by Whinston (1990). He
recognizes that Chicago School's criticism of leverage theory only applies when the
complementary good market is perfectly competitive and characterized by constant
4 See Evans and Salinger (2005) for efficiency-enhancing motive of tying.
returns to scale, and the primary good is essential for use of the complementary good. He
shows that in an oligopoly market with increasing returns to scale, tying of two
independent goods can deter entry by reducing the entrant's profits below the entry cost.
As was mentioned earlier, however, his model has a credibility problem since bundling is
not profitable if entrance actually occurs.
Nalebuff (2004) also shows that bundling can be used to deter entry, but without a
commitment problem since in his model the incumbent makes higher profits with
bundling than independent sale even when the would-be entrant actually enters.5 Carlton
and Waldman (2002a) focus on the ability of tying to enhance a monopolist's market
power in the primary market. Their model shows that by preventing entry into the
complementary market at the first stage, tying can also stop the alternative producer from
entering the primary market at the second stage.
Carlton and Waldman (2005a) shows that if the primary good is a durable good and
upgrades for the complementary good are possible, the monopolist may use a tying
strategy at the first stage in order to capture all the upgrade profits at the second stage.
Especially when the rival's complementary good is superior to the monopolist's, the only
way the monopolist sells second-period upgrades is to eliminate the rival's product in the
first period by tying its own complementary good with its monopolized primary good. By
showing that tying can be used strategically even when the primary good is essential for
use of the complementary good, it provides another condition under which the Chicago
School argument breaks down.
5 However, the optimal bundling price is higher when the entrant enters than the price
that is used to threaten the entrant. So there exists a credibility problem with the price of
the bundled good.
The model presented here also assumes the primary good is essential, but the
primary good is not necessarily a durable good and constant returns to scale prevail. So it
can be added to the conditions under which the Chicago School argument breaks down
that bundling can be used strategically when consumers buy the primary good and the
complementary good sequentially.
The rest of Chapter 2 is organized in the following way. Section 2.2 describes the
basic setting of the model. Sections 2.3 to 2.5 show and compare the cases of independent
sale, pricing with commitment, and bundling, respectively. Section 2.6 analyzes the
welfare effect of bundling. Section 2.7 is devoted to the effect of bundling on R&D
investments. The last section summarizes the results.
2.2 The Model
Suppose there are two goods and two firms in an industry. A primary good is
produced solely by a monopolist, firm 1. The other good is a complementary good that is
produced by both the monopolist and a rival, firm 2. The purchases of the primary good
and the complementary good are made sequentially, i.e., consumers buy the
complementary good after they have bought the primary good. Consumers buy at most
one unit of each good,6 and are divided into two groups. Both groups have same
reservation value vo for the primary good. For the complementary good, however, one
group has zero reservation value and the other group has positive reservation value v,,
where i= 1,2 indicates the producer.7 For modeling convenience, it is assumed that the
6 So there is no variable proportion issue.
7 Consumption of the complementary good may increase the reservation value of the
primary good. It is assumed that v, also includes this additional value.
marginal cost of producing each good is zero and there is no fixed cost for producing any
good.8
The PC software industry fits in this model, in which Microsoft Windows OS is the
monopolized primary good and other application programs are complementary goods.
Microsoft also produces application programs that compete with others in the
complementary good market. Sometimes Microsoft bundles application programs such as
an Internet browser and a media player that could be sold separately into Windows OS.
Consumers usually buy the Windows OS at the time they buy a PC, then buy application
software later.
Let the total number of consumers be normalized to one, and ac be the portion of the
consumers, group S, who have positive valuations for the complementary good. It is
assumed that the consumers in S are distributed uniformly on the unit interval, in which
the monopolist and firm 2 are located at 0 and 1, respectively.
The two complimentary goods are differentiated in a Hotelling fashion. A
consumer located at x incurs an additional transportation cost tx when she buys the
monopolist's complementary good, and t(1 x) when she buys firm 2's. So the gross
utility of the complementary good for the consumer is vl tx when she buys from the
monopolist, and v2 t(1 x) when she buys from firm 2. vl and v2 are assumed to be
greater than t in order to make sure that consumers in S cannot have a negative gross
utility for any complementary good regardless of their positions. Further, in order to
8 Unlike the models that explain tying as an entry deterrence device, the model in this
paper assumes constant returns to scale.
make sure that all the consumers in S buy the complementary goods at equilibrium, it is
assumed that9
v + V2 > 3t (2-1)
The model presented here allows a difference between vi and v2 in order to analyze
bundling decision when the monopolist produces inferior-or superior-complementary
good and the effect of bundling on R&D investments. But the difference is assumed to be
less than t, i.e.,
I v v2 < t (2-2)
since otherwise all consumers find one of the complementary goods superior to the other
good. 10
In the software industry, the primary good is the operating system (OS), and
application programs like an Internet browser or a word processor are examples of
complementary goods. The OS itself can be seen a collection of many functions and
commands. Bakos and Brynjolfsson (1999) show that the reservation values among
consumers of a large scale bundle converge to a single number, which justifies the
assumption that consumers have the same valuation for the primary good. A single
application program, however, is not as broadly used as an OS, so the valuation for the
9 The prices chosen by two firms could be too high so that some of the consumers in S
may not want to buy the complementary good. The assumption vl+v2 > 3t guarantees
that every consumer in S buy a complementary good at equilibrium.
10 This is also a sufficient condition for the existence of the various equilibria.
complementary good may vary among consumers. Furthermore, not all the application
programs are produced for all consumers. Some of them are developed for a certain
group of consumers such as business customers.
The game consists of two stages.11 At the first stage consumers buy the primary
good or bundled good at the price that the monopolist sets. The monopolist can set the
price of its own complementary good with or without commitment, or sell both goods as
a bundle. At the second stage, consumers buy one of the complementary goods, the prices
of which are determined by the competition between the two firms.
Letpo,pi, and p2 be the prices of the monopolist's primary good, the monopolist's
complementary good, and firm 2's complementary good, respectively. Then the net
utilities of the consumer located at x if she consumes the primary good only, the primary
good with the monopolist's complementary good, and the primary good with firm 2's
complementary good are, respectively,
Uo= vo -po
u = Vo+ vi- tx-po-pl
u2 = V+ V2- t(1 -x) -Po-P2
The consumer will buy only the primary good if
Uo > U1, Uo > u2, and uo > 0
She will buy the primary good and the monopolist's complementary good if
11 In section 2.7, an earlier stage will be added at which two firms make investment
decisions that determine v,'s.
ul > u2, ul > uo, and ul > 0
She will buy the primary good and firm 2's complementary good if
u2 u> u2 > uo, and u2 > 0
Lastly, she will buy nothing if
uo < 0, ul < 0, and u2 < 0
2.3 Independent Sale without Commitment
In this section, it is assumed that the monopolist cannot commit topl at the first
stage. Without commitment, pi must be chosen to be optimal at the second stage. That is,
in game-theoretic terms, the equilibrium price must be subgame perfect.
As in a standard sequential game, the equilibrium set of prices can be obtained by
backward induction. Let x* be the critical consumer who is indifferent between the
monopolist's complementary good and firm 2's good. One can find this critical consumer
by solving vi tx*-pl = v2 t(1 x*) -p2, which gives
1 v, v 2 2 1-
x --+ (2-3)
2 2t
There are two cases to be considered: when the monopolist sells the primary good
to all consumers, and when it sells its products to group S only. Consider first the case
that the monopolist sells the primary good to all consumers. At the second stage, the
monopolist will setpl to maximize oapx*, while firm 2 will setp2 to maximize ap2(l -x*).
By solving each firm's maximization problem, one can obtain the following best
response functions:
vl V + p +t
2
i,j= 1,2andi j
from which one can obtain the following equilibrium prices for the case of the
independent sale without commitment (IA case):
t V V
pA=v +t, i,j= ,2andi j
3
Plugging these into (2-3) gives the location of the critical consumer:
IA 1 V1 -V2
2 6t
At the first stage, the monopolist will set the price of the primary good equal to vo
since consumers outside of group S will not buy the good for the price higher than vo:
One needs to check whether consumers actually buy the goods for this set of prices.
This can be done by plugging the prices into the net utility of the critical consumer, i.e.,
A A +V2 3t
Po -p1 >0
2
(2-4)
U, (XIA) = Vg + V1 tXI
where the last inequality holds because of the assumption given in (2-1). As was noted in
footnote 10, this assumption guarantees that all consumers in S buy both goods at
equilibrium.
The profits of the firms at equilibrium are
A p t px(vI v2 + 3t)2
7Tj = P, + apDX, =VO +-a
18t
IA IA 1 a(v2- v + 3t)2
18t
The monopolist may find it profitable to sell the primary goods exclusively to
group S by charging the price higher than vo. If a consumer located at x have bought the
primary good at the first stage, the maximum prices she is willing to pay for the
monopolist's and firm 2's complementary goods at the second stage are v tx and v2
t(1 x), respectively, regardless how much she paid for the primary good at the first stage.
Since the payment at stage one is a sunk cost to the consumer, she will buy a
complementary good as long as the net utility from the complementary good is non-
negative. This implies that when the monopolist sells the primary good to group S only
without commitment to p (IS case), the equilibrium prices and the location of the critical
consumer at the second stage are exactly the same as in the IA case. 12 That is,
V -V
s= v- +t, i,j=1,2andi j
3
12 There may exist multiple equilibria because of the coordination problem among
consumers. For example, suppose consumers around at xs did not buy the primary good
at stage 1. Then at stage 2, the two firms will charge higher prices than p,s. At this
price set, consumers who did not buy the base good will be satisfied with their decision.
XIS 1 + 2
2 6t
When consumers buy the primary good at stage 1, they rationally predict that the
second period prices of the complementary goods are pf So the monopolist will set the
primary good price to make the critical consumer indifferent between buying the
complementary good and not buying, which yields the following equilibrium price:
IS v +v2- 3t
Po = Vo +
2
Note that the primary good price is higher than vo as is expected. By excluding the
consumers who buy only the primary good, the monopolist can charge a higher price in
order to capture some surplus that would otherwise be enjoyed by the consumers of the
complementary goods.
The monopolist's profits may increase or decrease depending on the size of a,
while firm 2's profits remain the same as in the IA case since the price and the quantity
demanded in IS case are exactly the same as in the IA case:
IJs = a(pis is) = a ( -v1 2)2 5v, +v2 -6t
01 1 1 18t 6
s = (1 I) a(v2 v + 3t)2
18t
By comparing tis and A, one can derive the condition in which the monopolist
prefers the IS outcome to the IA outcome:
2v ^Is
a> --- a
2v + (v + v, 3t)
^IS
Note that a lies between 0 and 1 since v +v2 3t > 0 is assumed in (2-1).
2.4 Independent Sale with Commitment
The results of the previous section may not be optimal for the monopolist if it can
choose bothpo and pi simultaneously at the first stage and commit topi. To see this,
suppose the monopolist can set both prices at the first stage with commitment. As in the
previous section, one can distinguish two cases depending on the coverage of the primary
good market. When the monopolist sells its primary good to all consumers with
commitment topl (CA case), the model shrinks to a simple game in which the
monopolist setpl at the first stage and firm 2 setp2 at the second stage since the primary
good price should be set equal to vo, i.e., pf = 0 The equilibrium prices of the
complementary goods can be derived using a standard Stackelberg leader-follower model.
The equilibrium can be found using backward induction. At the second stage, the
critical consumer who is indifferent between the monopolist's complementary good and
firm 2's good is determined by (2-3) with p2 replaced by firm 2's best response function
given by (2-4), i.e.,
3 Vl -v v -p p
x -+ (2-5)
4 4t
The monopolist will set p to maximize ~ap x*, which gives the following optimal
price:
CA V1 V2 +3t
2
The remaining equilibrium values can be obtained by plugging this into (2-4) and
(2-5):
CA 2 v + 5t
P2
CA 3 V1 V2
8 8t
CA (v1 vi + 3t)2
16t
CA a(v2 v1 + 5t)2
2 32t
The differences between the equilibrium prices of CA case and IA case are
CPA _pA v -v+ 3t > 0
6
CA IA v v2 3t>
P2 -p = > 0
12
The price differences are positive since the difference between vl and v2 is assumed
to be less than t. Since the monopolist's complementary good is a substitute for firm 2's
good, pi andp2 are strategic complements. If one firm can set its price first, it will set a
higher price so that the rival also raises its own price compared to the simultaneous move
game. With the increase in the prices, both firms enjoy higher profits as the following
calculation shows:
c -- a(vl v + 3t)2 > 0
144t
CA (vl v2 + 3t)[27t -7(v, -v)] > 0
288t
The profit of the monopolist must increase since it chooses a different price even if
it could commit to p1A at the first stage. Firm 2's profit also increases as both firms'
prices of complementary goods increase while the price of the primary good remains the
same.
When the monopolist covers only the consumers in group S with commitment topl
(CS case), the equilibrium can be found in a similar way as in the CA case. At the second
stage, firm 2's best response function is the same as (2-4) and the critical consumer is also
determined by (2-5). Since the monopolist will make the critical consumer indifferent
between buying and not buying the complementary good, po will be set to satisfy the
following condition:
po= vo+ vl-pi- tx' (2-6)
Using (2-5) and (2-6), the monopolist's profits can be rewritten as a function of p
in the following way:
S 3v, + v, 3t P (v, v2 P1)
Tr, = a(po + px) = a + +3t (V
Maximizing this profit function w.r.t. pi yields the optimal price for the
monopolist's complementary good, which is
CS V_ V2
2
Plugging this back to (2-4), (2-5) and (2-6), one can derive the remaining
equilibrium values:
s 3v, +5v2 6t
PoCS = V +-
8
cs v2 -v + 2t
P2 4
4
cs 3 v, -v2
x^ =-+
4 8t
cs (I -v )2 3v, + V -3t
S16t 4
2cs a(v2 +2t)2
32t
The differences between the equilibrium prices of CS case and IS case are as
follows:
Is Is V v2 + 6t
pCS <0
8
C-S 1 <- 6
cs is vI 6t
P2 p2 12
12
When the monopolist can commit to its complementary good price, it charges a
higher price for the primary good and a lower price for the complementary good. And the
rival firm also charges a lower price for its own complementary good. Sincepl andp2 are
strategic complements, the monopolist can induce firm 2 to decrease p2 by lowering pi,
which makes it possible for the monopolist to raise po for higher profits. This would not
be possible if the monopolist cannot commit topl at the first stage since the monopolist
has an incentive to raise the complementary good price at the second stage after
consumers have bought the primary good.
The difference between the profits of CS case and IS case are as follows:
s_ Is a(v,2 + 6t)2>
144t
cs I_ s a[18t-7(v v2)][6t -(v v2 ) 0
288t
The monopolist's profits increase when it can commit as in CA case. However, firm
2's profits decrease since the monopolist can capture some of the consumers' surplus
generated by firm 2's complementary good by charging a higher price for the primary
good.
Comparing TCS and z c, one can derive the following condition for the
monopolist to prefer the CS outcome to the CA outcome:
16vo ^cs
a > -a
16vo + (6v, +10v2 -21t)
CS
a lies between 0 and 1 since 6v, +10v2 21t = 8(v, +v2 -3t)+ 2(v2 -v)+3t > 0
^IS ^CS
from the assumptions given in (2-1) and (2-2). The difference between a and a is
^Is ^cs 2vo(2v 2v, +3t)
a =->0
(2vo + v, + v2 3t)(16v0 + 6v, +10v2 21t)
The critical level of a with commitment is lower than with independent sale since
the profit gain from commitment is higher in the CS case than in the CA case.13 That is,
the monopolist is willing to sell both goods to a smaller group of consumers when it can
commit to the price of its own complementary good sold in the second period.
The problem that the monopolist earns lower profits when it cannot commit to the
second period price of the complementary good is common in cases of durable goods
with aftermarkets.14 That is, rational consumers expect that the monopolist will set its
second period price to maximize its second period profit regardless of its choice in the
first period. The monopolist has an incentive to charge a higherpl after consumers in S
have bought the primary good at the first stage, since the price consumers have paid for
the primary goods is sunk cost at stage 2.15 If the monopolist cannot commit to p1cs
therefore, some consumers in S would not buy the primary good at the first stage. So the
monopolist would have to set a lowerpo (p- ) and a higher p (pf ) because of the hold-
up problem.
One of the problems in relation to the pricing with commitment is that the optimal
prices may not be implemented since ps is negative when vl < v2.16 The bundling
13 Note that (cs s)-( RCA A a[2(v2 v)+ 3t]
=t, -)T1 16 >0
16
14 See Blair and Herndon (1996)
15 After consumers have bought the primary goods at stage 1, the monopolist has an
incentive to charge p1s which is higher than pcs.
16 If the marginal cost of producing the complementary good is positive, the optimal price
strategy that will be presented in the following section can resolve this problem as well as
the commitment problem.
2.5 Bundling: An Alternative Pricing Strategy without Commitment
An alternative strategy for the monopolist when it cannot commit to the second
period price or implement a negative price is bundling. That is, it sells both the primary
good and its own complementary good for a single price. Note first that it is not optimal
for the monopolist to sell the bundled good to all consumers since the bundled price must
be equal to vo in that case. So the monopolist will sell the bundled good to group S only if
it chooses the bundling strategy.
It is assumed that tying is reversible, i.e., a consumer who buys a bundled good
may also buy another complementary good and consume it with the primary good.17
Further, suppose consumers use only one complementary good, so the monopolist's
bundled complementary good is valueless to the consumers who use firm 2's
complementary good.18
At the second stage, a consumer who has bought the bundled good earlier may buy
firm 2's good or not, depending on her location x. If she buys firm 2's complementary
good, her net gain at stage 2 is v2 t(1 x) -p2. If she does not buy, she can use the
monopolist's complementary good included in the bundle without extra cost, and get net
gain of v tx. So the critical consumer who is indifferent between buying firm 2's
complementary good and using the bundled complementary good is
can be positive even if v < v2.
17 In the software industry, a consumer who uses Windows OS bundled with Internet
Explorer may install another Internet browser.
18 As long as there is no compatibility problem, consumers will use only one
complementary good they prefer.
1 v, v, + p
x- +-- (2-7)
2 2t
Since the price paid for the bundled good is a sunk cost at the second stage, the
critical consumer is determined by p2 only. Firm 2 will choosep2 to maximize p2( -x*),
which yields the following optimal price for firm 2:
BS V 2- V, + t
P2 2
Plugging this into (2-7) gives the location of the critical consumer as follows:
xs 3 v, v2
x = -+
4 4t
For this critical consumer to exist between 0 and 1, it is required that -3tvi- v2
So the assumption of I vi 2 | < t given in (2-2) is also a sufficient condition for the
existence of a bundling equilibrium without the exit of the rival firm. If vi -v2 > t, then all
consumers buy the bundled good only so the rival firm will exit the market. If vi v2 <
-3t, on the other hand, all consumers buy both the bundled good and firm 2's
complementary good.
At stage 1, the monopolist will set the bundled good price, pb, that makes the
critical consumer indifferent between buying and not buying: 19
19 At the second stage, the monopolist may have an incentive to unbundle the product and
sell the primary good to the consumers outside of group S as long as the consumer's
second stage valuation for the good is positive, i.e., higher than the marginal cost.
Knowing this, some consumers in group S may want to wait until the second period,
S 3v, +v2 -3t
PBS = 2
Pb Vo+-
When consumers choose the monopolist's complementary good, the total price for
the primary good and the complementary good decreases compared to the IS case since
BPS _s r V j v -3t
PBS Sp 2 +P ) 1 < 0 (2-8)
12
If consumers buy firm 2's complementary good as well as the monopolist's
bundled good, the total price increases compared to the IS case since
BS + p) __ (pS + pI) =V -- 2 +- 3t
(p +BS + )= 1+ > 0 (2-9)
12
Comparing (2-8) and (2-9) one can find that the total price decrease for the
consumers of monopolist's complementary good is exactly the same as the total price
increase for the consumers of firm 2's good. With the decrease of the total price, the
number of consumers who choose to use the monopolist's complementary good increases
compared to the IS case as the following shows:
xBS _- 3t >0 (2-10)
12t
The profits of the firms are
which will lower the monopolist's profits. To avoid this, the monopolist will try to
commit to not unbundling. One way to commit is to make unbundling technologically
difficult or impossible, as Microsoft combined Internet Explorer with Windows OS.
S aBS 3v1 +v2 -3t]
BS BS (_BS) a(v-2_ + t)2
2 = ap2 (8-XBS)=t
8t
The following proposition shows that bundling increases monopolist's profits
compared to the IS case.
Proposition 2-1 Suppose the monopolist sells its goods to consumers in S only. Then the
monopolist's profit in the bundling equilibrium is strictly higher than under IS, but not
higher than under CS.
Proof. The difference between profits with bundling and IS case is
BS IS 3
S- (VI 2+ 3t)(v v2 t) >
18t 2
The inequality holds since | vl v2 < t.
On the other hand, the difference between profits with bundling and CS is
-BS C=s a(v 2 <0
S16t
16t
where the inequality holds steadily when vi # v2.
Q.E.D.
In most of the previous analysis of bundling based on the leverage theory, one of
the main purposes of the bundling strategy is to foreclose the complementary good
market. By lowering expected profits of the would-be entrants, bundling can be used to
deter entry. The difference between the previous models and the current one is that
bundling increases the profits of the monopolist even though the rival firm does not exit
the market. On the contrary, the existence of the rival firm helps the monopolist in some
sense since it creates demand for the monopolist's bundled good.
When compared to the CS case, bundling strategy generates the same profits for the
monopolist if vl = v2. Technically, bundling strategy is equivalent to settingpl =pb andp2
= 0. When vl = v2, the equilibrium commitment price for the monopolist's
complementary good, pS is zero, hence the monopolist's profits of bundling and CS
cases are equal.20 Since the optimal commitment price is either positive or negative if vl
/ v2, the monopolist's bundling profits is less than the CS case.
By comparing bundling case with IA case, one can find the critical level of a above
which the monopolist finds bundling is more profitable if commitment is not possible.
The difference between the monopolist's profits is
BS IA V (v, V + 5v, +72 -15t1
I 18t 12
20 If the marginal cost (MC) of producing the complementary good is positive (c), the
optimal commitment price for the good is c when vl = v2 since the monopolist can avoid
double marginalization problem by MC pricing for the downstream good. In this case,
bundling cannot generate same profits as the CS case even when vl = v2 since it
implicitly charges zero price instead of the one equal to MC.
And the critical level of a at which the monopolist is indifferent between bundling
and independent sale is
^BS 36v t
36vot + 3t(5v, +7v2 -15t)- 2(v, v2)2
^BS
If a is higher than a the monopolist can make higher profit by bundling both
goods together and selling it to group S only than by separately selling the primary good
to all consumers. That is, bundling is profitable if the complementary good is widely used
by the consumers of the primary good. In the software industry, Microsoft bundles
Internet Explorer into Windows OS, while it sells MS Office as an independent product
since Internet browser is a widely used product whereas the Office products are used by
relatively small group of consumers.
^BS
Note that a lies between 0 and 1 since
3t(5v, + 7v2 -15t)- 2(v v2 )2 > 3t(5v, + 7v2 -15t)- 2t2
= 15t(v, + v2 3t) + 2t(3v2 t) > 0
where the first and second inequalities hold because of the assumptions given in (2-2) and
SIS ^-BS
(2-1), respectively. The difference between a and a is
^s -BS 2vo (v v + 3t)[3t 2(v, v2)
[3t(12v + 5v, +7v2 -15t)- 2(v v2)2 ](2v + v, + 3t)
The inequality holds because of the assumptions (2-1) and (2-2). Since the
monopolist can make much higher profits by bundling than IS case, it is willing to sell its
goods to a smaller group of consumers than IS case if bundling is possible.
^CS ^BS
The difference between a and a is
^cs -Bs 4vo [4(v, v,) + 3t] [3t 2(v, v,)]
^CS ^BS
Using assumptions (2-1) and (2-2), one can find that a is higher than a except
3
when -t < v2- v1 < t. Even though sC' is not smaller than sTBS, the profit gain from
4
3
selling group S only is higher in bundling case than CS case except -t < v2- v < t as the
4
following shows:
(ZBS ZA CS ZCA) a[4(v v2) + 3t][3t 2(v -v2)]
144t
This explains why the monopolist is willing to sell the goods to a smaller group of
consumers than the commitment case.
2.6 Bundling and Social Welfare
Most previous analyses on bundling have ambiguous conclusions about the welfare
effect of bundling. It has been said that bundling could increase or decrease welfare. In
the model presented here, bundling decreases Marshallian social welfare except for an
extreme case.
Marshallian social welfare consists of the monopolist profits, firm 2's profits, and
consumers' surplus. When the monopolist bundles, its profits always increase compared
to the IS case. Firm 2's profits, on the other hand, decreases in bundling equilibrium since
BS Is a(v, -v2+ 3t)[5(v v2)-9t]
z <0
2 2 72t
Consumers' surpluses with bundling and IS are
BS p
CSs =a (v, +, tx-pb)dxBS +a( +v bt(l x) BS pBS)dx
a[(v- v2 2 + 2t(v1 v2)+5t2]
16t
IsS IS
CSs =aJ (vo + v tx -S ps)d x ++a j(v +v (1 x) S -
a[(v v2)2 + 9t2]
36t
The shaded area of Figure 2-1 shows consumers' surplus of each case when v1 < v2.
The difference between consumers' surplus with bundling and IS is
Sv + V2
ip Po + Pi
(a) Bundling (b) Independent sale (IS)
Figure 2-1. Consumers' surplus in bundling and IS cases when vi < V2
CSBS CSS = a(v, -v2 + 3t)[5(v v2) + 3t]
144t
which shows that consumers' surplus increases by the monopolist's decision to bundle
unless v2- v > (3/5)t. That is, unless firm 2's product is much superior to the
monopolist's complementary good, consumers' surplus increases as the monopolist
bundles. The consumers' surplus increases mainly because consumers who pay less in
bundling case than in IS case outnumber consumers who pay more in bundling
equilibrium. Unlike consumers' surplus, however, social welfare is more likely to
decrease with bundling strategy by the monopolist, as the following proposition shows.
Proposition 2-2 Suppose the monopolist sells its goods to consumers in S only. Then
Marshallian social welfare decreases with the monopolist's decision to bundle unless
3
-t
7
Proof. Marshallian social welfare is defined as the sum of consumers' surplus and
profits of all firms. So social welfare with bundling is
Bs = CSBS +BS BS =a vo 3(vl V2)2 10v +6v2 5t
;12 6t 16
And social welfare with IS is
W = CS + + = v o +
S36t 4
The difference between them is
WBS WIs 1= a (v -v2 +3t)[7(vl -v2)-3t]
144t
which is negative if -3t < v v2 < (3/7)t, and positive otherwise. Since I vl v2 < t, the
social welfare decreases except (3/7)t < v v2 < t. Q.E.D.
The above proposition shows that unless the monopolist's complementary good is
superior enough, the monopolist's bundling strategy lowers the social welfare. Especially,
the social welfare always decreases when the monopolist bundles an inferior good or a
good with the same quality as the rival's, i.e., v, < v2.
2.7 Bundling and R&D Incentives
One of the concerns about the bundling strategy by the monopolist of a primary
good is that it may reduce R&D incentives in the complementary good industry. This
section is devoted to the analysis of the effect of bundling on R&D incentives.
To analyze this, one needs to introduce an earlier stage at which two firms make
decisions on the level of R&D investments to develop complementary goods. The whole
game consists of three stages now. Let R(v) be the minimum required investment level to
develop a complementary good of value v. A simple form of the investment function is
R(v) = ev2, e > 0
Using this, the firms' profit functions can be rewritten as follows:21
is V (v V )2 5v, + v 6 2
I1 =a vo+ 18/ + 6 ev,
18t 6
is a(v2 -1 + 3t)2 2
22 = -ev2
18t
aBS v 3v, + V -3t 2
"1 =a 0 -ev,
4
-B a(v2 +t)2 2
"2 -- ev2
8t
The following proposition shows that the monopolist's bundling strategy reduces
not only the R&D incentive of the rival firm, but also its own incentive.
3
Proposition 2-3 Suppose the investment cost satisfies e > -. Then the equilibrium
8t
values of v, (i = 1, 2) are higher in the IS equilibrium than in the bundling equilibrium,
i.e., v2 > ys, and ys > BS Further, firm 2's incentive decreases more than the
monopolist's by bundling.
Proof. The first order conditions for profit maximization problems yield each firm's best
response functions from which one can obtain the following equilibrium levels of v,'s for
each equilibrium:
21 In previous sections, it is assumed that the complementary goods already have been
developed before the start of the game. The exclusion of the investment costs in profit
function does not affect equilibrium since they are sunk costs.
s a(90et 7a)
1 24e(9et a)
Is a(36et- 7a)
2 24e(9et a)
"BS 3a
S 8e
"BS a(8et- 3a)
8e(8et a)
3
Since 0 < c < 1, the assumption e > guarantees non-negative equilibrium values.
8t
Now the following comparisons prove the main argument:
s _BS a(9et + 2a)
V > 0
24e(9et a)
_is _Bs a a2(1et-a) >
v2 -v2 =-+ >0
24e 8e(9et- a)(8et-a)
2
(- BS _=S > 0 Q.E.D.
2 2 4(9et a)(8et a)
Firm 2 has a lower incentive to invest in R&D because part of the rents from the
investment will be transferred to the monopolist by bundling. The monopolist also has a
lower incentive to invest because the bundling strategy reduces competitive pressure in
the complementary good market.
2.8 Conclusion
It has been shown that the monopolist of a primary good has an incentive to bundle
its own complementary good with the primary good if it cannot commit to the optimal set
of prices when consumers buy the primary good and the complementary good
sequentially. Since the monopolist can increase its profits and the market share of its own
complementary good by bundling, the model provides another case in which the Chicago
School's single monopoly price theorem does not hold. While bundling lowers the rival
firm's profits and Marshallian social welfare in general, it increases consumers' surplus
except when the monopolist's complementary good is sufficiently inferior to the rival's
good. Bundling also has a negative effect on R&D incentives of both firms.
Since bundling may increase consumers' surplus while it lowers social welfare, the
implication for the antitrust policy is ambiguous. If antitrust authorities care more about
consumers' surplus than rival firm's profits, this kind of bundling may be allowed. Even
if total consumers' surplus increases, however, consumers who prefer the rival's
complementary good can be worse off since they have to pay higher price for both the
bundled good and the alternative complementary good. So bundling transfers surplus
from one group to another group of consumers.
In addition to the problem of a redistribution of consumers' surplus, bundling also
has a negative long-term effect on welfare since it reduces both firms' R&D incentives.
This long-term effect of bundling on R&D investment may be more important than
immediate effects on competitor's profit or consumers' surplus, especially for so-called
high-tech industries that are characterized by high levels of R&D investments. For
example, if a software company anticipates that development of a software program will
induce the monopolist of the operating system to develop a competing product and
bundle it with the OS, then the firm may have less incentive to invest or give up
developing the software. This could be a new version of market foreclosure.
A related issue is that if the risk of R&D investments includes the possibility of the
monopolist's developing and bundling of an alternative product, it can be said that
bundling increases social costs of R&D investments. Furthermore, since the monopolist
is more likely bundle a complementary good that has a broad customer base, bundling
may induce R&D investments to be biased to the complementary goods that are for
special group of consumers. A possible extension of the model lies in this direction.
Another extension could be to introduce competition in the primary good market,
which is suitable for the Kodak case.22 It has been pointed out that when the primary
good market is competitive, the anticompetitive effect of bundling is limited. In the
model presented here, firm 1 (the monopolist) could not set the bundling price so high if
it faced competition in the primary good market. However, if the primary goods are also
differentiated so that the producers of them have some (limited) monopoly powers,
bundling may have anticompetitive effects. The result can be more complicated-but
more realistic-if it is combined with the possibility of upgrade which is common in the
software industry.
22 See Klein (1993), Shapiro (1995), Borenstein, MacKie-Mason, and Netz (1995), and
Blair and Herndon (1996).
CHAPTER 3
COMPETITION AND WELFARE IN THE TWO-SIDED MARKET:
THE CASE OF CREDIT CARD INDUSTRY
3.1 Introduction
It is well known that a two-sided market-or more generally a multi-sided
market-works differently from a conventional one-sided market. In order to get both
sides on board and to balance the demands of both sides, a platform with two sides may
have to subsidize one side (i.e., set the price of one side lower than the marginal cost of
serving the side). In the credit card industry, cardholders usually pay no service fee or
even a negative fee in various forms of rebate. In terms of the traditional one-sided
market logic, this can be seen as a practice of predatory pricing. Several models of two-
sided markets, however, show that the pricing rule of the two-sided market is different
from the rule of the one-sided market, and a price below marginal cost may not be anti-
competitive. 1
Another feature of the two-sided market is that competition may not necessarily
lower the price charged to the customers. In the credit card industry, competition between
nonproprietary card schemes may raise the interchange fee, which in turn forces the
acquirers to raise the merchant fee. The interchange fee is a fee that is paid by the
acquirer to the issuer for each transaction made by the credit card. If the interchange fee
decreases as a result of competition, the cardholder fee is forced to increase. For the
1 Published papers include Baxter (1983), Rochet and Tirole (2002), Schmalensee (2002),
and Wright (2003a, 2003b, 2004a).
proprietary card schemes that set the cardholder fees and the merchant fees directly,
competition may lower one of the fees but not both fees.
The distinctive relationship between competition and prices raises a question about
the welfare effect of competition in the two-sided market. Even if competition lowers the
overall level of prices, it does not necessarily lead to a more efficient price structure.
Previous models about competition in the two-sided markets focus mainly on the effect
of competition on the price structure and derive ambiguous results on the welfare effects
of competition. I present a model of the credit card industry in order to show the effects
of competition on social welfare as well as on the price structure and level. The main
result is that while the effects of competition on the price structure are different
depending on the assumptions about whether consumers single-home or multi-home2 and
whether card schemes are identical (Bertrand competition) or differentiated (Hotelling
competition), the effects of competition on social welfare do not vary regardless of
different model settings. That is, competition does not improve the social welfare in the
various models presented here.
The main reason for this result is that competition forces the platforms to set the
prices) in favor of one side that is a bottleneck part, while a monopoly platform can fully
internalize the indirect network externalities that arise in the two-sided market.3 In order
to maximize the transaction volume (for nonproprietary schemes) or profits (for
proprietary schemes), the monopolist first needs to make the total size of the network
2 If a cardholder (or merchant) chooses to use (or accept) only one card, she is said to
single-home. If she uses multiple cards, she is said to multi-home.
3 In a two-sided market, the benefit of one side depends on the size of the other side. This
indirect network externality cannot be internalized by the end-users of the two-sided
market. See Rochet and Tirole (2005).
externalities as large as possible. Competing card schemes, on the contrary, set biased
prices since they share the market and try to attract single-homing consumers or
merchants.
Since the first formal model by Baxter (1983), various models of two-sided markets
have been developed. Many of them focus on the price structure of a monopolistic two-
sided market.4 It is in recent years that considerable attention has been paid to
competition in two-sided markets. Rochet and Tirole (2003) study competition between
differentiated platforms and show that if both buyer (consumer) and seller (merchant)
demands are linear, then the price structures of a monopoly platform, competing
proprietary platforms and competing (non-proprietary) associations are the same and
Ramsey optimal. They measure the price structure and Ramsey optimality in terms of the
price-elasticity ratio, so price levels and relative prices are not the same for different
competitive environments. While they assume that consumers always hold both cards, the
model presented here distinguishes cases with single-homing consumers and multi-
homing consumers and uses Marshallian welfare measure which includes platforms'
profits as well as consumers' and merchants' surpluses.
Guthrie and Wright (2005) present a model of competition between identical card
schemes. They introduce the business stealing effect by allowing competing merchants
and show that competition may or may not improve social welfare. I extend their model
to the case of the competition between differentiated card schemes as well as the cases of
proprietary card schemes, while removing the business stealing effect for simpler results.
4 The interchange fee is the main topic in these analyses of the credit card industry. See
Rochet and Tirole (2002), Schmalensee (2002) and Wright (2003a, 2003b, 2004a) for
the analyses of the credit card industry with monopoly card scheme.
Chakravorti and Roson (2004) also provide a model of competing card schemes
and show that competition is always welfare enhancing for both consumers and
merchants since the cardholder fee and the merchant fee in duopoly are always lower
than in monopoly. To derive the results, they assume that consumers pay an annual fee
while merchants pay a per-transaction fee and cardholder benefits are platform specific
and independent of each other. In contrast to their model, this paper assumes both
consumers and merchants pay per-transaction fees5 and cardholder benefits are either
identical or differentiated according to the Hotelling model, and concludes that
competition does not improve Marshallian social welfare. Further, it shows competition
may not always lower both the cardholder and merchant fees even for the proprietary
scheme as well as non-proprietary scheme.
The rest of Chapter 3 proceeds as follows. Section 3.2 sets up the basic model of
the non-proprietary card scheme. Section 3.3 and 3.4 show the effects of competition on
the price structure and welfare for the cases of single-homing consumers and multi-
homing consumers. Section 3.5 extends the model to the case of the proprietary card
scheme and compares the results with those of the non-proprietary card scheme. The last
section concludes with a discussion of some extensions and policy implications.
3.2 The Model: Nonproprietary Card Scheme
Suppose there are two payment card schemes, i = 1, 2, both of which are not-for-
profit organizations of many member banks. A cardholder or consumer receives a per-
transaction benefit bBk from using card i, which is assumed to be uniformly distributed
between (bB, bB). A merchant receives a per-transaction benefit, bs, which is also
5 The per-transaction fee paid by consumers can be negative in the various forms of
rebates.
uniformly distributed between (bs, bs). It is assumed that merchants find no difference
between two card schemes.
There are two types of member banks. Issuers provide service to consumers, while
acquirers provide service to merchants. Following Guthrie and Wright (2006), both the
issuer market and the acquirer market are assumed to be perfectly competitive. Card
schemes set the interchange fees in order to maximize total transaction volumes.6
For modeling convenience, it is assumed that there is no fixed cost or fixed fee. Let
c, and cA be per-transaction costs of a issuer and a acquirer, respectively. Then card
scheme i's per-transaction cardholder fee and merchant fee are, respectively,
Jf = c1 a
m, = CA +c
where a, is scheme i's interchange fee. Note that the sum of the cardholder fee and the
merchant fee is independent of the interchange fee since
f + m, =c +cA
In order to rule out the possibility that no merchant accepts the card and all
merchants accept the card, it is assumed that
6 Rochet and Tirole (2003) assume constant profit margins for the issuers and the
acquirers. Under this assumption, maximizing member banks' profits is same as
maximizing total transaction volume, and the sum of the cardholder fee and the
merchant fee is also independent of the interchange fee.
bB + b
Both the numbers of consumers and merchants are normalized to one. Consumers
have a unit demand for each good sold by a monopolistic merchant.7 Merchants charge
the same price to cash-paying consumers and card-paying consumers, i.e., the no-
surcharge-rule applies.
The timing of the game proceeds as follows: i) at stage 1, the card schemes set the
interchange fees, and the issuers and acquirers set the cardholder fees and merchant fees,
respectively; ii) at stage 2, consumers choose which card to hold and use, and merchants
choose which card to accept.
3.3 Competition between Identical Card Schemes: Bertrand Competition
In this section, two card schemes are assumed to be identical, i.e., bB, = bB2 (- bB).
Consumers can hold one or both cards depending on the assumption of single-homing or
multi-homing, while merchants are assumed to freely choose whether to accept one card,
both cards, or none.
One of the key features of the two-sided market is that there exist indirect network
externalities. As the number of members or activities increase on one side, the benefits to
the members of the other side also increase. In the credit card industry, cardholders'
benefits increase as the number of merchants that accept the card increases, while the
merchants' benefits increase as the number of cardholders who use the card increases.
Some of the previous analyses of the credit card industry did not fully incorporate
this network effect in their models by assuming homogeneous merchants, in which case
7 Since merchants do not compete with each other, the business stealing effect does not
exist in this model.
either all merchants or none accept the card.8 So at any equilibrium where transactions
occur, all merchants accept card and consumers do not need to worry about the size of the
other side of the network. The model presented here takes into account this indirect
network effect by assuming merchants are heterogeneous and the net utility of a
consumer with bB takes the following form:
UB, = (bB- f)Qs, = (b c + a )s, i= 1,2
where Qsi is the number of merchants that accept card i.
For modeling convenience, it is assumed throughout this section that the issuer
market is not fully covered at equilibrium, which requires
bB b > 2(b + bs -c)
3.3.1 Single-Homing Consumers
If consumers are restricted to hold only one card, they will choose to hold card i if
UB, > UB, and UB, > 0. Note that the cardholding decision depends on the size of the other
side as well as the price charged to the consumers. Even iff >f, a consumer may choose
card i as long as the number of merchants that accept card i (Qsi) is large enough
compared to the number of merchants accepting card (Qsj).
Merchants will accept card i as long as bs > m, since accepting both cards is always
a dominant strategy for an individual merchant when consumers single-home. So the
8 See Rochet and Tirole (2002) and Guthrie and Wright (2006).
number of merchants that accept card i (quasi-demand function for acquiring service) is9
bs -m, bs cA a-2)
Qs, = (3-2)
bs- b bs b
Using (3-2), the consumer's net utility can be rewritten as
(b, -c, + -a c )(bs-c-a)
bs -bS
Let bB* be the benefit of the critical consumer who is indifferent between card 1 and
2. One can obtain bB* by solving UB1 = UB2, which is
b = bs +c -cA -a1 -a
A consumer with low bB is more sensitive to the transaction fee, so she prefers the
card with lower cardholder fee (i.e., higher interchange fee). On the other hand, a
consumer with high bB gets a larger surplus for each card transaction, so she prefers the
card that is accepted by more merchants. Therefore, a consumer whose bB is higher than
bB* will choose a card with lower a,, and a consumer whose bB is lower than bB* will
choose a card with higher a,. If a, = a,, then consumers are indifferent between two cards,
so they are assumed to randomize between card 1 and 2. This can be summarized by the
following quasi-demand function of consumers:
9 Schmalensee (2002) calls Qs, and QB, partial demands, and Rochet and Tirole (2003)
call them quasi-demands since the actual demand is determined by the decisions of both
sides in a two-sided market.
b _bs -c, -a,
bB ifa,>a
be -bB be -b B
be -b* be -bs -c, +c, +a, +a
QB, = b if a < aJ (3-3)
be -b, be -b
bB c, +a .
2bB -C + if a =a
2(bs-bB)
At stage 1, the card schemes choose the interchange fees to maximize the
transaction volume which is the product of QB, and Qsi. The following proposition shows
the equilibrium interchange fee of the single-homing case of Bertrand competition.
Proposition 3-1 If two identical card schemes compete with each other and consumers
single-home,
(i) the equilibrium interchange fee is
bs 3 [2(bs c,)- (b- c,)]
(ii) ab maximizes total consumers' surplus
Proof. (i) Without loss of generality, suppose al > a2. Then scheme 2 will maximize the
following objective function:
((bs -a2 -c )(bB -bs -c, + c + a + a)
T (a; al) = Q(b Qs) =b b
(bB- b B)(s -bs)
from which scheme 2's best response function can be obtained as follows:
I1- -
R2(a) = (2bs -bB -a, +c1 -2c
Scheme l's objective function is
T,(a,; a,) =Q, Ql
(bs a, cA)(bs a c)
(b- bb,)(bs -b,)
Since the function is a linear function of al with negative coefficient, scheme 1 will
set al as low as possible, i.e., as close to a2 as possible. So the best response function of
scheme 1 is
Rl(a2)= a2
Solving Ri(a2) and R2(al) together, one can obtain the following Nash equilibrium:
a=a = [2(bs c)-(b c)]- ab"
3
The equilibrium transaction volume of scheme i when a, = a2 = as is
S(ab;a bs) (bB +bs c) 2 bT
9(bB -b,)(bs -bs)
Since scheme l's best response function seems to contradict the premise that
a1 > a2, it is necessary to show that card schemes do not have an incentive to deviate
from the equilibrium. To see this, suppose scheme 1 changes al by Aa. Then the
transaction volume of scheme 1 becomes
(bB + bs- c)(bB + bs -c 3Aa)i
if Aa > 0
9(b bB)(bs bs)
7(abs + Aa; a ") =
(bB + bs -c + 3Aa)(bB +bs -c- 3Aa)i
if Aa < 0
9(bB -b,)(bs -bs)
Both of them are less than Tb, so there is no incentive for scheme 1 to deviate
from as.
(ii) At symmetric equilibrium with common a, the consumers' demands for the
card services are given by (3-3). So the total consumers' surplus is
bB (b f (a))2 (bs -m(a))
TU (f (a)) = Qs, QB, df =
l 2f 2(bB -b,)(bs -bs)
_(bB c + a)2(bs -c a)
2(b bb)(bs -b,)
The optimal a that maximizes TUB is
1 -
a = [2(bs- c,)- (bB c)
which is same as a. .E.D.
which is same as abs. Q.E.D.
When consumers single-home, each card scheme has monopoly power over the
merchants that want to sell their products to the consumers. This makes the card schemes
try to attract as many consumers as possible by setting the interchange fee favorable to
consumers. The resulting interchange fee chosen by the card schemes is one that
maximizes total consumers' surplus.
An interchange fee higher than ab" may attract more consumers due to the lower
cardholder fee, but fewer merchants will accept the card due to the higher merchant fee.
Therefore, a card scheme can increase the transaction volume by lowering its interchange
fee, which attracts higher types of consumers who care more about the number of
merchants that accept the card. On the other hand, an interchange fee lower than ab may
attract more merchants, but fewer consumers will use the card. In this case, a card scheme
can increase the transaction volume by raising its interchange fee.
In order to see how competition in the two-sided market affects the price structure,
it is necessary to analyze the case in which the two card schemes are jointly owned by
one entity. As the following proposition shows, it turns out that joint ownership or
monopoly generates a lower interchange fee, which implies a higher cardholder fee and a
lower merchant fee. In other words, competition between card schemes when consumers
single-home raises the interchange fee.
Proposition 3-2 If two identical card schemes are jointly owned and consumers single-
home,
(i) the symmetric equilibrium interchange fee is
ab 2 [(bs- C)-(b -c,)]
(ii) the joint entity may engage in price discrimination in which one scheme sets
the interchange fee equal to abi and the other scheme sets the interchange fee at any level
above abJ, but the total transaction volume cannot increase by the price discrimination,
(iii) abJ maximizes the social welfare, which is defined as the sum of the total
consumers' surplus and the total merchants' surplus.
Proof. (i) Since the card schemes are identical, there is no difference between operating
only one scheme and operating both schemes with same interchange fees. So suppose the
joint entity operates only one scheme. Then the quasi-demand functions are
be -f be -c, +a
QB =-
bB bB bB b
bs -m bs -cA -a
bs b bs -bs
The joint entity will choose the optimal a in order to maximize the transaction
volume QBQs. The optimal interchange fee obtained from the first-order condition is
a* c)( c)] abj
--
a=2 [(bs c) -(b cr) ab
which is less than abs since
a-a =a bs +bB -c)>O
(ii) Without loss of generality, suppose al > a2. Then scheme 1 will attract low-type
consumers and scheme 2 will attract high-type consumers. The quasi-demand functions
are determined by (3-2) and (3-3). And the total transaction volume is
(bs c, + a2)(bs c, a,)
QBS +QB2QS2 = (3 -4)
(b bB)(bs -bs)
Note that (3-4) is independent of al, which implies al can be set at any level above
a2. The optimal a2 can be obtained from the first-order condition for maximizing (3-4):
1-j
a = [(bs c,) -(bB -c)] -a
It is not difficult to check that the total transaction volume at equilibrium is also the
same as in the symmetric equilibrium.
(iii) The sum of the total consumers' surplus and the total merchants' surplus is
TUs = TU + TU* = Qs bdf +Q dm
(bB -c, +a)(bs c -a)(bB +bs -c)
2(bB -b)(bs -b,)
The optimal a that maximizes TUb, is
ab= [(bs ) (b c,)]
which is same as ab~. So abi maximizes social welfare. Q.E.D.
The most interesting result of the proposition is that the joint entity, which acts like
a monopolist, chooses the socially optimal interchange fee. This is possible because both
the issuing and acquiring sides are competitive even though the platform is monopolized,
and the joint entity can internalize the indirect network externalities of both sides.
Comparing propositions 3-1 and 3-2, one can find that competition between card schemes
lowers social welfare as well as decreases total transaction volume. In a typical example
of prisoner's dilemma in game theory, competing firms choose higher quantity and/or
lower price, which is detrimental to themselves but beneficial to the society. But this
example of the two-sided market shows that competitive outcome can be detrimental to
the society as well as to themselves.
3.3.2 Multi-Homing Consumers
In this subsection, consumers are allowed to multi-home. Since there is no fixed fee
or cost, individual consumer is always better off by holding both cards as long as bB, >f.
So the number of consumers who hold card i is
be B bB -c, +a (35)
O=_ (3-5)
bB b bB b
On the other hand, since merchants have monopoly power over the products they
sell, they may strategically refuse to accept card i even if bs > m,.
If a merchant accepts card i only, it receives a surplus equal to
(b -CA- a)(bB -c +a,)
be -bB
(3-6)
If the merchant accept both cards, the surplus is
Usb = (bs ml)Qbl + (bs m2)Qb2 = (bs cA al)Qbl + (bs CA a2)Qb2
(3-7)
where Qb, is the number of consumers who will use card i if the merchant accepts both
cards. 10 When a consumer holding both cards buys from a merchant that accept both
cards, the consumer will choose to use the card that gives a higher net benefit, i.e., she
will use card i if b, f > bJ f,. And the consumer will randomize between card i andj
if b, / = b, fj .
If the two card schemes are identical (bBl = bB2), consumers will use the card that
has a lower consumer fee if merchant accepts both cards, i.e.,
QB,
Qb, 0
tI (/2)QB,
if a > a (f < f)
if a < a (f > f )
if a = :a (f = f )
A merchant with bs will accept card i only if Us, > Us, and Us, > Usb. It will accept
both cards if Usb > Usi, i = 1, 2. To see the acceptance decision by a merchant, suppose
10 Consumers' card-holding decision and card-using decision can be different since they
can hold both cards but use only one card for each merchant.
(3-8)
Us, = (bs M, )QB,
al > a2 without loss of generality. Then the net surplus to the merchant if it accepts both
cards is
Usb = (bs cA al)QB1 + (bs cA a2)O = Us'
Merchants are indifferent between accepting card 1 only and accepting both cards
since consumers will only use card 1 if merchants accept both cards. In other words, there
is no gain from accepting both cards if consumers multi-home. So merchants' decision
can be simplified to the choice between two cards. Let bs* be the critical merchant that is
indifferent between accepting card 1 only and card 2 only, which can be obtained by
setting Us, = Us2:
bs* =bB C + +al+ a2
Merchants with low bs will be sensitive to the merchant fee and prefer a card with
low merchant fee (low interchange fee), while merchants with high bs will prefer a card
with low consumer fee (high interchange fee) since they care more about the number of
consumers who use the card. Therefore, if ml > m2 (al > a2), merchants with bs smaller
than bs* (and greater than m2) will accept card 2 only and merchant with bs higher than
bs* will accept card 1.
If al = a2, all cardholders have both cards and it is indifferent for merchants
whether they accept card 1, card 2 or both. For modeling simplicity, it is assumed that
merchants will accept both cards if aa = a2. The following summarizes the number of
merchants that accept card i:
bs- bB +c -c, a
bs -bS
bB c +aJ
bs b,
bs -cA -a
bs -b,
if a >a
if a < a
if a, = a
(3-9)
Proposition 3-3 If two identical card schemes compete with each other and consumers
multi-home,
(i) the equilibrium interchange fee is
abm [(bs -c,) 2(b- c)
3
(ii) abm maximizes total merchants' surplus.
Proof. (i) Without loss of generality, suppose al > a2 (ml > m2). Then scheme l's best
response function can be obtained by solving the optimization problem of the scheme,
which is
R,(a) bs
2
2bB +2c, -c,
Scheme 2's objective function is
(bB -c, +a)(bB -c, +a2)
T2 (;2a1) QB2QS2 -
(bB b )(bs -bs)
bs
bs
bs
_bs-
bs
Since the function is linear in a2 with positive coefficient, scheme 2 will set a2 as
high as possible, i.e., as close to al as possible. So the best response function of scheme 2
is
R2(ai) = a
Solving Ri(a2) and R2(al) together, one can obtain the following Nash equilibrium:
a = a2 = bs A)- 2( -c)] ab
The equilibrium transaction volume of scheme i when a, = a2 abm is
T ab) (b+bs c)2 b
T(abm;a 9(bB b
9(b b_)(bs bs)
As in Proposition 3-1, it is necessary to show that the card schemes do not have an
incentive to deviate from abm in order to justify the equilibrium. To see this, suppose
scheme 1 changes al by Aa. Then the transaction volume of scheme 1 becomes
(bB+ bs c 3Aa)(bB +bs-c+ 3Aa)
if Aa > 0
9(bB -b)(bs -b,)
T,(abm +Aa;abm) -B -s
(bB +bs c)(bB +bs -c+ 3Aa)
if(b Aab < 0)(bs
9(bB -bB)(bs bs)
Both of them are less than Tb, so there is no incentive for the scheme to deviate
from abm.
(ii) At symmetric equilibrium with common a, the total merchants' surplus is
2 bs
-^ -
TUb, (M(a)) Q Qm
(bs -c, )2(bB
2(bB b,)(bs
(bs m(a)) (bs f(a))
2(bB -bB)(bs -b,)
-ct +a)
-bs)
The optimal interchange fee that maximizes TU)" is
a 3 = [(bs -c,)-2(bB -c,)
3=
which is equal to ab". So ab" maximizes total merchants' surplus.
Q.E.D.
When consumers multi-home, the card schemes care more about merchants since
they can strategically refuse to accept one card. By setting the interchange fee so as to
maximize the merchants' surplus, the card schemes can attract as many merchants as
possible. As in the single-homing case, an interchange fee higher or lower than ab" is
suboptimal and a card scheme can increase its transaction volume by changing the
interchange fee closer to ab".
The interchange fee in the multi-homing case is lower than in the single-homing
case since the fee is set in favor of the merchants. The following proposition shows that
the interchange fee is higher if the card schemes are jointly owned, which implies the
interchange fee decreases as a result of competition between card schemes when
consumers multi-home. It also shows that competition lowers social welfare as in the
single-homing case.
Proposition 3-4 If two identical card schemes are jointly owned and consumers multi-
home,
(i) the symmetric equilibrium interchange fee is
a =2 [(bs -A) (bB -c)]>abm
(ii) the joint entity may engage in price discrimination in which one scheme sets
the interchange fee equal to abj and the other scheme sets the interchange fee at any level
below ab but the total transaction volume cannot increase by the price discrimination,
(iii) abJ maximizes social welfare.
Proof. (i) Regardless whether consumers single-home or multi-home, there is no
difference for the joint entity between operating two card schemes with same interchange
fee and operating only one scheme since the card schemes are identical. So the proof is
the same as the first part of Proposition 3-2. And for multi-homing consumers, the
monopolistic interchange fee is higher than the competitive interchange fee since
'(b- -B
(ii) Without loss of generality, suppose al > a2. Then scheme 1 will attract low-
type merchants and scheme 2 will attract high-type ones. Then the total transaction
volume is
QBlQSl + QB2QS2
(bB -c1 +a)(bs -c -a,)
(b- -b,)(bs -b)
Note that (3-10) is independent of a2, which implies a2 can be set at any level
below al. The optimal al obtained from the first-order condition is
1 [( c (b c,)
az= (s cA -b -z
which is equal to ab.
The total transaction volume at equilibrium is
(b +bs -c)2
4(bB -b)(bs -bs)
which is the
same as in the symmetric equilibrium.
(iii) The proof is the same as in part (iii) of Proposition 3-2.
Q.E.D.
The optimal interchange fee for the joint entity is the same as in the single-homing
case since the card schemes do not compete for consumers or merchants. Unlike the
single-homing case, however, the interchange fee decreases as a result of competition
between the card schemes when consumers multi-home. Social welfare deteriorates since
(3-10)
competing card schemes set the interchange fee too low in order to attract more
merchants.
Figure 3-1 shows the results of this section. As is clear in the figure, competitive
equilibrium interchange fees maximize either consumers' surplus or merchants' surplus.
Since monopoly interchange fee maximizes total surplus, competitive outcome is
suboptimal in terms of social welfare.
3.4 Competition between Differentiated Card Schemes: Hotelling Competition
In this section, card schemes are assumed to be differentiated and compete a la
Hotelling. As in a standard Hotelling model, suppose consumers are uniformly
distributed between 0 and 1, and the card scheme 1 is located at 0 and scheme 2 is at 1. A
consumer located at x receives a net benefit of be tx (- bB,) if she uses card 1, and
b t(1 x) (- bB2) if she uses card 2. In order to comply with the assumption that
consumers' benefits from card usage is uniformly distributed between (bB bB ), the
transportation cost t is assumed to be equal to bB -bB.
TU = TUB + TUs
cI -bB abm ab ab
Figure 3-1. Welfare and interchange fees of Bertrand competition
The net utilities of a consumer located at x when she uses card 1 and 2 are
UB =(bB -tx- f)Q= (bB(1-x)+bBx-cI+a)Qs
UB2 (bB t(1-x)-f2)Q2= (bBx+bB(1-x) -c+a Q
The critical consumer, x*, who is indifferent between card 1 and 2 can be obtained
by solving UBs = UB2:
bB (b -c, + a)Qsl -(b -c +a2)Qs2 (3-11)
(b bB)(QsI Q2)
If the issuer market is not fully covered, each card scheme has a full monopoly
power over the consumers and the resulting equilibrium will be the same as in the
monopoly case of the previous section. In order to obtain competitive outcomes, the
issuer market is assumed to be fully covered at equilibrium. This requires the following
assumption:1
bB +bs > c
Depending on whether consumers single-home or multi-home, and whether card
schemes compete or collude, various equilibria can be derived. There may exist multiple
equilibria including asymmetric ones. For expositional simplicity, however, only
symmetric equilibria will be considered unless otherwise noted.
11 For the issuer market to be fully covered, the net utility of the consumer located at x =
12 must be non-negative for the monopolistic interchange fee abJ.
3.4.1 Single-Homing Consumers
When consumers are restricted to single-home, merchants will accept card i as long
as bs > m, as in the previous section. So Qs, is determined by (3-2). Since the issuer
market is fully covered, the number of consumers who choose card i is
QB =x* and Q = 1- x*
where x* is defined in (3-11).
The following proposition shows the symmetric equilibrium of the Hotelling
competition when consumers single-home.
Proposition 3-5 If two differentiated card schemes compete a la Hotelling and
consumers single-home,
(i) the symmetric equilibrium interchange fee is
c --(bBs+bB) if b -b, > 2(b +bs -c)
a0 = (3-12)
4 [2(bs-cA)-2(bB-c,)-(bB-bB)] if bB -bB <2(bB +bs -c)
(ii) ahs maximizes the weighted sum of total consumers' surplus and total
merchants' surplus, w TU1 + (1 w)TUs where the weight for consumers' surplus is
3(bB-bB,)+ 2(b, +bs-c)
w= B if bB-bB <2(bB +bs-c)
6(bB b)+2(b + bs c)
2(bs- bB)+ 4(b + bs -c)
w2 = if be-b B > 2(bB + bs- c)
3(bB-bB)+ 8(b +bs -c)
Proof. (i) For a given a2, card scheme 1 will set al to maximize its transaction volume.
The symmetric equilibrium interchange fee can be obtained from the first-order condition
in which al and a2 are set to be equal to each other for symmetry:
1-'
a=4 [2(bs-cA)-2(bB -c)- (b -bB)]
For this fee to be an equilibrium, net benefit of the consumer at x = 2 must be
nonnegative since the issuer market is assumed to be fully covered, which requires
t [B
bB 2 f(a)= 2(b,+bs-c)-(bB-bB)]>0
That is, a* is an equilibrium interchange fee if bB b < 2(bB + bs c).
If bB -bB > 2(b + bs c), the equilibrium interchange fee can be obtained by
setting consumer's net benefit at x = 12 equal to zero:
1-
a= c -(bB +bB)
For a** to be an equilibrium, it needs to be shown that the card schemes have no
incentive to deviate from a*. The transaction volume of card scheme 1 at a** is
**, bB+bB+2bs -2c
7T(a ,a )=- 4-
4(bs -bs)
The right and left derivatives of scheme 1's profit at al = a* are, respectively,
lim T(a- + Aa,a")- T7(a",a')
Aa-o+ Aa
lim T(a** +Aa,a**)- T(a**, a**)
Aa-^o Aa
2(bs + bB c) -(b b)
< 0
4(b bb)(bs -b,)
(b + bs -c)
(b B)( < 0)
(bs b,)(bs bs)
So a is an equilibrium when bB bB > 2(b, + b c). Note that a* a when
bB-bB =2(b +bs c).
(ii) First, note that QB = 12 at symmetric equilibrium since the market is fully
covered. The weighted sum of total consumers' surplus and merchants' surplus for
scheme 1 is
wTU +(1 w)TU = w f 2 UBdx+ I B2 1 b s Q, dm
; 22 ,2 i
(bs -cA -a)[w(3bB +b -4c +4a)+ 2(1 w)(bs c a)
4(bs -bs)
The optimal interchange fee that maximizes this weighted surplus is
4(2w 1)(bs cA)- w(3bB+bB- 4c,)
4(3w 1)
(3-13)
(3-14)
The size of the weight can be obtained by setting ah = a*, which is
3(bB-b b)+ 2(b, +bs-c)
w, = if bB-b <2(b, +bs -c)
6(bB b)+ 2(bB +bs c)
2(bB bB) + 4(b, + bs c)
w2 = -- if bB-bB >2(b + bs- c)
3(b -b)+ 8(b +bs- c)
Note that wl = w2 = 4/7 if bB b = 2(b +bs -c). Q.E.D.
When the card schemes compete a la Hotelling, they have some monopoly power
over the consumers. So unlike the Bertrand competition case, they do not need to set the
interchange fee so high as to maximize total consumers' surplus. While the weight for
consumers' surplus (w) in Bertrand competition is equal to 1, the weight in Hotelling
competition ranges between 4/7 and 1. If (bB bB) = 2(bB + bs -c), the weight is equal to
4/7. It becomes close to one as bB bB approaches zero. Note that bB bB is equal to the
transportation cost t. As in a standard Hotelling model, the monopoly power of a card
scheme weakens as t becomes smaller. Therefore, the card scheme will set the
interchange fee so as to maximize total consumers' surplus when the transportation cost
becomes zero.
The following proposition shows the monopoly interchange fee in the Hotelling
model also maximizes the social welfare as in the Bertrand model.
Proposition 3-6 If the two differentiated card schemes are jointly owned and consumers
single-home,
(i) the joint entity will set the interchange fee equal to
1-
a = -2 (b +b,)
(ii) ah maximizes the sum of the total consumers' surplus and the total merchants'
surplus.
Proof. (i) I will prove this proposition in two cases: (a) when the joint entity sets the
same interchange fees for scheme 1 and 2, and (b) when it sets two different fees (price
discrimination).
(a) When the joint entity sets the same interchange fees for both schemes, the joint
transaction volume is
bs cA -a
TM (aa) = QBQs + 0QB2S2 CA
bs -bs
where QB1 = QB2 = 12 since the issuer market is assumed to be fully covered.
Note that TM is decreasing in a, which implies that the optimal a is the minimum
possible level that keeps the issuer market covered. This fee can be obtained by setting
the consumer's net benefit at x = 12 equal to zero, which is ah.
(b) Now suppose the joint entity tries a price discrimination by setting al = aj + Aa
and a2 = a Aa, Aa > 0. The joint transaction volume when it charges same fee, ah, is
S( 2(bs +bB -c) +(bB b)
SM(a ",a 1)= -
2(bs -bs)
while the joint transaction volume of the price discrimination is
T +,2(bs b -c)(b -b) -2Aa
TM(ahj + Aa,ahj Aa) (bB
2(bs -bs)
It is not beneficial to engage in price discrimination since
T, (ah + Aa, ah Aa) T, (a", ah )
Aa
bs -bs
(ii) Since QB1 = QB2 = 12 at full-cover market equilibrium, the sum of total
consumers' surplus and total merchants' surplus is
TUh + TU~h
JU2ldx + UBdx+ 2lQsdm
(bs c a)(3bB + b 4c + 2bs 2cA + 2a)
4(bs -bs)
The optimal a that maximizes social welfare is12
a= c 3bB +bB
4L=7 ^^5^
12 The fee is equivalent to a* in (3-14) when w = /2.
Note that the market is not fully covered at a since a* < ah In other words, a
is not feasible. Therefore, ah maximizes the sum of the total consumers' surplus and the
total merchants' surplus when the market is fully covered. Q.E.D.
Note that ah = ahj if bB b > 2(bB + bs -c) and ah > ahj if
bB bB < 2(bB + bs -c). As in the Bertrand competition case, competition does not lower
the equilibrium interchange fee nor increase social welfare when the card schemes
compete a la Hotelling and consumers single-home.
3.4.2 Multi-Homing Consumers
If consumers are allowed to multi-home, they will hold card i as long as bBZ >f. So
the number of consumers who hold card i is the same as (3-5). If the issuer market is fully
covered and the merchants accept both cards, the critical consumer who is indifferent
between card 1 and 2 is obtained by solving b tx f = bB t(1 x) f2, which is
x f-+ -+J
X I + 1+ a1- a2
2 2t 2 2(b b,)
The number of consumers who use card i if merchants accept both cards is
Qbl = x*, and Qb2= 1 x*
Lemma 3-1 If a, > a,, merchants accept either card only or both cards, i.e., no
merchant will accept card i only.
Proof. Without loss of generality, suppose al > a2. The critical merchant that is
indifferent between accepting card 1 only and accepting card 2 only can be obtained by
setting Usi = Us2, where Us, is defined in (3.6):
bs = bB c + c+al+ a2
Merchants with low bs will be more sensitive to the merchant fee, while merchants
with high bs will care more about the number of consumers who use the card. So if
bs > bs, the merchant prefers card 1 to card 2 and vice versa.
The critical merchant that is indifferent between accepting card i only and
accepting both cards can be obtained by setting Usb = Us,, where
Usb = (bs m,)Qbl + (bs m )Qb2
Let bs, be the critical merchant. That is,
(a, a,)(a, + a + b b,)- 2a,(a + b + c )+ 2c (c bs)
2(c, -a -b,)
If bs > bs, accepting both cards is more profitable than accepting only card i since
merchants with high bs care more about the transaction volume. The difference between
bs and bs is
bs* _b a2 + ( + a2 )(bB + b -2c,)+ 2(bB c)(bB -cI)
2(c -a -bB)
Note that the numerator is independent of i and the denominator is positive.13 Since
al > a2, bs > bS2 > bs if the numerator is positive, and bs < bS2 < bs, if the numerator is
negative. Note also that bs is larger than m, since
bs m, = bB- f >0, i j,
and bs, is smaller than m\ since
(a, -a2,) (bs bB) (a -a2)]
bs- --m--- < 0
2(f -bB)
which implies bs > bS2 > bsl. Note that the difference between two interchange fees,
which is same as the difference between two cardholder fees, cannot exceed the
difference between bB and bB since _B< < / < bB.
As is shown in Figure 3-2, merchants will accept card 2 only if bs e [m2, b2 ), and
accept both cards if bs e [ bS2, bs ].14 Q.E.D.
13 C -a, -bB > 0 since it is equal to f -b and the cardholder fee must be higher than
b,.
14 S (a -0) [(b+ b)- (f +f2)]
Since b- m = b, < m2 (bbs2 > m ) if and only if
S2(f, b, _)
be+ b >f+ f2
accepting card 2 accepting both cards
I I I'I I I I
bs bs m2 m b* b bs
accepting card 2 accepting both cards
I I I I I I s
b, m, bb b2 m1 b
Figure 3-2. Merchants' acceptance decision when al > a2 (ml > m2)
Based on Lemma 3-1, the number of merchants that accept card i is
bs -b
bs b
bs -m
bs -b,
if a, >a, (m, >m)
if a, < a (m,
Let Qa, be the number of merchants that accept card i only, and QSb be the number
of merchants that accept both cards. That is,
Q s, -Qs if a
0 if a, > a (m, > m )
Qsb = Qs, where a, > a (m, > m )
The following proposition summarizes the equilibrium interchange fee of the
Hotelling competition with multi-homing consumers.
Proposition 3-7 If consumers can multi-home and card schemes compete a la Hotelling,
(i) the symmetric equilibrium interchange fee is
ahm { I2b
CA +I -b,
1-(
2 (bs + bB)
2
if bB
if bB
bB < 2(b +bs -c)
b > 2(b +bs -c)
where A = 2(bB -
bB)2 + (b+bs -c)2
(ii) ahm < ah where the equality holds when bB b > 2(bB + bs -c)
(iii) ahm maximizes the weighted sum of total consumers' surplus and total
merchants' surplus, w TUB + (1 -w)TU s, where the weight for the consumers' surplus is
2(b+bs -c)+2A
w, =----
2(b +bs c)- 3(b bs)+6A
2(bB bB) + 4(bB + bs c)
3(b- b )+ 8(bB +bs -c)
if bB -bB <2(bB +bs -c)
if b-bB >2(b+bs -c)
Proof. (i) Without loss of generality, suppose al > a2 (ml > m2). Then scheme 1 and 2's
transaction volumes are, respectively,
TI (a; a2) QblQSb
T2 (a2; a) = B2,2 + b2Sb
The symmetric equilibrium can be obtained by taking derivative of T, w.r.t. a, at
a, = a,, which yields
a bs-c +c -b -2(b -bB +(b+bs c)
At the symmetric equilibrium, all merchants accept both cards (i.e., Qae = 0) and
Qb = Qb2 = /2. So the transaction volume of each card scheme is
S bB + bs-c+ 2(bB-bB)2 +(b+bs )2
T(a ;a *)=
4(bs bs )
To see the card schemes do not have an incentive to deviate from a*, suppose
scheme 1 changes al by Aa. Then the transaction volume of the scheme becomes
T7(a + Aa; a) = Qb
QBO, + Qb2Qsb
if Aa > 0
if Aa <0
The transaction volume does not increase by changing a since
7T(a + Aa; a)- T7(a ;a )
Aa (2(bB b) + Aa)
--- <0
2(bB b)(bs b)(A (bs + b c))
Aa2 (3A 2(bB- b, )-(bs + bB -c)- 3Aa)
2(bB bB)(bs bs)(A (bs + b -c)- 2Aa)
if Aa > 0
<0 ifAa<0
So the card schemes do not have an incentive to deviate from a .
For a* to be an equilibrium, the issuer market must be fully covered at equilibrium.
The net benefit of the consumer located at x = /2 is
bB--t-(c-a)= (b+bs c) -2(bB b)2 ( bs c)2
2 2L
This is nonnegative if and only if b b < 2(bB + bs c) since
(bB +bs-c)2 -[2(bB b,) +(b+bs -c)2]= (b -BB)[2(B +bs -c)-(bB -b)].
If bB b > 2(bB + bs c), as in the single-homing case, the equilibrium
interchange fee can be obtained by setting consumer's net benefit at x = /2 equal to zero,
which is
S 1 1
a c -(bs +bB)
Note that, as in the single-homing case, a* a** when be bB = 2(bB + bs c).
(ii) If bB bB < 2(bB + bs -c), ah = 1[2(bs cA)- 2(bB c) -(bB bB)] and the
difference between the two equilibrium fees is
a a _hm ( 22(bB -bB)2 +(bB +bs -)2 3(bs -bB))
> 22(bB b)2 +[(1/2)( b)]2 -3(bB bB)) = 0
If bB -bB > 2(b +bs -c), both ah" and ahm are equal to c, -12(bB +b,).
(iii) The weighted sum of total consumers' surplus and total merchants' surplus is
the same as (3-13), hence the optimal interchange fee maximizing the weighted surplus is
also the same as (3-14). The level of the weight can be obtained by setting ah' = aW,
which is
2(b_ + bs c)+ 2A
w,= -- if bB -bB <2(b, +bs -c)
2(bB +bs -c)- 3(b bB)+6A
2(b- b, )+ 4(b +bs -c)
w = if be -b > 2(bB +bs -c)
3(b- bB )+ 8(b +bs -c)
Note that, as in the single-homing case, wl = w2 = 4/7 if bB bB = 2(b + bs c).
Q.E.D.
When consumers multi-home, the equilibrium interchange fee is lower than that of
the single-homing case. But unlike the Bertrand competition case in which card schemes
set the interchange fee so as to maximize the merchants' surplus, the card schemes do not
lower the fee enough. In the Bertrand competition with multi-homing consumers,
merchants accept only one card if the merchant fees set by two card schemes are different.
Therefore, a card scheme can maximize its transaction volume by attracting as many
merchants as possible. In Hotelling competition, however, each card scheme has its own
patronizing consumers since it provides differentiated service. This weakens merchant
resistance, which forces many merchants to accept both cards. 15 Therefore, card schemes
do not need to provide maximum surplus to the merchants.
If the card schemes are jointly owned, the result will be the same as in the single-
homing case since the joint entity will split the issuer market so that each consumer holds
15 See Rochet and Tirole (2002) for a discussion of merchant resistance.
only one card at equilibrium.
Figure 3-3 shows the relationship of various equilibrium interchange fees and
welfare, which is drawn for the case of be bB < 2(bB + bs c).16 The left side of ahi is
not feasible since the market cannot be fully covered. As is clear from the figure,
competition not only increases the equilibrium interchange fee but also lowers social
welfare. It also shows that allowing consumers to multi-home increases social welfare in
the Hotelling competition case, although it lowers total consumers' surplus.
3.5 Proprietary System with Single-Homing Consumers
The analysis of the previous sections has been restricted to the competition between
non-proprietary card schemes that set interchange fees and let the cardholder fees and
TU = TUB +TUs
TUB
bs LA
Figure 3-3 Welfare and interchange fees of Hotelling competition when
bB b, <2(bB +bs -c)
16 When bB -bB > 2(b + bs c), ahj = ahm = a h
merchant fees be determined by issuers and acquirers, respectively. Another type of
credit card scheme, a proprietary scheme, serves as both issuer and acquirer. It sets the
cardholder fee and merchant fee directly, so there is no need for an interchange fee. 17
3.5.1 Competition between Identical Card Schemes
One of the features of the proprietary card scheme is that competition may not only
alter the price structure but may also change the price level. In the previous sections, the
sum of the cardholder fee and merchant fee does not change even after the introduction of
competition between card schemes. When a card scheme sets both the cardholder fee
and the merchant fee, it may change one of the fees more than the other since the effects
of competition on two sides are not equivalent.
To see how competition affects the equilibrium fees of the proprietary card scheme,
the equilibrium of the monopoly case will be presented first. For the sake of simplicity,
only the case of single-homing consumers will be considered.
When the monopoly proprietary card scheme setsfand m, the quasi-demand
functions of consumers and merchants are
b f bs -m
QB = and Q = --
be b bs -bs
17 In the United States, Discover and American Express are examples of this type of card
scheme.
18 This feature of the non-proprietary scheme requires an assumption of perfect
competition among issuers and acquirers. If the perfect competition assumption is
removed, competition may alter the price level as well as the price structure in the non-
proprietary card scheme model.
and the profit of the scheme is19
Tr = (f +m-c)QBQ
From the first order condition for the profit maximization problem, one can obtain
the following equilibrium cardholder fee and merchant fee:
1---
f = 3 (2b -bs+c)
(3-15)
mnM =(2bs -b +c)
The following lemma shows that there does not exist a pure strategy equilibrium
when two identical proprietary card schemes compete with each other.
Lemma 3-2 If two identical proprietary card schemes compete in a Bertrand fashion, no
pure strategy equilibrium exists.
Proof. Note first that any set of prices that generates positive profit cannot be a
symmetric equilibrium. If an equilibrium set of prices is (f m) such that+ m > c, a card
scheme can increase profit by lowering the cardholder fee marginally while keeping the
merchant fee since the scheme can attract all consumers instead of sharing them with the
other scheme.
Second, a set of prices which satisfiesf+ m = c cannot be an equilibrium, either. To
see this, let the equilibrium set of prices is (f m) such that+ m = c. Without loss of
19 The proprietary card scheme maximizes profits instead of card transaction volume.
generality, suppose scheme 2 lower the cardholder fee by d and raise the merchant fee by
e, where e > d > 0. As in the Bertrand competition case of the previous section,
consumers whose bB is higher than b* will choose card 1 while consumers with bB lower
than b* will choose card 2, in which b* is defined as
f (bs m)d
b =f -d+
e
The quasi-demands of consumers and merchants for scheme 2's card service are
b2 (f -d) d(bs m)
be -bB e(bB -bB)
bs -(m+e)
9S2 =
bs -bs
The profit of the scheme 2 is
d(bs m)(bs -m -e)(f + m c + e -d)
)T2 = -- -- > 0
e(bB -bs)(bs -bs)
Since the scheme 2 can make positive profits by deviating from (f m), it cannot be
an equilibrium set of prices. Q.E.D.
The above lemma does not exclude the possibility of a mixed strategy equilibrium
or asymmetric equilibrium. As the following proposition shows, however, competition
cannot improve social welfare since the monopolistic equilibrium set of prices maximizes
social welfare.
Proposition 3-8 The equilibrium prices set by the monopolistic proprietary card scheme
in the Bertrand model maximize Marshallian social welfare which is defined as the sum
of cardholders' surplus, merchants' surplus and card schemes' profits.
Proof. Marshallian social welfare is defined as follows:
W =TU, +TUs r = QQsdf + Q,Qsdm + (f +m c)QQs
(3-16)
(bB f)(bs m)(bB + bs + f + m- 2c)
2(bB -b,)(bs -bs)
The optimal prices that maximize welfare are
f =3 (2b -bs+c)
mW = Ibs -b +c
3W (2bs b
These are same as fM and mM, respectively. Q.E.D.
For comparison with other models, one may derive a set of Ramsey-optimal prices
which is the solution of the following problem:
Max TU + TUs s.t. f+m = c
f,m
From the first-order condition of this maximization problem, the following
Ramsey-optimal prices can be obtained:
fR=1 -
fR (b -bs +c)
R=1 -
mR =(bs -bB +c
The differences between two different optimal prices are same for both cardholder
and merchant fees. That is,
61/- -b c z
fw fR =mW _mR = b +bs-c>0
6(
Ramsey-optimal prices are lower than the prices that maximize Marshallian welfare
since the former does not allow profits of the firms while the latter puts the same weight
on profits as on customers' surplus. If social welfare is measured by the Ramsey standard,
competition may increase the social welfare as long as competition lowers both
cardholder and merchant fees.
It is also worth noting that the Ramsey-optimal fees of the proprietary scheme is
equal to the consumer and merchant fees that are determined by the monopoly
interchange fee of the nonproprietary scheme, i.e., fR =c, -ab' and mR = cA + ab, which
confirms that abj maximizes both Marshallian and Ramsey social welfare.
3.5.2 Competition between Differentiated Card Schemes
When two proprietary card schemes are differentiated and compete a la Hotelling,
the critical consumer, x*, who is indifferent between card 1 and 2 is determined in the
same way as (3-11) except that the card schemes setf and m, instead of a,:
X*= (b f)(bs -m,)+(f2 -bB)(bs -m2)
(bB b)(2bs m -m2)
If the issuer market is not fully covered, each card scheme has a monopoly power
over its own consumers, so the equilibrium set of prices will be same as fM and mM in
(3-15).20 In order to obtain a non-trivial result, suppose the issuer market is fully covered
at equilibrium as in the previous section. This requires the following assumption:21
bB-bb
Using the first-order conditions, one can derive the best response functions of card
schemes from which the following equilibrium prices can be obtained:
fP = l5bB-3_bB 2bs +2c)
mp 4 =(2bs -bB-bB +2c)
20 Since merchants accept card i as long as bs > m,, the existence of competing card
schemes does not affect the equilibrium merchant fee.
21 For the issuer market to be fully covered, the net utility of the consumer located at x =
12 must be non-negative for the monopoly prices, fM and mM.
If two schemes collude and act like a monopolist, the joint entity will set the
cardholder fee such that the critical consumer who is located at x = 12 is indifferent
between using card and cash as well as between card 1 and 2. Since the transportation
cost is assumed to be equal to bB -bB, the cardholder fee that will be set by the joint
entity is
f (bB +bB) (3-17)
Given this cardholder fee, the joint profit can be rewritten as follows:
; Ti p( = (f bs m)(bB+ bB 2c + 2m)
r = ( f ^ + m-c)(QQ,,Qs + QzQs2) =
2(bs -bs)
The optimal merchant fee that maximizes this profit function is
m ph =(2bs -bB-b +2c)
Note that the merchant fee set by the joint entity is the same as the competitive
merchant fee, i.e., mph' = mp This is because the issuer market is fully covered in both
cases and the multi-homing merchants will accept any card as long as the merchant fee is
less than bs.
Proposition 3-9 When the two proprietary card schemes are differentiated in a Hotelling
fashion, competition does not improve Marshallian social welfare.
Proof. If two card schemes charge same prices and the issuer market is fully covered,
Marshallian social welfare is
W = TUB +TUs + T1 + 2
= UBl+dxf UB2d++ 1Jf rQsdm+ (f +m-c)QBQs,
2 m 1=i
(3bB+bB -4f)(bs -m) (bs -m)2 (f +m-c)(bs -m) (3-18)
+ +
4(bs -b ) 2(bs -bs) (bs -bs)
(bs -m)[3bB + bB + 2(bs 2c + m)]
4(bs -bs)
Note that social welfare is independent off That is, the cardholder fee has no effect
on the welfare as long as the fee is low enough for the issuer market to be fully covered.
An increase in the cardholder fee just transfers surplus from consumers to the card
schemes.
Since the social welfare is only affected by the merchant fee and the equilibrium
merchant fees of the competitive case and the monopoly case are equal to each other,
competition does not improve the social welfare. Q.E.D.
The cardholder fee cannot affect social welfare since the issuer market is fully
covered, i.e., the consumers' quasi-demand is fixed regardless of the cardholder fee.
When the cardholder fee changes, it does not affect the demand of the issuer market, but
