HIAS-E-37

A Leverage Theory of Tying in Two-Sided Markets
Jay Pil Choi
Department of Economics, Michigan State University
Hitotsubashi Institute for Advanced Study, Hitotsubashi University

Doh-Shin Jeon
Toulouse School of Economics, University of Toulouse Capitole (IDEI)
CEPR

November 2016

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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A Leverage Theory of Tying in Two-Sided Markets!
Jay Pil Choiy

Doh-Shin Jeonz

November 11, 2016

Abstract
Motivated by the recent antitrust investigations concerning Google, we develop aleverage theory of tying in two-sided markets. In a setting where the "one monopoly
proÖt result" holds otherwise, we uncover a new channel through which tying allows a monopolistic Örm in one market to credibly leverage its monopoly power toanother competing market if the latter is two-sided. In the presence of the nonnegative price constraint, tying provides a mechanism to circumvent the constraint
in the tied product market without inviting an aggressive response by the rivalÖrm. We identify conditions under which tying in two-sided markets is proÖtableand explore its welfare implications. In addition, we show that our model can beapplied more widely to any markets in which sales to consumers in one market cangenerate additional revenues that cannot be competed away due to non-negative
price constraints.

JEL Codes: D4, L1, L5
Key Words: Tying, Leverage of monopoly power, Two-sided markets, Zero pricing,Non-negative pricing constraint!
We thank Federico Etro, Bruno Jullien, Andras Niedermayer, Volker Nocke, Martin Peitz, Patrick
Rey, Kyoungwon Rhee, Chengsi Wang and participants in various conferences and seminars for valuable
discussions and comments. Taeyoon Hwang and Sungwook Koh provided excellent research assistance.
This work was supported by the Ministry of Education of the Republic of Korea and the National
Research Foundation of Korea (NRF-2016S1A5A2A01022389). Jeon acknowledges the Önancial supportof the Jean-Jacques La§ont Digital Chair.
yDepartment of Economics, Michigan State University and Hitotsubashi Institute for Advanced Study,
Hitotsubashi University. E-mail: choijay@msu.edu.
z
Toulouse School of Economics, University of Toulouse Capitole (IDEI) and CEPR. E-mail:dohshin.jeon@gmail.com.

1

Introduction

Tying has a long history of scrutiny by antitrust laws and has been harshly treatedby courts both in the US and in Europe. Such harsh treatment is based on the belief
in the leverage theory of tying, according to which tying provides a mechanism for amonopolist to leverage its monopoly power to monopolize another market where it faces
competition.1 The theory has been heavily criticized by scholars of the Chicago School(Bork (1954), Director and Levi (1956), Bowman (1957), Posner (1979)), whose argument
can be summarized as the well-known "one monopoly proÖt result"2 . Even if there havebeen attempts to restore the leverage theory against the criticism (such as Whinston 1990
and Nalebu§ 2004), these constructions require some relatively strong assumptions (asexplained later) and the existing leverage theory does not deal with tying in two-sided
markets. Motivated by the recent Google case in Europe and other countries where Google
is accused of an abuse of dominance with its practice of bundling applications for Android
mobile phones,3 we develop a leverage theory of tying in two-sided markets. In a settingin which the "single monopoly proÖt result" holds if the market of the tied product isone-sided, we uncover a new channel through which a monopolistic Örm in one market
can credibly leverage its monopoly power to another competing market if the latter is
two-sided, which can invalidate the one monopoly proÖt result.
Our analysis applies when platforms in a two-sided market are constrained to set nonnegative prices on the consumer side. In two-sided markets the need for all sides of themarket to engage creates a ìchicken and eggî problem (Caillaud and Jullien, 2003) inthat members of each group are willing to participate in the market only if they expect
many members from the other side to participate. The literature on two-sided markets
has analyzed the optimal pricing structure to coordinate the demands of distinct groupsof customers and shows that below-cost pricing naturally arises on one side in order toenhance participation because the loss from the below-cost pricing can be recouped on theother side of the market (see Armstrong (2006) and Rochet and Tirole (2006)). When themarginal cost is low as in digital markets, this implies that the optimal pricing strategy
entails negative prices. However, we can imagine situations in which negative prices may
1
For instance, the U.S. Supreme Court invoked the leverage theory in declaring block booking (i.e.,bundling of movies) illegal in U.S. v. Paramount Pictures (1948) and in U.S. v. Loewís (1962).
2
See, for instance, Areeda, Kaplow and Edlin (2013) for a discussion of the single monopoly proÖt
theory.
3
Russian antitrust o¢cials Öned Google $6.8 million on August 11, 2016 for abusing its market position
with its mobile operating system Android. Google is also under investigation in Korea and India.

1

be impractical due to adverse selection and opportunistic behaviors by consumers (Farrell
and Gallini (1988) and Amelio and Jullien, 2012).4
We show that tying provides a mechanism to circumvent the non-negative price constraint in the tied product market without inviting an aggressive response by the rival
Örm if the rival Örmís price response to tying faces the non-negative constraint. In ourmodel, the non-negative price constraint plays two roles: 1) it limits competition in thetied good market, which creates additional surplus to extract through tying, and 2) it limits the rival Örmís response to tying. In addition, our theory Ölls the gap in the literatureby showing that tying is credible without any commitment mechanism and forecloses therival in the tied product market regardless of whether the two products tied together are
independent or complementary. The credibility of tying can explain the use of contractual
tying (such as Googleís contract with handset manufacturers) for the leverage purpose.5
To see the incentives to leverage monopoly power, consider a monopolist in one market
that also competes in a two-sided market with a more e¢cient rival Örm (i.e., a rival
platform). The two-sided market consists of consumers on one side and advertisers on
the other. We envision a situation in which consumers are competitive bottleneck in the
two-sided market (see Armstrong (2006) and Armstrong and Wright (2007)); a platform
can extract extra revenues from the advertiser side when it attracts a consumer from thebottleneck side. First, in the absence of tying, the non-negative price constraint limits
competition on the consumer side of the two-sided market and generates additional proÖt
for the more e¢cient rival Örm which cannot be competed away. This creates incentives forthe monopolistic Örm to tie and e§ectively engage in a negative price in order to capture
this extra surplus. Second, when the monopolistic Örm tries to steal the additional proÖt
in the two-sided market, tying will induce a more aggressive response by the rival Örm
as in Whinston (1990). However, the rival Örmís ability to respond aggressively can bemuted as the non-negative price constraint begins to bind. Thus, the non-negative priceconstraint in two-sided markets plays the dual role of creating additional surplus andlimiting aggressive response by the rival Örm, which provides incentives to tie by themonopolistic Örm.
4
To quote Farrell and Gallini (1988), "[a]t a negative price, people could take computers at a negative
price and use them for landÖll (p.679)."
5
The tying literature distinguishes contractual tying from technical tying: the former can be undone
ex post with a relatively low cost while the latter can be undone only with a signiÖcant cost. Therefore, technical tying can be used as a device to pre-commit to tying when tying is not ex post credible(Whinston, 1990).

2

The paper is partly motivated by the recent antitrust investigations concerning Google
in Europe and elsewhere. In one of the investigations, the European Commission (EC)is concerned with Googleís conduct in relation to applications and services for smartphones and tablets.6 In particular, Google has been alleged by rival Örms to "illegally
hinder the development and market access of rival applications and services by tying or
bundling certain Google applications and services distributed on Android devices withother Google applications, services and/or application programing interfaces of Google."7
After preliminary investigations, the EC issued a statement of objections on April 20,
2016, informing Google of its view that "the company has, in breach of EU antitrust
rules, abused its dominant position by imposing restrictions on Android device manufacturers and mobile network operators."8 Even though Googleís Android operating system
for mobile devices is supposed to be "open," Google e§ectively implements bundling of its
apps through Androidís Mobile Application Distribution Agreements (MADA) contracts
that force manufacturers to include them and their respective terms in an all-or-nothingfashion.9 According to MADA contracts, Android device manufacturers must preinstall
all or none of Google apps, depriving them of the ability to mix and match Google apps
considered essential with other third-party apps which may be superior to some of thelisted Google apps. Even though they are allowed to preinstall competing apps in addition to the listed Google apps with similar functionalities, they are reluctant to do so
because of the limited screen space available on devices. In addition, Google requires its
apps to be the default in key app categories such as search and location and be placed
in prominent positions on the screen.10 As a result, the potential for installing multiple
apps (i.e., multihoming) to mitigate the e§ects of tying is somewhat limited (Edelman,
6

The ECís other investigations on Google concern Googleís search engine, which has been alleged tobias its search results to promote other revenue-generating Google services such as comparison shopping. See the European Commission Fact Sheet entitled "Antitrust: Commission sends Statementof Objections to Google on comparison shopping service," released on April 15, 2015. Available at
http://europa.eu/rapid/press-release_MEMO-15-4781_en.htm
7
See the European Commission Fact Sheet entitled "Antitrust: Commission opens formal investigation
against Google in relation to Android mobile operating system," released on April 15, 2015. Available
at http://europa.eu/rapid/press-release_MEMO-15-4782_en.htm
8
See the European Commission Press Release entitled "Antitrust: Commission sends Statement of Objections to Google on Android operating system and applications," available at
http://europa.eu/rapid/press-release_IP-16-1492_en.htm
9"Devices may only be distributed if all Google Applications [listed elsewhere in the agreement] ... are
pre-installed on the Device." See section 2.1 of the MADA between Google and HTC.
10
For instance, the phone manufacturer must set ìGoogle Search ... as the default search provider for
all Web search access points.î See MADA Section 3.4(4). The same requirement applies to Googleís
Network Location Provider service.

3

2015).11 The ECís analysis in relation to the investigation of Google also indicates that"consumers rarely download applications that would provide the same functionality as
an app that is already pre-installed (unless the pre-installed app is of particularly poor
quality)."12 The tying mechanism we uncover can be applied to the Google Android caseand can explain why Google has an incentive to tie. We further show that such tying
arrangements can lead to the exclusion of (more e¢cient) rival Örms, providing a theoryof harm associated with Googleís practices in the MADA contracts.13
In Section 2, we present our mechanism of tying in the baseline model of homogenous
consumers. In a setting in which the well-known "single monopoly proÖt result" holds if
the tied market are one-sided (or in the absence of the non-negative price constraint), weshow that the interplay between two-sidedness of the tied market and the non-negative
pricing constraint breaks the result and restores an incentive to tie to leverage monopoly
power into the tied two-sided market. In addition, this mechanism is robust even when
consumers can multi-home (section 2.2) and when the bundled products are perfect complements instead of having independent values (section 2.3). Moreover, our insight canbe more widely applied to any markets in which sales to consumers in one market cangenerate additional revenues that cannot be competed away due to non-negative priceconstraints. SpeciÖcally, even if the tied market is one-sided, additional revenue can begenerated in an intertemporal context from the same market in the future in the presenceof network e§ects or switching costs (section 2.4). We also brieáy discuss how we canmodify the model to be applied to the Google case (section 2.5) because app developers
can make payments to original equipment manufacturers (OEMs) to have their apps to bepre-installed, which makes the non-negative price assumption less applicable to OEMS.
In Sections 3 and 4, we extend the baseline model by incorporating heterogenous
consumers in the tying market. This allows us to check the robustness of our insightobtained from the baseline model and also addresses the question of how the degree of
two-sidedness of the tying market a§ects the incentive to tie.14 Sections 3 and 4 di§er
11

For a discussion of how to apply antiturst law to zero-price markets, including the issues of market
deÖnition and market power, from the legal perspective, see Newman (2015, 2016).
12
See the European Commission - Fact Sheet entitled "Antitrust: Commission sends Statement of
Objections to Google on Android operating system and applications,í released on April 20, 2016. Available
at http://europa.eu/rapid/press-release_MEMO-16-1484_en.htm.
13
For a detailed discussion of the key aspects of Googleís practices and their potentially exclusionarye§ects in the mobile phone industry from a legal perspective, see Edelman and Geradin (2016).
14
In the baseline model of homogenous consumers, the incentive to tie does not depend on the degreeof two-sidedness of the tying market.

4

in that we introduce intergroup network e§ects in the tied market in Section 4 while no
network e§ect is introduced in Section 3.
More speciÖcally, in section 3 we assume that consumers are bottleneck both in thetying and in the tied two-sided markets. For example, the other side interacting with
consumers in both markets could be advertising. In such a case, we Önd that the incentive
to tie increases with the degree of two-sidedness of the tied market but decreases with thedegree of two-sidedness of the tying market, where the two-sidedness is represented by theimportance of advertising revenues. For instance, when the tying market is su¢ciently
two-sided while the tied market is close to being one-sided, tying reduces both Örmsí
proÖts as in Whinston (1990). Under tying, consumers whose valuations for the monopoly
product are low enough prefer buying the competing tied product instead of the bundle.
This means that the tying Örm loses the advertising revenue in the tying market from these
consumers. This loss increases with the degree of two-sidedness of the tying market. As
our model in Section 3 includes tying in one-sided markets as a special case, it also captures
various motives for tying identiÖed by the previous literature in one-sided markets.
Section 4 introduces cross-group network e§ects in the tied market while maintaining
heterogenous consumers in the tying market. For instance, the other side in the tied market could be applications. Consumersí beneÖts increase with the number of applications
and application developersí beneÖts increase with the number of consumers. When there
is no tying, the network e§ects generate a tipping equilibrium toward the more e¢cient
rival platform in the tied market. However, tying can generate a tipping toward thetying platform and such tipping is more likely as the two-sidedness of the tying market
increases. Intuitively, when the tying market becomes more two-sided, the tying platform
becomes more aggressive in pricing in order to capture more consumers in the tying market, which generates a larger network e§ect and hence makes tipping more likely in thetied market. When tipping occurs, there is no loss of advertising revenue in the tying
market and therefore the incentive to tie increases with the degree of two-sidedness of thetying market. In addition, we show that this insight is robust to an interior equilibrium
in which the tying platform occupies a large consumer market share. We expect that asimilar insight can be applied to when the tying market (or both the tying and the tiedmarkets) exhibits cross-group network e§ects.
Our paper contributes to the literature on the leverage theory of tying, which has
been extensively studied. In a classical paper on tying, Whinston (1990) shows thatselling a monopolized primary product and an unrelated di§erentiated product together5as a bundle may allow the monopolist to commit to a more aggressive pricing strategy,
preventing entry in the di§erentiated market. In Carbajo, De Meza, and Seidman (1990),
on the other hand, bundling is used as a strategy to segment the market and relax pricecompetition.15 By considering a model in which each of the tying and the tied markets canbe one-sided or two-sided, our paper encompasses both Whinstonís and Carbajo et al.ís
results as special cases. In particular, when the tied market is two-sided, we uncover a newchannel through which tying is proÖtable at the expense of rival Örms. We emphasize that
our leverage mechanism does not have any credibility issue in sharp contrast to Whinston(1990) and Nalebu§ (2004). Our mechanism thus does require neither technical tying as
a commitment device as in Whinston (1990) nor the controversial assumption that the
tying Örm is a Stackelberg leader in setting prices as in Nalebu§ (2004).16 We show that amonopolistís tying is credible and reduces the rivalís proÖt if the tied market is two-sided,
which provides a justiÖcation for the use of contractual tying. We further show that tying
is credible and forecloses the rival even in the case of perfect complements. This result
is noteworthy because it is known that a monopolist selling a perfectly complementary
product has no incentive to practice tying to exclude the rival (Proposition 3 of Whinston,1990).
Choi and Stefanadis (2001) and Carlton and Waldman (2002) provide a dynamic
leverage theory of tying. In Choi and Stefanadis (2001), the probability of successful
entry depends on the level of R&D expenditures. When an incumbent monopolist in
two complementary components practices bundling, it makes entry in one component
completely dependent upon success in the other. By making the prospects of investment
less certain, bundling discourages rivals from investing and innovating. Carlton andWaldman (2002) construct a model of bundling with dynamic entry deterrence. In their
two-period model, the incumbent initially has a monopoly position in both the primary
and the complementary market. There is a potential rival that has the opportunity
to enter the complementary market in period 1 and the primary market in period 2.
15

See also Chen (1997).
A notable exception is Peitz (2008), who points out the credibility issue and builds an example
in which a monopolistís tying is credible and reduces the rivalís proÖt. Hurkens, Jeon and Menicucci(2016) restores credibility by considering that the tying Örm is not a monopolist but a dominant Örm
facing competition in the tying market. In the context of competition among product portfolios in acommon agency setting, Jeon and Menicucci (2006, 2012) Önd that pure bundling is credible as it reducescompetition among the products within a Örmís portfolio but that competitive bundling leads to e¢cient
outcome if the buyer has no budget constraint (otherwise, bundling builds an entry barrier since it allowsÖrms with large portfolios to capture all the budget).
16

6

Furthermore, the potential rival can recoup its Öxed cost of entry into the complementary
market only by actively operating in both periods. By practicing bundling, the incumbent
prevents the competitorís complementary component from having access to customers inperiod 1, permanently blocking the productís introduction. As a result, the rival also
refrains from entering the primary market; entry without a complementary component
would not be proÖtable. The incumbent maintains its monopoly proÖts in both periods.
However, most papers in the literature consider one-sided markets. We show how thetwo-sidedness of markets in conjunction with the non-negative price constraint generates a
new channel through which monopoly power in one market can be leveraged into another.
Amelio and Jullien (2012) analyze the e§ects of tying in two-sided markets. They
consider a situation in which platforms would like to set prices below zero on one sideof the market to solve the demand coordination problem in two-sided markets, but are
constrained to set non-negative prices. In the analysis of Amelio and Jullien, tying canserve as a mechanism to introduce implicit subsidies on one side of the market in orderto solve the aforementioned coordination failure in two-sided markets. As a result, tying
can raise participation on both sides and can beneÖt consumers in the case of a monopoly
platform. In a duopoly context, however, tying also has a strategic e§ect on competition.
They show that the e§ects of tying on consumer surplus and social welfare depend on
the extent of asymmetry in externalities between the two sides. Their paper and ours arevery di§erent in terms of focus and mechanisms through which tying a§ects competition.
Their main focus is to compare the e§ects of tying across di§erent market structures(monopolistic vs. duopolistic) whereas our focus is on the leverage of monopoly power in
two-sided markets. Choi (2010) also analyzes tying in two-sided markets. However, his
focus is on the role of multi-homing. In particular, he shows that tying can be welfareenhancing if multi-homing is allowed, even in cases where its welfare impacts are negativein the absence of multi-homing.
Finally, Iacobucci (2014) develops a similar idea to our mechanism. In particular,
he argues that the elimination of competition via tying of two platforms in one sideof the market can expand the set of customers to whom the tying Örm can sell in theother side of the market. This is similar to our mechanism in which capturing thecustomer side provides additional revenue from the advertiser side. However, he does
not provide a formal analysis as his argument is based on a numerical example. In
addition, his logic breaks down when the non-negative price constraint is not binding as
the same e§ects can be achieved by an appropriate pricing scheme without the use of7

tying. This is due to his implicit assumption that Örms set prices equal to marginal
costs in a competitive equilibrium. However, in the context of complementary products
envisioned in his model, the equilibrium price in each component market need not be equal
to marginal cost because it is the total price, not the two individual prices, that mattersfor complementary products. In contrast, we provide a rigorous and formal analysis oftying that fully accounts for intricacies of two-sided markets.
The rest of the paper is organized in the following way. In section 2, we present thebaseline model of homogenous consumers to highlight the main essence of the leveragemechanism in two-sided markets and the importance of the non-negative price constraint.
After establishing the robustness of our mechanism to multihoming and perfect complements, we demonstrate the applicability of our framework to other types of markets with
auxiliary revenues. In section 3, we extend the baseline model by introducing heterogenous consumers in the tying market and investigate how the two-sidedness of both thetying and tied product markets interact to draw policy implications. In Section 4, weconsider a model with inter-group network e§ects in the tied good market. We provide
concluding remarks in section 5. Detailed proofs are relegated to the Appendix.

2

The Baseline Model of Homogenous Consumers

To show how the interaction between the two-sidedness of markets and the non-negative
price constraint generates new incentives to tie, we start by analyzing a simple baselinemodel of homogenous consumers. In order to highlight the leverage mechanism in twosided markets, we consider a setting in which the well-known "single monopoly proÖt"
result holds and hence the extension of monopoly power from one market to another is
not possible in the benchmark without two-sided markets.
Consider two markets, A and B. We build a simple framework to analyze tying as aleverage mechanism when both markets are characterized as two-sided. More speciÖcally,in each market there are three classes of agents. There are two distinct customer groups(consumers and advertisers) that interact with each other and intermediaries which enable
these two groups to ìmeetî with each other. In the example of the Internet search market,
the two customer groups can be described as advertisers and consumers (who search for
information through the Internet). The intermediaries are search engine providers such
as Google and Bing. We assume that the two markets are independent in the sense thatthe value that each consumer (or advertiser) derives from participating in a market is8independent of whether he or she participates in the other market.
To analyze incentives to leverage market power, we assume that market A is servedby Örm 1, a monopolistic platform, and entry to market A is not possible. In contrast,
two platforms, Örm 1 and Örm 2, compete in market B. More speciÖcally, we envisiona competitive bottleneck situation in which platforms provide services (such as Internet
search and video streaming) to consumers and use the customer base to derive advertising
revenues from advertisers who need access to consumers. There is a mass one of identicalconsumers, who have a unit demand for each service/product. To focus on the strategic
motive for bundling, we assume that there is no cost advantage or disadvantage associatedwith bundling.
In market A, each consumerís reservation value is denoted by u. In market B, eachconsumerís willingness to pay for each Örmís product is given by v1 and v2 , respectively,
where ! $ v2 % v1 > 0. For most of the paper, we assume that each consumer buys only
one of the two products in market B and normalize Örmsí production costs to zero inall markets.17 This single-homing assumption is relaxed in Section 2.2. Each consumer
yields additional proÖt of & & 0 and ' & 0 to the advertiser side in market A and B,
respectively. If & = 0, the tying market is one-sided and if ' = 0, the tied market isone-sided.
2.1

The Leverage Mechanism with the Non-Negative Price Constraint in
Two-Sided Markets

Consider the benchmark in which both markets are one-sided (i.e., & = ' = 0). If
these products are sold independently without tying, Örm 1 will charge pA
1 = u in its
monopolized market A. In market B, both Örms compete in prices and the equilibriumB
prices are given by pB
1 = 0 and p2 = !, where ! $ v2 % v1 > 0. All consumers buy fromÖrm 2. Each Örmís proÖts are given byB$1 = ) A
1 + ) 1 = u + 0 = u;

)B
= !:
2
17

This is not an innocuous normalization assumption with the non-negative price constraint because
it also implies no below-cost pricing. However, as long as production costs are not too high (as would
be the case for digital goods), our results would be robust even if we allow below-cost pricing. More
importantly, we can derive qualitatively similar results when we introduce asymmetry between the twoÖrms in market B in terms of production costs rather than product qualities.

9Now suppose that Örm 1 ties its monopolized product A with product B1: Let Pe andpe2 denote the bundle price and Örm 2ís price for product B under tying.18 We assume
that a consumer buys either the (A; B1) bundle or product B2: this assumption is relaxedby allowing multi-homing in section 2.2. Then, Örm 1 will be able to sell its bundled
products only if
u + v1 % Pe & v2 % pe2 :Since Örm 2 will be willing to set the price as low as its marginal cost, which is zero,
the maximum price Örm 1 can charge for its bundle in order to make sales is given by
Pe = u % !: Under tying, Örm 1ís proÖt is given by
e 1 = max[u % !; 0];$e 1 = max[u % !; 0] < u = $1 .19 Thus, evenwhich is less than the proÖt without tying as $if tying provides a mechanism to capture the tied good market, it is a Pyrrhic victory as
it reduces Örm 1ís proÖts. Thus, the monopoly Örm has no incentives to tie to extend its
market power to the other market. This is the essence of the Chicago schoolís criticism of
the leverage theory of tying. However, this conclusion can be overturned if the tied goodmarket B is two-sided.
Suppose now that market B is two-sided while market A is one-sided. Assume thatthe advertising revenue is large enough in market B such that ' > ! holds.20 Due tothis additional revenue source, now each Örm in market B is willing to set its price below
its marginal cost on the consumer side up to '. We, however, put a non-negative pricerestriction that the market prices cannot be negative. Thus, in the absence of tying, onceBagain Örm 1 charges a price of pB
1 = 0 and Örm 2 will set the price of p2 = !. Firm 1ís
proÖt in market B is zero. However, note that Örm 2í proÖt is given by ) B
2 = ! + ' dueto the additional revenue from the advertiser side.
Now suppose that Örm 1 ties its monopolized product with product B on the consumer
side. Assume u > ! such that Örm 1 can conquer market B by tying. Once again, themaximum bundle price that enables the tying Örm to make sales is given by Pe = u % !:However, due to the additional revenue source in market B, the tying Örmís proÖt now
18

We denote variables under the tying regime with a tilde (~).
If v1 & v2 , Örm 1ís proÖt with tying is the same as the one without tying.
20
According to the estimates by Önancial analyst Richard Windsor, Google made about $11 billion in2015 from advertising sales on Android phones through its apps such as Maps, Search and Gmail. See
Chee (2016).
1910

e 1 = u + (' % !) > u = $1 : Hence, tying is proÖtable if the tied good market
becomes $is two-sided.
This simple model also illustrates the importance of the non-negative price constraint
for the two-sidedness of the market to restore incentives to tie.21 To see this, supposethat Örms can charge a negative price. Then, after tying Örm 2 would be willing to set
the price as low as %': Thus, the maximum bundle price that enables the tying Örm to
make sales is now given by Pe = u % ! % ': As a result, the tying Örmís proÖts is now
e 1 = u % ! < u = $1 as in the Chicago school benchmark case above.
reduced to $Finally, we can consider the case in which the tying market is two-sided (i.e., & > 0).
This would increase Örm 1ís proÖt by & regardless of tying as long as it sells its monopoly
product. Hence, whether the tying market is one-sided or two-sided does not a§ect the
incentive to tie.
Summarizing, we have:
Proposition 1. Consider the baseline model of homogenous consumers in which products
have independent values.
(i) Tying is never proÖtable and hence the single monopoly proÖt result holds either if
the tied-market is one-sided or if Örms can charge negative prices.
(ii) Suppose u > !. Tying is proÖtable if the tied-market is two-sided such that ' > !
holds and Örms cannot charge negative prices. Then, tying reduces both Örm 2ís proÖt
and social welfare, but increases consumer surplus.
(iii) Whether the tying market is one-sided or two-sided does not a§ect the above
results.
Our simple model demonstrates two important di§erences between the case of a onesided tied product and that of a two-sided tied product, both of which are generated by
the non-negative price constraint. First, in the absence of tying, Örm 2 realizes a proÖt
beyond its competitive advantage: its proÖt per consumer is larger than ! by ', whichcannot be competed away due to the binding non-negative price constraint that Örm 1
faces. This creates an incentive for Örm 1 to practice tying in order to subsidize its productB by circumventing the constraint of non-negative pricing. Second, when Örm 1 practices
tying and is e§ectively engaging in negative pricing for the tied product, Örm 2 would
respond more aggressively against tying. However, its aggressive response can be limited21
Our leverage theory can also be applied to other kinds of price restrictions such as price regulations
that prohibt a price below a certain level.

11as Örm 2 hits the zero price áoor even though it is willing to set a price as low as %':Therefore, tying in two-sided markets can be immune to an aggressive price response by
the rival Örm because the non-negative price constraint limits an aggressive response.22
Tying reduces welfare as it induces consumers to consume the inferior product of Örm
1 instead of the superior product of Örm 2 in market B. This force will be preserved in any
extension. However, tying may increase welfare by expanding the sale of the monopoly
product, which we examine in a model of heterogenous consumers in market A, in thenext section.
Two remarks are in order. First, the result in Proposition 1(ii) that tying increases
consumer surplus has to do with the fact that tying reduces Örm 2ís price from ! tozero and results from the assumption of homogenous consumers. Instead, if consumers
are heterogenous in terms of their valuations of Örm 2ís product, then Örm 2 may Önd it
optimal to charge a zero price in the absence of tying because of the advertising revenue.
In this case, tying will not a§ect the price charged by Örm 2 and hence will not increase
consumer surplus. Second, for a similar reason, Proposition 1(iii) is likely to be an artifactof the assumption of homogenous consumers. In order to investigate how the degree of
two-sidedness in the tying market (captured by &) a§ects Örm 1ís incentive to tie, we
consider heterogenous consumers in the tying market in Sections 3 and 4.
2.2

Extension to Multihoming

Up to now, we have assumed that multihoming is impossible. We here allow consumers
to do multihoming at some cost. Let m denote the multihoming cost. Consumers areheterogenous in terms of the multihoming cost, which is assumed to be distributed over[0; m] according to distribution G(:) with strictly positive density g(:), where m = 1 isallowed. In addition, assume that the density function g(:) is log-concave andm > !, G(!)=g(!) ( ':With a log-concave density function, G(:)=g(:) is increasing. Thus, G(!)=g(!) ( ' meansthat the revenue from the advertising side is large enough in market B compared to thedegree of superiority of Örm 2ís product !. Note that the introduction of multihoming
22

In the actual Google case, competition in search engines takes place mainly with payments to OEMsfor pre-installation of the applications (not to consumers), which can be at odds with our assumption of
non-negative pricing. We address this issue in sectioin 2.5.

12

does not a§ect the analysis of no tying as there is no multihoming. We below extend thetying analysis of Section 2.1. Consider & = 0 and ' > 0.23
Our analysis shows that the equilibrium prices with multi-homing are the same as thosein the previous analysis where multihoming is not feasible. We thus have the following
proposition.
Proposition 2. Consider the baseline model of homogenous consumers in which products
have independent values. Suppose that multi-homing is possible.
(i) Without tying, all consumers use the superior product of Örm 2 in market B andno consumers multi-home.
(ii) With tying, the equilibrium prices are given by Pe = u % ! and pe2 = 0: Consumers#
whose multi-homing cost is less than ! multi-home. Tying is proÖtable if ' > [1!G(#)];
that is, ' is relatively large compared to !. Social welfare always decreases with tying eventhough consumers are better o§.
Proof. See the Appendix.
Proposition 2 indicates that Proposition 1(ii) naturally extends and is robust to thepossibility of multi-homing. Introducing multihoming does not a§ect the equilibrium
prices and those whose multihoming cost is above ! buy only the bundle (while all
the others multihome). Tying reduces welfare because single-homing consumers use theinferior tied product while those who multihome incur the cost of multihoming. However,tying increases every consumerís surplus at the expense of Örm 2, as Örm 2 charges azero price and captures a smaller market share. More precisely, in the absence of tying
each consumer has a surplus of v1 . Under tying with multi-homing, a consumer receives
a surplus of v1 + ! + max[0; ! % m]: Firm 2ís proÖts decrease from (! + ') to G(!)'.
In the rest of the paper, we maintain the single-homing assumption.
2.3

Extension to Perfect Complements

We show that our leverage mechanism can also be applied when the tying and the tied
products are complements. Suppose that the monopoly product A of Örm 1 and productB are perfect complements that need to be used together. Consumers can use one of
the two system products, (A; B1) and (A; B2), depending on which Örmís product B is
23

Analyzing the case of # > 0 does not a§ect the result because Örm 1ís proÖt increases by the sameamount of # with or without tying.

13

used. Consumersí valuation for system (A; Bi) is given by u + vi , where i = 1; 2 and! $ v2 % v1 > 0. Once again, we demonstrate that the non-negative price constraint (inconjunction with the two-sidedness of the tied good market) can create incentives to tie
even for the perfect complements case. Assume & = 0.24
As a benchmark, consider Örst the case in which the non-negative price constraint is notbinding. Without tying, Örms are willing to cut the price down to %' for product B andÖrm 2ís proÖt can never be larger than !. In fact, there is a continuum of Nash equilibria
due to Örm 1ís ability to "price squeeze" and extract a portion of the surplus ! (Choiand Stefanadis, 2001). More precisely, there is a continuum of equilibria parameterizedby 3 2 [0; 1], which represents the degree of price squeeze exercised by Örm 1:
pbA
bB
bB
1 = u + v1 + ' + 3!; p
1 = %', p2 = %' + (1 % 3)!

In equilibrium, consumers purchase the system that includes B2.25 Firm 1ís proÖt is$1 = pbAbB
1 = u + v1 + ' + 3! and Örm 2ís proÖt is ) 2 = p
2 + ' = (1 % 3)!. It can beeasily veriÖed that no Örm has an incentive to deviate because they jointly extract thefull surplus from consumers.
With tying, only system (A; B1) is available to consumers since products A and B are
e 1 = u + v1 + ' (< $1 ) while Örm
perfect complements. Firm 1ís proÖt under tying is $2ís proÖt is zero. Therefore, for any 3 2 [0; 1], Örm 1 has no incentive to practice tying
to exclude the superior complementary product of the rival Örm. This result reconÖrmís
Whinston ís (1990) Önding (i.e., Proposition 3, p.851) and lends support to the Chicago
schoolís criticism of the leverage theory.26
Consider now the case in which the non-negative price constraint is binding. Without
tying, once again, we have a continuum of equilibria parameterized by 3 2 [0; 1]:B
B
pA
1 = u + v1 + 3!; p1 = 0, p2 = (1 % 3)!:24

As in the analysis of multi-homing, adding # > 0 does not a§ect the analysis.
We adopt the standard tie-breaking convention that if a consumer is indi§erent between buying
product B from Örms 1 and 2, he will buy from Örm 2 (the high-quality product producer) because thisconvention is equivalent to deÖning price equilibrium as the limit in which prices must be named in somediscrete unit of account of size % as % ! 0:26
For instance, according to Posner (1976, p. 173), "[A fatal] weakness of the leverage theory is its
inability to explain why a Örm with a monopoly of one product would want to monopolize complementary
products as well. It may seem obvious ... but since the products are by hypothesis used in conjunction
with one another ... it is not obvious at all."25

14

The non-negativity constraint binds for Örm 1ís product B (i.e., pB
1 = 0). As a result,consumers prefer buying B2 at any price pB
2 smaller than !. This means that Örm 2
captures the advertising revenue ' while Örm 1 can extract 3 2 [0; 1] fraction of ! fromÖrm 2. Hence, each Örmís proÖt without tying is given by:$1 = u + v1 + 3!; ) 2 = pB
2 + ' = (1 % 3)! + '.
e 1 = Pe + ' = u + v1 + 'With tying, Örm 1 can charge Pe = u + v1 and receives a proÖt of $and Örm 2ís proÖt is zero. Hence, as long as ' > !, Örm 1 has an incentive to tie
regardless of the value of 3.
Summarizing, we have:
Proposition 3. Consider the extension of the baseline model of homogenous consumers
to perfect complements.
(i) If Örms can charge negative prices, tying is never proÖtable regardless of whether
the tied-market is one-sided or two-sided.
(ii) If Örms cannot charge negative prices and the tied-market is two-sided enough(i.e., ' > !), tying is proÖtable regardless of the degree of price squeeze prevailing in theabsence of tying.
We provide a novel mechanism for leveraging monopoly power though tying for thecase of perfect complements; to the best of our knowledge, such theory has not beendeveloped yet in the previous literature. For instance, our theory may be applied to theMicrosoft tying cases in which the operating system was bundled together with othercomplementary applications such as a web browser and a media player. Our theory also
exposes the áaw in the reasoning of Iacobucci (2014) who proposes a theory of tying for
perfect complements in two-sided markets: the two-sidedness is a necessary condition fortying to be credible, but not a su¢cient condition because it should be combined with
the non-negativity constraint in pricing as we have shown above.
2.4

Applications to Other Markets with Ancillary Revenues

We have developed a leverage theory of tying in the context of two-sided markets. However, our insight can be applied to any market in which sales to consumers in one market
can generate additional revenues that cannot be competed away due to non-negative priceconstraints. This additional revenue came from the other side of the market in the case15

of a two-sided market. The additional revenues can also be generated in an intertemporalcontext with the other market being the same market in the future. To illustrate thisidea, we present two variations of the baseline model with & = ' = 0.
2.4.1

Tying in Markets with (Direct) Network E§ects

Market A is the same as in the baseline model. Market B is characterized with networke§ects. We adopt a simpliÖed and modiÖed version of Katz and Shapiroís (1986) dynamic
technology adoption model with network e§ects for market B. Let us assume a stationary
environment with respect to both Örmsí technologies.27 There are two periods. In periodt (= 1; 2), N t consumers of generation t make a purchase decision. For simplicity, assume
N 1 = N 2 . Each period, two Örms compete with incompatible products B1 and B2. A
consumer who purchases product B1 in period t derives net beneÖts of v1 +6(x1 +x2 )%pt1 ,and the corresponding value from product B2 is given by v2 +6(y 1 +y 2 )%pt1 ; where xt andy t denote the quantities of product B1 and B2, respectively, sold in period t. Consumers
have the same preferences and coordinate on the Pareto-e¢cient outcome when there
are multiple equilibria. Once again, Örmsí production costs are zero in all markets and! = v2 %v1 > 0.
In the absence of tying, Örm 1 charges a price of pA
1 = u in its monopolized market A.
In market B, suppose that Örm 1 has captured market B in the Örst period. This confers
Örm 1 an installed-base advantage in period 2. Let us assume that this installed-baseadvantage is su¢ciently large such that 6b % 62 > !, where 6b = 6(N 1 + N 2 ) and 6t= 6(N t ). Then, Örm 1 will be able to maintain its market dominance with a price of(6b % 62 ) % !. If Örm 2 has captured market B in the Örst period, it can maintain thedominance at a price of (6b % 62 ) + !:Now consider market competition in the Örst period. If Örm 2 decides to capture themarket in period 1, its price needs to satisfy
v1 + 6b % p11 ( v2 + 6b % p12 :Firm 1 is willing to charge a price equal to %9[(6b % 62 ) % !] < 0. However, the nonnegative price constraint limits its price at zero. Given this, Örm 2 is able to sell at a27

In contrast, Katz and Shapiro (1986) assume a situation in which one technology is better than theother in the Örst period but becomes inferior in the second period to derive interesting dynamics.

16

price of ! in the Örst period. As a result, each Örmís overall proÖt is given by$1 = (1 + 9)u;)B
= ! + 9[(6b % 62 ) + !]:2
Now suppose that Örm 1 ties its monopolized product A with product B1. Consider
a subgame in which all consumers bought the bundle. Let Pe2 and pe22 denote the bundle
price and Örm 2ís price for product B in period 2 under tying. Then, Örm 1 will be able
to sell its bundled products if
u + v1 + 6b % Pe2 & v2 % pe22 + 62 :Since Örm 2 will be willing to set the price as low as its marginal cost, which is zero,
the maximum price Örm 1 can charge for its bundle in order to make sales is given by
Pe2 = u % ! + (6b % 62 ). Under tying, Örm 1ís proÖt in the second period is given by
e 2 = u % ! + (6b % 62 ) > u:$1

If consumers bought from Örm 2 in period 1, the maximum price Örm 1 can charge for its
bundle in order to make sales is given by Pe2 = u % ! % (6b % 62 ), which is assumed to bepositive with a su¢ciently high u. Firm 1ís proÖt in period 2 in this subgame is given bye 21 = u % ! % (6b % 62 ):$Consider competition in period 1 under tying. If Örm 1 decides to capture the market
in period 1, its price needs to satisfy
u + v1 + 6b % Pe1 & v2 + 61 % pe12 :Note that 6b is present on the LHS but 61 on the RHS because Örm 1 captures the second
period market regardless of who wins the Örst-period one. Therefore, the highest pricethat allows Örm 1 to capture the market in period one is Pe1 = u % ! + (6b % 61 ). In thiscase, Örm 1ís overall proÖt is given by
e1 = $
e1 + $
e 2 = (u % !) + (6b % 61 ) + 9[u % ! + (6b % 62 )]
$
1
1
17

e1 = $
e1 + $
e2 =In contrast, if Örm 1 gives up the market in period 1, its proÖt becomes $11b2
0 + 9(u % ! % (6 % 6 )). Thus, Örm 1 chooses the Örst option. Therefore, tying can be
proÖtable if the following condition holds

that is,e 1 = (u % !) + (6b % 61 ) + 9(u % ! + (6b % 62 )) > (1 + 9)u = $1 ;$(6b % 61 ) + 9(6b % 62 ) > (1 + 9)!:If N 1 = N 2 with 61 = 62 , this condition simpliÖes to
6b % 62 > !:Carlton and Waldman (2002) develop a related model of tying for complementary
products in the presence of network e§ects. However, in their model tying is strategicallyused to preserve monopoly power rather than extend monopoly power to another market.
In their model, for some parameter constellations the monopolist need not actually tie its
products and can achieve the same goal with a virtual tie by charging a "high" price for thetying product and a "low" price for the tied product because they consider complementary
products. They point out that a real tie is needed when the non-negativity constraint
for the tied product is binding but do not systematically explore implications of the nonnegativity constraint as we do in the current paper.28
2.4.2

Tying in Markets with Switching Costs

Market A is the same as in the baseline model, monopolized by Örm 1 with consumers
having valuation of u for the product. Market B is one-sided but characterized with aswitching costs of s (Klemperer, 1990). We consider a two-period model with a discountfactor of 9: Once again, Örmsí production costs are zero in all markets.
In the absence of tying, Örm 1 charges a price of pA
1 = u in its monopolized market
A. In market B, we apply backward induction to derive the equilibrium. Suppose thatÖrm 1 has captured market B in the Örst period. In the second period, consumers needto pay a switching cost of s to buy from Örm 2. To illustrate the idea in the simplest
28
They also provide a model with network e§ects in which strategic tying is used to extend monopoly
power to the newly emerging market. However, their logic applies only when the tying and the tied
products are complementary and the new product in the emerging market is a perfect substitute for asystem consisting of the tying and the tied products.

18

manner, let us assume that s > ! = v2 %v1 > 0 and (s % !) < u: In this case, Örm 1 will
be able to maintain its market dominance with a price of (s % !): If Örm 2 has captured
market B in the Örst period, it can maintain the market at a price of (s + !):Now consider market competition in the Örst period. For analytical simplicity, we
assume myopic consumers. Firm 1 is willing to charge a price which is %(s % !) < 0:However, the non-negative price constraint limits its price at zero. Given this, Örm 2 isable to sell at a price of ! in the Örst period. As a result, each Örmís overall proÖts are
given by
$1 = (1 + 9)u)B= ! + 9(s + !)2
Now suppose that Örm 1 ties its monopolized product A with product B1: Consider
a subgame in which all consumers bought the bundle. Let Pe2 and pe22 denote the bundle
price and Örm 2ís price for product B2 in period 2 under tying. Then, Örm 1 will be able
to sell its bundled products if
u + v1 % Pe2 & v2 % pe22 % s:

Since Örm 2 will be willing to set the price as low as its marginal cost, which is zero,
the maximum price Örm 1 can charge for its bundle in order to make sales is given by
Pe2 = u % ! + s: Under tying, Örm 1ís proÖts in the second period are given bye 21 = u % ! + s > u:$

If consumers bought from Örm 2 in period 1, the maximum price Örm 1 can charge for its
bundle in order to make sales is given by Pe2 = u % ! % s: Firm 1ís proÖts in period 2 inthis subgame is given by
e 2 = u % ! % s:$
1
Consider competition in period 1 under tying. If Örm 1 decides to capture the market
in period 1, its price needs to satisfy
u + v1 % Pe1 & v2 % pe12 :This implies that the highest price Örm 1 can charge in period 1 and still capture the19market is given by Pe1 = u % !: In this case, Örm 1ís overall proÖt is given by
e1 = $
e 11 + $e 21 = (u % !) + 9(u % ! + s):$

e1 = $
e1 + $
e2 =In contrast, if Örm 1 gives up the market in period 1, its proÖt becomes $1
1
0 + 9(u % ! % s): Thus, Örm 1 chooses the Örst option. Therefore, tying can be proÖtable
if the following condition holds
e 1 = (u % !) + 9(u % ! + s) > (1 + 9)u = $1 ;$which can be written as

s>
2.5

1+9
!:
9

Duality Model with the Non-Positive Price Constraint in the Tying Good
Market: Application to the Google Android Case

We have developed a leverage theory of tying in two-sided markets with the non-negative
price constraint in the tied good market, motivated by the Google Android case . However, actual antitrust cases are fact-intensive and our model does not perfectly Öt withthe current Google case in one important aspect. More speciÖcally, the assumption of
non-negative prices is more appropriate for consumers, but in reality mobile phone manufacturers make decisions on which apps to pre-install and app developers can makepayments to OEMs to have their apps to be pre-installed, which e§ectively constitutes anegative price for consumers if the payments to OEMs are passed along to consumers.
To address this issue, we develop a dual model with the non-positive constraint in thetying good market, as opposed to the non-negative price constraint in the tied good market. For the sake of concreteness, consider YouTube or Google Play as the (monopolized)tying good (as there are no e§ective competitors) and Google search as the tied good (forwhich there are some potential competitors).29 Suppose that for some reason (outside of
the model) Google is not charging or unable to charge any positive price to consumers or
OEMs for YouTube or Google Play.
In reality, consumers can use these applications for free and as their pre-installation by
OEMs simply facilitates consumersí usage, it seems hard for Google to charge OEMs forthe pre-installation. To explain why Google o§ers such killer applications for free, we can29
See Edelman and Geradin (2016) and the ECís
http://europa.eu/rapid/press-release_MEMO-16-1484_en.htm.

20

press

releaseavailableatthink of the following microfoundation based on reference-dependent preferences.30 Whenthese apps were Örst introduced, they needed to be o§ered for free for strategic reasons
such as market penetration and building an installed base of consumers. Even when they
have established a dominant position with insurmountable lead over competitors, they
may not be able to charge a positive price if consumers are accustomed to free app andbalk at paying any positive price. In other words, consumersí loss aversion with the zeroreference price creates a non-positive price constraint for Google.
This implies that Google has a "surplus slack" in the tying good market (due tothe non-positive price constraint), which can be leveraged to the tied good market via
bundling. Note that in the Chicago school critique of the leverage theory, the tying Örm is
extracting all surplus in the tying good market in the absence of bundling, which renders
bundling unproÖtable. If the tying Örm is not extracting all surplus in the tying goodmarket (as in the case of YouTube), we can restore incentives to tie.
To illustrate this idea, we consider the same model as in section 2.1, but we allow
negative prices in the tied good market whereas there is non-positive price constraint
in the tying good market. This implies that the tying Örm can use the surplus of u incompetition for the tied good because its price for the tying good is constrained to be
zero (pA
1 = 0). Without tying, the only source for Örm 1ís proÖt in market A is fromthe advertiser side with ) A
1 = &: In market B, Örm 1 is willing to set its price below its
marginal cost on the consumer side up to ' with pB
1 = %': In response, Örm 2 charges
BB
a price of p2 = %' + ! with a proÖt of ) 2 = !:Now suppose that Örm 1 ties its monopolized product with product B on the consumer
side. Then, Örm 1 is able to sell its bundled products only if
u + v1 % Pe & v2 % pe2 :As Örm 2 is willing to charge a price as low as pe2 = %', the maximum bundle price thatenables the tying Örm to make sales is Pe = u % ! % ': The tying Örmís proÖt undere 1 = (u % ! % ') + (& + ') = u % ! + &: Thus, if u > !, tying isbundling is given by $proÖtable, restoring incentives to tie.
30

See, for instance, Koszegi and Rabin (2006) for reference-dependent preferences.

21

3

Heterogenous Consumers in the Tying Market

In this section, we consider heterogeneous consumers in market A. To derive a closed-form
solution, we assume that u is uniformly distributed on [0; 1], which yields a linear demand
function in market A.31 Market B is modeled as in the baseline model.
In the absence of tying, the two platform markets can be analyzed independently. In
market B, the analysis is the same as in Section 2. In market A, Örm 1 chooses p on the
consumer side to maximizemax (p + &) (1 % p):p

Under the non-negativity constraint on price, the monopolistic platformís optimal priceis given by(
1!)
if & < 1;m2p =0 otherwise.
The optimal price in the consumer side decreases with the importance of advertising
revenue (&).32
Therefore, without tying, each Örmís proÖt is given bym$1 = ) =((1+))2
4

if & < 1;& otherwise(1)) 2 = ! + ':We assume ! < 1=2. Under the assumption, if both Örms charge zero price undertying, Örm 1ís market share is larger than a half. Hence, this assumption implies thatÖrm 1 has a signiÖcant ability to leverage its monopoly power in market B.
3.1

Equilibrium under Tying

Now suppose that Örm 1 engages in tying on the consumer side. Consumers have two
choices: they can buy a bundled product/service from Örm 1 or buy only product/service
B from Örm 2. Let Pe and pe2 be the bundle price by Örm 1 and product B price by Örm
31
The uniform distribution assumption is made for simplicity to derive a closed-form solution in ourmodel. We can derive qualitatively similar results with a more general distribution of u:32
We can assume a more general distribution of u. If u is distributed according to cdf G(:), themaximization problem becomes max (p + #) [1 % G(p)] and the optimal price is characterized by pm =p
1$G(pm )
%
#:
With
a
monotonehazardrate condition, it can be shown that pm (#) is decreasing in #:g(pm )

22

2, respectively. The consumer type u" who is indi§erent between the bundled o§ering andÖrm 2ís single product o§ering satisÖes the following condition:

which is equivalent to(u" + v1 ) % Pe = v2 % pe2 ,u" = (Pe % pe2 ) + !:As the bundle will be bought by consumers with valuations u & u" , the demand forthe bundle is given by
8
"
e
e
>
< 1 % u = 1 % [(P % pe2 ) + !] if 0 ( (P % pe2 ) + ! ( 1
D(Pe; pe2 ) =1
if 0 ( Pe < pe2 % !
>:0
if Pe > 1 + pe2 % !

(2)Firm 2ís demand is 1 % D(Pe; pe2 ). We have:

Lemma 1. Under tying(i) Firm 1ís best response is given by
BR1 (ep2 ) =((1!#)!()+*)+ep2
2

max[0; pe2 % !]for ! + [1 % (& + ')] & pe2 > ! % [1 % (& + ')] ;otherwise.

(ii) Firm 2ís best response is given by(! % ') + Pe
BR2 (Pe) = max[
; 0]:
2

Therefore, we have to distinguish four cases depending on whether or not 1%! > &+'and/or ! > '. Let (Pe# ; pe2 # ) be the solution of(1 % !) % (& + ') + pe2
Pe =;2(! % ') + Pe
pe2 =
:
2

23

That is,2(1 % &) % !
Pe# = %' +
;
3
1%&+!
pe2 # = %' +:3Then, depending on the relative magnitudes of two-sidedness parameters (& and ')and vertical di§erentiation parameter !, we have the following four cases for equilibrium
prices under tying.
Case 1 (Both & and ' Small): 1 % ! > & + ' and ! > '.
In this case, the non-negative price constraints are not binding and we have Pe" =Pe# (> 0) and pe"2 = p# (> 0):
Case 2 (Both & and ' Large): 1 % ! < & + ' and ! < '.
Pe" = 0; pe"2 = 0:Case 3 (Small & and Large '): 1 % ! > & + ' and ! < '.
In this case, it can be easily shown that Pe# > pe2 # : Thus, the non-negative priceconstraint is binding Örst for the rival Örmís price under tying. We then have the following
result.
(i) If ' < (1!))+#;
3
Pe" = Pe# > 0; pe"2 = pe2 # > 0:(ii) If ' > (1!))+#,
3(1 % !) % (& + ') "Pe" = BR1 (0) =; pe2 = 0:2Case 4 (Large & and Small '): 1 % ! < & + ' and ! > '.
This is the opposite of case 3 and the non-negative price constraint is binding Örst for
the tying Örm as Pe# < pe2 # : We have the following result.
(i) If ' < 2(1!))!#;
3
Pe" = Pe# > 0; pe"2 = pe2 # > 0:(ii) If ' > 2(1!))!#
,
3

24

!%'
Pe" = 0; pe"2 = BR2 (0) =:2
The equilibrium price conÖgurations under tying are summarized in Figure 1.
3.2

Incentives to Tie in Two-Sided Markets

We now analyze the monopolistic Örmís incentives to tie for each price conÖguration
regime.
+ Case of Pe" = 0 and pe"2 = 0This case corresponds to a situation in which both Örmsí non-negative price constraints
are binding under tying because both & and ' are relatively large compared to ! (i.e.,
when 1 % ! < & + ' and ! < '). With these prices, each Örmís market demand undertying is given by
e 1 = 1 % ! and de2 = !:D
Each Örmís corresponding proÖt under tying is given by
e 1 = (1 % !) (& + ') ;$ProÖt comparisons lead to:)e2 = !':

Lemma 2. Consider the case when the equilibrium prices under tying are given by Pe" =0; pe"2 = 0 (i.e., & + ' > 1 % ! and ' > !).
(i) Tying unambiguously reduces the rival Örmís proÖt.
(ii) Firm 1ís incentive to tie increases with ' and decreases with &. More precisely,
2 +4)#
tying is always proÖtable for & < (1 % 2!) and increases Örm 1ís proÖt if ' > (1!))
4(1!#)#for (1 % 2!) < & < 1 and if )* > 1!#
for & & 1:
Tying reduces Örm 2ís proÖt by reducing its market share and its price. Firm 1ís
e 1 increases with ' but Örm 1ís proÖt without
incentive to tie increases with ' since $tying does not depend on '. This is consistent with the Önding from the baseline model.
In addition, Lemma 2(ii) shows that Örm 1ís incentive to tie decreases with & for given'. This is very clear when & > 1: Then, in market A, tying generates a loss of !& since25

Figure 1: Equilibrium Price ConÖgurations under Tying

26

consumers with low valuations for the monopoly product buy Örm 2ís product instead of
the bundle.
+ Pe" > 0; pe"2 = 0This price regime arises when both & < 1 % 2! and 1 % ! % & > ' > (1 % & + !)=3
hold. The latter condition implies ' > !. Essentially, market A is close to being one-sided
while market B is quite two-sided. This case is fundamentally the same as the baselinemodel and we have:
Lemma 3. When the equilibrium prices under tying are given by Pe" > 0 and pe"2 = 0 (i.e.,
when both & < 1 % 2! and 1 % ! % & > ' > (1 % & + !)=3 hold), tying is always proÖtablefor Örm 1 whereas tying always reduces the rival Örmís proÖt.
+ Case of Pe" > 0 and pe"2 > 0This price regime arises under tying when both & and ' are relatively small compared
to !: More precisely, the following condition needs to hold:(1 % &) + !
if & < 1 % 2!;32(1 % &) % !
' <if & > 1 % 2!:
3' <In this case, the non-negative price constraint is binding for neither Örm as Pe" = Pe# >0; pe"2 = pe2 # > 0:
Firm 1 and Örm 2ís demands are respectively given by
e 1 = 2 + & % ! ; de2 = (1 % &) + ! :D
3
3

The corresponding proÖts are given by

(
)2
(
)2
2+&%!
(1 % &) + !
e
$1 =
; )
e2 =
:
3
3

Since each Örm reduces its price by ', ' is competed away and therefore both Örmsí
proÖts under tying are independent of '. Note that the highest & that satisÖes theconditions above for this price regime is & = 1 % #2 < 1: Hence, Örm 1ís proÖt under no2
e 1 > $1 if and only if & < 1 % 2!: The comparison of thetying is $1 = (1+)). We have $427

rival Örmís proÖts shows that )e2 > ) 2 if ' <lemma.

h(1!))+#
3i2% !: We thus have the following

Lemma 4. Consider the case when the equilibrium prices under tying are given by Pe" > 0
and pe"2 > 0 (i.e., either ' < (1!))+#
and & < 1 % 2! hold or ' < 2(1!))!#
and & > 1 % 2!
33hold).
(i) Tying is proÖtable for Örm 1 if and only if & < 1 % 2!:hi2(ii) Tying also increases the rival Örmís proÖts if and only if ' < (1!))+#% !:3(iii) When & = ' = 0, tying is proÖtable for both Örms if ! . 0:14590:
Note that the possibility of tying to relax competition and increase both Örmsí proÖtspcan arise only when ! ( 7!32 5 ' 0:14590: Lemma 4 implies that when ! is small, tying
increases both Örmsí proÖts if both & and ' are close to zero (i.e., when both markets
are essentially one-sided). For instance, when & = ' = 0, Lemma 4(iii) shows that tying
increases both Örmsí proÖts if and only if ! . 0:14590. This result replicates the insightprovided by Carbajo et al. (1990) for the case of standard one-sided markets. WhenÖrms compete in prices, tying/bundling provides a mechanism to di§erentiate products.
High valuation consumers for the tying product buy the bundled product whereas low
valuation consumers just buy the stand-alone product provided by the rival Örm in thetied good market. This relaxes competition in the tied good market. This intuition comesout clearly if we consider an extreme case of ! = 0. In this case, the tied good productis homogeneous and the equilibrium price will be driven down to zero (or more generally
to MC if MC>0) and both Örms receive zero proÖts in market B in the absence of tying.
With tying, both Örms can increase proÖts by segmenting the market.
+ Case of Pe" = 0 and pe"2 > 0This price regime arises when the following condition holds.
2(1 % &) % !
< ' < !:
3

The parameter restriction of ' < ! implies that market B is not two-sided enoughsince the assumption in the baseline model ' > ! is violated. Moreover, the condition
implies 1 % 2! < & (i.e., market A can be quite two-sided). Therefore, tying is not
proÖtable for Örm 1. Tying reduces Örm 2ís proÖt as well by reducing its price and its
market share.
28

Lemma 5. When the equilibrium prices under tying are given by Pe" = 0 and pe"2 > 0 (i.e.,
when 2(1!))!#< ' < ! holds), tying reduces both Örmsí proÖts.
3
For & large and ' small, tying makes both Örms more aggressive since the prices undertying are lower than the prices without tying. It thereby reduces both Örmsí proÖts, whichis reminiscent of the Önding of Whinston (1990).
All cases taken together, we can summarize our analysis so far in the following proposition.
Proposition 4. Consider the model in which consumers are heterogenous in terms of
their valuations for the monopoly product. Assume 1=2 > !.
(i) Firm 1ís incentive to tie increases with ' and decreases with &. SpeciÖcally, Örm
1 has incentives to tie if & < (1 % 2!): For & 2 (1 % 2!; 1); tying is proÖtable if ' >(1!))2 +4)##
: For & > 1; tying is proÖtable if )* > 1!#:4(1!#)(ii) Tying is more likely to reduce Örm 2ís proÖt as & increases and as ' increases.
hi2SpeciÖcally, tying reduces Örm 2ís proÖt unless ' < (1!))+#% ! and & < (1 % 2!).
3
The area in which tying is not proÖtable for Örm 1 is represented by the shaded areain the right bottom corner of Figure 2. In all other areas tying is proÖtable. For the rival
Örm, the area in which tying is proÖtable is represented as the triangle area near the origin
in Figure 2. In all other areas tying reduces the rival Örmís proÖt.
We make two remarks from our analysis. First, tying is proÖtable and reduces therivalís proÖt if ' in the tied good market is su¢ciently large. As a result, our leveragemechanism does not have any credibility issue, in sharp contrast to the tying literature in
one-sided markets (Whinston (1990) and Nalebu§ (2004)). More speciÖcally, our mechanism does require neither technical tying as a commitment device as in Whinston (1990)
nor the controversial assumption that the tying Örm is a Stackelberg leader in setting
prices as in Nalebu§ (2004). The use of contractual tying, even if it can be easily undone,
is su¢cient for credible leverage. Second, tying reduces both Örmsí proÖts if & in the tying
good market is su¢ciently large (relative to '). In the next section, we will investigate
how this second result is a§ected by the presence of inter-group network e§ects.
3.3

Welfare Analysis

The welfare e§ects of tying in our model depend on whether or not the monopoly pricein market A is already zero (i.e., & & 1) in the absence of tying.
29

Figure 2: Incentives to Tie in Two-Sided Markets

30

Case 1. (Full Market Coverage Case without Tying) & & 1
For & & 1; the price in market A is zero and all consumers enjoy the service without
tying, which is an e¢cient outcome. The equilibrium in market B is also e¢cient without
tying because all consumers use the more e¢cient service provided by Örm 2. Therefore,tying can only reduce social welfare. From Proposition 4, we know that for & & 1, Örm 1
has incentives to engage in tying only for the parameter values such that the post-tying
equilibrium prices are characterized by Pe" = 0; pe"2 = 0 (see Figure 2). In this case, there
are two sources of ine¢ciencies. First, any consumer whose valuation in market A is lessthan ! purchases from Örm 2 and consumes only product B. Second, consumers whose
valuation in market A is more than ! ine¢ciently switch from Örm 2ís more e¢cient
product to the bundled product. Note that consumer surplus increases in this case due toa decrease in the price of Örm 2ís product from ! to zero. Consumers whose valuation inmarket A is more than ! (u > !) are indi§erent between the two regimes, but low typeconsumers in market A (whose u < !) gain with tying because their utility is v2 withtying while it was u + v1 without tying.
Case 2. (Partial Market Coverage Case without Tying) & < 1
When & < 1; the price in market A is positive without tying and the market outcome
is ine¢cient whereas the market outcome in B is e¢cient. As in the previous case, tying
can introduce ine¢ciency in market B when Örm 1 ine¢ciently serves consumers with abundled product, but there can be a positive market expansion e§ect in market A as thebundle price is reduced. As a result, social welfare can be ambiguous. However, for social
welfare to increase with tying a necessary condition is that market A expands with tying.
The next lemma derives the condition under which market A expands with tying.
Lemma 6. Market A expands with tying if and only if & < 1 % 2!:
The immediate corollary of lemma 6 is that tying decreases social welfare if & > 1%2!:
Now let us consider the case where & < 1 % 2! and hence market A expands withtying. More generally, the social welfare change with tying (!W ) in this case can be
written as
Z 1!!
2

!W =

e 1!
(u + &)du
%
D
;
| {z
}
|
{z
}
Social Welfare Loss in Market B
Welfare Gain in Market A
due to Market Expansion
e1
1!D

31

e 1 denotes the demand for the bundle after tying. After analyzing the welfare
where De§ects of tying depending on the equilibrium price conÖgurations (see the proof of Proposition 5), we Önd that tying increases welfare for ! small enough as in this case the welfare
loss in market B is small.
Proposition 5. Tying is welfare-reducing if & > 1 % 2! since market A contracts andthere is ine¢cient switching in market B. If & < 1 % 2!; market A expands and tying
can enhance welfare if ! is su¢ciently small since ine¢ciency associated with tying inmarket B becomes arbitrary small as ! ! 0:4

Heterogenous consumers with intergroup network e§ects

Up to now, we assumed two-sided markets in which the consumer side serves as a competitive bottleneck but did not derive any direct utilities (or disutilities) from the other sidewhich can be considered as advertisers. This was to capture the advertising revenue-based
business model of many Internet platform markets such as Google. In this section, we extend our analysis by explicitly considering intergroup network e§ects from the other side
to consumers. For instance, we can interpret the other side as app producers or content
providers. In the presence of network e§ects, consumersí choices can exhibit strategic
complementarity and coordination plays an important role. In such a situation, tying canbe used as a strategic instrument to help consumers coordinate and coalesce around thebundled product. We investigate how the presence of network e§ects a§ects the previous
results on the relationship between the incentive to tie and & (and ').
For simplicity, we assume that the monopolistic platform market A is the same as in
Section 3. We modify market B as follows. Let pci be the price charged by platform i on
the consumer side in market B. Then, a consumer subscribing to platform i in market B
derives a utility ofvi + ' c nai % pci ;where ' c (> 0) is a parameter representing intergroup network e§ects and nai is the measure of applications/content available at platform i. We assume ! $ v2 %v1 > 0. On
the application side, let pai be the price an application developer needs to pay to haveaccess to consumers on platform i. An application developerís payo§ from developing anapplication or creating content for platform i is32

' a nci % pai ;where ' a > 0 and nci is the measure of consumers on platform i. Marginal costs of serving
consumers are once again assumed to be zero.
We consider the following sequential game.
1. Each Örm (i.e., platform) sets the prices for consumers.
2. Consumers make purchase or subscription decisions
3. (After observing consumer choices at stage 2), each platform chooses the price forapplication developers
4. (After observing consumer choices at stage 2 and price choices at stage 3), application developers choose subscription decisions.
Stages 1 and 2 of this sequential game are identical to the games we studied in theprevious sections. The main di§erence is that the advertising revenue per consumer inmarket B was exogenously given in the previous sections whereas its equivalent is now
endogenous: solving for Stages 3 and 4 allows us to compute the additional proÖt thata platform can obtain from the application side by attracting one more consumer. We
assume that consumers single home while application developers can multihome. As usual,
we analyze the game by using backward induction.
4.1

Equilibrium on the Application Side

Suppose that platform i has nic measure of consumers on board. Let us study the gamethat starts at stage 4.
In order to endogenously derive the number of applications available, we assume thateach developer incurs a platform-speciÖc Öxed cost f . The Öxed cost is distributed overthe positive line with density equal to one. Given the heterogeneity in the Öxed costof development, an application developer will subscribe to platform i if the following
condition holds:' a nci % pai & f:Hence, given nci and pai , the total measure of app developers who subscribe to platform iisnai (pai ; nci ) = ' a nci % pai :Therefore, at stage 3 with nai determined in the previous stage, platform i solves the33following problem.
max
pai nai (pai ; nci ) = pai (' a nci % pai ):apiFrom the Örst-order condition for platform iís proÖt maximization, platform i chooses
pai =' a nci:2* nc

The measure of app developers who subscribe to platform i also is given by nai = a2 i witha uniform distribution of f with density one. As a result, given the number of consumers
nci ; platform iís proÖt on the application side is) ai (nci ) =1' a nci222:Note that this result does not depend on whether or not there is tying on the consumer
side.
4.2

Equilibrium on the Consumer Side

We now analyze the e§ects of tying on the consumer side.
4.2.1

No Tying

Consider platform competition at stage 1 without tying. As we assume that all consumers
are homogenous in market B, there can potentially be multiple equilibria for some price
conÖgurations due to positive intergroup network e§ects. In such cases, we assume that
all identical consumers coordinate on the Pareto dominant equilibrium.33
Hence, if pc2 < pc1 +!, all consumers coordinate on platform 2 as long as their individual
rationality constraint is satisÖed. Therefore, in equilibrium, we have pc1 = 0; pc2 = ! andall consumers join the more e¢cient platform 2, that is, nc1 = 0 and nc2 = 1. In this* nctipping equilibrium with nc2 = 1, Örm 2 charges pa2 = a2 2 = *2a on the application sideand induces a measure of na2 = *2a application developers to be on its platform. Thus,
each consumerís payo§ on platform 2 isv2 + ' c na2 % pc2 = v2 + ' c

1

'a
2

2

% ! = v1 + A;

33
This is a standard equilibrium selection criterion in the presence of multiple equilibria with networke§ects. See, for instance, Katz and Shaprio (1986) and Waldman (1993).

34

where A $ * c2* a represents the network beneÖts in the tipping equilibrium. As in theanalysis of the previous section, each consumer receives a surplus of v1 due to platformcompetition on the consumer side. In addition, each consumer receives additional beneÖt
of A = * c2* a from intergroup network e§ects.
4.2.2

Tying

Now consider the equilibrium in the tying regime in which platform 1 ties its monopolized
product A with another product in competitive two-sided market B, and charges Pe forthe bundled product. We make the following assumption:
Assumption 1. (stability) ' c ' a < 1 , A < 1=2
Without this assumption, there cannot exist any interior equilibrium in which both
platforms share the market in our two-sided market. This is because the feedback mechanism due to intergroup network e§ects magniÖes any initial advantage of one platform
and leads to exploding demands.
We further assume that ! < 1=2 for tying to have a leverage e§ect in the tied goodmarket. If ! is too large (i.e., ! > 1=2), the disadvantage of Örm 1 vis-‡-vis Örm 2 is
too large to overcome in the tied good market; tying has no leverage e§ect. Note that
! < 1=2 along with the stability assumption implies that purchasing the bundle is thedominant strategy for the highest consumer type in market A (whose u = 1) if the bundleis o§ered for free. For this type of consumer, the valuation of product A is so high thatthe bundled product is preferred even if all other consumers purchase stand-alone productB2 and there are no application developers for B1.
Therefore, in what follows, by using iterated elimination of dominated strategies, westudy a cut-o§ equilibrium: we consider a u" 2 [0; 1] such that all consumers whose
valuation from the monopoly product is higher than u" join platform 1 while all therest join platform 2, which implies that nc1 = 1 % u" ; nc2 = u" on the consumer side andna1 = *2a (1 % u" ); na2 = *2a u" :Let Pe and pe2 be the bundle price by Örm 1 and Örm 2ís price for product B2 in thetying regime, respectively. In the (interior) cut-o§ equilibrium, u" is implicitly determined
as follows:[u" + v1 ] + A(1 % u" ) % Pe = v2 + Au" % pe2 :(3)
The LHS of (3) represents type u" consumerís utility when he purchases the bundled
product whereas the RHS of (3) represents type u" consumerís utility when he purchases
35only product B supplied by Örm 2. The critical type u" is indi§erent between these two
options. Hence,nc2 (Pe; pe2 ) = u" =(#+Pe!ep2 !0;(1!* c * a )0;

if Pe > pe2 + A % !;if Pe ( pe2 + A % !:The expression for nc2 above reveals new e§ects of tying in the presence of intergroup
network e§ects. If there are no network e§ects (i.e., ' c ' a = 0), Örm 2 can always
guarantee a positive market share unless its price does not exceed the bundle price by!. For instance, if Örm 2 charges a price of zero, Örm 2 will always have some positivemarket share (nc2 > 0). However, because of the existence of network externalities (i.e., as
the number of consumers who choose platform 1 increases, the applications available onthat platform increases), tying can create an advantage for platform 1 if tying provides
a mechanism for Örm 1 to capture more consumers than Örm 2. As the network e§ects
term (' c ' a ) increases, Örm 2ís market share decreases down to zero3and3 eventually tying
3 @nc 3@nc
leads to the foreclosure of Örm 2 in the market. Note that @ Pe2 = 3 @ pe22 3 = 1!*1 * > 1 isc aa multiplier in the two-sided market. It shows that the demand changes more sensitively
to price as ' c ' a increases.
Consider the case where nc2 (Pe; pe2 ) > 0. Then, we can decompose the market share of
the tying Örm into two terms as follows:
nc1 (Pe; pe2 ) = 1%nc2 (Pe; pe2 ) =hi
1 % (Pe % pe2 ) + !
|{z}

Demand in the Absence ofIntergroup Network E§ects(i)' c' a1 h e+% (P % pe2 ) + ! ;(1 % ' c ' a ) 2|{z}Induced Demand Due toIntergroup Network E§ects(4)ie
where 1 % (P % pe2 ) + ! is a hypothetical demand for Örm 1 in the absence of intergroup
network e§ects (i.e., ' c ' a = 0), which we have seen in Section 3 (see equation (2)).
In order to understand equilibrium pricing, it is useful to decipher the economic(* a )2e1 =meaning
of
4
$
1
%
'
'
%
. As Örm 1ís proÖt under tying is given by $
c
a
2
h
i
2 c
(*
)
n
Pe + & + a4 1 nc1 , its Örst-order condition for proÖt maximization is given by:
h

i
e1
@$
@nc1 h e=
P + & % nc1 4 = 0;
@ Pe
@ Pe
36

@nc

where @ Pe1 = % 1!*1 * . We can rewrite the condition above in a modiÖed Lerner index
c a
form:
Pe + &
nc 4
= 1 :
Pe
Pe

If the tying Örm did not take into account the e§ect of its bundle pricing decision on(* a )2
the proÖt from the application side, 4 would not contain3 c 3 the term 2 and we would
3 @n 3have 4 = 1 % ' c ' a , which is equal to the inverse of 3 @ Pe1 3. Therefore, we can state theÖrst-order condition as the standard Lerner formula with the interpretation of (%&) as theopportunity cost of the bundle. 4 corrects the standard formula by taking into accountthe fact that an increase in market share increases the proÖt from the application market
through the network e§ects. If this e§ect is large such that 4 ( 0, Örm 1 Önds it optimal
to choose Pe = 0 for any & & 0. The same remark applies to Örm 2. Therefore, %4 can beconsidered a measure of degree of competition induced by intergroup network e§ects, which
increases with the network e§ects and the proÖt per consumer from the app market.
We can have two types of equilibria: an interior equilibrium in which Örm 1 and Örm 2
share the market with nc2 > 0 and the tipping equilibrium in which the tying Örm captures
the whole market with nc2 = 0.34 The following proposition characterizes the equilibrium
conditions.
Proposition 6. We have a tipping equilibrium (with nc1 = 1 and nc2 = 0) if A & ! and2& & &b $ (! % A) + 4, where 4 = 1 % ' c ' a % (*2a ) : Otherwise, we have an interior
equilibrium with both nc1 > 0 and nc2 > 0:
In the appendix, we not only prove the proposition above but also further characterize
equilibrium prices in each type of equilibrium. Proposition 6 shows that if ! > A,
that is, the quality advantage of Örm 2 is su¢ciently large compared to the strength of
network e§ects, we always have an interior equilibrium. However, if the network e§ects
are relatively stronger than the quality advantage of Örm 2 (i.e., ! < A), tying can bea mechanism for Örm 1 to leverage its market power in A to B and lead to foreclosureof the rival Örm. Note that the condition for tipping is more likely to be satisÖed as &increases. For instance, if & > 1, we always have a tipping equilibrium as long as ! < A:
34

With the maintained assumption of , < 1=2 and the stability assumption, there is no tipping
equilibrium in which Örm 2 captures the whole market.

37

4.3

Incentives to Tie with Intergroup Network E§ects

We now analyze incentives to tie in the presence of intergroup network e§ects. Proposition
6 shows that when ! ( A holds, a tipping equilibrium is more likely as & increases. Thisis because a high & induces the monopolist to price more aggressively in the consumer
market, providing more surplus to inframarginal consumers, which can be leveraged to
attract all consumers in market B through network e§ects (see the demand decompositionin (4)). If tipping occurs, Örm 1 captures all advertising revenue in market A, which breaks
the result obtained in the previous section that there exists a critical value of &" such thattying is not proÖtable if & & &" .
4.3.1

Tipping Equilibrium

As shown in Proposition 6, we have a tipping equilibrium if ! ( A and & & 1 % [(' c ' a +(* a )2)+(A %!)]: In the tipping equilibrium, p" = 0, P " = A %!. Thus, Örm 1ís proÖt with
2
4 52
6
7
e "1 = &+ * a +(A% !) while its proÖt without tying is $"1 =max 1 (1 + &)2 ; & :tying is $2

4

When & & 1, Örm 1 serves all consumers in the tying market regardless of tying. Therefore,tying is proÖtable as it increases the proÖt from the tying market from & to & + A % !
and generates the proÖt from the apps. By continuity, tying is proÖtable for & close to
one and we have the following result.
Proposition 7. Whenqthe tipping equilibrium prevails under tying, Örm 1 has incentives
to tie if & > &" $ 1 %(' a )2 + 4(A % !):Proof. When & & 1, Örm 1ís proÖt without tying is &: Since * c2* a & ! in the tipping
equilibrium, it immediately follows that tying is proÖtable when & & 1: For & < 1, Örm1ís proÖt under tying is 14 (1 + &)2 : A simple comparison of Örm 1ís proÖts under eachregime yields the result."
The proposition indicates that tying is more likely to be proÖtable for & high, ' c highand ' a high. Propositions 6 and 7, taken together, yield the following Corollary.
Corollary 1. If A & ! and & > max[b&; &" ] (where max[b&; &" ] < 1), Örm 1 engages intying and the resulting equilibrium is a tipping equilibrium in which Örm 2 is foreclosed
from the market.
We point out the di§erence the introduction of intergroup network e§ects creates withrespect to the result in the previous section. In section 3, for a given '; tying becomes38

unproÖtable at some point as & increases. The reason is that without network e§ects,
the rival Örm can always induce some marginal consumers in market A to buy product
B when tying takes place. As a result, tying entails greater loss as & increases. However,this e§ect can disappear in the model with intergroup network e§ects if tying induces atipping equilibrium, which is more likely when & is high.
4.3.2

Interior Equilibrium

Due to the conáicting e§ects of &, the e§ects of tying on the tying Örmís proÖts can be
subtle in the interior equilibrium. As explained earlier, a higher & induces the tying Örm toprice more aggressively in the consumer market, providing more surplus to inframarginal
consumers of A, which can be used to compete against Örm 2 in market B when tying
takes place. In the interior equilibrium, however, capturing a certain share of market B
comes at the expense of the share of market A, which can be very costly when & is veryhigh. We thus can show that tying cannot be a proÖtable strategy if & is su¢ciently large
as in Section 3.
In the case of an interior equilibrium, let us focus on the case in which Pe" = 0. This
holds either if 4 < 0 and A < ! (see Lemma 7(ii) in Appendix) or if 4 & 0 but nc"
1 4 isbounded above by & (i.e. (6) is satisÖed) (see Lemma 9(i) in Appendix). In both cases,Örm 1ís market share under tying is independent of & and is given by
nc"
1 =

(

#!0
1 % (1!0)
if 4 < 0;#!0
1 % -+(1!0)
if 4 & 0.

DeÖne the di§erence between Örm 1ís proÖt under tying and Örm 1ís proÖt without tying
as follows(
9:2(* a )2 c"
nc"
n1 ) % 1+); if & 2 [0; 1] ;1 (& +
42!$ (&) $(* a )2 c"c"n1 (& + 4 n1 ) % &,if & > 1:
Suppose that nc"
1 is close enough to one that !$ (1) > 0. Note Örst that !$ (&) is concave2c"for & 2 [0; 1]. For & > 1, as !$ (&) linearly decreases, there exists & = (*4a ) nc"
1 =(1 % n1 )at which !$ (&) = 0. We below distinguish the case of !$ (0) & 0 from !$ (0) < 0. Suppose !$ (0) & 0. Then, tying is strictly proÖtable for any & 2 (0; &). If !$ (0) < 0, there
exists some &2 (0; 1) at which !$(&) = 0. Hence, tying is proÖtable for any & 2 (&; &).
As & tends to 1 when nc"
1 goes to one, the result obtained when !$ (0) < 0 extends theresult on the incentive to tie in the tipping equilibrium, described in Proposition 7, to an
39

interior market share.
4.4

Welfare Analysis

We now analyze welfare implications of tying. First consider the case where tying leads
to tipping. If & > 1, all consumers in market A were served before tying. There isthus no market expansion e§ect in this case. The only e§ect is that consumers switchfrom Örm 2ís product to Örm 1ís product, which is inferior. Thus, tying unambiguously
reduces welfare in this case, with the welfare loss being !: If & < 1, but tying induces
tipping, there is a positive market expansion e§ect in market A. The welfare gain fromthis market expansion can be expressed as
Z 1!!
2

E=

udu +

0

1

1%&
2

2

&=

(1 + 3&)(1 % &)8This welfare gain needs to be measured against the loss in welfare in market B, which
is !: Tying can be welfare reducing if E < !: E is highest at 1=6 when & = 1=3:
Therefore, a su¢cient condition that tying reduces welfare is ! > 1=6 if tying induces
tipping towards the bundled product.
If tying does not lead to tipping, there is an additional type of welfare loss due to a
fragmented market structure which does not fully exploit network beneÖts.
We can also analyze the e§ects of tying on consumer surplus. For instance, supposethat & > 1 and tying leads to tipping. If Pe" = 0, consumers are indi§erent between
tying and no tying. However, if Pe" > 0, consumers lose from tying. Without tying, each2
consumer enjoys (*2a ) of surplus from Örm 2 in market B because of price competition inmarket B. Tying removes the level playing Öeld and allows the tying Örm to extract the
surplus from network e§ects.

5Concluding Remarks

In this paper, we have developed a leverage theory of tying in two-sided markets. The
basic premise of our theory is that the optimal pricing and competition strategies in a
two-sided market often lead to below-cost pricing on one side of the market whose loss
is often recouped on the other side. When the below-cost pricing is constrained by the
non-negative price constraint, competition is relaxed and supra-competitive proÖts can be40

created. A monopolist in another market can extract this surplus via a tying arrangement
which e§ectively allows the tying Örm to engage in negative pricing. Furthermore, as thecompeting Örm in the tied two-sided market is constrained by non-negative pricing, its
response to tying cannot be so aggressive, which in turn makes tying even more proÖtable.
We also demonstrated that this mechanism can also be applied to other market contexts
such as dynamic competition with direct network e§ects and switching costs.
Our analysis has focused on the e§ects of tying on short-run price competition and theirimplications on consumer surplus and welfare. In our baseline model with homogenous
consumers, for instance, tying is welfare-reducing because it induces ine¢cient switching
to the inferior product of the tying Örm in the tied market. Nonetheless, consumers are
not worse o§ because they are compensated for the consumption of the inferior product by
the tying Örm. However, if we consider rival Örmsí incentives to develop new products inthe tied market, we may Önd further harm in consumer welfare because more innovative
products may not be introduced by the rival Örms due to the lack of market access.
However, we also point out that there could be many e¢cient reasons to engage in tying
such as integration of di§erent apps which enables the tying Örm to provide better andseamless services to consumers. Antitrust investigations should weigh these potential
e¢ciencies against the potential harm identiÖed in this paper.

41

References[1] Amelio, Andrea and Bruno Jullien, "Tying and Freebie in Two-Sided Markets,"
International Journal of Industrial Organization, 2012, pp. 436-446.
[2] Armstrong, Mark, ìCompetition in Two-Sided Markets,î Rand Journal of Economics, 2006, 37 (3), pp. 668-691.
[3] Armstrong, Mark and Julian Wright, ìTwo-Sided Markets, Competitive Bottlenecks
and Exclusive Contracts,î Economic Theory, 2007, 32, pp. 353-380.
[4] Areeda, Phillip E., Louis Kaplow, and Aaron S. Edlin, Antitrust Analysis: Problems,
Text, and Cases, 7th Edition, Aspen Casebook, 2013.
[5] Bork, Robert, "Vertical Integration and the Sherman Act: The Legal History ofan Economic Misconception," University of Chicago Law Review, 1954, 22 (1), pp.
157-201[6] Bowman, Ward S. Jr., ìTying Arrangements and the Leverage Problem,î Yale Law
Journal, 1957, 67, pp. 19-36[7] Caillaud, Bernard and Bruno Jullien, ìChicken and Egg: Competition among Intermediation Service Providers,î Rand Journal of Economics, Summer 2003, pp.
309-328.
[8] Carbajo, Jose, David De Meza, and Daniel J. Seidman, "A Strategic Motivation
for Commodity Bundling," Journal of Industrial Economics, March 1990, 38, pp.
283-298.
[9] Carlton, Dennis, W. and Michael Waldman, ìThe Strategic Use of Tying to Preserve and Create Market Power in Evolving Industries,î Rand Journal of Economics,
Summer 2002, pp. 194-220.
[10] Chee, Foo Yun, "EU hits Google with second antitrust charge," Reuters,
April 20, 2016, available at http://www.reuters.com/article/us-eu-google-antitrustidUSKCN0XH0VX.
[11] Chen, Yongmin, ìEquilibrium product bundling,î Journal of Business, 1997, 70, pp.
85-103.
[12] Choi, Jay Pil, "Tying in Two-Sided Markets with Multi-Homing," Journal of Industrial Economics, 2010, 58 (3), pp. 607-626[13] Choi, Jay Pil and Christodoulos Stefanadis, ìTying, Investment and the Dynamic
Leverage Theory,î Rand Journal of Economics, Spring 2001, pp. 52-71.

42

[14] Director, Aaron and Edward H. Levi, ìLaw and the Future: Trade Regulation,îNorthwestern University Law Review, 1956, 51, pp. 281-296[15] Edelman, Benjamin, "Does Google Leverage Market Power Through Tying and
Bundling?," Journal of Competition Law & Economics, 2015, pp.365-400.
[16] Edelman, Benjamin and Damien Geradin, "Android and Competition Law: Exploring and Assessing Googleís Practice in Mobile," October 2016, unpublished manuscript, available at http://www.benedelman.org/publications/google-mobile-201610-24.pdf.
[17] Farrell, Joseph and Nancy T. Gallini, "Second-Sourcing as a Commitment: Monopoly
Incentives to Attract Competition," Quarterly Journal of Economics, 1988, pp. 673694.
[18] Hurkens, Sjaak, Doh-Shin Jeon, and Domenico Menicucci, "Leveraging Dominancewith Credible Bundling," CEPR Discussion Paper, 2016.
[19] Jeon, Doh-Shin and Domenico Menicucci, ìBundling Electronic Journals and Competition among Publishers," Journal of the European Economic Association, 2006,.4(5), pp. 1038-83.
[20] Jeon, Doh-Shin and Domenico Menicucci, ìBundling and Competition for Slots",
American Economic Review, 2012, 102, pp. 1957-1985.
[21] Iacobucci, Edward M., "Tying in Two-Sided Markets, with Application to Google,"
unpublished manuscript, September 2014.
[22] Katz, Michael and Carl Shapiro, "Technology Adoption in the Presence of Network
Externalities," Journal of Political Economy, 1986, 94, pp.822-841.
[23] Klemperer, Paul, ìCompetition when Consumers have Switching Costs,î Review of
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43

[28] Posner, Richard A., ìThe Chicago School of Antitrust Analysis,î The University ofPennsylvania Law Review, 1979, 127, pp 925-948[29] Rochet, Jean-Charles and Jean Tirole, ìTwo-Sided Markets: A Progress Report,î
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44

Appendix
Proof of Proposition 2
We show that no Örm has incentives to deviate at the putative equilibrium prices of
Pe = u % ! and pe2 = 0. When consumers are indi§erent between buying the bundle
only and buying Örm 2ís product only, we break the tie in favor of the bundle to avoid
the open set problem: this is because Örm 1 can decrease its price by " > 0; but Örm 2
cannot because it is constrained by non-negative pricing. At these prices, a consumeríspayo§ is v2 if he single-homes and buys only the bundle while his payo§ is v2 + ! % m if
he multihomes. Therefore, only those consumers with m ( ! multihome in equilibrium.
Consider Örst the deviation of Örm 2 to pe2 > 0. A deviation to pe2 > ! is obviously
not optimal as Örm 2 would have no demand. If pe2 ( !, the marginal consumer typewho is indi§erent between multihoming and not is given by m" = ! % pe2 . With thisdeviation, Örm 2ís proÖt is given by) 2 (ep2 ; Pe = u % !) = (ep2 + ') G(! % pe2 ):The Örst-order derivative is@) 2= G(! % pe2 ) % (ep2 + ') g(! % pe2 ) ( 0;
@ pe2when it is evaluated at pe2 = 0 with the assumption of G(!)=g(!) ( ': The log-concavity
2of g guarantees that @2< 0 when evaluated at all positive pe2 . Therefore, there is no@ pe2
proÖtable deviation for Örm 2.
Consider now a deviation by Örm 1. Its equilibrium proÖt from Pe = u % ! is (u % !) +[1 % G (!)] '. It has no incentive to decrease its price from the equilibrium price since it
sells the bundle to all consumers. Consider a deviation to Pe > u%!. Then, all consumers
prefer buying Örm 2ís product to buying the bundle. If a consumer multi-homes and alsobuys the bundle, his payo§ increases by (u % Pe % m). Therefore, only consumers whose
cost of multi-homing is less than (u % Pe) multi-home. Firm 1ís proÖt upon the deviation
is PeG(u % Pe), which cannot be larger than the proÖt under no tying, u. Hence, if Örm 1
has an incentive to tie, such deviation is never proÖtable. Firm 1 has incentives to tie if
e 1 = (u % !) + [1 % G (!)] ' & u = $1 , that is, if [1 % G (!)] ' > !:$Proof of Lemma 1
45

(i) If Pe 2 [ep2 % !; 1 + pe2 % !], Örm 1ís proÖt is$1 (Pe; pe2 ) = (1 % [(Pe % pe2 ) + !])(Pe + & + ')The Örst-order condition with respect to Pe is equivalent to(1 % !) % (& + ') + pe2
Pe =
:
2

p2
For this to be the optimal, the condition Pe = (1!#)!()+*)+e& maxf0; pe2 % !g needs to2
be satisÖed.
When 0 ( Pe < pe2 % !, Örm 1ís proÖt is$1 (Pe; pe2 ) = (Pe + & + ')Hence, it is optimal to choose Pe = maxf0; pe2 %!g given the non-negative price constraint.
p2
for ! + [1 % (& + ')] & pe2 >Therefore, the best response is Pe = (1!#)!()+*)+e2! % [1 % (& + ')]. If 1 ( & + ' or for pe2 2= [! + [1 % (& + ')] ; ! % [1 % (& + ')], it ise
optimal to choose P = maxf0; pe2 % !g.
(ii) Firm 2ís proÖt is zero if 0 ( Pe < pe2 % !: If Pe & pe2 % !, its proÖt is) 2 (Pe; pe2 ) = [(Pe % pe2 ) + !](ep2 + '):The Örst-order condition with respect to pe2 yields
which can be rewritten as[(Pe % pe2 ) + !] % (ep2 + ') = 0;pe2 =(! % ') + Pe:
2

Pe
For this to be the optimal, the condition for pe2 = (#!*)+( Pe + ! needs to be satisÖed,
2i.e., Pe & %(! + ');which holds always. We can thus write the best response of Örm 2 asBR2 (Pe) = max[(! % ') + Pe
; 0]2

Proof of Lemma 2
46

It is omitted as it is straightforward from the explanations in the main text.
Proof of Lemma 3
In this case, it can be easily shown that Pe# > pe2 # : Thus, the non-negative priceconstraint is binding Örst for the rival Örmís price under tying. We thus have
(1 % !) % (& + ') "Pe" =; pe2 = 0:2

Firm 1 and Örm 2ís demands are respectively given by

e 1 = (1 % !) + (& + ') ; de2 = (1 + !) % (& + ') :D
2
2

The corresponding proÖts are given by

(
)2
(
)
(1 % !) + (& + ')(1 + !) % (& + ')
e
$1 =
; )
e2 = ':
22

9:2
Since & < 1%2! in the case we consider, Örm 1ís proÖt under no tying is $1 = 1+):2
e 1 > $1 , that is, tying is proÖtable for the tying Örm.
Since ' > ! in this case, we have $In contrast, tying reduces the rival Örmís proÖt. To see this,h note thati the rival Örmís
proÖt in the absence of tying is ) 2 = ' + ! > ' > ' (1+#)!()+*)= )e2 because2(1 + !) % (& + ') < 2 as ' > !:Proof of Lemma 4
It is omitted as the proof of (i) and (ii) is provided in the main text and the proof of(iii) is straightforward.
Proof of Lemma 5
In this case, it can easily be shown that Pe# < pe2 # : Thus, the non-negative priceconstraint is binding Örst for the tying Örm We thus have!%'
Pe" = 0; pe"2 =:247

Firm 1 and Örm 2ís demands are respectively given by
e 1 = 1 % ! + ' ; de2 = ! + ' :D
2
2

The corresponding proÖts are given by

(
)
(
)2
!+'
!+'
e
$1 = (& + ') 1 %
; )
e2 =
:
2
2

We need to show that(
) ( 9 1+) :2
if & < 1!
+'2
e 1 = (& + ') 1 %$<2& if & > 1

For & < 1;we use the following fact
1

x+y
2

22

& xy for any x and y:

Let x = & + ' and y = 1 % #+*: Then, we have
2
(
) 1
22
!+'
2 + 2& + ' % !
(& + ') 1 %
<
:
2
4
9
:2
9
:2
e 1 = (& +Since ' < ! in this case, we have 2+2)+*!#< 1+): Therefore, $42;< 9 1+) :2#+*') 1 % 2 < 2= $1 if & < 1:;<
For & > 1; tying is proÖtable only when (& + ') 1 % #+*> &: This condition can be2
rewritten as
2'
!<
% ':
&+'
2*
Since ' < !, the condition above can be satisÖed only when ' < )+*% ', or (& + ') < 1;which is impossible because & > 1: Therefore, tying cannot be proÖtable for & > 1, either.
Tying obviously hurts Örm 2 as tying reduces its price and market share.

Proof of Lemma 6
In market A, all consumers whose valuations satisfy u & 1!)
purchase product A.
2e 1 ) consume product A,
With tying, all consumers whose valuations satisfy u & (1 % D
e 1 denotes the bundle demand for Örm 1. Thus, for market A to expand we need
where D48

1!)e 1 ): We can easily verify that this condition is equivalent to & < 1 % 2! by> (1 % D
2checking all possible equilibrium price conÖgurations with tying.

Proof of Proposition 5+ Pe" = 0; pe"2 = 0

Consider a situation in which the post-tying equilibrium prices are zero (Pe" = 0; pe"2 =e 1 = 1 % ! in this case, the social welfare change with tying (!W ) can be0): Because D
written as
Z 1!!
2

!W =

#

(u + &)du % (1 % !)!

We can easily see that if ! is su¢ciently small, then social welfare loss in market B
from tying is negligible. Thus, if ' is su¢ciently large, Örm 1 has incentives to capture
market B with tying, and tying can be socially beneÖcial as the market expansion e§ects
in market A outweigh social welfare loss in market B.35
Consumer welfare in this case always increases because now both goods are provided
for free.
+ Pe" > 0; pe"2 = 0

e 1 = (1!#)+()+*) : By proceeding in a similar way as above, we can verify
In this case, D2
that if ! is su¢ciently small, tying can increase welfare. Otherwise, tying decreases
e1welfare. It can also be shown that market A always expands with tying as 1!)> 1%D
2
in this case.
Concerning consumer welfare, we can easily verify that Pe" = (1!#)!()+*)< pm = 1!),
2
2i.e., the bundled product is even cheaper than the stand-alone A product price before
tying. As a result, consumers beneÖt from tying.
+ Pe" > 0; pe"2 > 0e 1 = 2+)!# : Once again, if ! is su¢ciently small, tying can increase
In this case, D3
welfare. Otherwise, tying decreases welfare.
Proof of Proposition 6
35

We can also show that if , > 1=7, tying reduces welfare regardless of the value of #:49

To derive the equilibrium prices under tying, note that Örm 1ís and Örm 2ís total
proÖts are respectively given bye 1 (Pe; nc1 (Pe; pe2 )) = (Pe + &)nc1 +$1;

ProÖt from the App Side)e (ep2 ; nc2 (Pe; pe2 )) = pe2 nc2 +222' a nc1
2| {z }

1' a nc22

22Firm 1ís Örst-order condition for proÖt maximization is given by

@nc

"
#
2 c
c
e1
@$
('
)
n
1 @n1
= nc1 + Pe + & + a
2
@ Pe
@ Pe
"
#
2
1
('
)
a
c
= %
Pe + & % n1 (1 % ' c ' a %)1 % ' c' a
2
2

@nc

where @ Pe1 = % 1!*1 * (= % @ Pe2 ): Using 4 $ 1 % ' c ' a % (*2a ) , we can rewrite the expression
c aabove as
h
i
e1
@$
1
=%
Pe + & % nc1 4 :
1 % ' c' a
@ Pe
e1
If 4 < 0, we always have @@/Pe < 0, implying that Pe = 0 is optimal in the interior
equilibrium where nc2 > 0. In a tipping equilibrium with foreclosure where nc2 = 0, there
is no market share to gain and hence it is not optimal to reduce the price further and wewill have Pe = pe + (A % !).

Similarly, Örm 2ís Örst-order condition for proÖt maximization is given by
"
#
@e
)2
(' a )2 nc2 @nc2c
= n2 + pe2 +
pe2
2
pe2
= %

1
[e
p2 % nc2 4]
1 % ' c' a@nc

22
where @ pe22 = % 1!*1 * . Once again, if 4 < 0, we have @e< 0, implying that pe2 = 0 is
pe2
c a
optimal.
Note that the second-order derivatives for Örm 1 and Örm 2 are the same and given

50

by:

"
#
e1
@2$
@2)
e2
1
(' a )2
1
1
=
=%
% 1%
:
2
@ pe2
1 % ' c' a
2 1 % ' c' a 1 % ' c' a
@ Pe2They are negative if(' a )2 < 4 (1 % ' c ' a ) ;which is equivalent to

1(' )2 < 1:
4 a
We consider two cases depending on the magnitudes of intergroup externality parameters (i.e., the sign of 4).
2
Case 1. 4 < 0 (i.e., ' c ' a + (*2a ) > 1)
22
In this case, the equilibrium prices are pe"2 = 0 and Pe" = max f0; A % !g as @e< 0: Wepe2
can have two types of equilibria depending on the relative magnitude of A and !: a tipping
equilibrium or an interior equilibrium. We have the following lemma that characterizes
equilibrium in this case.
' c' a +Lemma 7. Suppose that 4 < 0: Then, we have pe"2 = 0.
e"(i) If A > !; we have a tipping equilibrium in which nc"
2 = 0 and P > 0.
(ii) If A < !, we have an interior equilibrium with nc"
2 > 0 and both prices being driven""edown to zero (P = pe2 = 0), where!%A> 0;(1 % ' c ' a )1 % (! + A)
= 1 % nc"
> 0:
2 =(1 % ' c ' a )

nc"
=
2
nc"
1

2

Case 2. 4 & 0 (i.e., ' c ' a + (*2a ) ( 1)
We also consider two types of equilibria under tying in this case: a tipping equilibrium
and an interior equilibrium. Note that in this case (i.e., when 4 & 0); the second-ordercondition is always satisÖed in each Örmís maximization problem.
Tipping Let us Örst analyze the possibility of a tipping equilibrium. In a tipping
equilibrium with nc"
2 = 0; we have the following Örst-order condition for Örm 2:
@e
)2
pe2
=%
( 0:
pe2
1 % ' c' a51

Hence, pe"2 = 0. Then, the highest Pe consistent with tipping is Pe" = max f0; A % !g.
At this price, Örm 1 must have no incentive to deviate; the Örst-order derivative must be
zero or negative.
If A < !; the equilibrium prices in a tipping equilibrium need to be Pe" = pe"2 = 0:
However, at these prices Örm 2 has a strictly positive market share. We thus have acontradiction and there cannot be a tipping equilibrium if A < !:If A & !; then Pe" = A % ! and the following condition must hold:which is equivalent to

e1
@$
1
=%
[A % ! + & % 4] ( 0;
1 % ' c' aPe
& & (! % A) + 4:

Summarizing, we have:
Lemma 8. Suppose that 4 & 0. There is a tipping equilibrium in which Pe" = A % ! &2p" = 0 if A & ! and & & (! % A) + 4, where 4 = 1 % ' c ' a % (*2a ) :Interior Equilibrium Let us look for an interior equilibrium (i.e., each Örm has apositive market share): 0 < nc"
For an interior equilibrium, Örm 2ís Örst-order2 < 1.
condition for proÖt maximization is given bype"2 = nc"
2 4 > 0:However, Pe" can be zero. From Örm 1ís Örst-order condition, we have:
Pe" = max f0; %& + nc"
1 4g :

Consider the case of Pe" = 0: Then, from the demand function and (5), we havenc"
2 =(! % A) % nc"
2 4(1 % ' c ' a )By collecting terms in nc"
2 , we have
nc"
2 =

!%A
:
4 + (1 % ' c ' a )52(5)

Pe" = 0 is an equilibrium price for Örm 1 ifnc"
1 4 ( &;

which is equivalent to

(
1%

)
!%A
4 ( &:
4 + (1 % ' c ' a )(6)

If (6) is violated, thenPe" = %& + nc"
1 4 > 0:Then,

Pe" % pe"2 = (1 % 2nc"
2 )4 % &:Putting this into the market share equation leads toc"nc"
2 (1 % ' c ' a ) = (! % A) + (1 % 2n2 )4 % &;which can be rewritten as
nc"
2

"

(' )2
34 + a2#= (! % A) % & + 4:

Therefore,(! % A) % & + 4ih2
3 4 + (*6a )hi(* a )2
3
4+% 12 + ! % &
2
6
h
i
=
2
3 4 + (*6a )

nc"
=
2

=

1 1 + 2& % 2!
%
2
64 + (' a )2

Hence, Örm 1ís market share isc"nc"
1 = 1 % n2 =1 1 + 2& % 2!
+:
2
64 + (' a )22

Because we assume 4 = 1 % ' c ' a % (*2a ) > 0, the denominator in the market share53

expression is strictly positive. Under the maintained assumption of ! < 1=2, Örm 1ís
market share is larger than 1=2, which is the market size in the monopolized market
A without tying when it is one-sided (i.e., when & = 0). The market share for Örm 1
increases with & and decreases with !, as expected. The market shareís response to & or
! is greater as the intergroup externalities in market B becomes more important (i.e., as' c or ' a increases).
Firm 1ís equilibrium price isPe" = %& + nc"
1 4 & 0:4 (1 + 2& % 2!) 4= %& + +2
64 + (' a )2 :We Önd@ Pe"
24= %1 +
2@&
64 + (* a )2

44 + (' a )2
= %< 0:
64 + (' a )2
@ Pe"
< 0:
@!

@ Pe"
@4
@nc"
= nc"
+40 1
1
@' c
@' c
@' c
(
)
1 1 + 2& % 2!
1 + 2& % 2!
= %' a++ 4 0 'a h
i
2
2 2
2
64 + (' a )6 4 + (*6a )2"#3
1 1 + 2& % 2!
4
5 < 0:
i 1%= %' a 4 + h
2
2(*)
2 6 4+ a
4 + (* a )66

Firm 2ís equilibrium price ispe"2 = nc"
4
12
2
1 1 + 2& % 2!
=%
4:2
64 + (' a )2
For a given market share, ' c ' a > 0 induces Örm 2 to charge a lower price to consumers
54

for product B than when ' c ' a = 0. In addition, as long as 1 + 2& % 2! > 0, Örm 2ísmarket share is smaller when ' c ' a > 0 than when ' c ' a = 0. Hence, both factors would
induce Örm 2 to become more aggressive.
Summarizing, we have:
Lemma 9. Suppose that 4 & 0 and the conditions for the tipping equilibrium in Lemma
8 are not met. Then, there is an interior equilibrium such that(i) if (6) is satisÖed, Pe" = 0, pe"2 = nc"
2 4 > 0 and
nc"
2 =

!%A

24 + (*2a )2:(ii) if (6) is violated, Pe" = %& + (1 % nc"
e"2 = nc"
2 )4 > 0; p
2 4 > 0 andnc"
2 =1 1 + 2& % 2!
%
2
64 + (' a )2

All the results in lemmas 7-9 we derived under di§erent parameter conÖgurations canbe summarized as in Proposition 6.

55