ESSAYS IN INDUSTRIAL ORGANIZATION

By

Hyunsoo Kim

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Economics – Doctor of Philosophy

2018

ABSTRACT

ESSAYS IN INDUSTRIAL ORGANIZATION

By

Hyunsoo Kim

The first essay "Tying and Platforms’ R&D Incentives in Two-sided markets" analyzes how the tying
arrangements can affect platforms’ R&D incentives in two-sided markets under the possibility of
multi-homing. The model shows that when all consumers single-home, the tying distorts platforms’
R&D incentives because the tying acts as a commitment device to invest aggressively in R&D,
leading to rival firm being foreclosed in the R&D decision stage even if the tying does not have
exclusionary effect in the price competition. However, when exclusive contents are offered to
each platform so that some of the consumers can engage in multi-homing, the tying raises the
rival firm’s R&D incentives as well as the tying firm’s R&D incentives. This is because i) tying
induces more consumers to multi-home so that total demand of consumers for both platforms can
be increased and ii) the strategic effect in R&D competition disappears in the multi-homing case on
the consumer side, implying that the rival firm’s R&D incentives is not affected by more aggressive
R&D investment of tying firm. Thus, the anti-competitiveness of tying involved with innovation
can vary depending on the possibility of consumer’s multi-homing.

The second essay "Information Sharing and R&D Incentives" investigates how sharing of cost
information affects firms’ incentives to invest in cost reduction and the role of the observability of
rivals’ R&D investment level. I study duopoly price competition with cost reducing R&D in three
cases:
the complete information case, unobservable investment case and observable investment
case. The opponents’ cost information is unknown in both the unobservable investment case
and the observable investment case, but the investment level is unobservable and observable,
respectively. I find that firms have identical incentives of investment in the complete information
and the unobservable investment case, whereas they will tend to underinvest in cost reduction
when the investment level is observable because a negative strategic effect makes investment
less profitable. Due to this underinvestment, welfare and the consumer surplus decrease in the

observable investment case. These results have implications for the analysis of information sharing
in markets where cost reduction activities are important.

The third essay "The Grandfather of Price Discrimination", coauthored with Brady Vaughan
and Aleks Yankelevich, examines firms’ motivations for implementing grandfather clauses that
allow certain consumers to continue access to a service at a favorable, but no longer available
price. We find that when consumers are fully cognizant of their valuations for available product
alternatives, firms are typically better off offering all potential consumers the optimal uniform price.
However, if grandfathered consumers are made complacent, failing to reevaluate the service over
time, grandfather clauses may permit firms to profitably price discriminate between early adopters
and new consumers in exchange for forfeiting the right to optimally set prices for early adopters.

ACKNOWLEDGEMENTS

It would not have been possible to write this doctoral thesis without the help and support of my
advisors, committee members, and family.

Above all, I would like to thank my co-advisors, Prof. Jay Pil Choi and Prof. Thomas Jeitschko
for their help, support, and patience. Despite all of my weaknesses, they have guided me with
invaluable advice and persistent support throughout the graduate studies. I am also grateful to other
members of my dissertation committee. Prof. Christian Ahlin has always given kind and detailed
comments and the passionate guidance and help of Dr. Aleks Yankelevich has been invaluable.

I would like to thank to the academic, financial and technical support of the department of
Economics in Michigan State University and its staffs, Lori Jean Nichols and Margaret Lynch for
their friendly support.

Last but by no means least, I would like to thank my family. I am indebted to My parents,
Sangmin Kim and Eunyoung Jung for their endless and unconditional love. Also my gratitude goes
to my wife, Eunsun So and my son, Aaron. They are a source of motivation and encouragement
throughout my grad studies.
I could finish all process of this dissertation with their love and
sacrifice.

iv

TABLE OF CONTENTS

LIST OF FIGURES .

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CHAPTER 1 TYING AND PLATFORMS’ R&D INCENTIVES IN TWO-SIDED MAR-

CHAPTER 2

1.4 Conclusion .

KETS .
Introduction .

Introduction .

2.1
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2.2 Model
2.3 Basic Analysis .

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1.2.1
1.2.2
1.2.3 Welfare Analysis

1.3.1
1.3.2
1.3.3 Welfare Analysis
.

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1.1
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1.2 Tying in Two-sided Markets with Single-Homing . . . . . . . . . . . . . . . . . .
Platform Competition without Tying . . . . . . . . . . . . . . . . . . . . .
Platform Competition with Tying . . . . . . . . . . . . . . . . . . . . . . .

1
1
6
8
9
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1.3 Tying in Two-sided Markets with Multi-homing . . . . . . . . . . . . . . . . . . . 14
Platform Competition without Tying . . . . . . . . . . . . . . . . . . . . . 15
Platform Competition with Tying . . . . . . . . . . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
INFORMATION SHARING AND R&D INCENTIVES . . . . . . . . . . . 27
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2.3.1 Complete Information Case . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Unobservable Investment Case . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.3 Observable Investment Case . . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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2.6.1 Asymmetric Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6.2 Asymmetric Firms
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CHAPTER 3 THE GRANDFATHER OF PRICE DISCRIMINATION . . . . . . . . . . . 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Firms and Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Consumer Complacency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
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3.3.1 Welfare .
3.4 Price Unfairness .
3.4.1 Welfare .
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3.5 Conclusion .
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2.4 Welfare Analysis
2.5 Robustness . .
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2.6 Discussion .

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3.2.1
3.2.2 Grandfather Clauses

3.1
Introduction .
3.2 Baseline Model

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2.7 Conclusion .

APPENDICES .

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APPENDIX A PROOFS IN CHAPTER 1 . . . . . . . . . . . . . . . . . . . . . . . . 80
APPENDIX B PROOFS AND DERIVATIONS IN CHAPTER 2 . . . . . . . . . . . . 82
APPENDIX C PROOFS IN CHAPTER 3 . . . . . . . . . . . . . . . . . . . . . . . . 109
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

BIBLIOGRAPHY .

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vi

LIST OF FIGURES

Figure 1.1: Two-sided Market with Single-homing on the Consumer Side . . . . . . . . . .

8

Figure 1.2: Equilibrium in R&D Investment with Single-homing . . . . . . . . . . . . . . . 11

Figure 1.3: Two-sided Market with Multi-homing on Both Side . . . . . . . . . . . . . . . 17

Figure 1.4: Two-sided Market with Multi-homing under Tying . . . . . . . . . . . . . . . . 20

Figure 1.5: Equilibrium in R&D Investment with Multi-homing . . . . . . . . . . . . . .

. 21

Figure 2.1: Pricing Strategy in Observed Investment Case . . . . . . . . . . . . . . . . . . 40

Figure 2.2: Mixed Pricing Strategy in Observed Investment Case

. . . . . . . . . . . . . . 47

Figure 2.3:

Investment Level and Price Level in Competition between Asymmetric Firms

. 49

Figure 2.4: Production Inefficiency and Profits in Competition between Asymmetric Firms . 49

Figure 2.5: Consumer Surplus and Total Welfare in Competition between Asymmetric Firms 50
Figure 3.1: Area where Grandfather Clause is Useful (a = 1) . . . . . . . . . . . . . . . . . 73
Figure 3.2: Area where Consumers are Better Off under GFC (a = 1)

. . . . . . . . . . . . 76

vii

CHAPTER 1

TYING AND PLATFORMS’ R&D INCENTIVES IN TWO-SIDED MARKETS

1.1 Introduction

This paper analyzes the effect of tying arrangements on platforms’ R&D competition as well
as price competition and social welfare in two-sided markets. Especially, I analyze how the effect
of tying arrangements on platforms’ R&D competition can vary across the possibility of multi-
homing; that is, some of the comsumers have an incentive to join multiple platforms in order to
enjoy more indirect network externalities. The motivation of this analysis is particularly antitrust
cases involving tying, from Microsoft to Google. Recently, Yandex, the largest search engine
company in Russia, has opened the case against Google, claiming the violation of antitrust law.
Yandex alleges that Google has been inhibiting smartphone manufacturers from preinstalling other
competing applications in handheld devices that use Google’s Android. The lawsuit explains
that if smartphone manufacturers choose to preinstall other applications, instead of one of Google’s
applications, say YouTube, they cannot load any of Google’s applications, such as Google Search and
Google Maps. That is, Google has been offering all or nothing option to smartphone manufacturers
and this kind of contract might be seen as a de facto tying of its applications to Android.

Google’s current practice is in many ways similar to Microsoft’s strategies in the late 1990s
and early 2000s; especially to tying Window Media Player with its Windows operating system.
Firstly, both cases concern tying in high-tech sectors where products are frequently altered and the
boundaries of markets are constantly changing. Dynamic aspects are very important in high-tech
sectors; one of main arguments of both plaintiff and defendant in the antitrust cases is the effect of
tying on innovation. Plaintiffs claimed that tying would stifle innovation in the tied good market as it
gives competitors lower incentives to innovate, while defendants argued that tying itself is a way of
innovation as it improves the value of tying product to consumers and to makers of complementary
goods. This debate shows that innovation plays a key role in the antitrust cases regarding tying.

1

Secondly, the markets of bundled product are characterized by two- sidedness, where the two
sides of the market are content providers on one side and consumers on the other side. This market
is characterized by indirect network externalities between two sides of the market. The utility of
agents on each side depends on the number of agents on the other side. Since two types of agents
participate and interact in platforms, platforms charge two prices, one on each side. According to
Rochet and Tirole (2006), “the market is said to be two-sided if the volume of transaction varies
with not only the overall price level but also the price structure.” Especially, one important factor
when analyzing the effect of tying in two-sided markets is whether comsumers single-home or
multi-home. Multi-homing in the case of consumers of media players means that consumers use
more than one media player and multi- homing in the case of content providers means that content
providers encode their media content in several formats and offer to more than one media player.
This multi-homing insures against a tipping in the market, which is a typical concern of antitrust
authorities in tying issues. Moreover, Tirole (2005) concludes that if the costs of multi-homing on
the tying side of the market are small, tying might induce more agents to multi-home. The literature
on two-sided markets has shown that the standard economic models used in antitrust analysis often
yield incorrect results and implications when applying them to two-sided markets. Thus, antitrust
authorities should care about how the results from traditional economic models can vary in the
two-sided market context.

In light of these distinguishing features of recent antitrust issues regarding tying, I investigate
how the economic theory held in the traditional one-sided market can vary in a two-sided market
context. In this paper, I examine how the effect of tying on platforms’ incentives to invest in R&D
depends on the possibility of multi-homing on the consumer side. More specifically, I analyze the
interaction between the price competition and R&D competition as in Choi (2004). I show that
when all consumers single-home, the tying distorts platforms’ R&D incentives because the tying
acts as a commitment device to invest aggressively in R&D, leading to rival firm being foreclosed in
the R&D decision stage even if the tying does not have exclusionary effect in the price competition.
However, when exclusive contents are offered to each platform so that some of the comsumers can

2

engage in multi-homing, the tying can raise the rival firm’s R&D incentives as well as the tying
firm’s R&D incentives. This is because i) tying induces more consumers to multi-home so that
total demand of consumers for both platforms can be increased and ii) the strategic effect in R&D
competition disappears in the multi-homing case on the consumer side, implying that the rival
firm’s R&D incentives is not affected by more aggressive R&D investment of tying firm. Thus,
the anti-competitiveness of tying involved with innovation can vary depending on the possibility of
multi-homing.

Choi (2004) shows that tying can lower the rival firm’s incentives to invest and innovate when
considering R&D incentives of the tying firm and the rival firm in the tied good market. The
model demonstrates that tying distorts each firm’s R&D incentives due to the strategic effect of
the R&D competition; the tying firm’s R&D investment level increases while the rival firm’s R&D
investment level decreases with tying.
It might induce the rival firm’s foreclosure with R&D
competition even when it is not foreclosed in the absence of R&D competition. This effect lends
weight to the argument that tying by the monopolist can stifle the rival firm’s innovation incentives
in the tied good market. Moreover, in the model, tying can be a profitable strategy even if the rival
firm is not foreclosed with the R&D competition. It is analogous to the result in the case of the
single-homing in my model. Nevertheless, I extend the mechanism to the multi-homing case, and
it becomes a different story.

Choi and Stefanadis (2001) also show that tying can discourage the rival firm’s R&D investment
by analyzing how tying can protect the incumbent’s position in the market. The model assumes
that the success of stochastic R&D investment is required for potential entrants to enter the market.
When the incumbent monopolist of two complementary products ties, the potential entrant in each
market can enter the market only if the both entrants have to succeed in innovation. Thus, tying
lowers incentives for investment and innovation of potential entrants. This model, however, focuses
on the situations where one firm has monopoly power in both of two complementary goods markets
whereas my model deals with the cases where one of the competing platforms is the monopolist in
another market. In addition, in their paper, the success in innovation in two complementary markets

3

is assumed to be stochastic depending on its initial R&D investment while my model assumes that
R&D outcomes in the tied good market are deterministic.

Despite the importance of innovation in high-tech sectors where two- sidedness is prevalent, the
changes in platforms’ incentives to invest in R&D when tying is practiced have not been analyzed
before. Hagiu (2007), Belleamme and Peitz (2010), Zhao (2010), and Lin, Li, and Whinston (2011)
investigate sellers’ (content providers in this paper) incentives to invest in the quality of the products
they sell, rather than in the platform. Casadesus and Llanes(2012) study incentives to invest in
the platform’s quality, but their focus is on the comparison between open-source platforms and
proprietary platforms.

This paper is also related to the leverage theory of tying. Traditionally antitrust authorities
had regarded tying as anti-competitive; they have mainly been concerned about the leverage effect
of tying, implying that the monopolist in one market might have incentive to bundle its good to
unrelated product in order to monopolize the market. While the leverage theory has been used
as the ground rule in many juridical decisions since its advent as an informal concept in the law
literature, it has faced heavy and influential criticism from a number of economists. According to
standard economic theory argued by the Chicago school, tying would not be a profitable strategy
for the monopolist if the monopolist tries to exclude rival firms from the complementary market.
They explicitly show that the tying firm can obtain higher profits under separate selling if rival
firms are more efficient. But from the early 1990s, the leverage theory of tying has been resurrected
in many economics literatures. Whinston (1990) shows that tying can be a profitable strategy
for the monopolist as tying has the exclusionary effect if the tied good market is oligopoly and
characterized by economies of scale.

Li (2009) studies about tying of independent products in two-sided markets. The model
concludes that tying of the CD and the magazine can be a profitable strategy for the monopolist.
If the positive network externality that magazine readers exert on advertisers is large, prices are
strategic substitutes. This means that the competing platform increases its price in reaction to more
aggressive pricing behavior of the tying platform, implying that more magazine readers choose the

4

tying platform. Then, tying may be profitable strategy if gains in the magazine market outweigh
losses in the CD market. Li (2009), however, considers the magazine market where there exists
only one-side network externality and the model does not allow the possibility of multi-homing on
the consumer’s side.

Choi (2010) is closely related to this paper; he investigates the effect of tying of two complemen-
tary goods on platforms’ competition and social welfare. Furthermore, the model allows consumers
on both sides to multi-home. Choi (2010) creates incentives for consumers to multi-home by as-
suming that there exists exclusive content for each platform. In the model, if the monopolist ties
the two products and captures the tied good market, the number of multi-homers on the consumer’s
side increases relative to the no-tying case. Given that the monopolist charges its price so that
every consumer will purchase the bundle, tying is a profit-maximizing strategy. In addition, social
welfare increases under tying in the model due to the increase in the number of multi-homers.
However, the model is concerned only with pricing implications of tying, not with dynamic aspects
in the markets.

Amelio and Jullien (2010) are also worth mentioning in relation to this paper. They analyze
the rationale for bundling, especially mixed bundling, in the monopoly and duopoly context,
when platforms are constrained to set non-negative prices.
In the model, bundling can be a
tool to introduce implicit subsidies between bundling and bundled products and hence relax the
non-negativity constraints. They show that the effect on social welfare depends on the degree
of asymmetry in network externalities between two sides.
In the case of symmetric network
externalities, for example, profits of both platforms increase under bundling while consumer surplus
and social welfare decrease. However, Amelio and Jullien (2010) mainly deal with mixed bundling
and hence illustrate price discrimination implemented through bundling.

The remainder of the paper is organized as follows. In section 1.2, I set up a basic model of
tying in two-sided markets in the absence of consumer’s multi- homing. Section 1.3 extends the
analysis by allowing multi-homing. In each section, the role of tying in the R&D competition and
welfare implications are discussed. Conclusion follows in Section 1.4.

5

1.2 Tying in Two-sided Markets with Single-Homing

The model I lay out in this paper is an extension of the model presented in Choi (2004) and
Choi et al (2017). In this paper, I analyze R&D competition and price competition between two
platforms when one platform ties the platform with unrelated monopoly product. As in Choi (2004),
I focus on tying of independent products in order to avoid multiple equilibria problem. If one of
producers is a monopolist for one of the products, price competition with complementary goods
yields multiple price equilibria because the monopolist practices a price squeeze, depending on the
degree of price squeeze practiced by the monopolist. As Choi (2004) points out, however, once the
degree of price squeeze is assumed, the analysis and implication of this model can carry over to the
case of complementary products.

Suppose that there are two symmetric platforms, i = A, B. Platforms deal with two distinct
groups of agents; consumers on side 1 and content providers on side 2. These two groups of agents
wish to interact on the platform and platforms coordinate the possible matches between the two
groups. Platforms compete for market share in each agent group and charge prices pi
2 to
consumers and content providers respectively with i = A, B. I further assume that marginal costs
of serving another consumer and content provider are c1 and c2 respectively. Ni
1 is the number of
consumers participating in platform i, and Ni

2 is the number of content providers on platform i.

1 and pi

In order to analyze consumers’ choice of platform, I adopt a Hotelling model of horizontal
product differentiation. Platforms are located at the endpoints of the Hotelling line, that is platform
A is located at 0 and platform B is located at 1. Consumers are uniformly distributed on the interval
[0,1]. The utility that a comsumer derives from participating in a platform depends on the number
of content providers in the platform. Consumers obtain additional utility α1 from each additional
content provider. The utility of a consumer who is located at x from joining platform A is given by

uA = α1N A

2 − pA

1 − tx

(1.1)

where t is the transportation costs in order to reach platform i. Similarly, the utility of a consumer

6

who is located at x from joining platform B is given by

uB = α1N B

2 − pB

1 − t(1− x)

(1.2)

I assume that content providers view the platforms as homogenous as in Armstrong and Wright
(2007). The potential number of content providers is normalized to 1. Fixed costs for producing
content are considered to be zero. Content providers obtain additional profits α2 from each
additional consumer who has access to their content. Profits for a content provider participating in
platform i are hence given by α2Ni
2. A content provider is willing to participate in platform i
if α2Ni

1 − pi
In this section, I focus on the single-homing case where all consumers single-home and this
gives an incentive for sellers to multi-home given no fixed costs for producing content. Thus, as in
“competitive bottleneck” equilibria in Armstrong and Wright (2007), the consumer side completely
single-homes while the content provider side completely multi-homes.

2 ≥ 0.

1 − pi

In order to analyze the effects of tying on platform competition in two- sided markets, assume
that the platform A is the monopolist for the product M with unit production cost of cM. All
consumers have valuation of vM, which is greater than cM, for the product M. I further assume that
entry to market M is not feasible.

I analyze the following four-stage game in order to introduce the R&D investment in the platform
competition. In the first stage, platform A decides whether to tie. In the second stage, the two firms
engage in R&D activities for reducing the marginal cost. I assume that platform i can reduce c1 by
∆i with the investment costs of I(∆i), implying that the R&D outcomes are deterministic as long as
platforms make an investment. The cost function of R&D investment, I(·), is characterized by I(cid:48)(·)
and I(cid:48)(cid:48) > 0. In the third stage, platforms set simultaneously their prices pA
1 on the consumer
side and consumers decide which platform they would join. In the fourth stage, platforms set their
prices pA
2 on the provider side and content providers decide which platform they would join.
To focus on the impact of tying arrangements on R&D competition on the consumer side, I ignore
the possibility of R&D activities in the market M as well as on the content provider side.

1 and pB

2 and pB

7

Content
Provider

Side

Non-exclusive Content

Platform

Platform A

Platform B

User
Side

Single-homing A

Single-homing B

Figure 1.1: Two-sided Market with Single-homing on the Consumer Side

1.2.1 Platform Competition without Tying

If the two products, the platform and the product M, are not bundled, they can be analyzed separately.
Given that all consumers have valuation of vM for the product M, platform A can charge vM for the
product M and hence have profits of vM − cM ≡ sM.

2 ≤ α2Ni

From the participation constraints for content providers, we have Ni

2 = 1 as long as platform i
charge the price pi
1. In platform competition on the consumer side, we have the Hotelling
type competition. We can derive the number of consumers in platform i by considering the utility
of consumers from joining each platform given above.
1 − pi
pj
2t

Ni
1 =

(1.3)

1

1
2

+

Given that ∆A, ∆B, and the rival firm’s price pj

1, platform i chooses pi

1 to maximize its profits

in the third stage.

max
pi
1

(pi
1 − c1 + ∆i)Ni

1 + αNi

1 − c2 − I(∆i)

The first order condition for platform i is given by

1 = Ri
pi

1(pj

1; ∆i) =

1
2

1 + c1 − ∆i + t − α2)
(pj

8

(1.4)

(1.5)

In Nash equilibrium, prices for platform i on each side are given by

1 (∆i, ∆j) = c1 + t − α2 − 2
pi∗
∆i − ∆j
pi∗
2 (∆i, ∆j) = α2

+

6t

(cid:16)1

2

(cid:17)
3 ∆i − 1
3 ∆j

The number of consumers joining in platform i can be written as

1(∆i, ∆j) =
Ni

1
2

+

∆i − ∆j

6t

(cid:16)1

(cid:17)

(cid:16)

(cid:17)(cid:16)1

+

Therefore, in the second stage, platform i chooses ∆i to maximize
∆i − ∆j

∆i − ∆j

∆i − ∆j

t − α2 +

max
∆i

3

2
The first order condition in the second stage is given by
∆i − ∆j

6t

2

+ α2

+

6t

I(cid:48)(∆i) =

1
3

+

9t

(cid:17) − c2 − I(∆i)

(1.6)

(1.7)

(1.8)

(1.9)

From (1.9), we can derive the reaction function of the platform i in the R&D competition,
denoted as ∆i = ρi(∆j). For the second order conditions for the platform’s maximization problem
9t − I(cid:48)(cid:48)(∆i) < 0. In this model, cost reducing R&D activities by two
to be satisfied, I assume that 1
competing platform have the strategic effect. We can see the strategic effect between platforms’
R&D investments by totally differentiating (1.9).

(cid:110)

I(cid:48)(cid:48)(∆i)(cid:111)−1

ρi(cid:48)(∆j) =

1− 1
9t

(1.10)

which has a negative value. This implies that R&D activities of two platforms are strategic
substitutes. I also assume that | ρi(cid:48)(∆j)| < 1, or I(cid:48)(cid:48)(∆j) > 2
9t for ensuring the stability of the Nash
equilibrium. In symmetric equilibrium in the R&D stage, the optimal level of investment is given
by ∆A∗ = ∆B∗ = ∆∗ where I(cid:48)(∆∗) = 1
3.

1.2.2 Platform Competition with Tying

Under tying, platform A ties the sale of the platform and the product M and sells them for a price
˜pA
1 .1 In this case, consumers have two options; to purchase the bundle from platform A at the price
1Variables corresponding to tying are denoted with a tilde.

9

1 , or to subscribe to platform B only. In the Hotelling line, we can derive the point ˜x where a

of ˜pA
consumer who is indifferent between two options is located by the following equation.

1 − t(1− ˜x)
We can derive the number of consumers joining in each platform.

1 − t ˜x = α1 ˜N B

vM + α1 ˜N A

2 − ˜pA

2 − ˜pB

˜N A
1 =

˜N B
1 =

1
2
1
2

1

+

1 − ˜pA
vM + ˜pB
2t
1 − ˜pA
− vM + ˜pB
2t

1

(1.11)

(1.12)

Substituting the number of consumers into each platform’s profit function and maximizing with

respect to ˜pA

1 and ˜pB

1 , respectively, yield the following reaction function in the second stage.
˜pA
1 = RA
˜pB
1 = RB

( ˜pB
1 + vM + cM + c1 − ˜∆A + t − α2)
( ˜pA
1 − vM + c1 − ˜∆B + t − α2)

1 ; ˜∆A) =
1 ; ˜∆B) =

1 ( ˜pB
1 ( ˜pA

1
2
1
2

(1.13)

Then, the Nash equilibrium prices with tying and corresponding demand equations on the

consumer side are given by

˜∆A− 1
3
˜∆A− 2
3

˜∆B

˜∆B

(1.14)

2

3

3cM − 2
(cid:17)
3cM − 1
(cid:17)

3

+

2

(cid:16)1
(cid:16)1

1
˜pA∗
1 = c1 + t − α2 +
3 vM +
1 = c1 + t − α2 − 1
1
˜pB∗
3 vM +
sM + ˜∆A− ˜∆B
˜pA∗
2 = α2
− sM + ˜∆A− ˜∆B
˜pB∗
2 = α2
2
sM + ˜∆A− ˜∆B
1
˜N A∗
1 =
2
− sM + ˜∆A− ˜∆B
1
˜N B∗
1 =
2

6t

6t

6t

6t

+

Note that in the absence of R&D investment, platform B is driven out from the market if
sM > 3t.To emphasize the role of R&D competition under tying, I assume that sM < 3t to ensure
that platform B does not exit from the market after tying.

In the second stage, platform A chooses ˜∆A to maximize its profits.

(cid:16)

sM + ˜∆A− ˜∆B

3

t +

max
˜∆A

(cid:17)(cid:16)1

2

(cid:17) − c2 − I( ˜∆A)

sM + ∆A− ∆B

6t

+

10

(1.15)

∆B

ρA(∆B)

˜ρA(∆B)

ρB(∆A)

˜ρB(∆A)

∆A

Figure 1.2: Equilibrium in R&D Investment with Single-homing

The first order condition in the second stage is given by

I(cid:48)( ˜∆A) =

1
3

+

sM
9t

+

˜∆A− ˜∆B

9t

Similarly, we have the optimal level of R&D investment for platform B.

I(cid:48)( ˜∆B) =

1
3

− sM
9t

+

˜∆A− ˜∆B

9t

(1.16)

(1.17)

Equation (1.16) and (1.17) give each platform’s reaction function in the second stage with
tying, denoted as ˜∆i = ˜ρi( ˜∆j). When comparing equation (1.16) and (1.17) to equation (1.9), the
assumptions on the cost structure of R&D investment, I(cid:48)(·) > 0 and I(cid:48)(cid:48)(·) > 0, allow us to verify
easily that ˜∆A > ∆A whereas ˜∆B < ∆B for any given level of the rival firm’s R&D investment.

Proposition 1.1. Suppose that all consumers join a single platform. If firm A decides to tie, firm
A’s R&D investment level increases while firm B’s R&D investment level decreases under tying.

Figure 1.2 shows that platform A’s reaction function ˜ρA( ˜∆B) shifts outwards while platform B’s
reaction function ˜ρB( ˜∆A) shifts inwards. The reason why tying firm’s R&D incentives increase
while rival firm’s incentive decrease with tying is that R&D incentives in this model depend on
the market share. The platform can spread out the R&D costs over more consumers as its market

11

share increases. As shown in equation (1.14), ˜N A∗
1 decreases under tying. In
1
this context, tying distorts R&D incentives of two competing platforms. Thus, when considering
R&D competition prior to price competition to allow platforms to benefit from dynamic rent, tying
arrangements might yield additional anti-competitive effects.

increases but ˜N B∗

I assume that the cost function of R&D is given by I(∆) = k

In the first stage, platform A decides whether to tie, comparing its profits with tying and without
2 ∆2, where k stands for the
tying.
R&D cost parameter. This specific cost function of R&D gives us the closed-form solutions for the
optimal level of R&D investment as well as platform’s profit-maximizing prices. In the case of no
tying, the optimal level of R&D investment for each platform is given by

∆A∗ = ∆B∗ = ∆∗ =

1
3k

The equilibrium profit for platform A is

Π A = sM +

t
2

− c2 − 1
18k

(1.18)

(1.19)

In the case of tying, the optimal levels of R&D investment for platform A and B are given by

˜∆A∗ =
˜∆B∗ =
By substituting (1.20) with (1.14), we have

1
3k
1
3k

+

sM
9tk − 2
− sM
9tk − 2

2sM

3(9tk − 2)

1
3 sM +
2sM
3(9tk − 2)

(cid:16)1
(cid:16)1

2

˜pA∗
1 = c1 + t − α2 + cM − ˜∆A∗ +
(cid:17)
1 = c1 + t − α2 − ˜∆B∗ − 1
˜pB∗
3 sM −
(cid:17)
˜pA∗
sM
3t(9tk − 2)
2 = α2
˜pB∗
3t(9tk − 2)
2 = α2
1
˜N A∗
sM
1 =
2
1
˜N B∗
1 =
2

sM
6t
− sM
2
6t
sM
6t
− sM
6t

2t(9tk − 2)
2t(9tk − 2)

sM

+

−

+

−

sM

+

+

(1.20)

(1.21)

I assumed that sM < 3t to ensure that platform B does not exit from the market after tying in
3k < sM < 3t, however, platform B is foreclosed

the absence of R&D competition. If ˜N B∗

1 , or 3t − 2

12

with R&D competition even in the case that it is not in the absence of R&D competition. This
illustrates one of the potential anti-competitive effects of tying on platform competitions.

Since the model is set up based on Hotelling type platform competition on the consumer side
and quadratic R&D cost function, the sum of two platforms’ R&D investment is always equal to
2
3k .

The equilibrium profit with tying for platform A is given by

(cid:16)

(cid:17)

(cid:16)

˜Π A =

t
2

+

1
3

sM +

2sM
9tk − 2

+

1
18t

sM +

2sM
9tk − 2

(cid:17)2 − c2 − k

(cid:16) 1

2

3k

+

sM
9tk − 2

(cid:17)2

(1.22)

Platform A chooses to tie if either tying itself is profitable ( ˜Π A > πA), or tying gives rise to the
exclusion of rival firm in R&D competition (3t − 2
3k < sM < 3t). One interesting fact is that, as
pointed out in Choi (2004), the profitability of tying can be achieved even without the exclusionary
effect.

1.2.3 Welfare Analysis

In this section, I analyze social welfare implications of tying. Social welfare without tying can be
written as

W =sM +(α1 − c1) +(α2 − c2)
1 ∆B∗

1 + N B∗

(cid:110)(N A∗

1 ∆A∗

+

1 )− k
2

(cid:2)(∆A∗)2 +(∆B∗)2(cid:3)(cid:111) −(cid:110)Þ N A∗

1

Social welfare with tying can be written as
˜W =sM ˜N A∗

(cid:110)( ˜N A∗

1 +(α1 − c1) +(α2 − c2)
˜∆B∗
1 )− k
2

˜∆A∗
1 + ˜N B∗

(cid:2)( ˜∆A∗)2 +( ˜∆B∗)2(cid:3)(cid:111) −(cid:110)Þ ˜N A∗

+

1

1

1

0

0

Þ N B∗

1

0

Þ ˜N B∗

1

0

(cid:111)

(cid:111)

txdx

txdx

(1.23)

(1.24)

txdx +

txdx +

If we substitute (α1 − c1) +(α2 − c2) by (vB − cB), expressions above are exactly the same as in

Choi (2004). He explicitly shows that ˜W < W unambiguously.

Proposition 1.2. Social welfare decreases with tying when all consumers join a single platform.

13

There are several reasons of this strong implication. First, asymmetric demand on each platform
causes the increase in the total transportation in the Hotelling type competition. Second, as shown
above, the sum of R&D investment of two platforms are constant. This implies that the distortion
effect of tying on R&D investment reduces the net benefit from innovation in the market. The terms
in the curly brackets in (1.23) and (1.24) represent the dynamic rents from R&D stage. We can easily
check that the dynamic rents from R&D are maximized under the symmetric R&D investment.
This shows a direct negative effect of tying on innovation. Lastly, some of the consumers do not
choose to buy the bundled product under tying, leading to the loss of sM.

The result in this section is the same as in Choi (2004) because if we focus on single-homing
case on the consumer side in addition to no product differentiation case on the content provider side,
“competitive bottleneck” equilibrium would occur and hence the platform’s maximization problem
boils down to the same as in the one-sided market. This leads to same welfare implications as well
as R&D investment and prices decisions of each platform.

In sum, I apply the model in Choi (2004) to the two-sided market setting and verify that the
It shows that tying
results in Choi (2004) are robust to the introduction of the two-sidedness.
can have an anti-competitive effect that many antitrust authorities have concerned; it reduces the
rival firm’s incentives to invest in R&D while strengthens the tying firm’s R&D incentives. In
this section, however, I focus on the single-homing case where all consumers join in only one
platform. But in reality, especially in media content markets or smartphone application markets,
multi-homing equilibrium is commonly observed. Thus, I will modify the analysis to reflect this
reality in the next section.

1.3 Tying in Two-sided Markets with Multi-homing

In this section, I extend the analysis of tying to the multi-homing case. As mentioned above,
in the media content markets, many consumers have or subscribe to several platforms and many
In section 2, I assumed that
content providers provide multiple platforms with their content.

14

content providers regard platforms as homogenous while consumers view them as differentiated.
This assumption yields a typical “competitive bottleneck” equilibrium; all consumers single-home
whereas all content providers multi-home. Given that all content providers multi-home, however,
consumers have no incentive to join in multiple platforms because they can meet and interact with
all content providers once they subscribe to one platform. But if each platform deals with exclusive
content, some of the consumers are willing to join in more than one platform to enjoy the exclusive
content so that multi-homing equilibrium can occur.

As in Choi (2010), in order to introduce the possibility of multi-homing on the consumer side, I
assume that there are two types of content; exclusive and non-exclusive content. λ ∈ [0,1] is defined
as the amount of exclusive content available on each platform and hence 1− λ is the amount of non-
exclusive content which is available for both platforms. Let me assume that λ is exogenously given.
This assumption can be justified by the fact that some content is more suitable for the specific
platform while other content is easily made compatible with both platforms for technical reasons.
For example, some platforms are able to support high-definition multimedia content whereas other
platforms are not. Multi-homing on the content provider side in setting means that some of the
non-exclusive content providers choose to offer both platforms with their content.

As in section 1.2, I assume that the cost function of R&D investment is given by I(∆) = k

2 ∆2,
where k stands for the R&D cost parameter. In addition, for the expositional simplicity, let me
assume c2 = 0 in this section.

1.3.1 Platform Competition without Tying

Since I focus on the multi-homing equilibrium on both sides, let me first assume that non-exclusive
content providers multi-home and I will derive the condition for this to hold later in A1. Under this
assumption, each platform has exclusive content of λ and non-exclusive content of 1− λ available.
It means that the total number of content available when users multi-home is 1 + λ.

The utility of a consumer who is located at x from joining platform A and platform B is the
2 = 1. In addition, the utility of a consumer who

same as (1.1) and (1.2), respectively with N A

2 = N B

15

is located at x from multi-homing can be written as

uAB = α1(1 + λ)− pA

1 − pB

1 − t

(1.25)

Equation (1.25) shows that given pA

1 , the utility of a multi-homer is the same regardless
of her location. Then the location of a consumer who is indifferent between single-homing on A
and multi-homing can be obtained as

1 and pB

x = 1− α1λ− pB

1

t

(1.26)

Similarly, we can derive the location of a consumer who is indifferent between single-homing

on B and multi-homing.

α1λ− pA
1

t

y =

(1.27)

Let’s denote nA

1 and nm

1 by the number of single-homers joining in platform i and the number of
1 . Then, the number of agents participating

1 = nA

1 + nm

multi-homers, respectively, implying that N A
in platform i is given by

ni

1

t

1 = 1− α1λ− pj
2α1λ−(pi
t
α1λ− pi
1

1)
1 + pj

nm
1 =

Ni
1 =

t

− 1

(1.28)

Comparing to the single-homing case, it is a noteworthy feature in multi-homing case that there
is no interaction between optimal prices of two platforms on the consumer side anymore when
the consumer’s demand for each platform is determined. As we can see in equation (1.28), the
total number of consumers subscribing to platform i, Ni
1, depends only on its own price while the
number of single-homers on platform i, ni
1, depends only on the rival’s price. It implies that the
total demand for platform i is determined by its price whereas the rival’s price is a determinant for
the proportion of multi-homers.

Next, I look at the incentives of content providers to participate in each platform. Since profits
2, an exclusive content

for a content provider participating in platform i are hence given by α2Ni

1− pi

16

Content
Provider

Side

Exclusive Content

for Platform A

λ

Non-exclusive Content

1− λ

Exclusive Content

for Platform B

λ

Platform

Platform A

Platform B

User
Side

Single-homing A

Multi-homing

Single-homing B

Figure 1.3: Two-sided Market with Multi-homing on Both Side

1 − pi

2 ≥ 0. However, the incentives for non-exclusive
provider will participate in platform i if α2Ni
content providers to provide platform i with their content depend on whether they already participate
in platform j. If a non-exclusive provider already offers to platform j, she will provide to platform
2 ≥ 0 due to the existence of multi-homers. If she has not provided to
i as well only if α2ni
2 ≥ 0. Thus, platform i can
platform j, however, she would participate in platform i if α2Ni
attract both exclusive and non-exclusive content providers if it charges pi
1, or it can attract
only exclusive content providers if it charges pi
1. Since I assumed that non-exclusive content
1. Then, ni
providers multi-home, platform i will charge pi

2 = α2Ni
2 = α2ni

2 are given by

2 = α2ni

2, nm

2 , and Ni

1 − pi

1 − pi

ni
2 = λ
2 = 1− λ
nm
Ni
2 = 1

+ λ(cid:0)(2α1+α2)λ−2c1

(cid:1)

4

< t

(1.29)

Assumption 1. α1λ−c1

2

+ 1
2k

λ(cid:0)2(1−λ)α1−α2λ(cid:1)−2(1−λ)c1

1

If k >

, the parameter space in which the multi-homing condition
holds is non-empty. Assumption 1 ensures that serving both types of content providers is more
profitable for each platform, which is derived in the Appendix A. In addition, it ensures that both
x and y are located within Hotelling line. Assumption 2 guarantees the existence of multi-homers.

17

Assumption 2. t < α1λ− c1

Similarly to the analysis in section 1.2, given ∆A, ∆B, and the rival firm’s price pj

1, platform i

chooses pi

1 to maximize its profit in the third stage.
(pi
1 − c1 + ∆i)Ni
1 + pi
2Ni
max
pi
1,pi
2
1 − c1 + ∆i) α1λ− pi
(pi

1

= max
pi
1

2 − I(∆i)

(cid:16)

1− α1λ− pi

1

t

+ α2

(cid:17) − I(∆i)

Then, the optimal prices for platform i on each side are given by

The number of consumers joining in platform i can be rewritten as

t

(cid:16)

1
pi∗
1 (∆i) =
2
pi∗
2 (∆j) = α2

(α1λ + c1 − ∆i)
1− α1λ− c1 + ∆j

2t

(cid:17)

2t

1(∆j) = 1− α1λ− c1 + ∆j
ni
1 (∆i, ∆j) =
nm
1(∆i) =
Ni

α1λ− c1 + ∆i

2t

2t

2(α1λ− c1) + ∆i + ∆j

− 1

(1.30)

(1.31)

(1.32)

In the second stage, platform i chooses ∆i to maximize its profits and the first order condition

in the second stage is given by

I(cid:48)(∆i) =

α1λ− c1 + ∆i

2t

(1.33)

Equation (1.33) shows that there is no strategic effect of R&D investment between two platforms
anymore. In single-homing case, we already saw that R&D investment of two competing platforms
are strategic substitutes; the increase in one platform’s R&D investment leads to the decrease in
the rival platform’s R&D incentives. Contrary to the single-homing case, one platform’s R&D
investment does not affect the rival platform’s incentives to invest in R&D.

1.3.2 Platform Competition with Tying

Under tying, platform A ties the sale of the platform and the product M and sells them for a price
˜pA
1 . If platform A sells the bundled product, incentives for platform B to serve content providers

18

are changed. In the absence of tying, I assumed that it is more profitable for platforms to serve
both exclusive and non-exclusive content providers. Under tying, however, serving both types of
content providers cannot achieve equilibrium anymore. This is because under the condition in
which serving both types of content providers brings more profits to platform B, tying cannot be
a profitable strategy for platform A. I thus consider an equilibrium in which all non- exclusive
content providers offer platform A with their content. Let me assume A3 for this equilibrium to be
sustained.2

A3

sM > 2t − 1
k

In this equilibrium, platform i charges ˜pi

1 and then, ˜Ni

−(α1λ− c1)− λ[(2α1 + α2)λ− 2c1]
2
2 = α2 ˜Ni
2 is given by
˜N A
2 = 1
˜N B
2 = λ

(1.34)

Let me denote the location of a consumer who is indifferent between single-homing on A
and multi-homing under tying by ˜x and the location of a consumer who is indifferent between
single-homing on B and multi-homing under tying by ˜y. Then, ˜x and ˜y are given by

˜x = 1− α1λ− ˜pB
1
vM + α1 − ˜pA
1

t

˜y =

t

Then, we have the number of consumers joining in each platform.

t

t

1

1

˜nB

˜nA

1 = 1− α1λ− ˜pB
1 = 1− vM + α1 − ˜pA
vM + α1 − ˜pA
1
˜N A
1 =
t
α1λ− ˜pB
1
vM + α1(1 + λ)− ˜pA

˜N B
1 =

t

˜nm
1 =

1 − ˜pB

1

− 1

t

2The detailed derivation of A3 is in the Appendix A.

19

(1.35)

(1.36)

Content
Provider

Side

Exclusive Content

for Platform A

λ

Non-exclusive Content

1− λ

Exclusive Content

for Platform B

λ

Platform

Platform A

Platform B

User
Side

Single-homing A

Multi-homing

Single-homing B

Figure 1.4: Two-sided Market with Multi-homing under Tying

Substituting the number of consumers into each platform’s profit function and maximizing with
1 , respectively, yield the following optimal prices of each platform in the second

1 and ˜pB

respect to ˜pA
stage.

1
˜pA∗
1 ( ˜∆A) =
2
1
˜pB∗
1 ( ˜∆A) =
2
˜pA∗
2 ( ˜∆B) = α2
˜pB∗
2 ( ˜∆A) = α2

(vM + cM + α1 − α2 + c1 − ˜∆A)
(α1λ− α2λ + c1 − ˜∆B)

(cid:16) sM + α1 + α2 − c1 + ˜∆A
(cid:16) α1λ + α2λ− c1 + ˜∆B
(cid:17)

(cid:17)

2t

2t

The number of consumers participating in each platform can be rewritten as

2t

1 = 1− α1λ + α2λ− c1 + ˜∆B
˜nA
1 = 1− sM + α1 + α2 − c1 + ˜∆A
˜nB
sM + α1 + α2 − c1 + ˜∆A
˜N A
1 =
α1λ + α2λ− c1 + ˜∆B
˜N B
1 =
sM +(α1 + α2)(1 + λ)− 2c1 + ˜∆A + ˜∆B

2t

2t

˜nm
1 =

2t

2t

− 1

(1.37)

(1.38)

Solving the profit-maximizing problem with respect to ∆i gives us the first order condition in

20

∆B

∆A∗

˜∆A∗

˜∆B∗

∆B∗

∆A

Figure 1.5: Equilibrium in R&D Investment with Multi-homing

the second stage for each platform.

I(cid:48)( ˜∆A) =
I(cid:48)( ˜∆B) =

sM + α1 + α2 − c1 + ˜∆A
α1λ + α2λ− c1 + ˜∆B

2t

2t

(1.39)

Comparing equation (1.39) to equation (1.33), we can easily verify might yield additional

pro-competitive effects in the multi-homing equilibrium.

Proposition 1.3. Suppose that a part of consumers have an incentive to join multiple platforms.If
firm A decides to tie, the tying decision raises the rival firm’s R&D investment level as well as the
tying firm’s R&D investment level.

We focus on the situation where the value of bundled product is sufficiently high. Sufficiently
high value of bundled products induce more consumers to multi-homing so that not only the demand
for platform A but also the demand for platform B can be increased with tying. As mentioned
earlier, the benefit from innovation in this model is proportional to the demand for the platform
so both platforms have more incentives to invest in R&D. Moreover, when consumers engage
in multi-homing, strategic effect of investment disappears, implying that a platform’s innovation
incentives are not eroded by more aggressive investment by the rival.

21

In the first stage, platform A decides whether to tie, comparing its profits with tying and without
tying. In the case of no tying, the optimal level of R&D investment for each platform is given by

∆A∗ = ∆B∗ = ∆∗ =

α1λ− c1
2tk − 1

The equilibrium profits for platform A and B are

(cid:16) k(α1λ− c1)
(cid:17)2
(cid:16) k(α1λ− c1)

2tk − 1

2tk − 1

(cid:17)2
(cid:16)

(cid:16)
(cid:17) − k
1− k(α1λ− c1)
2tk − 1
+ α2
1− k(α1λ− c1)
2tk − 1

2

(cid:17) − k
(cid:16) α1λ− c1
(cid:17)2
(cid:16) α1λ− c1

2tk − 1

2
2tk − 1

+ α2

Π A = sM + t

ΠB = t

(cid:17)2

(1.40)

(1.41)

In the case of tying, the optimal levels of R&D investment for platform A and B are given by

(1.42)

(1.43)

(1.44)

By substituting (1.42) with (1.38), we have

˜∆A∗ =
˜∆B∗ =

sM + α1 + α2 − c1
2tk − 1
α1λ + α2λ− c1
2tk − 1

˜N A∗
1 =
˜N B∗
1 =

k(sM + α1 + α2 − c1)
k(α1λ + α2λ− c1)

2tk − 1
2tk − 1

Unlike the single-homing case, the aggregate level of R&D investment is not constant; the

aggregate level is increased with tying. The equilibrium profit for platform A is

(cid:16) k(sM + α1 + α2 − c1)

2tk − 1

(cid:17)2 − k

2

(cid:16) sM + α1 + α2 − c1

(cid:17)2

2tk − 1

˜Π A = t

Platform A chooses to tie if ˜Π A > Π A. It implies that the parameter space is not empty set for
the tying arrangement to be a profitable strategy. Similarly to the single-homing case, tying can be
a private optimum for the monopolist even in the absence of the exclusion of its rival.

1.3.3 Welfare Analysis

In order to analyze welfare implications, I compare total welfare under tying and no tying. Social
welfare without tying can be written as
W =sM + α1(1 + λnm∗
1 ∆A∗ + N B∗

1 ) + 1− λ(cid:9)

1 )c1 + α2

(1.45)

+

(cid:8)λ(1 + nm∗
(cid:8)(∆A∗)2 +(∆B∗)2(cid:9)(cid:105)
(cid:105)

txdx + nm∗
1 t

(cid:104)(N A∗
(cid:104)Þ 1−N B∗

1

+

0

1 )−(1 + nm∗
Þ 1−N A∗
1 ∆B∗)− k
2
1

txdx +

0

22

(cid:8)λ(1 + ˜nm∗

1 ) +(1− λ) ˜N A∗

1

(cid:9)

1 )c1 + α2

where N A∗

1 = N B∗

1 = k(α1λ−c1)
2tk−1

Social welfare with tying can be written as
˜W =sM ˜N A∗

.

and nm∗

1 (1− λ)−(1 + ˜nm∗

1 = 2k(α1λ−c1)
2tk−1
(cid:8)( ˜∆A∗)2 +( ˜∆B∗)2(cid:9)(cid:105)
(cid:105)
txdx + ˜nm∗
1 t
1 = k(α1λ+α2λ−c1)
, ˜N B∗

1 )− α1 ˜nB∗
Þ 1− ˜N A∗
˜∆B∗)− k
2
1

2tk−1

0

1

1 + α1(1 + λ ˜nm∗
˜∆A∗ + ˜N B∗
1

(cid:104)( ˜N A∗
(cid:104)Þ 1− ˜N B∗
1 = k(sM +α1+α2−c1)

txdx +

1

0

+

+

where ˜N A∗

, and ˜nm∗
Hence, the changes in social welfare due to tying is given by

2tk−1

∆W = ˜W −W
=(˜nm∗

2tk−1

1 = k(sM +(α1+α2)(1+λ)−2c1)
1 )(cid:2)sM +(α1 + α2)(1− λ)(cid:3)
(cid:8)( ˜∆A∗)2 +( ˜∆B∗)2(cid:9)
(cid:8)(∆A∗)2 +(∆B∗)2(cid:9)(cid:105)
(cid:105)
(cid:105)

txdx + ˜nm∗
1
txdx + nm∗
1

1 )(α1λ + α2λ− c1)−(1− ˜N A∗
˜∆A∗ + ˜N B∗
1
1 ∆A∗ + N B∗

˜∆B∗)− k
2
k
2
1

Þ 1− ˜N B∗
1 ∆B∗) +
Þ 1−N B∗

0

1

txdx +

+

1 − nm∗

1
−(N A∗

(cid:104)( ˜N A∗
−(cid:104)Þ 1− ˜N A∗
(cid:104)Þ 1−N A∗

1

0

1

+

0

txdx +
1 > N A∗

1

0
and ˜N B∗

(1.46)

− 1.

(1.47)

1 + N B

1 = N A

1 − nm∗

1 > N B∗

1 . From the fact that nm

We can easily check that ˜N A∗

1 −1, we
can derive ˜nm∗
1 > 0, which means that tying gives rise to more multi-homers and ensures that
exclusive content is exposed to more consumers. In the assumption A2, I assumed that t < α1λ− c1.
Given the positive value of transportation cost t, α1λ + α2λ − c1 > 0. The first term in equation
(1.47) represents net benefits from the increase in multi- homers because multi-homers are able to
have access to exclusive content, and this is also beneficial to exclusive content providers. The term
(1− ˜N A∗
1 )(α1 + α2)(1− λ) stands for the loss from changes in incentives for platform B to serve two
types of content providers. Under tying, it is more profitable for platform B to serve only exclusive
content providers. It reduces the chances for non-exclusive content providers to be exposed to more
consumers, as well as the utility of users who single-home on platform B. The terms in the last
brackets are the total transportation costs of consumers. Tying may increase overall transportation

23

costs due to the increase in the number of multi-homers. The terms in curly brackets represent
the net benefits from R&D activities. From equation (1.40), (1.41), and (1.42), (1.43), we have
Ni∗
1 = k∆i∗ and ˜Ni∗
1 with k∆i∗ and k ˜∆i∗, respectively, we can
rewrite the net benefits from R&D as

1 = k ˜∆i∗. By substituting Ni∗

1 and ˜Ni∗

( ˜N A∗
˜∆A∗ + ˜N B∗
1
1
(N A∗
1 ∆A∗ + N B∗

˜∆B∗)− k
2
1 ∆B∗)− k
2

(cid:8)( ˜∆A∗)2 +( ˜∆B∗)2(cid:9) =
(cid:8)(∆A∗)2 +(∆B∗)2(cid:9) =

(cid:8)( ˜∆A∗)2 +( ˜∆B∗)2(cid:9)
(cid:8)(∆A∗)2 +(∆B∗)2(cid:9)

k
2
k
2

(1.48)

which explicitly shows that the net benefit from R&D investment increases with tying because
˜∆i∗ > ∆i∗ for i = A, B.

In order to facilitate comparison between ˜W and W, let me manipulate equation (1.47) as

follows:

∆W =(˜nm∗

1 )(α1λ + α2λ− c1 − t)−(1− ˜N A∗
˜∆A∗ + ˜N B∗
1
1 ∆A∗ + N B∗

˜∆B∗)− k
2
k
2
1

Þ 1− ˜N B∗
1 ∆B∗) +

1 )(cid:2)sM +(α1 + α2)(1− λ)(cid:3)
(cid:8)( ˜∆A∗)2 +( ˜∆B∗)2(cid:9)
(cid:8)(∆A∗)2 +(∆B∗)2(cid:9)(cid:105)
Þ 1−N B∗
(cid:104)Þ 1−N A∗

(cid:105)

1

+

1 − nm∗

(cid:104)( ˜N A∗
−(cid:104)Þ 1− ˜N A∗

1
−(N A∗

1

txdx

+

txdx +

0

txdx +

0

0

0

(1.49)

(cid:105)

1

txdx

Proposition 1.4. The overall effects of tying on social welfare are ambiguous in the multi-homing
case. But if sM is sufficiently high so that every consumer purchases the tying product, social
welfare increases with tying.

1 > 1− ˜Ni∗

By the assumption A2, α1λ + α2λ− c1−t > 0. The last two terms, which stand for the difference
between the total transportation costs with tying and without tying, also have positive value because
1− Ni∗
1 for i = A, B. In addition, we already show that the expression in curly brackets is
positive. Hence, ∆W > 0 as 1− ˜N A∗
1 goes to zero. That is, tying is welfare-enhancing in
this model if the surplus of the product M is sufficiently high so that platform A can capture almost
whole market on the consumer side.

1 , or ˜nB∗

24

1.4 Conclusion

I analyze the effects of tying on R&D incentives as well as price competition between two com-
peting platforms in a two-sided market, especially with the possibility of multi-homing. Previous
literature has discovered the role of multi-homing in anti-trust issues regarding tying. As shown in
Choi (2010), it plays a role against tipping and the lock-in effects of tying, preventing the rival firm
from being excluded from the market. It further makes tying welfare-enhancing because more users
have an incentive to multi-home and hence have access to exclusive content in the multi- homing
equilibrium. In addition to welfare-enhancing effects of tying in static analysis, my analysis may
provide a new aspect of the role of multi-homing. If multi-homing is allowed, tying can be welfare-
enhancing even in a dynamic setting. The fact that tying increases the number of multi-homing
consumers leads to the increase in total demand of consumers for not only tying firm but also rival
firm. This implies that both platforms have greater incentives to innovate under tying because
they can spread out R&D costs over more units relative to no tying case. On top of that, more
aggressive R&D investment by one platform does not discourage the rival’s investment, as there is
no strategic effect of reducing rival’s incentives to invest in R&D in the multi-homing case. Hence,
my model shows that tying can stimulate the rival’s R&D incentives rather than stifle innovation,
so consumers do not miss an opportunity to have better products or services when multi-homing
is allowed. In order to highlight the importance of multi-homing, I analyze the effect of tying on
innovation in the single-homing equilibrium on the consumer side as well. In the single-homing
case, I show that tying has exactly the reverse effects; it has a chilling effect on innovation efforts by
the rival platform and hence reduces social welfare. This contrary result suggests that competition
authorities should consider the possibility of multi-homing on both sides when analyzing cases of
tying in two-sided markets.

One limitation of the model in this paper is the assumption on the number of exclusive content
providers. As in Choi (2010), I assume that the number of content providers, λ, is exogenously
given. If content providers are allowed to choose whether to provide their content exclusively, it
is likely that content providers have fewer incentives to provide exclusive content to the non-tying

25

platform under tying. Provided that the non-tying platform has less exclusive content available, the
number of multi-homers in turn decreases on the user side. If this is the case, incentives for the
non-tying platform to invest in R&D might be reduced with tying. As in Jeitschko and Tremblay
(2015), the existence of duplication costs for encoding additional content in order to multi-home
can allow for λ to be endogenous. This would be one possible extension of the model in this paper.

26

CHAPTER 2

INFORMATION SHARING AND R&D INCENTIVES

2.1 Introduction

In this paper, we study firms’ incentives to invest in cost reducing R&D when information
about R&D investment is shared. More specifically, we consider a situation in which firms
are uncertain about opponents’ R&D investment level and the outcome of R&D, i.e., reduced
production cost and private information about the realized cost can be exchanged. The exchange of
private information between competitors is a common practice in many industries. It is typically
implemented by common agencies such as industrial trade associations which collect, aggregate,
and disseminate data on behalf of their members. Another example of information exchange
is the mandatory disclosure requirements by regulatory agencies, such as the SEC and FASB.
Recently, regulatory agencies have increasingly emphasized disclosure requirements in addition to
the financial statements.1 In the US and Canada, for instance, publicly traded firms are required
to augment GAAP mandated disclosures, referred to as Management Discussion and Analysis
(MD&A). In MD&A, firms must discuss and analyze the results and trends of operations, including
cost information.2

We formally study a simple duopoly model with homogeneous products to address the issues
related to information sharing. Each firm first makes cost reducing R&D investment and its outcome
is stochastically realized. Before setting a price, each firm has private information regarding the
reduced production cost. That is, in the absence of information sharing, firms face ex ante
uncertainty and ex post information asymmetry about the result of R&D investment.

Interestingly, even when private cost information is not shared, firms may observe opponents’
R&D investment level, which can be a signal of the R&D outcome in many cases. For example,
1SEC undertakes a comprehensive review of disclosure requirements on public company filings
and recommend changes. MD&A is the most frequent area in the comment letter in 2014 and 2015.

2See Bryan (1997) for the details of MD&A requirement in the US.

27

the US Statement of Financial Accounting Standards No.2 (SFAS2) requires firms to reveal R&D
expenditure in its financial statement. However, in several European countries, such as Germany,
France, and Italy, public disclosure of R&D expenditure is not required.3 The observability of the
R&D investment level is likely to affect strategic incentives to invest and the subsequent market
outcomes. This situation leads to important questions about the effect of information sharing on
technological improvement and the role of the observability of the investment level.

To answer these questions, we compare firms’ investment incentives and the eventual market
outcomes under three different information structures: complete information, unobservable invest-
ment, and observable investment. In the complete information case, the outcome of cost reducing
investment is shared before firms choose prices. In the unobservable investment case, realized cost
information is concealed and the rival’s investment level is also unobservable. An intermediate
case, the observable investment case, concerns a situation where realized cost information is private
but firms can observe the rival’s investment level prior to the price competition. We find that both
investment incentives and expected market outcomes are identical in the complete information case
and the unobservable investment case. In contrast, investment in cost reduction is discouraged in
the observable investment case, in which both consumers and society may be hurt compared to
the other two cases. This implies that the effect of information sharing on investment incentives
crucially depends on the observability of opponents’ investment level. The intuition for this result
is as follows.

The exchange of cost information allows the most efficient firm to fully exploit the market power
by choosing its price at the marginal cost of the second most efficient firm, whereas the other firm
sets prices at its own marginal cost and earn zero profit. The cost uncertainty, however, moderates
this concentration and every firm in the market has positive expected profits. Despite this difference
of pricing between information structures, the firms’ expected payoff is identical in the equilibrium
for both cases as long as firms invest the same amount and the symmetry between firms is sustained.
3In these countries, government agencies or central banks collect data about R&D expenditures,
but such data is kept confidential by the institutions and is not easily accessed.(see Hall and Oriani
(2006))

28

This is analogous to the "Revenue Equivalence Theorem" in auction theory, which states that the
expected revenue (which equals the expected price in this paper) is the same regardless of whether
the auction is the second price or sealed-bid first price (which is in this paper interpreted as whether
or not cost information is private).4 The implication of the Revenue Equivalence Theorem for the
situation addressed here is that firms’ expected payoff boils down to the probability of winning the
market given c in winner-takes-all type markets. Although firms have private cost information, the
symmetric firms set the same price for each realized cost and like the complete information case,
the most efficient firm always captures the market and hence the probability of winning the market
given c is equalized in both cases. On top of that, if opponents’ investment level is unobservable,
there is no room for a strategic element of R&D expenditures because firms’ moves are not based
on opponents’ actual spending but their beliefs in the price competition stage. The equivalence
of the mechanism of winning the market is sustained and thus firms have an identical incentive to
invest in the complete information and the unobservable investment case.

However, the observability of opponents’ investment level introduces a strategic effect in firms’
equilibrium behavior. It induces firms’ pricing adaptations in the price competition stage following
opponents’ deviation in the R&D competition stage. From the comparative statics analysis, we find
that the more a firm invests in the first stage, the lower the price set by the rival in the following stage.
Although the effect of one firm’s additional investment on its own pricing strategy varies across
specifications of R&D technology, it turns out that under the new configuration of pricing strategies,
investment becomes less profitable, leading to the reduction in investment incentives.5 In other
words, cost reducing investment has a negative strategic effect, which results in the underinvestment
in the observable investment case.

Our results suggest different policy implications from those of the existing literature. Previous
studies such as Sakai (1986), Sakai and Yamato (1989), and Amir et al (2010) find that if firm
4On the revenue equivalence theorem, see, for example, Klemperer (1999) or Krishna (2002)

for the detailed explanations.

5In the baseline model and discrete type model in section 6.1, a investor chooses higher price
given c after making investment but in the oligopoly model, the changes in the investor’s pricing
strategy cannot be tractable.

29

specific cost information is private in a price competition, firms have an incentive to conceal the
information, which is beneficial to the social welfare compared to instances when firms exchange the
cost information. This result justifies a laissez-faire approach to the exchange of cost information
and our analysis obtains weakly same result when investment is unobservable.
If investment is
observable, however, firms’ private incentives to share information are not aligned with social
welfare. Firms are still willing to conceal the information while the exchange of information
increases social welfare.

The economic literature has extensively studied information sharing.6 Previous studies have
investigated firms’ incentives to share information and its welfare impacts. The key elements of
these analyses are the type of information shared (demand or cost), characteristics of information
(common value or private value), and the type of competition (price or quantity). Most early
works, such as Vives (1984) and Gal-Or (1985), investigate firms’ incentives to share common
value information in Cournot competition and show how the effect of asymmetric information
can vary across the type of information. Gal-Or (1986), Sakai (1986), and Raith (1996) address
cost information sharing in Bertrand competition, which is closely related with our paper. Gal-
Or(1986) and Sakai(1986) find that concealing information is a dominant strategy for firms if
each firm’s cost is independently drawn from the distribution, whereas Raith(1996) shows that
this result is not robust in terms of correlated costs. Sakai and Yamato (1990) and Amir et al
(2010) examine the welfare implications of information sharing and show that the exchange of cost
information in price competition reduces social welfare. Unlike our work, they analyze the effect
of asymmetric cost information with differentiated products. Further, the present paper introduces
cost-reducing investment prior to Bertrand competition so that the effect of information disclosure
can be examined not only with respect to the investment outcome but also the level of investment.
Thomas (1997) and Arozamena and Cantillon (2004) study about incentives to reduce produc-
tion costs in auctions and find that the firm has lower incentives to invest in cost reduction when the
investment level is observable. Similar to our results, they find that less incentives to invest in cost
6European Commission (1995) provides an overview of theoretical literature and competition

policies in various countries associated with information sharing.

30

reduction result from the firm’s anticipation that it will face fiercer price competition.7 However,
the analysis in Thomas and Arozamena and Cantillon assume that only one firm has an investment
opportunity at a time, whereas the present paper analyzes investment incentives when investment
is simultaneous.

Our paper is organized as follows: the next section describes the model; in Section 2.3, we
derive the investment equilibria and compare the investment incentives in each information struc-
ture; welfare implications are analyzed in Section 2.4; in Section 2.5, we modify R&D technology
in order to assess the robustness of our results; Section 2.6 discusses other concerns related to
observable investment, and Section 2.7 concludes the paper.

2.2 Model

There is a mass of homogenous consumers normalized to one, each of whom wishes to purchase
one unit of product from the market. Consumers have valuation v > ¯c for the product. Two firms,
labeled 1 and 2, produce homogeneous products. They compete in prices and interact only once
in the market. Firms have identical marginal costs of production ex ante8, denoted by ¯c, which
may be reduced through cost reducing R&D investment.9 Before competing in prices, the two
firms engage in R&D activities to lower costs. Cost reductions are stochastic. Specifically, if
firm i invests Ii, its marginal cost of production is uniformly distributed over [ ¯c − Ii, ¯c] and for
convenience, let me denote by fi the probability density and Fi the cumulative density, i ={1,2}.
That is, R&D investment can stretch out the lower bound of the support of distribution of marginal
7Interestingly, Thomas (1997) and Arozamena and Cantillon (2004) highlight the magnitude of
the negative strategic effect of investment by providing numerical examples: in some parameter
space, the firm which is allowed to invest may not want to reduce its cost even when the cost of
investment is zero.

8If firms are ex ante asymmetric, they may signal to the rival about their initial cost type through
R&D investment. Aoki and Reitman (1992) analyze the strategic use of R&D investment for
signaling.

9Because the ex ante marginal cost is assumed to be ¯c, firms would earn zero profit if they do not
engage in R&D investment. Therefore, in contrast to Thomas (1997) or Arozamena and Cantillon
(2004), disincentives of investment is not an issue in this model.

31

cost. We interpret the present model as representing the shift to a more reliable technology. If
I1 > I2, F1 first order stochastically dominates F2 and has a higher hazard rate than F2. In contrast
to much previous literature which covers information sharing about cost, such as Gal-Or(1986)
and Raith(1996), each firm’s cost is drawn independently. Because our focus is on the interaction
between R&D and pricing competition, not a common shock to the industry, we assume, for clarity
of exposition, that firms’ costs are not correlated. Additionally, the present model rules out any
spillover effect of technologies. The investment cost is Φ(I), where Φ(cid:48)(I) > 0, Φ(cid:48)(cid:48)(I) > 0, Φ(cid:48)(cid:48)(cid:48)(I) > 0,
Φ(cid:48)(0) = 0, and Φ(cid:48)( ¯c) = ∞.

The timing of the game is as follows.

In the first stage, firms simultaneously decide how
much to invest. The level of investment by the rival may be observable. After observing its own
cost realization, firms set prices simultaneously in the second stage. Lastly, consumers’ purchase
decisions are made by choosing the firm which offers the lower price. If cost information is public
in the second stage, a standard two stage Nash game would be played. If cost information is private,
a two stage Bayesian-Nash game would be played. That is, a noncooperative Nash game is played
in the R&D stage, given that firms calculate their expected profits from the second stage. In the
second stage, firms play a Bayesian-Nash game.

2.3 Basic Analysis

Three information cases are considered in this section: complete information, unobservable
investment, and observable investment. In the complete information case, firms commit to share cost
information at the outset of the game. We assume that the information will be truthfully revealed and
all information is publicly known.10 In the unobservable investment and the observable investment
case, the rival’s cost information is not shared and remains unknown. In the observable investment
case, however, firms can observe the rival’s investment level in the first stage. For each case, we
10In the present model, we assume away partial revelation as in Gal-Or (1986) and focus only on

full revelation for simplicity.

32

analyze how much firms choose to invest in the equilibrium. By comparing these three cases, we can
examine how the effect of cost uncertainty on investment incentives depends on the observability
of R&D investment.

2.3.1 Complete Information Case

When the rival’s marginal cost is public, firms play the standard Bertrand game ex post in the
second stage. That is, the lowest cost firm wins the market and sets its price at the rival’s marginal
cost. The other firm charges its marginal cost and earns zero profit. Thus, a firm can earn positive
profits by the difference in both firms’ costs, cj − ci, when it turns out that the firm is more efficient
in production than the rival. The profit maximization problem for firm i in the first stage is given
by

Pr[ci < cj]· E[cj − ci|ci < cj]− Φ(Ii).

max
Ii

Once a firm invests more than the rival, it is possible for the firm to have an exclusive level of
efficiency that the rival cannot beat. Thus, the shape of profit function depends on the relative level
of investment. Assume that I1 ≥ I2. The expected profit for firm 1 is

E ¯π1(I1) = Pr[c1 < c2]· E(c2 − c1|c1 < c2)− Φ(I1).

Substituting Pr[c1 < c2] = 1− I2
2I1

,11 firm 1’s expected payoff can be rewritten as

I2
2
6I1
Similarly, if I1 ≤ I2, we have firm 1’s expected profit as

E ¯π1(I1) =

2 I1 − 1

2 I2 +

1

− Φ(I1).

11The probability of winning the market for firm 1 can be calculated by

Þ ¯c

Þ z

¯c−I2

¯c−I1

Pr[c1 < c2] =

1− I2

I1
2I2

2I1

if
if

I1 ≥ I2
I1 ≤ I2

E π1(I1) =

− Φ(I1).

I2
1
6I2

dF1(x) dF2(z) =

33

E ¯πi(Ii) and E πi(Ii) show that the expected profit function is continuous and differentiable at all

Ii ∈ [0, ¯c]. The following Lemma describes firms’ best responses in the first stage.
Lemma 1. Define ¯Ii(Ij) as a function which implicitly solves 1
− Φ(cid:48)(Ii) = 0. Then, firm i’s best response, Θi(Ij) is given by
Ii
3Ij

2 − I2
j
6I2
i

− Φ(cid:48)(Ii) = 0 and Ii(Ij) solving



Θi(Ij) =

¯Ii(Ij)

if Ij ≤ Io

Ii(Ij)

if Ij ≥ Io

where Iosatisfies Φ(cid:48)(Io) = 1
3

Proof. See Appendix B.

(cid:3)

The firm i’s best response intersects with firm j’s best response at Ii = Ij = Io and in order to

ensure the stability condition at the symmetric equilibrium, we assume the following condition.
Assumption 1. Φ(cid:48)(cid:48)(Io) > 2
3Io

i(Ij)| < 1 and |I(cid:48)

The assumption is equivalent to | ¯I(cid:48)

i(Ij)| < 1 at I1 = I2 = Io. Although the
assumption above is implicitly determined by the shape of R&D cost function, it does not require
the tight condition. Suppose that, for instance, the R&D cost function is given by Φ(I) = kI2/( ¯c− I).

Then, Io is derived as ¯c(1−(cid:112)3k/(3k + 1)) and we can check that the condition above is satisfied

for all positive k. Also, from ∂2E ¯πi(Ii)/∂Ii∂Ij < 0 and ∂2E πi(Ii)/∂Ii∂Ij < 0, we can confirm that
firms’ investments in the R&D stage are strategic substitutes.

CI) = 1
3.

Proposition 2.1. If information about cost is shared, there exists a symmetric equilibrium I1 = I2 =
CI such that Φ(cid:48)(I∗
I∗
Proof. Without loss of generality, assume I1 > I2. The stability condition guarantees that ¯I1 is
steeper than I2 at I1 = I2 = Io. Given this, the uniqueness of the equilibrium can be shown by
looking at the changes in slope of best responses. The changes in the slope of firm 1’s best
2/I1
response can be obtained as ¯I(cid:48)(cid:48)
2/I1)2 < 0 and for firm 2, it can be derived as

1 Φ(cid:48)(cid:48)(I1)+I2
1 (I2) = − 3I2
1 Φ(cid:48)(cid:48)(I1)−I2
(3I2

34

1 Φ(cid:48)(cid:48)(I2)−1)
2I2(9I2
I(cid:48)(cid:48)
2(I1) =
1 Φ(cid:48)(cid:48)(I2)−I1)2 > 0. In other words, firm 1’s best response gets steeper as I2 decreases
(3I3
and firm 2’s best response becomes flatter as I1 increases. Therefore, the two firms’ best responses
do not interact in the area of I1 > I2. We can make a similar argument in the area of I1 < I2 by
imposing symmetry of the two firms’ reaction functions.

The symmetric equilibrium in the R&D stage is given by I1 = I2 = I∗

CI where I∗

defined by Φ(cid:48)(I∗

CI) = 1
3.

CI is uniquely
(cid:3)

2.3.2 Unobservable Investment Case

We next consider the unobservable investment case where information about costs is not disclosed
and the rival’s investment is unobservable. Firm i invests in cost-reducing R&D and privately
observes the realized production cost which is drawn from the uniform distribution on [ ¯c− Ii, ¯c].
The more a firm invests in the first stage, the greater the chance that a low production cost is
realized, leading to a high probability of capturing the market. The information about cost is
private, but it is commonly known that c1 and c2 are independently and uniformly distributed
on [ ¯c − I1, ¯c] and [ ¯c − I2, ¯c], respectively. The equilibrium of the two stage game consists of an
R&D investment strategy profile {I1, I2} and a pricing strategy profile {P1(c1), P2(c2)}, where
{I1, I2} is a noncooperative Nash equilibrium in the first stage and {P1(c1), P2(c2)} is a Bayes-Nash
equilibrium in the second stage. On the equilibrium path, Pi(c) is determined by the firm’s beliefs
about the equilibrium investment strategy, ˆIi, for i = 1,2. For ease of exposition, we define φi(p)
the inverse function of firm i’s equilibrium pricing strategy and ϕji(c) := φ j(Pi(c)) the composite
function of Pi and φ j. That is, ϕji(c) is firm j’s realized cost when the two firms’ prices are the
same, given that firm i’s cost is c. Plum(1992) shows that the equilibrium pricing strategy Pi for
i = 1, 2 is continuous and strictly monotonically increasing on [ ¯c− Ii, ¯c] so the functions φi(p) and
ϕji(c) can be defined and are continuous, strictly monotonically increasing.

Before proceeding to the equilibrium analysis, we first examine the second stage subgame where
firms compete with price based on the private cost information and beliefs about the rival’s price.

35

In the second stage, Pi(ci) solves

(cid:0)p− ci

(cid:1)(cid:0)1− Fj(φ j(p))(cid:1)

Max

p

for each ci ∈ [ ¯c− Ii, ¯c].

Taking logarithm and differentiating with respect to p, we obtain

F(cid:48)
j(φ j(p))
(cid:48)
j(p) =
1− Fj(φ j(p)) φ

1

p− φi(p) .

(2.1)

The solution to the system of differential equations gives each firm’s equilibrium pricing
strategy but in general, the system of differential equations above does not have an explicit solution.
However, the cost distributions in the present model comes from a special class of distributions
which allows for closed form solutions to the system in (2.1).

Lemma 2. Given the beliefs on rival’s investment ˆIi and ˆIj, firm i’s equilibrium price strategy in
the second stage is

Pi(c) = ¯c−

Proof. See Appendix B.

1 +(cid:114)1−( 1

¯c− c
− 1
ˆI2
j

ˆI2
i

)( ¯c− c)2

(2.2)

(cid:3)

Equation (2.2) gives P1( ¯c − I1) = P2( ¯c − I2) = ¯c − I1I2
I1+I2

≡ p and P1( ¯c) = P2( ¯c) = ¯c, which
means that a firm with the highest value of cost cannot expect positive profits. Given the pricing
equilibrium {P1(c), P2(c)}, firm i’s ex ante expected payoff is calculated as

E πi(Pi, c) = −

Fi(c) d
dc

(Pi(c)− c)[1− Fj(ϕji(c))]dc− Φ(Ii).

Using the Envelope Theorem, we obtain

(Pi(c)− c)[1− Fj(ϕji(c))] = −[1− Fj(ϕji(c))].

d
dc

(2.3)

36

E πi(Pi, c) =

(Pi(c)− c)[1− Fj(ϕji(c))]dFi(c)− Φ(Ii),

which, after integration by parts, can be rewritten as

¯c−Ii

Þ ¯c
Þ ¯c

¯c−Ii

Substituting this expression into the integral, firm i’s profit maximization problem can be written

Þ ¯c

¯c−Ii

as

max
Ii

Fi(c)[1− Fj(ϕji(c))]dc− Φ(Ii).

(2.4)

Proposition 2.2. Suppose that the rival’s investment is unobservable. If information about cost
is not disclosed, there exist a symmetric equilibrium, where i) firms invest I1 = I2 = I∗
UI in the
first stage and ii) use the same pricing strategy P(c) = c+ ¯c
UI is
2
identical to I∗
CI.

in the second stage. Moreover, I∗

Proof. See Appendix B.

(cid:3)

The incentives to invest in the complete information and the unobservable investment case are
identical because the expected payoffs in the equilibrium are the same in both cases. The reason for
the same expected payoff in both cases is related to the Revenue Equivalence Theorem in auctions.
Although our model considers not only of winner-takes-all type price competition, which is similar
to an auction, but also the interaction of the competition with cost reducing R&D investment,
the principle of the Revenue Equivalence Theorem is helpful to explain our result. The Revenue
Equivalence Theorem states that, under the assumptions of a risk-neutral bidder, and a privately
known type which is independently drawn from a distribution, expected payoffs are determined
by a mechanism of winning the object and the expected payoff for a bidder who has the lowest
feasible type regardless of the type of auctions. That is, the expected payoff depends on the winning
probability of the market. To see this, refer to (3), which shows that changes in profits at c as
cost increases are equal to the winning probability at c. Thus, the profit at c is the sum of the
winning probability from c to ¯c.12 Proposition 2 implies that the effect of additional investment on
the winning probability is identical in both cases. The reasons for this are two-fold: first, in the
equilibrium, firms invest symmetrically so that costs are drawn from identical distribution function
in both cases; second, given that the cost distributions are symmetric, firms use the same pricing
12The expected profit in firm i’s profit maximization problem in (4) is indeed equivalent to the

expected value of the sum of the winning probability from each feasible c to ¯c.

37

strategy for all c in the unobservable investment case. If firms’ prices are the same for each c, a
firm that has more efficient production technology wins the market as in the complete information
case.
Investment cannot affect this equilibrium pricing strategy because when competing with
price, firms cannot adjust to the rival’s actual investment level but instead choose a pricing strategy
based on their belief about the rival’s investment. Therefore, in this case, cost uncertainty does not
diminish the efficiency of the investment relative to the complete information case.

We show that if firms are ex ante symmetric, the information structure about the investment
outcome does not affect investment incentives. However, this result relies on the assumption in the
unobservable investment case that the investment level is unobservable. If the investment level by
the rival can be observed prior to the price competition, this equivalence of investment incentives
may no longer hold. We will analyze this case in the next subsection.

2.3.3 Observable Investment Case

When cost information is private, cost reducing investment prior to the price competition has
two effects in the present model - a direct effect and an indirect effect. The investment directly
improves the investor’s cost distribution so that the investor can obtain the advantage in production
efficiency over the rival. We have shown that the complete information case and unobservable
investment case have the same direct effect. If the investment level is observable, however, the
investment forces firms to modify pricing in accordance with the new cost distribution. We will
show that investment has a negative indirect effect, or strategic effect, in the observable investment
case which results in less incentive to invest in cost reduction. This result may not be surprising
because investment acts as a "tough" commitment to the rival by allowing the investor to have a
higher chance of getting a low cost. With strategic complementarity of price competition, one may
anticipate that investment makes the competition more fierce and harms expected profits. However,
the following example shows that pricing strategies are not strategic complements when the rival’s
cost information is unknown. Suppose that two firms compete in price for a homogeneous product.
The information about cost is private, but it is commonly known that each firm’s cost is drawn from

38

p

p

b

˜P1(c1) = a+bci

for i = {1,2}. Suppose that firm 1 changes its pricing strategy:

a uniform distribution on [0,1]. Under the symmetry, the equilibrium pricing strategy for each firm
is Pi(ci) = 1+ci
a+b for
2
all c ∈ [0,1], where a > 0, b > 0, and a (cid:44) b. Note that if a > b, ˜P1(c) > P1(c) for all c ∈ [0,1] and
(p− c)(1− (a+b)p−a
).
(p− c)(1− F1( ˜φ1(p))), or Max
vice versa. Then, firm 2 reacts by solving Max
The new best response for firm 2 is still P2(c) = 1+ci
2 . Regardless of whether firm 1 changes pricing
strategy more aggressively or less aggressively, firm 2’s response does not change. We can also
construct other examples in which a firm reacts either more aggressively or less aggressively against
a change in the rival’s behavior. This is because firms do not care about the expected absolute
level of a rival’s price, but instead, it is the responsiveness of winning probability to a change in
price that matters. Thus, it is still worthwhile to analyze how observable cost-reducing investment
interacts with the price competition.

In the observable investment case, the nature of equilibrium pricing strategy is similar to
the one in the unobservable investment case discussed in Lemma 2, but here, firms can adjust
pricing strategies based on the observed investment by the opponent after the first stage. Thus, the
equilibrium pricing strategy in Lemma 2 will be used by substituting ˆI with I. Note that firm i’s
pricing strategy depends on not only its realized cost and the rival’s investment level, but also its
own investment level. When cost information is private, firms which play Bayes-Nash game set
prices by considering the rival’s equilibrium pricing strategy. Because each firm’s cost distribution
can affect how the rival will compete in prices, Pi(c) depends on Ii.

The equilibrium pricing strategy in (2.2) shows that for all c in the common support of two
firms, firm i’s price increases, whereas firm j’s price decreases in firm i’s investment level. This
result may seem counterintuitive to the nature of price competition. It is not surprising that firm
j’s responds to the rival’s more aggressive investment by lowering its price. However, firm i’s
reaction is to set a less aggressive price not merely relative to the rival, but in absolute terms.
As seen in the example above, the pricing strategies in the incomplete information case are not
generally strategic substitutes and are determined by how a new pricing strategy by the rival makes
the winning probability responsive to price changes. To see this, refer back to (2.1). Given that c is

39

P(c)

P1(c)

P2(c)

Figure 2.1: Pricing Strategy in Observed Investment Case

c

uniformly distributed, (2.1) can be rewritten as φ(cid:48)
j(p)/( ¯c− φ j(p)) = 1/(p− φi(p)), which shows that
changes in firm i’s pricing strategy after investment depends on how investment affects the hazard
rate of firm j’s pricing. Due to the fact that the directions of the two firms’ responses are different,
the effect of the observability of investments on the R&D competition is not easily predictable
when firms invest simultaneously.

The adjustment to the rival’s deviation in the first stage is reflected through the term ϕji(c) in
(2.3). Given the pricing equilibrium (P1(c|I1, I2), P2(c|I1, I2)), the first order condition with respect
to I1 in the first stage can be obtained as

Φ(cid:48)(I1) =

=

Þ ¯c
Þ ¯c

¯c−I1

¯c−I1

∂(cid:8)F1(c)[1− F2(ϕ21(c|I1, I2))](cid:9)
(cid:104) ∂F1(c)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

[1− F2(ϕ21(c|I1, I2))]

∂I1

∂I1

direct effect

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

= 0

(cid:105)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
+ F1(c) ∂[1− F2(ϕ21(c|I1, I2))]

∂I1
indirect effect

dc−(−1) F1( ¯c− I1)

[1− F2(ϕ21( ¯c− I1|I1, I2))]

dc

(2.5)

The first term and second term of RHS in (2.5) represents the direct effect and indirect effect,
respectively, of additional investment on the investor’s profit. In this subsection, we focus primarily
on the symmetric equilibrium, where I1 = I2 and P1(c) = P2(c).

40

Proposition 2.3. Suppose that the rival’s investment is observable. If information about cost is
not disclosed, both firms choose to invest I1 = I2 = I∗
OI is
less than the equilibrium investment level in the complete information and unobservable investment
cases.

OI in the symmetric equilibrium, where I∗

Proof. First, we determine whether the second order condition for the firms’ maximization problem
is satisfied. By differentiating the profit function twice in terms of I1, we have
· B dc− Φ(cid:48)(cid:48)(I1)

Þ ¯c

= −

2 +( ¯c− c)2)
( ¯c− c)2(I2
√
I5
1 I3
2

5
A

¯c−I1

∂2E π1
∂I12

where

− 1
I2
2

)( ¯c− c)2

2 + 2(I2

A = 1−( 1
I2
1

and B = 2I2

1 −( ¯c− c)2)− 3( ¯c− c)(I1 −( ¯c− c))

1( ¯c− c)2/I2
Because I1 ≥ ¯c − c for all c ∈ [ ¯c − I1, ¯c], 2(I2
1 −( ¯c − c)2) is greater than 3( ¯c − c)(I1 −( ¯c − c)), and
hence the second order condition is satisfied. It can be shown that R&D investments are strategic
substitutes by looking at the sign of ∂2E π1/∂I1∂I2.13 Furthermore, the slope of the best response
is less than 1 at I1 = I2, implying that the stability condition of the symmetric equilibrium holds.14.

By imposing symmetry, we obtain from (2.5)
− 1
I∗
OI

( ¯c− c)( ¯c− c)

OI) =

Φ(cid:48)(I∗

I∗3
OI

Þ ¯c

¯c−I∗
OI
− 1
20

1
3

=

(1− ¯c− c
I∗
OI

) ∂ϕ21(c|I1, I2)

∂I1

|I1=I2=I∗

OI

dc

(2.6)

=

17
60

Þ ¯c

Because the R&D cost function,Φ(I), is assumed to be convex, I∗
13The term ∂2E π1/∂I1∂I2 is given by

OI is less than I∗

CI or I∗

UI. (cid:3)

∂2E π1
∂I1∂I2

=

( ¯c− c)2(3( ¯c− c)3
√

I3
1 I4
2

A

+

1
√
I2
1 I2
2

A

)( ¯c− c
I1

− 1)dc

¯c−I1

Because ¯c−c
I1
14The slope of best response, |Θ(cid:48)

− 1 is negative for all c ∈ [ ¯c− I1, ¯c], ∂2E π1/∂I1∂I2 < 0.
13/84I∗
OI
OI +Φ(cid:48)(cid:48)(I∗

OI)| = | ∂2E π1
1(I∗
∂I1∂I2

/ ∂2E π1
∂I12 | =

71/84I∗

OI) < 1

41

The proposition states that the strategic effect of investment is negative in the symmetric
equilibrium. From equation (2.2), we know that the investor’s price increases in investment,
whereas the rival’s price decreases in investment for all c in the common support. The strategic
effect of investment through firms’ pricing may make the investor’s markup rise but, as mentioned
in the previous subsection, the expected payoff boils down to the probability of winning the market.
Obviously, the probability of winning the market for the investor is lower than the rival in the
common support. Moreover, unlike the situation in the complete information and unobservable
investment cases, the investor cannot capture the market for sure in the stretched support because
the lowest prices of their equilibrium strategies are the same.

The effect of the observability of investment information is reminiscent of the influence of
the observability of contract information in vertical delegations. Fershtman and Judd (1987) and
Bonanno and Vickers (1988) found that by delegating downstream pricing, firms can establish a
Stackelberg leadership advantage vis-a-vis their competitors, which can lead to higher prices and
profits than that present without vertical delegation. Katz (1991) and Coughlan and Wernerfelt
(1989) then responded, effectively saying that this result critically depends on the level of informa-
tion that competitors have about vertical contracts among their competitors. If competitors don’t
have any contract information about the cost and structure of how upstream divisions sell to down-
stream divisions, then the "Stackelberg advantage" results of Fershtman and Judd and Bonanno and
Vickers are not robust.

2.4 Welfare Analysis

In the previous section, incentives to innovate were analyzed. These can be ranked as follows:

I∗
CI = I∗

UI > I∗

OI

If all agents have the same valuation for the product, social welfare is a function of the minimum
actual cost, corresponding to the investment level. As shown above, whereas the investment level
in the complete information and unobservable investment cases are the same, the observable invest-

42

ment case induces underinvestment. This implies that social welfare decreases in the observable
investment case.

WCI = WUI > WOI

We next analyze how consumer surplus may vary across information regimes. Because demand
is assumed to be inelastic, it is sufficient to derive the expected price that consumers are charged in
order to calculate consumer surplus.

Lemma 3. If cost information is public, the expected price that consumers are charged is given by

E(p) =

F2(c) + cF(cid:48)

Þ ¯c
(cid:104)
2(c)− F1(c)F2(c)(cid:105)
2(c) + F1(ϕ12(c))(cid:8)ϕ12(c)F(cid:48)

¯c−I2

2(c)(cid:9)(cid:105)

2(c)− F2(c)− cF(cid:48)

(cid:104)

Þ ¯c

¯c−I2

E(p) =

F2(c) + cF(cid:48)

If cost information is private, the expected price that consumers are charged is given by

dc

(2.7)

dc

(2.8)

(cid:3)

(2.9)

(2.10)

Proof. See Appendix B.

At the symmetric equilibrium, (2.7) can be rewritten as

(cid:104)
(cid:104)

Þ ¯c
Þ ¯c

¯c−I∗

¯c−I∗

E(p∗) =

F2(c) + cF(cid:48)

E(p∗) =

F2(c) + cF(cid:48)

2(c)− F1(c)F2(c)(cid:105)
2(c)− F1(c)F2(c)(cid:105)

dc = ¯c− I∗
3

dc = ¯c− I∗
3

And because ϕ12(c) = c in the symmetric equilibrium, (2.8) becomes

= I∗

Equations (2.9) and (2.10) show that if firms are symmetric, expected prices depend on in-
vestment. Because I∗
UI, the expected prices in the complete information and unobservable
CI
investment cases are identical, implying that consumer surplus in both cases is the same as well.
Due to underinvestment, however, the expected price in the observable investment case is greater
than in the other two cases, leading to higher profits but lower consumer surplus.

E πCI = E πUI < E πOI

CSCI = CSUI > CSOI

43

Proposition 2.4. In the complete information and unobservable investment cases, total welfare is
identical. In the observable investment case, each firm’s expected profit increases but both social
welfare and consumer surplus decrease.

Proposition 2.4 contrasts the implication of asymmetric information about costs and investment.
The asymmetric information about costs does not affect competition at all in this environment. Of
course, it should be kept in mind that this implication results from the features of the model
of inelastic demand and independently drawn costs. Nonetheless, this strong result about the
implication of cost uncertainty highlights the impact of observability of investment. Even in
an environment where asymmetric information about costs plays no role in competition, welfare
implications can vary depending on whether information about investment is revealed.

This is an important point. As noted above, previous research dealing with cost information
sharing concluded that a firm’s private incentive to share the information is aligned with social
welfare. Specifically, if firms compete in prices, cost information sharing harms social welfare,
but firms also do not have an incentive to share the information about costs. This implies that
there is no reason to forbid cost information sharing if it is not part of a collusive agreement.
However, our result suggests that an additional criterion – observability of investment – should be
considered when the anti-competitiveness of cost information sharing practices in industries where
R&D activities for reducing costs play a significant role in competition.

2.5 Robustness

In the baseline model, we analyze cost reducing R&D incentives and other market outcomes
under each information structure. Yet, notwithstanding the attractive feature of the baseline model
that it yields a closed-form solution to the system of differential equations in (1), the model has
some limitations:
its applicability is limited to duopoly competition between symmetric firms
with specific R&D technology.
In this section, we extend the baseline model to another R&D
specification so that we can check the robustness of our main results.

44

Suppose that n firms compete with prices in the market and they have ex ante the same level
of efficiency in production, i.e, ci = ¯c for ∀i. To analyze the oligopoly competition, we modify the
R&D technology as follows. Suppose that every firm has the same support of the cost distribution,
say [c, ¯c] after making investment. For convenience, define ∆ := ¯c− c. The cost distribution after
R&D investment is assumed to be

G(c; I) = 1−(cid:104)

(cid:105) I

1− c

∆

(2.11)

This is a common specification of modeling R&D in previous literature. In this specification, I
can be interpreted as the number of R&D activities and c is the lowest number among independently
drawn R&D results. The result of investment is bounded at a certain level, but the cost distribution
of a firm with greater I first-order stochastically dominates rivals’ distributions. All other settings,
including the assumptions on R&D cost function, the timing of the game and the equilibrium
concepts in each case, are the same as in the baseline model. The following proposition presents
the comparison of investment incentives in each case.15

Proposition 2.5. In oligopoly competition, there exists a symmetric equilibrium in the complete
information and unobservable investment cases. The symmetric investment equilibrium in the
complete information and unobservable investment cases are identical. However, in the symmetric
investment equilibrium in the observable investment case, firms invest less compared to complete
information and unobservable investment case.

As with the baseline model, if investment is unobservable, firms choose to invest symmetrically
in the unobservable investment case and hence the equivalence between the complete information
case and unobservable case holds. Observable investment also induces underinvestment in the
equilibrium, implying that investment has a negative strategic effect. Unlike the baseline model,
however, the direction of strategic effect can clearly be derived. Recall Lemma 2 in the previous
section: if a firm engages in additional investment, it always makes the rival price lower, whereas

15The detailed derivation is in Appendix B.

45

the investor prices less aggressively given any c in the common support. The effect of investment
on each firm’s pricing strategy is not tractable here. But what we do know from the proposition
above is that the marginal benefit of investment is reduced due to the adjustment of firms’ pricing
strategies to investment. In other words, if a firm invests more in the first stage, it will have a lower
chance to win the market. This shows the robustness of our main result and yields the same welfare
implications as in the baseline model.

Proposition 2.6. In oligopoly competition, the complete information case and unobservable invest-
ment case implement the socially optimum level of total welfare. In the observable investment case,
however, both social welfare and consumer surplus decrease compared to the complete information
and unobservable investment case.

2.6 Discussion

2.6.1 Asymmetric Equilibria

In previous sections, we have focused on the symmetric equilibrium in the analysis. Indeed, the
complete information and unobservable investment case in both models have a unique symmetric
equilibrium16 and as mentioned in section 2.3.2, this is the crucial reason why market outcomes
are identical in those cases. However, considering the nature of a negative strategic effect, we
can anticipate that the post-investment symmetry between firms can be broken when investment
is observable. Specifically, firms may have an incentive to deviate to a lower level of investment
because, compared to the unobservable investment case, a strategic effect allows the firm to have a
relatively higher chance of winning the market. This explanation can be confirmed by modifying
R&D technology as follows.

16The uniqueness of the symmetric equilibrium is sustained even under a wider class of cost

distributions than the one used in our model.

46

K(p)

K2(p)

K1(p)

p

Figure 2.2: Mixed Pricing Strategy in Observed Investment Case

As is the case in the baseline model, firm 1 and 2 compete with price for a homogeneous
product. Two firms are initially symmetric, that is, ci = ¯c for i = {1, 2}. Consider the following
simple R&D technology. In the R&D stage, a firm can engage in R&D activities to reduce costs
to the lower level, c. Specifically, if a firm invests I ∈ [0, 1], its marginal costs can be lowered to c
with probability of I. Let’s denote a firm’s monopoly profits as ˆπ and assume that the R&D cost
function is Φ(I) = k
2 I2, where k represents the R&D cost parameter. The detailed analysis can be
found in appendix B.17

In the complete information and unobservable investment cases, there exists a unique symmetric
equilibrium for investment and they are identical. Furthermore, they achieve the socially optimal
level as the baseline model. The interesting case is the observable investment case. In the discrete
type case, firms face two types of payoff functions in the price competition stage, depending on
whether they invest more or less than the rival. This induces two strategies for firms in the R&D
stage: the aim to invest more than the rival or, inversely, less than the rival. In the equilibrium,
one firm chooses the former strategy, while the other firm chooses the latter, implying that only
asymmetric equilibria exist in the R&D stage.

As noted above, this is due to a strategic effect of investment. Suppose that I1 > I2. When
17The implications for private incentives of information sharing and investment are present in

the case where firms face a general form of downward sloping demand.

47

cost information is private, pure pricing equilibrium does not exist and firms use mixed strategies
in the equilibrium. Unobservable investment cannot break the symmetry between firms, and firms
therefore use the identical pricing strategy. If firms can observe the rival’s investment level, however,
firm 2 uses the same mixed strategy, K2(p) as in the unobservable investment case, whereas firm 1
modifies the pricing strategy, K1(p) by putting more weight on high prices. Specifically, there exists
a mass point at the highest price in order to prevent more fierce competition as shown in Figure 2.
It can be interpreted as a mixed strategy version of the strategic effect of investment. The strategic
effect enables firm 2 to deviate from the symmetric equilibrium in the unobservable investment
case. Due to this strategic effect, firm 2 can earn more profit at the lower level of investment
because the reduction in the probability of winning the market is lower than in an environment
without the strategic effect. Coupled with the savings in R&D cost, asymmetric equilibria arises
in the observed investment case. This is another possible effect of observable investment.

2.6.2 Asymmetric Firms

Up until now, we have assumed that the costs of firms are drawn from the identical distribution.
If we focus on price competition between symmetric firms without the consideration of cost
reducing investment, information disclosure plays no role in the competition and expected value of
market outcomes are identical regardless of information structures. However, if firms’ production
efficiencies are ex ante heterogeneous, information disclosure could induce additional effects on
market outcomes. In order to explore this possibility, we will study an asymmetric version of the
baseline model, using numerical methods.

The model we lay out in this subsection is the same as the baseline model, except for firms’
initial production technologies. Specifically, before engaging in cost reducing investment, firm i’s
cost is drawn from a uniform distribution on [ ¯c− δi, ¯c]. If δ1 > δ2, firm 1 is more efficient from an
ex ante perspective. For convenience, define δi + Ii := ˜δi. Then, ¯c− ˜δi is the lower bound of firm
i’s cost distribution after the investment. The c.d.f of firm i’s cost after the investment are denoted
˜Fi(c).

48

Investment

Price

1.5

IOI
ICI

4.6

pOI
pCI

I

1.3

1.1

p

4.4

4.2

1.2

1.4

1.6

δ2

1.8

2

1.2

1.4

1.6

δ2

1.8

2

Figure 2.3: Investment Level and Price Level in Competition between Asymmetric Firms

Production Inefficiency

Aggregate Profits

Prob c2 is lower
Prob p2 is lower

0.8

πOI
πCI

π

0.6

0.4

I

0.8

0.7

0.6

0.5

1.2

1.4

1.6

δ2

1.8

2

1.2

1.4

1.6

δ2

1.8

2

Figure 2.4: Production Inefficiency and Profits in Competition between Asymmetric Firms

The model is analyzed by using numerical methods, assuming v = 6, ¯c = 5, and Φ(I) = I2/( ¯c− I).
The results show changes in outcomes when firm 1’s efficiency is increased from δ1 = 1.1 to δ1 = 2,
whereas firm 2’s efficiency is fixed at δ2 = 1. We compare results under the complete information
and observable investment cases so that we can confirm whether the anti-competitiveness of
incomplete cost information can be carried over to the asymmetric firms’ case. The details can be
found in appendix B.

Figure 2.3 shows asymmetric firms’ investment level and price level. The solid curves represent
the aggregate level, whereas dotted and dot-dashed curves graph the individual level in the observ-

49

Consumer Surplus

Total Welfare

1.8

CSOI
CSCI

S
C

1.6

1.4

WOI
WCI

W

2.4

2.2

2

1.8

1.2

1.4

1.6

δ2

1.8

2

1.2

1.4

1.6

δ2

1.8

2

Figure 2.5: Consumer Surplus and Total Welfare in Competition between Asymmetric Firms

able investment case and complete information case, respectively. As in the symmetric firm case,
complete information provides the socially optimal level of investment incentives to both firms, but
the aggregate level of investment in the observable investment case is consistently lower than in the
complete information case. This result mainly comes from the underinvestment by firm 1 which is
ex ante more efficient.

When cost information is public, the expected prices of firm 1 and firm 2 are the same, but they
are different in the observable investment case. At any initial degree of asymmetry, firm 1’s price
is lower than firm 2’s, implying that firm 1 sets the price more aggressively in a stochastic sense,
although firm 2 sets the price lower at any given c in the common support.

Figure 2.4 graphs the probabilities that firm 1 has a lower price and lower cost in the observable
investment case and aggregate profits. The fact that those curves are not identical indicates that the
market is sometimes served by a firm which has a higher cost. This is because firm 2 prices more
aggressively than the rival at ∀c ∈ [ ¯c− ˜δ2, ¯c]. Thus, the difference between the two curves in the
first figure in figure 2.4 represents the product inefficiency due to incomplete information.

Figures 2.5 compares consumer suplus and welfare in the observable investment case and
complete information case. The second figure in Figure 2.5 shows that the information disclosure
has a positive effect on total welfare, which is consistent with the symmetric case. However, our

50

main result about consumer surplus and industry profits is not sustained under the assumption of ex
ante asymmetry. In the asymmetric firm case, there are two effects of cost uncertainty in addition
to the underinvestment. On the one hand, if cost information is private, it is possible for a firm with
higher cost to win the market in the asymmetric case, as shown in figure 2.4. Taken toghther with
the underinvestment as in the symmetric case, this possibility induces additional welfare loss. On
the other hand, the more efficient firm cannot fully exploit the market power and set a price less
than the rival’s cost in the observable investment case because the advantage on efficiency does not
guarantee capture of the market. This effect becomes larger as the degree of asymmetry is greater,
and most surplus is taken by the more efficient firm as shown in figure 2.4.

The result implies that there is no clear cut policy implications when firms are ex ante asym-
metric. When firms are somewhat symmetric, our main results are robust. That is, when the degree
of asymmetry between firms is small, information disclosure increases both consumer surplus and
social welfare. When one firm is much stronger than the rival, however, information disclosure may
hurt consumers, whereas it remains beneficial to society. Therefore, judgments about information
disclosure depends on whether authorities give greater priority to consumer or society when the
difference in initial technology between firms is huge.

2.7 Conclusion

This paper explores the effect of information sharing practices on R&D investment incentives
and implications of observability of the investment level. Although the assumption of investment
observability is appropriate in many situations, our understanding of its effect has been largely
limited to date.

In sections 2.3 and 2.4, we analyzed a pure Bertrand duopoly model in a situation where firms
have private information about the stochastic R&D outcome and showed that market outcomes are
identical when the investment level is unobservable regardless of whether or not cost information
If, however, firms can observe the opponents’ investment level, we show that this
is private.

51

observability of investment distorts firms’ pricing strategy and it induces a negative strategic effect
when cost information is not shared. Firms in turn invest less than in the complete information case,
leading to lower consumer surplus and total welfare. We found in section 2.5 that this effect is robust
in another type of R&D technology and oligopoly competition. However, under the assumption
of asymmetry between firms prior to R&D competition, our results about industry profits and
consumer surplus are reversed when the production efficiency of one firm initially dominates the
rival’s.

In this paper, we constructed a simple model with inelastic demand on homogeneous product
which induces identical market outcomes when investment level is not observable. However,
this result can be altered if the model embraces downward-sloping demand and/or horizontal
differentiated product. It would be interesting to investigate how market outcomes are changed
under each information structure when assumptions on demand and product type are relaxed.
Nevertheless, the present paper formalizes the mechanism through which the observability of
R&D investment introduces strategic incentives to invest in a situation where firms face ex ante
uncertainty and ex post information asymmetry about the R&D outcome. It also highlights what
the crucial elements to consider in evaluating the welfare effects of information sharing are.
Furthermore, beyond the observability of the investment level, the degree of ex ante asymmetry
between competitors should be an important criterion.

52

CHAPTER 3

THE GRANDFATHER OF PRICE DISCRIMINATION

3.1 Introduction

Grandfather clauses substantially fall into two categories:

A grandfather clause exempts certain groups of people from a wider change in circumstances.
Historically, grandfather clauses were found in a variety of legal contexts.1 However, as we discuss
in greater detail below, grandfather clauses have become common in retail, particularly in online and
telecommunications services. In retail, grandfather clauses typically exempt existing subscribers
of a service from a price hike or allow them to continue consuming a discontinued service. Thus,
grandfather clauses permit firms to discriminate between early adopters and new consumers of their
service, albeit at a cost: discrimination prevents the firm from pricing optimally to early adopters.
(i) clauses where grandfathered
customers are permitted to continue to pay the same price that they paid in the past, possibly for
a product generally viewed as an improvement, whereas new customers are forced to pay a higher
price, and (ii) clauses where grandfathered customers end up consuming a distinctly different
product from that made available to new consumers.
In this manuscript we focus primarily on
the profit and welfare implications of the former category with the aim of understanding when
businesses might be able to profitably rely on grandfather clauses and whether doing so makes
consumers better off. The advantage of focusing on category (i) is that it enables us to hone in on
the direct ramifications of offering a grandfather clause rather than on potential consumer switching
behavior between products produced by the same firm.2 Our model of grandfather clauses is
1For instance, at the turn of the twentieth century, grandfather clauses were used to circumvent
the fifteenth amendment. The grandfather clauses exempted poor whites from poll taxes and literacy
requirements used to disenfranchise Southern blacks (Schmidt 1982). Grandfather laws were also
occasionally used following state increases in the legal minimum drinking ages to exempt those
already permitted to drink from the change in law (Williams et al. 1983).

2From a modeling perspective, category (i) only differs from category (ii) insomuch that grand-
fathered consumers might opt to purchase a product that they value differently (the product made

53

motivated by the seminal qualitative choice frameworks of McFadden (1974) and Perloff and Salop
(1985). This modeling framework is apt for the analysis of grandfather clauses, which are generally
found in product categories that can reasonably be categorized by oligopolistic competition and
discrete choice: for instance, the markets for mobile wireless service, health insurance, and certain
on-line content. To make the application of grandfather clauses meaningful within our framework,
we allow products and consumer tastes to evolve over time. Thus, a consumer who subscribed
to one service in the past might choose a different service today, especially if faced with a higher
relative price. In this case, grandfather clauses have bite: they may allow firms to retain existing
customers while raising prices for product variants generally viewed as improvements over previous
versions.

We find that when consumers are fully cognizant of their valuations for available product
alternatives, a firm would prefer to offer all of its potential consumers a uniform price rather than
price discriminate via grandfather clauses. In our setting, we suppose that individual consumers’
idiosyncratic preferences for products are independent over time. In such a setting, a firm faces
the same maximization problem for both, its fully informed existing customers, and potential, but
likewise fully informed consumers and should therefore charge both groups of customers the same
price. Thus, a grandfather clause, which is by design intended to allow a firm to charge different
prices to these two groups of customers would not serve.

This negative result begs the question, “what must a business believe about its customers to want
to offer them a grandfather clause?” Although, we consider a number of potential explanations,
our primary focus in this manuscript is on an environment in which consumers are complacent
with regard to their idiosyncratic preference and alternative price discovery. That is, suppose
that because potential consumers are engaged in hundreds or thousands of markets, they do not
necessarily make the effort to reconsider their individual specific preferences for products or to
price shop, even if general changes in their previously chosen product suggest that idiosyncratic
preferences and alternative product prices should have changed as well. However, consumers might
available to new consumers) for a different price from the same firm.

54

be induced to preference and price discovery if their attention to a market is drawn by say a price
increase for the service they subscribe to or a general, relative improvement to a competitor’s
product. We find that in an environment where consumers are complacent, a firm might prefer to
offer grandfather clauses in order to prevent preference and price discovery if doing so outweighs
the potential gain from charging a higher uniform price to grandfathered customers who would
remain with the firm anyway. Even though prevention of preference and price discovery can be
detrimental to consumers, consumer surplus goes up if grandfathered customers pay a low enough
price for a sufficiently improved product. Yet, even if consumers benefit, total welfare is bound to
fall because some consumers don’t end up purchasing from the firm that would have left them with
a product they value more.

To motivate our analysis, we present various recent examples where grandfather clauses have
been used in a retail setting. We suspect that an example that is familiar to most readers is in the U.S.
mobile wireless industry, where nationwide service providers have been known to grandfather cus-
tomers into previously contracted plans that are not made available to new customers. For instance,
both AT&T and Verizon Wireless have previously grandfathered existing customers on unlimited
data plans.3 Moreover, in its 2014 Open Internet Order, the U.S. Federal Communications Com-
mission pointed to its concern over Verizon Wireless attempts to limit the speeds of customers on
grandfathered unlimited plans in its discussion of exceptions to its “bright line” anti-discrimination
rules (see U.S. FCC 2014). Although in the case of these unlimited plans, grandfathered mobile
wireless customers end up consuming what is effectively a different product from those marketed
to new customers, other customers are simply locked into a lower price for plans that remain on
the market. Although some such customers may have initially purchased what is now deemed a
slower service, as long as they upgrade their handsets to those capable of operations within the

3See Bea,

F.

“AT&T Permitting

in Exist-
iPhone
ing Unlimited Data Plans.” Digital Trends.
Retrieved
August, 2015.
<http://www.digitaltrends.com/mobile/att-letting-iphone-5-owners-keep-their-
old-dataplans/#ixzz3XsSqKJNp>; Dragani, R. “Verizon Nixes Unlimited Data Grand-
father Clause.” E-Commerce Times.
2015.
May 17,
<http://www.ecommercetimes.com/story/75141.html>.

Retrieved August,

5 Buyers

to Grandfather

September 12,

2012.

2012.

55

latest technology, grandfathered customers end up with an improved product at a lower price.

Netflix presents another well publicized example of a grandfather clause where existing con-
sumers retained access to the same service available to new customers at a discounted price. In
May 2014, Netflix announced plans to raise prices to subscribers receiving HD quality service
by $1 a month while allowing existing customers to keep their current price for two years.4 In
September 2014, Apple offered certain existing iCloud storage customers the opportunity to keep
their current storage plan, which is priced very closely to the lowest paid storage plan available to
new customers, but with an additional 5GB of storage.5 Yet another example of retail grandfather
clauses occurs when a retailer begins charging for a service that was previously free. For instance,
in 2012 Google ceased offering the free edition of its cloud computing Google Apps software and
e-commerce platform Ecwid decreased its product sales limit for new users of its free plan, while
permitting prior registered users to retain their existing level of service.6

In the U.S., grandfathering in retail has also been the consequence of legislative mandate. The
2010 Patient Protection and Affordable Care Act permitted health insurance providers to continue
to offer service to consumers enrolled in certain individual grandfathered plans while discontinuing
further enrollment as long as the insurer updated certain provisions of the plans in accordance with
the act and notified consumers of these plans that they may not get some rights and protections that
new plans satisfying the requirements of the act offer.7 These grandfathered plans were allowed to

4See Fung, B. “Netflix Prices are Rising Today.

But Existing Subscribers Will
Get a 2-Year Reprieve.” The Washington Post.
Retrieved Au-
gust, 2015. <https://www.washingtonpost.com/news/the-switch/wp/2014/05/09/netflix-prices-are-
rising-today-but-existing-subscribers-will-get-a-2-year-reprieve/>.

May 9,

2014.

5See Campbell, M. “Apple Offering Existing iCloud Storage Customers Grandfa-
Retrieved Au-
<http://appleinsider.com/articles/14/09/10/apple-offering-existing-icloud-storage-

thered Capacities, Cheaper Rates.” AppleInsider.
gust, 2015.
customers-grandfathered- capacities-cheaper-rates>.

September 10, 2014.

6See Google, Google Apps Administrator Help. Google Apps Free edition (legacy). Decem-
ber 6, 2012. Retrieved August, 2015. <https://support.google.com/a/answer/2855120?hl=en&ref
topic=6043588>; Ecwid, Blog. Ecwid announces new service plans. No price changes for our
existing users. September 5, 2012. Retrieved August, 2015. <http://www.ecwid.com/blog/ecwid-
news/ecwid-announces-new-service-plans-no- price-changes-for-our-existing-users.html>.

7See Pub.L. 111148, 124 Stat.

119; HealthCare.gov, Health Coverage Rights and Pro-

56

persist as long as they were not changed in ways that would substantially cut benefits or increase
costs to consumers.

To our knowledge, this is the first study that attempts to explicitly analyze the profitability
and welfare consequences of using grandfather clauses in retailing. Unlike many other price
discrimination strategies commonly used in retail (e.g., sales, coupons, quantity discounts, price-
matching guarantees, and add-on pricing8) grandfather clauses are best characterized by direct
market segmentation—a firm knows precisely who its grandfathered customers will be and can
segment the market accordingly.

More precisely, grandfather clauses segment the market based on past purchase history. Previous
studies have shown that when it is possible to discriminate among customers ac- cording to purchase
history, rather than offer existing customers lower prices, firms may use information inherent in
past purchase behavior to induce rivals’ customers to switch by offering them lower prices instead
(Villas-Boas 1999, Fudenberg and Tirole 2000).9 A related form of pricing sometimes results when
consumers face switching costs. When consumers must pay (either implicitly or explicitly) to switch
to a rival producer, firms may exercise market power over their existing consumers by charging them
more than they paid as new customers—so called “bargains-then-ripoffs” pricing (see for instance
Klemperer 1987a, 1987b, 1995; Padilla 1992)—and more than to the existing customers of rival
firms—“poaching” (see Chen 1997, Taylor 2003).10 These models entail vigorous competition for
market share through bargains early on followed by ripoffs to customers who would have to pay to
tections.
Retrieved August,
2015. <https://www.healthcare.gov/health-care-law-protections/grandfathered-plans/>. Job-based
grandfathered plans could continue to enroll individuals.

Grandfathered health insurance plans. March 23, 2010.

8The formulation of the price discrimination inherent in these strategies is well described in
respectively: Varian (1980), Narasimhan (1984), Dolan (1987), Png and Hirshleifer (1987), and
Ellison (2005).

9Acquisti and Varian (2005) also examine firms’ ability to condition prices on past purchase
history via a purchase tracking technology (e.g., HTTP cookies and related devices). As in Fu-
denberg and Tirole (2000), firms may induce switching behavior, albeit via service personalization
instead of through lower prices. Fudenberg and Villas-Boas (2006) survey the literature on price
discrimination according to customer recognition.

10Farrell and Klemperer (2007) provide a detailed overview of the literature on switching costs.

57

switch brands (potentially with inducements offered to rivals’ customers).

A closely related literature considers endogenous switching costs via loyalty or reward pro-
grams. As in the case with grandfather clauses, and unlike in the literature on exogenous switching
costs, loyalty programs allow existing customers to pay lower prices than newcomers. For instance
Banerjee and Summers (1987) examine loyalty inducements in a homogenous good setting with se-
quential pricing and find that loyalty programs can facilitate collusion. In their model, firms benefit
from increases in a rival’s “loyalty coupon” because the coupon deters the rival from undercutting.
In contrast, working in a horizon- tally differentiated model with simultaneous pricing, Caminal
and Matutes (1990) find that precommitments to charge returning customers lower prices lead to a
declining price path for all consumers and lower profits than without precommitments.11

A driving feature of most exogenous and endogenous switching cost models is forward looking
strategic behavior whereby firms use foresight of future prices and market shares in setting prices
before the market matures. Instead, in this manuscript, rather than treat grandfather clauses as a
strategic device that firms anticipate when setting prices early on, we focus on whether grandfather
clauses should be used in a mature market setting.
In contrast to much of the literature on
exogenous switching costs, Shaffer and Zhang (2000) find that when the customers of a firm are
substantially more loyal to that firm than the rival’s customers are to that rival, both firms should
offer inducements to the rival firm’s customers, which entails the rival offering its own customers a
lower price than to newcomers. As in Shaffer and Zhang (2000), our model takes past behavior as
given and asks how competition today is affected if firms can price-discriminate. However, aside
from focusing on a particular form of price discrimination not expressly contemplated by Shaffer
and Zhang, our model neither relies on switching costs nor on aggregate differences in preferences
or costs across groups of consumers who frequent particular firms.

The remainder of this manuscript is organized as follows. In Section 3.2 we present a base-
line model in which we introduce grandfather clauses in a setting where consumers are perfectly
11Caminal and Matutes (1990) also examine a framework where firms can reward returning
customers using coupons instead of precommitments, but find that coupons do not survive when
the choice of precommitment or couponing is endogenized.

58

informed about their preferences and the prices for product alternatives. In Section 3.3 we model
grandfather clauses when consumers are complacent and show that in this setting a higher quality
firm might wish to grandfather. Section 3.4 concludes with a discussion of alternative settings in
which grandfather clauses might raise profits.

3.2 Baseline Model

3.2.1 Firms and Consumers

Two firms, labeled 1 and 2, offer differentiated, competing subscription services. Firms face no
capacity constraints and have an identical constant cost of 0 of offering one unit of their respective
services. There is a unit mass of consumers with idiosyncratic tastes over these services described as
follows. Each consumer values consumption of a unit of service i according to some nonstochastic
average quality µi combined with a stochastic preference parameter 
i ∈ [
i, ¯
i] that represents the
consumer’s idiosyncratic preference for brand i.12

Consumers freely observe both firms’ prices and each consumer subscribes to a single unit of

the service that maximizes his utility, or:

ui = µi + 
i − pi

(3.1)

where pi is the price of service i and ui is the consumer’s utility. Following Perloff and Salop
(1985), we exclude “outside services” from the analysis by assuming that each consumer purchases
that service among those offered by firms 1 and 2 that gives him the highest utility regardless of
the actual cardinal level of utility.

We assume that individual consumers’ idiosyncratic preferences are distributed independently
and identically with mean zero within brands and that aggregate preferences for the two brands are
distributed independently according to density fi.13
12We do not explicitly rule out the possibility that 
i = −∞ or that ¯
i = ∞.
13Perloff and Salop (1985) also assume that preferences are symmetric across brands whereas

59

For a given consumer, u1 ≥ u2 if and only if µ1− µ2− p1 + p2 + 
1 ≥ 
2. Then, letting differen-
tiable function Fi represent the distribution associated with density fi, the probability that u1 ≥ u2
is given by F2(µ1 − µ2 − p1 + p2 + 
1). We can now represent the proportion of consumers who
purchase from brand i—alternatively, the quantity of services ordered from firm i—as a function
of firm prices and average qualities by:

Þ ¯
i


i

Qi(p1, p2; µ1, µ2) =

Fj(µi − µj − pi + pj + 
i) fi(
i)d
i

(3.2)

and because there is no outside option, Q j(p1, p2; µ1, µ2) = 1− Qi(p1, p2; µ1, µ2).

For ease of exposition, define 
 ≡ 
2 − 
1 and let 
 F(
) where F is a differentiable distribution
function with a density that is symmetric about zero to which we assign additional restrictions
as necessary throughout the manuscript.
If, for instance, aggregate preferences are distributed
identically and independently according to the Type I extreme value distribution, then F becomes
the widely used logistic distribution. An example that we will turn to throughout the manuscript
will assume that 
 is distributed uniformly.14

Define ∆ ≡ µ1− µ2. Then, the quantity of services ordered from firm 1 is simply F(∆ + p2− p1)

and firm expected profit functions are given by:

π1(p1; p2, ∆) = p1F(∆ + p2 − p1)

π2(p2; p1, ∆) = p2(1− F(∆ + p2 − p1))

(3.3)

(3.4)

Assumption 1. Suppose that firms engage in single-period differentiated Bertrand competition.
Then there exists a unique interior equilibrium of the duopoly game described above.

A necessary condition for existence of equilibrium is that both first order conditions are satisfied.
Suppose that ∆ = 0. Then if a single price equilibrium exists, it is readily shown that the equilibrium
Chen and Riordan (2008), who use a related model to study price-increasing competition, assume
that aggregate preferences are symmetric, but not necessarily independent across brands. Chen and
Riorden (2008) additionally permit consumers to have an outside option.

14Note that in this case, aggregate preferences for individual brands cannot be identically dis-

tributed. We impose additional assumptions as necessary (e.g., to calculate welfare).

60

price equals, p = 1/(2 f(0)) where f is the density associated with F. Perloff and Salop (1985) show
that when aggregate preferences are distributed symmetrically across brands, a unique single price
equilibrium exists and and there are no multi-price equilibria (see Perloff and Salop Proposition
4). However, as seen in the example below, because we allow average quality to differ across firms
(so that ∆ does not necessarily equal zero), a unique interior equilibrium need not result in a single
price.
Example. Suppose that 
 is distributed uniformly on [−a, a]. Solving for equilibrium in our
baseline model yields prices pi, quantities qi, and profits πi as follows:

(cid:16)

p1 = a +
1
2a
1
2a

1 +

1 +

(cid:16)

(cid:17)
(cid:17)2

∆
3 ,
∆
3
∆
3

,

,

q1 =

π1 =

(cid:16)
(cid:16)

p2 = a− ∆
3
1− ∆
3
1− ∆
3

1
2a
1
2a

q2 =

π2 =

(cid:17)
(cid:17)2

(3.5)

(3.6)

(3.7)

Importantly, the prices, quantities, and profits of firm 1 increase and those of firm 2 decrease as
firm 1’s average quality advantage relative to firm 2 increases.

Even when ∆ (cid:44) 0, this framework allows us to place reasonable bounds on prices in a single-
period differentiated Bertrand game, which we rely on later in the manuscript. In particular, as
we show below, in equilibrium, a firm with higher average quality will not set a lower price than
its competitor. Moreover, that price cannot exceed the price of its competitor by more than the
difference in average qualities.
Lemma 1. Suppose that ∆ > 0. Then 0 ≤ p1 − p2 ≤ ∆.

Proof. The first order conditions for firms 1 and 2 respectively are:
F(∆ + p2 − p1)− p1 f(∆ + p2 − p1) = 0
1− F(∆ + p2 − p1)− p2 f(∆ + p2 − p1) = 0

Solving Equations (3.8) and (3.9) for f(∆ + p2 − p1) and rearranging yields:

1 = F(∆ + p2 − p1) +

p2

p1 + p2

61

(3.8)

(3.9)

(3.10)

Because F(0) = 1/2, it follows that if p2 > p1, then both terms on the right-hand side of Equation
(3.10) are greater than 1/2, a contradiction. If instead p1− p2 > ∆, then both terms on the right-hand
side of Equation (3.10) are less than 1/2, which is likewise a contradiction.
(cid:3)

Inspection of Equation (3.5) in our uniform distribution example will show that p1− p2 = 2∆/3.

Thus, both bounds hold.

3.2.2 Grandfather Clauses

We study grandfather clauses using a single-period mature market in which half of all consumers
have previously subscribed to firm 1’s service and the other half subscribed to that of firm 2. The
idea that both firms initially hold half the market may be justified by supposing that prior to market
maturity, firms provided services with the same initial quality at the same initial price. We further
suppose that the initial price prior to market maturity, po, equaled the single-period, single price
equilibrium price, po = 1/(2 f(0)).15 Thus, if a firm wishes to grandfather its existing consumers,
it must offer them a price of po that is different from the price charged to potential new customers.
Going forward, we refer to the time prior to market maturity as the initial period and we refer to the
stage to be studies as the mature market period.

If neither consumer tastes nor firm quality changes between the initial and mature market
periods, then the use of grandfather clauses does not have a real world counterpart in the following
sense: grandfathered customers must be made to feel that they are consuming a better product than
that available to newcomers or they must be able to obtain a better price than what newcomers would
have to pay. Thus, suppose that between the initial and mature market periods, firm 1 announces
an average quality improvement of ν > 0. As a result, ∆ rises from 0 to ν. Because our focus in
this manuscript is on grandfather clauses, we suppose that the quality improvement is exogenous
and costless for firm 1.

15We note that because we are not solving a two-stage pricing game, equal market shares in this
setting imply that the actual (not just expected) realization of idiosyncratic preferences was such
that half of consumers initially preferred each service.

62

For simplicity, we suppose that only firm 1 can offer grandfather clauses. The game proceeds as
follows: at the outset of the mature market period, firm 1 decides whether or not it will grandfather
its existing consumers and simultaneously chooses a new price while firm 2 simultaneously sets its
own price.16 Neither firm can engage in any other form of price discrimination and grandfathered
consumers are free to choose the rival firm’s product.17 Moreover, grandfathered consumers benefit
from the average quality improvement, and will choose the new price if that price is lower. After
firms set prices, consumers make their purchasing decisions and profits and welfare are realized.

At this juncture, the astute reader may ask why we do not study grandfather clauses in a rational
expectations framework in which firms account for the potential competitive effects of grandfather
clauses when setting prices in the initial period. As discussed in Section 1, such frameworks
are common in the switching cost literature. In the subgame perfect Nash equilibrium of such a
game, there is no a priori reason to think that initially symmetric firms would set the same price
and end up with the same market share in the initial period if grandfather clauses stipulate that
these prices will directly impact profitability in the mature market period. However, mathematical
tractability aside,18 there are two reasons to fix initial prices and market shares regardless of whether
grandfather clauses will be used in the mature market period. First, we interpret the time that passes
between the initial and mature market periods as sufficiently lengthy to allow for average quality
to change and to justify price adjustment. However, without investigating investment in quality in
greater detail, it is difficult to determine which of two initially symmetric firms will end up with the
relative quality improvement in the mature market period, and consequently, how resulting price
16The simultaneous move structure of our game is motivated by various of our examples in
Section 1, where a mature market firm simultaneously resets its price while grandfathering existing
consumers. However, within the setting of our main model in the next section, it will become
apparent that the results are robust to a setting where firm 1 informs its rival and consumers of its
intent to grandfather prior to setting its price.

17So, for instance, we suppose that firm 2 cannot specifically offer firm 1’s existing, potentially

grandfathered, customers incentives to switch.

18In a rational expectations framework, subgame perfection would require us to calculate mature
market period prices for any level of initial market shares, and in light of grandfather clauses, it
would additionally require prices to be contingent on any initial period price that might be offered
to grandfathered consumers.

63

adjustments and the application of grandfather clauses in the future should affect prices in the initial
period. Second, our primary interest is in informing managers of firms in mature markets who are
contemplating implementing a grandfather clause rather than in any gaming (e.g., for the purposes
of collusion or exclusion) that grandfather clauses may permit firms to undertake.19

In this subsection, we assume that consumers are always cognizant of prices, average qualities,
and their idiosyncratic preferences for both brands. We suppose that individuals’ idiosyncratic
preferences are independent over time—effectively positing that enough time has passed such that
the initial period preferences do not convey any idiosyncratic information to the individual when
the market has matured. Moreover, we suppose that the 
i are redrawn in the mature market period
according to the same distribution as in the initial period—that is, we suppose that individual
preferences may have changed, but that aggregate preferences have not.20

It turns out that under the assumptions in this section, firm 1’s optimal strategy is to offer a
uniform price to both its initial period customers and to potential consumers that it could poach
from firm 2. As shown in Proposition 1, because this is the case for any potential price that firm 2
may offer, grandfather clauses are not played in equilibrium.

Proposition 3.1. Suppose that consumers are aware of firm prices, average qualities, and their
idiosyncratic preferences prior to choosing which service to purchase. Then, in equilibrium, firm
1 will choose to offer a uniform price rather than to grandfather its initial period customers by
offering them a lower price than to potential consumers.

Proof. Suppose that firm 2 chooses price p2. When setting prices, by offering a grandfather clause,
firm 1 can price discriminate between its initial period customers and those of firm 2. Given a
19It seems to us sensible that firms would account for say potential consumer switching in the
future when setting present prices. However, unlike switching costs, which are estimable in the
present, the decision to apply grandfather clauses is contingent on changing firm, consumer and
overall market characteristics that appear to us very difficult to predict up front, so that we believe
that firms do not necessarily account for the possibility that they may choose to grandfather certain
customers in the future when setting their prices in the present.

20Thus, our setting is similar to the independent preference setting of Fudenberg and Tirole
(2000), who also separately look at fixed preferences. In a future manuscript, we hope to study
preferences that are either fixed, or in some sense correlated over time.

64

price p1, firm 1’s initial period customers will remain with firm 1 if ν− p1 + p2 ≥ 
. Thus, firm 1’s
expected profit in the mature market period from its initial period customers is given by:

(ν + p2 − p1)

p1
2

(3.11)

Because idiosyncratic preferences for firm 2’s initial period customers are redrawn from the same
distribution, Expression (3.11) likewise represents firm 1’s expected profit from poaching firm
2’s initial period customers. Consequently, the value of p1 that maximizes expected profits from
former customers is the same as the value that maximizes expected profits from new customers. As
a result, firm 1 will not wish to price discriminate by offering returning customers one price and
(cid:3)
new customers a higher price.21

When preferences are independent over time, under the assumptions above, knowing whether
a potential customer had previously purchased from firm 1 or 2 does not convey any useful infor-
mation to firm 1 at either the individual consumer level or at any level of consumer aggregation.
Because in our setup, firm 1 has no other means of price discrimination at its disposal, it prefers
to set a uniform price. In fact, this straightforward result is more broad. Because firm 1’s profit
maximizing price for the two groups of customers does not depend on initial period market shares
or prices, firm 1 would prefer a uniform price even if the initial period equilibrium were asymmet-
ric—that is, when preferences are independent over time, under Assumption 1, uniform pricing
turns out to be the outcome of the unique subgame perfect Nash equilibrium of an alternative
rational expectations, two-stage pricing game.

3.3 Consumer Complacency

Our findings in Proposition 1 suggest that when consumers are perfectly informed about prices
charged by firms 1 and 2 as well as regarding their own idiosyncratic preferences, then in the
21More broadly, this proof also rules out price discrimination whereby returning customers are

asked to pay a higher price than new customers.

65

mature market setting being studied here, a firm would not be compelled to grandfather returning
customers. This leads us to wonder why various firms facing oligopoly competition in a real world
mature market setting would offer grandfather clauses. In this section, we set up a simple model
of complacent consumers to offer one potential explanation. In the next section, we contemplate
some alternative explanations.

Suppose that consumers are complacent in the following sense: they do not reevaluate their
idiosyncratic preferences for the service that they consume, nor seek updated price information
from other firms unless one of two things happens: (i) they are faced with a higher price for the
service they currently consume or (ii) they discover that a rival service provider offers a higher
relative quality alternative.22 As such, complacency here is not interpreted so much as a cost
to uncover additional price and idiosyncratic product match information (as in Wolinsky 1986 or
Anderson and Renault 1999), but as a failure to contemplate that this information changed after
some period of time.

As in the previous section, we suppose that idiosyncratic preferences are independently and
identically distributed in the initial and mature market periods. We also suppose that consumers are
perfectly informed of all average quality improvements—that is, all consumers learn ν at the outset
of the mature market game. One way to think about this is to suppose that quality improvements
that are expected to affect all consumers are broadly advertised by the firms, but that it is more
difficult to convey concrete price and individual preference information via advertising.23

Assumption 2. Consumers are complacent in the mature market period.

Proceeding with our model from Section 2, modified by Assumption 2, we now see that if
22For a firm i, this happens if µj rises relative to µi.
23Because our core interest is in grandfather clauses, we do not explicitly model the process
by which firms convey average quality information to consumers. A standard approach would be
to suppose that the cost of advertising is increasing and strictly convex in the proportion of the
population that receives an update of average quality (see for instance the price advertising models
of Butters 1977 and Robert and Stahl 1993). Alternatively, we could suppose that there is a fixed
cost to advertise to the entire population (e.g., Janssen and Non 2008). Our model implicitly
presumes that advertising quality is costless.

66

firm 1 offers to grandfather its initial period consumers, these consumers will not reevaluate their
idiosyncratic preferences nor prices for the two service alternatives and remain with firm 1. They
will continue to pay po = 1/(2 f(0)) for firm 1’s service, but also benefit from firm 1’s general
quality improvement ν. Conversely, because firm 1 offers a higher quality product in the mature
market period, firm 2’s initial period customers will undertake price and idiosyncratic preference
discovery (even if they were grandfathered). Thus, under Assumption 2, firm 1 can either choose a
uniform price, in which case a price above po will lead both firms to compete for the entire market,
or it can grandfather its initial period consumers and restrict competition in the mature market
period to firm 2’s initial period customers, but sacrifice potential profit gains from raising its price
to its initial period customers.

As we show in Proposition 2, under some reasonable assumptions on f, firm 1 will want to
offer a grandfather clause as long as ν is not “too high.” The intuition is straightforward. If firm
1’s relative quality advantage over firm 2 is sufficiently large, firm 1 does not need to worry that
its initial period customers will engage in information discovery following a price increase because
firm 1’s quality advantage will bring most of these customers back. On the other hand, if ν is
relatively low, firm 2 remains competitive for firm 1’s initial period customers and firm 1 can
improve its profit by using a grandfather clause to prevent firm 2 from poaching its customers.
Recall that this would not have worked in the previous section—what is crucial is that grandfather
clauses may prevent information discovery.

Proposition 3.2. Suppose that F is twice continuously differentiable and that f is symmetric about
zero and single-peaked. Then under Assumption 2, there exists ˆν such that for any average quality
improvement ν < ˆν, firm 1 will wish to offer a grandfather clause, and for any ν > ˆν it will not.

Let π∗

1(ν) represent firm 1’s equilibrium expected profit when it chooses a uniform price in the
mature market period and let π∗
2(ν) represent firm 2’s equilibrium expected profit when it believes
that firm 1 will play a uniform price (and these beliefs are correct). The key to the proof of
Proposition 2 is that under Assumption 2 the firms’ expected profit functions when firm 1 chooses
to grandfather its initial period customers are simply affine transformations of profits under uniform

67

pricing.24 In particular, because firms’ initial period profits equaled 1/(4 f(0)), firm 1’s and 2’s
respective equilibrium expected profits in the mature market period when firm 1 does grandfather
are 1/(4 f(0)) + π∗
2(ν)/2. Thus for firm 1, expected profit when it offers to grandfather
exceeds non-grandfathered expected profit if and only if π∗
1(ν) < 1/(2 f(0)). In the Appendix, we
show that π∗
1(ν) is continuously increasing in ν and that there are values of ν low enough to satisfy
and high enough to reverse the previous inequality, completing the proof.

1(ν)/2 and π∗

In Proposition 2, we placed a number of restrictions on the forms of F and f in order to derive
our “cut-off” result. However, as the following example shows, the single-peakedness condition
(which we rely on to assure continuity of π∗
1(ν)) is not necessary.
Example. Suppose that 
 is distributed uniformly on [−a, a].
1(ν) is given by
(ν) represent firm 1’s expected profit
replacing ∆ in Equation (3.7) with ν. Similarly, letting πGFC
when it chooses to grandfather its initial period customers in the mature market period, we obtain:

In this case, π∗

1

(cid:16)

(cid:17)2

ν
3
Firm 1 will want to offer a grandfather clause whenever πGFC

πGFC
1

a +

(ν) =

a
2

+

1
4a

(3.12)

1

√
2− 1)
ν < 3a(

(ν) > π∗

1(ν), which simplifies to:

(3.13)

That is, as in Proposition 2, firm 1 will want to offer a grandfather clause as long as its average

quality advantage relative to firm 2 does not exceed 3a(√

2− 1).

Observe that in the example above, the higher a—that is, the greater the variance of the
distribution of the difference in idiosyncratic preferences—the greater the range of ν for which
firm 1 would want to offer a grandfather clause under consumer complacency. This suggests
that grandfather clauses are more likely to prove useful if tastes are more dispersed. In the next
proposition, we explore this result in a more general setting.

For any distribution F consider the family of distributions Gα such that Gα(
) = F(α
). Observe
that Gα is a distribution for any α ∈ (0, ∞). When α < 1, Gα stretches F horizontally, behaving
24Thus, as suggested in footnote 16, it does not matter if firm 1 informs everyone that it will
commit to grandfather before firms set prices or not because under Assumption 2, the prices that
solve firms’ first order conditions will be the same with or without grandfathering.

68

similarly to the original distribution, but with larger variance. Conversely, when α > 1, Gα
horizontally compresses F.

Proposition 3.3. Suppose that F is twice continuously differentiable and that
is symmetric
about zero and single-peaked. Then under Assumption 2, if the distribution of the difference in
idiosyncratic preferences is given by Gα, for a fixed improvement ν, there exists some α∗(ν) > 0
such that firms will wish to offer a grandfather clause if and only if α ≤ α∗(ν).

f

Proposition 3.3 states that for any ν > 0, there always exists a level of dispersion in idiosyncratic
preferences (α low enough) that justifies offering a grandfather clause under Assumption 2. Greater
dispersion in tastes in the manner defined above makes ν less effective in retaining customers. Thus,
using grandfather clauses to make customers complacent becomes that much more valuable.

Together Propositions 3.2 and 3.3 characterize sufficient conditions for a firm to wish to offer a
grandfather clause. The firm wishes to grandfather when either the quality improvement it offers is
slight enough so as to keep it from having an overwhelming competitive advantage, or if consumer
tastes are so dispersed as to make the quality improvement relatively insignificant. Corollary 1
relates these two propositions by telling us that the larger the improvement in quality, the more taste
dispersion is needed to justify offering a grandfather clause.
Corollary 1. α∗(ν) is decreasing in ν.

3.3.1 Welfare

Under Assumption 2, grandfather clauses are unambiguously bad for total welfare. This can be
seen without any explicit calculation. Recall that because we have assumed that consumers have
no outside option, the price paid for each consumer’s chosen service is a pure transfer and may be
discounted in total welfare calculations. Thus, only the realizations of idiosyncratic preferences and
whether or not a consumer ends up purchasing from the firm that offers the best match (accounting
for average quality) matter. When consumers are complacent, grandfather clauses distort optimal
matching by keeping firm 1’s customers from potentially realizing a better match with firm 2.

69

As discussed earlier in this section, grandfather clauses are beneficial for firm 1 as long as ν is
not too high relative to the level of dispersion in idiosyncratic preferences. Conversely, grandfather
clauses are also unambiguously bad for firm 2 because they effectively reduce its market share.
Perhaps of greater interest is whether or not grandfather clauses increase consumer surplus. On
the one hand, grandfather clauses reduce the average price paid by consumers. On the other hand,
they cause some consumers to be matched poorly and to receive a service that they might not enjoy
as much relative to the price paid. As shown in a continuation of our uniform distribution example
below, the effect is ambiguous and depends on the value of ν relative to the level of dispersion in
tastes.
Example. Suppose that 
 is distributed uniformly on [−a, a].
In order to calculate consumer
surplus, we additionally need to know how 
i is distributed for each individual firm along with
firms’ initial period average service quality levels. Thus, for the purpose of this example, suppose
that 
1 is distributed uniformly on [−a, a] whereas 
2 is a point mass at zero and suppose that
a > ν/3. As will become evident in Equation (3.14), the inequality states that there is sufficient
taste dispersion relative to the value of ν for some customers to find it worthwhile to frequent firm
2 in the absence of grandfather clauses. Additionally, suppose that initially, µ1 = µ2 = µ. Then, if
firm 1 chooses not to offer a grandfather clause in the mature market period, from Equation (3.5),
we know that consumer surplus is given by:25

(cid:104)

µ + ν + 
1 −(cid:16)

Þ a

− ν
3

CS =

(cid:17)(cid:105) 1

a +

ν
3

d
1 +

2a

Þ − ν

3

−a

(cid:104)

µ−(cid:16)

(cid:17)(cid:105) 1

2a

a− ν
3

d
1

(3.14)

where the first integral is for firm 1’s consumers and the second is for firm 2’s. This expression
simplifies to:

CS = µ− 3a
4

+

ν
2

+

ν2
36a

(3.15)

Under Assumption 2, if firm 1 offers a grandfather clause, it retains the half of the market that it
possessed in the initial period. Firm 1’s initial period customers then pay it a price of a for a product
25Note that the bounds of integration follow because consumers strictly prefer to purchase from
firm 1 if and only if 
1 > −ν/3.

70

with mature market average quality of µ + ν. Although these consumers are complacent when firms
reset their mature market prices, these consumers ultimately realize idiosyncratic preferences as
distributed above. Consumer surplus for the other half of the market is represented by Equation
(3.14). Thus, when firm 1 offers a grandfather clause, consumer surplus is represented by:

µ + ν− a

2

+

CS
2

CSGFC =

(3.16)
√
Clearly, CSGFC > CS if and only if µ + ν− a > CS, which occurs whenever ν > 3a(3− 2

2) ≈
0.51a. That is, when ν is sufficiently large relative to a, the price savings to grandfathered customers
(which are proportional to ν) are larger than the foregone gains from price and idiosyncratic
preference discovery (which are proportional to a). Recall that firm 1 will only wish to offer
2 − 1) ≈ 1.24a. Thus, given some level of idiosyncratic
preference dispersion and considering the range of ν under which firm 1 would find it profitable
to offer a grandfather clause, consumers are worse off when ν ∈ (0, 0.51a) and better off when
ν ∈ (0.51a, 1.24a).

a grandfather clause when ν ≤ 3a(√

3.4 Price Unfairness

An alternative explanation for the use of grandfather clauses is that they may serve as a useful
mechanism to mitigate adverse consumer reactions to perceived price unfairness over increases in
prices (e.g., see Bolton et al. 2003). When customers view a price increase by their existing firm
as unfair, they may be more inclined to try a rival service. Grandfather clauses allow firms to avoid
angering former customers using lower prices while offering newcomers (who might direct their
anger at a rival firm) a higher price.26

In this section, we return to the baseline model in Section 3.2, but modify consumer utility to
suppose that any individual who encounters a price increase from their current service provider
26We suspect that the Netflix decision to grandfather their existing subscribers in 2014 may
have been driven in part by adverse subscriber reactions to an earlier substantial price increase
in 2011. See Fung, B. “Netflix Prices are Rising Today. But Existing Subscribers Will Get
a 2-Year Reprieve.” The Wall Street Journal. September 16, 2011. Retrieved October, 2011.
<http://www.wsj.com/articles/SB10001424053111904060604576572322651549428/>.

71

finds the increase unfair and faces a reduction in utility equal to φ > 0. To simplify, we suppose
that individuals only perceive price unfairness when subjected to a price increase from their current
service provider, and not from a provider with which they have no experience. Moreover, we
suppose that all consumers who perceive price unfairness face the same exogenously given level of
disutility. We can now rewrite the mature market utility for service i of a consumer who consumed
service i in the initial period as:

 µi + 
i − pi − φ

µi + 
i − pi

ui =

if pi > a

otherwise

(3.17)

Assumption 3. consumers perceive price increases as “unfair”.

As in Section 3.2, suppose that firm 1 announces an average quality improvement of ν > 0.
Moreover, for ease of exposition, in this section and the next, let us suppose that 
1 is distributed
uniformly on [−a, a] whereas 
2 is a point mass at zero, meaning that 
 is distributed uniformly on
[−a, a]. Thus, from Equations (3.5) to (3.7) it follows that firm 1 is the only one that would want
to offer a grandfather clause.

Throughout this section, we will assume that φ < 2ν: that is, the level of disutility from price
unfairness that consumers perceive is not so high as to induce firm 1 to set its price equal to or
below a for the purpose of avoiding an adverse reaction from consumers whenever it opts not to
offer a grandfather clause. In other words, in equilibrium, firm 1 sets p1 > a even when it does not
offer a grandfather clause (whereas firm 2 sets p2 < a). In the appendix, we conduct the analysis for
the case φ ≥ 2ν. When φ < 2ν, firms’ mature market period profit maximization problems when
firm 1 does not offer a grandfather clause are:

max
p1

2a

p1
2

(cid:20) v + p2 − p1 −(−a)
(cid:21)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:21)
(cid:20)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
1− v + p2 − p1 −(−a)

profit from new customers

profit from existing customers

p2
2

2a

p1
2

(cid:20) v + p2 − p1 − φ−(−a)
(cid:21)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:20)
(cid:21)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
1− v + p2 − p1 − φ−(−a)

profit from existing customers
p2
2

2a

2a

profit from new customers

+

+

max
p2

(3.18)

(3.19)

72

φ

φ = 2v

(cid:113) 2

φ = 2(3 + v)− 6

49 v2 + 4

7 v + 1

Figure 3.1: Area where Grandfather Clause is Useful (a = 1)

v

Simultaneously solving profit maximization problems (3.18) and (3.19) yields mature market period
prices, quantities, and profits:

(cid:19)
(cid:19)2

− φ
v
6
3
− φ
v
3
6
− φ
6

,

,

,

(cid:18)
(cid:18)

+

p2 = a− v
φ
6
3
a− v
3
a− v
3

1
2a
1
2a

q2 =

π2 =

(cid:19)
(cid:19)2

+

+

φ
6

φ
6

(3.20)

(3.21)

(3.22)

q1 =

a +

π1 =

a +

v
3

p1 = a +

(cid:18)

(cid:18)

1
2a
1
2a

As Equations (3.20) through (3.22) show, price unfairness dampens firm 1’s price, quantity, and
profit while increasing those of firm 2 relative to the baseline. This is because when φ < 2ν, in
equilibrium, only firm 1’s initial period consumers face disutility from price unfairness; firm 2’s
initial period consumers face a lower price in the mature market period and so do not perceive any
additional disutility by purchasing from firm 2 again.

This result suggests that firm 1 may be able to raise its profit by offering a grandfather clause in
order to avoid upsetting its initial period consumers while charging firm 2’s initial period consumers
a higher price for its higher quality. When firm 1 offers a grandfather clauses by charging its initial
period customers price a in the mature market period, in equilibrium, no consumer perceives price

73

unfairness because firm 2 also sets a price p2 < a and firm 2’s initial period customers who switch
to firm 1 do not perceive unfairness concerning a product with which they had no prior experience.
Firms’ mature market period profit maximization problems when firm 1 offers a grandfather clause
are therefore:

a
2

2a

(cid:20) v + p2 − a−(−a)
(cid:21)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:21)
(cid:20)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
1− v + p2 − a−(−a)

profit from existing customers

p2
2

profit from new customers

2a

+

2a

p1
2

(cid:20) v + p2 − p1 −(−a)
(cid:21)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:20)
(cid:21)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
1− v + p2 − p1 −(−a)

profit from new customers

profit from existing customers

p2
2

2a

+

max
p1

max
p2

Simultaneously solving profit maximization problems (3.23) and (3.24), we can confirm that the
price that firm 1 charges firm 2’s initial period customers, p1 = a +2v/7, is greater than a, whereas
firm 2’s price, p1 = a− 3v/7, is lower than a. Substituting p1 and p2 into firm 1’s profit equation
yields mature period profit:

πGFC
1

=

a
2

+

2ν
7

+

ν2
49a

(3.23)

(3.24)

(3.25)

(3.26)

Comparing Equation (3.25) with firm 1’s equilibrium profit from Equation (3.22) yields the fol-
lowing set of inequalities:

− π1 > 0
πGFC
1
⇔ 2(v + 3a)− 6

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

4
7av +

2
49 v2

a2 +

< φ < 2(v + 3a) + 6

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

4
7av +

2
49 v2

a2 +

(cid:114)

˜φ

(cid:114)

ˆφ

Because ˜φ > 2ν, ˜φ > φ always holds when, as assumed, φ < 2ν. In other words, when φ < 2ν,
grandfather clauses are profitable whenever consumer disutility from price unfairness exceeds
threshold ˆφ. In fact, as Proposition 3.4 indicates, this is also the case when φ > 2ν.

Proposition 3.4. Suppose that 
 ∼ U[−a, a]. Then, when consumers perceive price unfairness,
there exists ˆφ such that for any average quality improvement ν, firm 1 will wish to offer a grandfather
clause if and only if φ > ˆφ.

74

3.4.1 Welfare

Unlike the case in which consumers are complacent, the effect of grandfather clauses on welfare
is not obvious. As mentioned earlier, grandfather clause induces bad matching between firms and
consumers but under Assumption 3, grandfather clause can mitigate disutility from consumers’
negative perception against price increases. The following example with a continuation of our
uniform distribution shows that the latter effect may exceed the former effect so that grandfather
clause can increase welfare.
Example. As in the example in section 3.3.1, suppose that 
 is distributed uniformly on [−a, a].
Also, suppose that 
1 is distributed uniformly on [−a, a] whereas 
2 is a point mass at zero. Given
that φ < 2v, if firm 1 chooses to set the uniform price in the mature market period, we know that
consumer surplus is given by:

(cid:104)

µ + v + 
1 −(cid:0)a +

6 φ(cid:1)(cid:105) 1

2a

1

3 v − 1

d
1 +

Þ − v

−a

3 + φ
6

(cid:104)

µ−(cid:0)a− 1

3 v +

6 φ(cid:1)(cid:105) 1

1

2a

Þ a

CS =

− v
3 + φ
6

where the first integral is for firm 1’s consumers and the second is for firm 2’s. This expression can
be written as:

d
1

(3.27)

(3.28)

(cid:111)

d
1

(3.29)

If firm 1 chooses to offer a grandfather clause, we know that consumer surplus is given by:

CS = µ− 3a
4

+

+

v
2

v2
36a

+

vφ
18a

− 5φ2
144a

(cid:110)Þ a
Þ a

−4
7 v

+

−2
7 v

(cid:104)
(cid:104)

µ + v + 
1 − a

(cid:105) 1
µ + v + 
1 −(cid:0)a +

2a

7 v

d
1 +

Þ −4
7 v(cid:1)(cid:105) 1

−a

2a

2

d
1 +

(cid:104)
7 v(cid:1)(cid:105) 1
µ−(cid:0)a− 3
Þ −2
(cid:104)
7 v(cid:1)(cid:105) 1
µ−(cid:0)a− 3

d
1

2a

7 v

−a

2a

CSGFC =

1
2

where the first integral is for those who remain firm 1 among firm 1’s original consumers and the
second integral is for those who move to firm 2 among firm 1’s original consumers. Similarly, the
third integral is for those who move to firm 1 among firm 2’s original consumers and the last integral
is for those who remain firm 2 among firm 2’s original consumers. Equation (3.27) simplifies to:

CSGFC = µ− 3a
4

+

9v
14

+

5v2
98a

75

(3.30)

(cid:104)

4v +

φ = 1
5

(cid:113) 1604

49 v2 − 720
7 v

(cid:105)

φ

φ = 2v

Figure 3.2: Area where Consumers are Better Off under GFC (a = 1)

v

Then, we know that CSGFC > CS if and only if

(cid:110)

(cid:114)(cid:16)

φ >

1
5

4av +

16a2 +

820
49

(cid:17)

(cid:111)

v2 − 720

7 av

(3.31)

Given the level of idiosyncratic preference dispersion and considering the range of ν and φ,

figure 3.2 depicts the parameter space for ν and φ in which consumers are better off.

Whether or not offering the grandfather clause improves social welfare can also be confirmed.
Let me donote WGFC social welfare when the grandfather clause is offered and W social welfare
when firms use uniform prices. When φ < 2v, the difference between WGFC and W is given by

(cid:17)

CSGFC − CS

+

(cid:16)

i=1

(πGFC
i

− πi)(cid:111)
(cid:110) 2

(cid:16)− 1
7 v − φ2
φ2
144a

vφ
18a

36a

+

+

+

+

v2
882a

vφ
9
43v2
1764a > 0

WGFC −W =

=

=

(cid:17)

(cid:16)1

+

7 v +

(cid:17)

(3.32)

41v2
1764

− vφ
18a

+

5φ2
144

for all positive a,v, and φ. Contrary to the consumer complacency case, social welfare increases with
grandfather clause, implying that the effect of mitigating consumers’ disutility from the negative
perception outweighs welfare loss from bad matching between firms and consumers.

76

3.5 Conclusion

In this manuscript we seek to understand firm price discrimination via the use of grandfather
clauses. Using a discrete choice setting in which consumer preferences for individual alternatives
can vary over time, we found that when individual consumers are perfectly informed, a firm that
achieves a relative quality improvement will not find it profitable to grandfather its former customers
while charging potential new customers a higher price. Our finding was observed in a setting in
which individual preferences were distributed independently (and identically) from one period to
the next, but we suspect that this result is more general. For instance, consider a setting in which
individual preferences for the same product are positively correlated over time. Then, if a price
increase for individuals who consumed a different firm’s product in the past is optimal, the price
offered to existing customers—who hence value the product more highly in the present—should be
no lower than that offered to potential consumers. We hope to formalize this point in a future draft.
Cognizant of the fact that grandfather clauses are present in various retail service settings, we
consider an alternative framework in which firms might wish to offer them. Our chosen framework
is a “behavioral” model in which grandfather clauses act as a potential barrier to information
discovery.
In this setting, we find that the higher quality firm will want to grandfather early
adopters as long as its relative quality improvement is not so high that the grandfather clause
keeps the firm from realizing a large gain in inframarginal profit. Moreover, the range of quality
improvements that makes a grandfather clause worthwhile grows with the level of dispersion in
idiosyncratic preferences.

Although we rationalize the use of grandfather clauses in a setting with consumer complacency,
we believe that alternative models could also serve to explain their recent proliferation. One
framework that comes to mind is a model where individual preferences are negatively correlated
over time—perhaps because consumers get tired with their initially sought out service. In such a
setting, we suspect that the opposite intuition conjectured in the positive correlation setting holds:
that is, grandfather clauses permit firms to set a higher price to consumers who are expected to
value the service more highly.

77

An alternative explanation for the prevalence of grandfather clauses is that they permit a firm to
maintain a requisite level of market penetration to exploit potential positive network externalities.
This explanation appears particularly cogent in cases where grandfathered consumers get to continue
to consume a service for free. A related explanation is one where a multi-product firm allows
grandfathered consumers to use its (potentially free) service as a loss leader.

78

APPENDICES

79

APPENDIX A

PROOFS IN CHAPTER 1

Derivation of Assumption 1

In an equilibrium with multi-homing on both sides, no platform should have an incentive to
deviate by charging ˆpi
1 to content providers and serving only exclusive content.1 Hence,
assumption 1 requires that the profits from serving only exclusive content are lower than the profits
from serving both exclusive and non-exclusive content.

2 = α2Ni

Deviation profits ˆΠi are given by ˆΠi = (pi

1 − c1 + ∆i)Ni

1 + α2λNi

1 − I(cid:48)(∆i). Plugging in the

expression for Ni

1 given in equation (1.28), we have:
1 − c1 + ∆i) α1λ− c1

ˆΠi = (pi

t

α1λ− c1

t

− I(cid:48)(∆i)

(A.1)

+ α2λ

From the first order condition on ˆΠi with respect to pi

1, I obtain the optimal price for platform

i and the number of users on platform i:
pi∗
1 =
Ni∗
1 =

1
(α1λ− α2λ + c1 − ∆i)
2
1
(α1λ + α2λ− c1 + ∆i)
2t

(A.2)

(A.3)

1 and Ni

After substituting pi

1 in the deviation profit function with the expression in equation
(A.2) and (A.3), we can derive the optimal level of R&D investment from the first order condition
on ˆΠi with respect to ∆i:

(A.4)

(A.5)

(A.6)

I(cid:48)(∆i∗) =

(α1λ + α2λ− c1 + ∆i∗)

1
2t

Given that I(cid:48)(∆) = k∆2/2, the optimal level of R&D investment ∆i∗ is:

∆i∗ =

α1λ + α2λ− c1

2tk − 1

Then, the optimal deviation profits for platform i can be written as:

2(2tk − 1)(α1λ + α2λ− c1)2
1Variables corresponding to deviation are denoted with a hat.

ˆΠi∗ =

k

80

Comparing ˆΠi∗ in equation (A.6) to Πi∗ in equation (1.41), assumption 1 can be derived as no

deviation condition (Πi > ˆΠi).

Derivation of Assumption 3

2 = α2 ˜nB

In an equilibrium with single-homing on the content provider side under tying, platform B
should not have an incentive to deviate by charging ˆpB
1 to content providers and serving
both types of content. Hence, assumption 3 requires that the profits from serving both exclusive and
non- exclusive content are lower than the profits from serving only exclusive content. If platform
B charges ˆpB
2 = 1,
1 )/t. The profit maximization problem
implying that ˜nB
in the price competition gives the optimal price, ˆpB∗

1 so that it can attract both types of content providers, ˜nB

2 = α2 ˜nB
1 = 1−(vM + α1λ− ˜pA

2 = λ and ˜N B

1 )/t and ˜N B

1 = (α1λ− ˆpB

1 = (α1λ + c1 − ˆ∆B∗)/2.

After solving the profit-maximizing problem with respect to ˆ∆B, we have the first order condition
in the second stage for platform B, I(cid:48)( ˆ∆B∗) = (α1λ− c1 + ˆ∆B∗)/(2t). Provided that I(cid:48)(∆) = (k∆2)/2,
the optimal level of R&D investment, ˆ∆B∗ is:

ˆ∆B∗ =

α1λ− c1
2tk − 1

Then, the optimal deviation profits for platform B can be written as:

ˆΠB∗ =

k

2(2tk − 1)(α1λ− c1)2 − α2

1−

k

2tk − 1

(cid:110)

(sM + α1λ− c1)(cid:111)

(A.7)

(A.8)

On the other hand, from Equation (1.37) and (1.42), the optimal profits for platform B is given

(A.9)
From the comparison of ˆΠB∗ in Equation (A.8) to ˜ΠB∗ in Equation (A.9), we obtain assumption

2tk − 1

˜ΠB∗ =

k

(α1λ + α2λ− c1)2

by:

3.

81

APPENDIX B

PROOFS AND DERIVATIONS IN CHAPTER 2

Proof of Lemma 1.
Proof. First, consider optimal Ii when Ij ≤ Io. Note that Φ(cid:48)(Ij) < 1/3 for all Ij ∈ [0, Io). If firm i
invests less than the rival, dE πi(Ii)/dIi = Ii/(3Ij)−Φ(cid:48)(Ii). Because dE πi(0)/dIi = 0, dE πi(Ij)/dIi >
0, and Φ(cid:48)(Ii) is convex, dE πi(Ii)/dIi > 0 for all Ii ∈ [0, Ij], implying that firm i’s profit increases as
i )− Φ(cid:48)(Ii).
Ii increases up to Ii = Ij. If firm i invests more than the rival, dE ¯πi(Ii)/dIi = 1/2− I2
i ) is concave, whereas MC = Φ(cid:48)(Ii) is convex for all Ii. Coupled with
Note that MR = 1/2− I2
dE ¯πi(Ij)/dIi > 0, there exists a unique ¯I∗
= argmax E ¯πi ∈ (Ij, ¯c). Therefore, firm i’s profit is
maximized at ¯Ii(Ij) when Ij ≤ Io.

j /(6I2

j /(6I2

i

Second, suppose Ij ≥ Io. Note that Φ(cid:48)(Ij) > 1/3 for all Ij ∈ (Io, ¯c]. If firm i invests more
than the rival, MR < MC at Ii = Ij and its profit declines further as Ii increases because of the
convexity of MC and concavity of MR for all Ii. If firm i invests less than the rival, dE πi(0)/dIi = 0,
dE πi(Ij)/dIi < 0. Then, there exists a unique I∗
= argmax E πi ∈ (0, Ij), implying that firm i’s best
response is Ii(Ij) when Ij ≥ Io.
(cid:3)

i

Proof of Lemma 2.
Proof. The method of derivation is mainly based on Plum(1992). Substituting p = Pj(c) and
rearranging Equation (2.1), we have:

From the definition of ϕji and P(cid:48)

Pj(c)− ϕi j(c)

P(cid:48)
j(c) =
¯c− c
j(ϕji(c)) = Pi(c)−c
¯c−ϕji(c):
P(cid:48)
i(c)
(cid:48)
ji(c) =
P(cid:48)
j(ϕji(c))
ϕ
Pi(c)− ϕji(c)
Pi(c)− c

=

· ¯c− ϕji(c)
¯c− c

82

(B.1)

The equilibrium can be described by Equation (B.1) and the following boundary value:

Pi( ¯c) = ϕji( ¯c) = ¯c,

ϕji( ¯c− Ii) = ¯c− Ij

(B.2)

Taking the logarithm of Equation (B.1) and differentiating with respect to c, we have:

+

∂
∂c

ln

¯c− c

i(c)− 1
Pi(c)− c

(cid:18) ¯c− ϕji(c)
(cid:19)
i(c)− ϕ(cid:48)
P(cid:48)
ji(c)
Pi(c)− ϕji(c) − P(cid:48)
(Pi(c)− c)( ¯c− c) − Pi(c)− ϕji(c)−( ¯c− c)
¯c− ϕji(c)
1
∂
(Pi(c)− c)( ¯c− c)
¯c− c
∂c
ϕji(c)− c + Pi(c)− Pi(c)
= − Pi(c)− ϕji(c)
(cid:18) ¯c− ϕji(c)
(Pi(c)− c)( ¯c− c)

(Pi(c)− c)( ¯c− c) +

1
¯c− c

(cid:19)

−

+

=

+

ln

= 3 ∂
∂c

ln

¯c− c

(cid:18) ¯c− ϕji(c)
(cid:19)
(cid:18) ¯c− ϕji(c)

¯c− c

(cid:19)

+

∂
∂c

ln

¯c− c

(cid:48)
ji(c)) =
ln(ϕ

∂
∂c

Therefore, we obtain:

where A is some positive constant. Solving the differential equation given by (B.1) and applying

(cid:16) ¯c− ϕji(c)

¯c− c

(cid:17)3

(cid:48)
ji(c) = A
ϕ

1 +(cid:114)1−( 1

¯c− c
− 1
I2
j

I2
i

the boundary condition, we derive the solution:

Pi(c) = ¯c−

Proof of Proposition 2.2.

)( ¯c− c)2

(B.3)

(cid:3)

Proof. Suppose that I1 > I2. In firm 1’s profit maximization problem, the first order condition with
respect to I1 yields:

Φ(cid:48)(I1) =

( ¯c− c)( ¯c− ϕ21(c))

I2
1 I2
Similarly, the first order condition for firm 2 is given by:

¯c−I1

dc

dc

(B.4)

(B.5)

Þ ¯c
Þ ¯c

Φ(cid:48)(I2) =

¯c−I2

( ¯c− c)( ¯c− ϕ12(c))

I1I2
2

83

(cid:48)
12(t)dt
ϕ

(B.6)

Substituting ϕ21(c) with t in Equation (B.4), we have:

( ¯c− ϕ12(t))( ¯c− t)

Φ(cid:48)(I1) =

¯c−I2

Þ ¯c
12(c) = 1/(cid:114)
12( ¯c− I2) =(cid:112)I1/I2 and ϕ(cid:48)

I2
1 I2
− 1
I2
1

1−( 1
I2
2

(cid:112)I1/I2 and I1/(I2 · ϕ12( ¯c)) = 1. Because(cid:112)I1/I2 > 1 and ϕ(cid:48)(cid:48)

From Equation (2.2), we have ϕ(cid:48)
)( ¯c− c)2. Dividing the expression inside the
integral sign in right-hand side of Equation (B.5) by the one in right-hand side of Equation (B.6), we
have I1/(I2 · ϕ12(c)). From ϕ(cid:48)
12( ¯c) = 1, we obtain I1/(I2 · ϕ12( ¯c− I2)) =
)( ¯c − c)/(1−( 1
−
I2
)( ¯c − c)2)3/2 < 0 for all c ∈ [ ¯c − I2, ¯c], we have I1/(I2 · ϕ12(c)) ≥ 1 for all c ∈ [ ¯c − I2, ¯c], which
1
I1
implies that right-hand side of Equation (B.5) is greater than right-hand side of Equation (B.6).

12(c) = −( 1
I2

− 1
I1

Given I1 > I2, right-hand side of Equation (B.5) is greater than right-hand side of Equation (B.6)
but it leads to a contradiction because Φ(cid:48)(I1) > Φ(cid:48)(I2). Similarly, both Equation (B.5) and (B.6)
cannot be satisfied if I1 < I2. Therefore, firms choose I1 = ˆI1 = I2 = ˆI2 = I∗
UI in the equilibrium.
In the symmetric equilibrium, pricing strategy in the second stage can be derived as P(c) = c+ ¯c
2

from Equation (2.2) and in the first stage, firms profit maximizing problem becomes:

Þ ¯c

¯c−I∗
UI

Φ(cid:48)(I∗

UI) =

( ¯c− c)2
I∗3
UI

dc =

1
3

(cid:3)

which implies that I∗

UI and I∗

CI are identical.

Proof of Lemma 3.

Proof. When firms’ cost information is observable, the expected price when firm 1 wins the market
can be calculated as:

E(p1|p1 < p2)· Pr[p1 < p2] = E(c2|c1 < c2)· Pr[c1 < c2]

Þ c

f1(x)dx f2(c)dc

c

¯c−I1
F1(c)cF(cid:48)

2(c)dc

Þ ¯c
Þ ¯c

¯c−I2

¯c−I2

=

=

84

Similarly, the expected price when firm 2 wins the market is:
E(p2|p1 > p2)· Pr[p1 > p2] = E(c1|c1 > c2)· Pr[c1 > c2]

Þ c

Þ ¯c
Þ ¯c
Þ ¯c

¯c−I2

¯c−I2

=

=

=

c

¯c−I1

¯c−I2

f2(x)dx f1(c)dc

cF2(c) f1(c)dc

(cid:104)(cid:8)1− F1(c)(cid:9) d

dc

(cid:8)cF2(c)(cid:9)(cid:105)

(using integration by parts)

dc

Therefore, the expected price is given by:

E(p) = E(p1|p1 < p2)· Pr[p1 < p2] + E(p2|p1 > p2)· Pr[p1 > p2]

Þ ¯c

(cid:104)

=

¯c−I2

2(c)− F1(c)F2(c)(cid:105)

F2(c) + cF(cid:48)

dc

(B.7)

When firms’ cost information is private, given c, the expected price when firm 1 wins the market

can be written as:
E(p1|p1 < p2)· Pr[p1 < p2] =

p[1− F2(φ2(p)]dF1(φ1(p))

(using integration by parts)

F1(φ1(p)) d
dp

{p[1− F2(φ2(p))]}dp

From Equation (2.1), we have d

dp{p[1− F2(φ2(p))]} = −φ1(p)F(cid:48)

2(φ2(p))φ(cid:48)

(B.8)
2(p). Equation (B.8)

Þ ¯c
Þ ¯c

p

p

= −

can be rewritten as:

E(p1|p1 < p2)· Pr[p1 < p2] =

Substituting p = P2(c), we obtain:

F1(φ1(p))φ1(p)F(cid:48)

(cid:48)
2(φ2(p))φ
2(p)dp

(B.9)

E(p1|p1 < p2)· Pr[p1 < p2] =

F1(ϕ12(c))ϕ12(c)F(cid:48)

2(c)dc

Þ ¯c
Þ ¯c

p

¯c−I2

85

Similarly, the expected price when firm 2 wins the market can be derived from Equation (B.9):

E(p2|p1 > p2)· Pr[p1 > p2] =

F2(φ2(p))φ2(p)F(cid:48)

(cid:48)
1(φ1(p))φ
1(p)dp

Þ ¯c

p

Substituting p = P2(c), we obtain:
E(p2|p1 > p2)· Pr[p1 > p2] =

Þ ¯c
Þ ¯c

¯c−I2

=

¯c−I2

(cid:48)
1(ϕ12(c))ϕ
12(c)dc

cF2(c)F(cid:48)

(cid:104)(cid:8)1− F1(ϕ12(c))(cid:9) d

(cid:8)cF2(c)(cid:9)(cid:105)

dc

(using integration by parts)

dc

Therefore, the expected price is given by:

Þ ¯c

E(p) = E(p1|p1 < p2)· Pr[p1 < p2] + E(p2|p1 > p2)· Pr[p1 > p2]
2(c)− F2(c)− cF(cid:48)

2(c) + F1(ϕ12(c))(cid:8)ϕ12(c)F(cid:48)

F2(c) + cF(cid:48)

2(c)(cid:9)(cid:105)

(cid:104)

=

¯c−I2

dc

(B.10)

(cid:3)

Derivation of Equilibrium in the Baseline Model: Complete Information Case

In the complete information case, given that each firm’s investment level, firm i’s expected

payoff is EΠi(ci) = E(cj − ci|ci < cj < ck, ∀k (cid:44) i, j)· Pr[ci < cj < ck, ∀k (cid:44) i, j]− Φ(Ii):
E(cj − ci|ci < cj < ck, ∀k (cid:44) i, j) =
Þ ¯c
where gi(x|∆k < ∆j < ∆i, ∀k (cid:44) i, j) = gi(x)· Pr[∆k < x, ∀k (cid:44) i]/Pr[∆k < ∆i, ∀k (cid:44) i]. Also, the
probability of having the lowest ∆ is given by:

Þ ¯c

(B.11)

0

0

∆i · gi(∆i|∆k < ∆j < ∆i, ∀k (cid:44) i, j)d∆i −
Þ ¯c


Pr[∆k < ∆i, ∀k (cid:44) i] =

n
Þ ¯c

E(∆i|∆k < ∆i, ∀k (cid:44) i) =

0

k(cid:44)i
k=1 Ik
Ii

0

Simliarly, we have:

0

gk(x)dxgi(z)dz =

∆j · gj(∆j|∆k < ∆j < ∆i, ∀k (cid:44) i, j)d∆j
Þ z
· Ii ·(cid:16) z
(cid:17)
n

k(cid:44)i Ik

k(cid:44)i Ik + 1

Ii
n
¯c(
n

n
k=1 Ik)
k=1 Ik + 1

· ¯c(
n

n

k=1 Ik)
k=1 Ik + 1

k=1 Ik dz =

k=1 Ik

¯c

E(∆j|∆k < ∆j < ∆i, ∀k (cid:44) i, j) =

86

(B.12)

(B.13)

(B.14)

=−2 ¯c/((
k(cid:44)i Ik+1)(
n

i

k=1 Ik+1)3)−

Therefore, Equation (B.11) can be rewritten as:
E(cj − ci|ci < cj < ck, ∀k (cid:44) i, j) =

¯c(
n
(
k(cid:44)i Ik + 1)(
n

k=1 Ik)

k=1 Ik + 1)

Firm i’s expected payoff is given by:

EΠi(ci) =

k=1 Ik + 1) − Φ(Ii)

¯c· Ii

(
k(cid:44)i Ik + 1)(
n
(
n

¯c

k=1 Ik + 1)2

and the first order condition for firm i is:
Φ(cid:48)(Ii) =

Equation (B.14) implicitly solves the best response, Θi(Ij). From ∂2EΠi/∂I2

Φ(cid:48)(cid:48)(Ii) < 0, we can see that the second order conditions for the firm’s maximization problem are
satisfied. The uniqueness of symmetric equilibrium can be easily checked when n = 2. The slope
of the reaction function is:

i(Ij)| = | ∂2E πi/∂Ii∂Ij
|Θ(cid:48)
2 | =
∂2E πi/∂Ii

2 ¯c

(Ii+Ij +1)3
2 ¯c

(Ii+Ij +1)3 + Φ(cid:48)(cid:48)(Ii) < 1

which is a sufficient condition for the uniqueness of the equilibrium. In the symmetric equilibrium,
firms’ optimal R&D investment level is:

Φ(cid:48)(I∗) =

¯c

(nI∗ + 1)2

(B.15)

In complete information case, the expected price at which the product is sold is calculated as:

EP∗

CI = E(cj|ci < cj < ck, ∀k (cid:44) i, j) =

(2n− 1)I∗ + 1

((n− 1)I∗ + 1)(nI∗ + 1) ¯c

(B.16)

Derivation of Equilibrium in the Baseline Model: Unobservable Investment Case

Next, we consider the situation where firms’ investment and costs are unobservable. Given that
firm i’s marginal cost is ci, suppose that firm i’s pricing strategy is Pi(ci) and let φi be the inverse

87

pricing function. Denote H(c) ≡ 1− c/ ¯c. Firm i’s profit is given by:

(cid:1) · Pr[ck > φk(Pi(ci)), ∀k (cid:44) i]− Φ(Ii)

E πi =

0

Þ ¯c


= G( ¯c; Ii)

Þ ¯c

k(cid:44)i

k(cid:44)i
G(c; Ii) d
dc

−

0

k(cid:44)i

k(cid:44)i

πi =(cid:0)Pi(ci)− ci
=

(cid:2)1− Gk(ϕki(ci))(cid:3)(cid:0)Pi(ci)− ci
(cid:1) − Φ(Ii)
=

(cid:1) − Φ(Ii)
H(ϕki(ci))Ik(cid:0)Pi(ci)− ci
(cid:1)dG(ci; Ii)− Φ(Ii)
H(ϕki(ci))Ik(cid:0)Pi(cI)− ci


H(ϕki( ¯c))Ik(cid:0)Pi( ¯c)− ¯c(cid:1)
(cid:124)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)


H(ϕki(c))Ik(cid:0)Pi(c)− c(cid:1)dc− Φ(Ii)
H(ϕki(c))Ik(cid:0)Pi(c)− c(cid:1) = −

Þ ¯c
(cid:2)1− H(z)Ii(cid:3)

Þ ¯c
H(z)Ii


−G(0; Ii)
= 0




Φ(cid:48)(Ii) = −

E πi =

d
dc

k(cid:44)i

k(cid:44)i

k(cid:44)i

k(cid:44)i

= 0

0

k(cid:44)i

H(ϕki(c))Ik

H(ϕki(z))Ik dz− Φ(Ii)

H(ϕki(z))Ik ln H(z)dz

0

k(cid:44)i

By using the Envelope Theorem, we have:

So, firm i’s expect profit can be rewritten as:

(B.17)

(B.18)

(B.19)

Then, the ex ante expected profit for firm i is:

H(ϕki(0))Ik(cid:0)Pi(0)− 0(cid:1)

The first order condition for firm i in R&D investment stage follows as:

Let’s consider the case of two firms. Assume that I1 > I2. Then, from Maskin and Riley (2000)
(Proposition 3.5. (ii)), we have φ1(p) < φ2(p) for all p ∈ (p, ¯c), where p = Pi(0),i = {1,2}. From
Equation (B.19), we obtain the following equation:

Þ ¯c

0

Φ(cid:48)(I1)− Φ(cid:48)(I2) =

Þ ¯c

0

H(z)I2H(ϕ12(z))I1 ln H(z)dz−

88

H(z)I1H(ϕ21(z))I2 ln H(z)dz

Since I1 > I2 and Φ(cid:48)(cid:48)(·) > 0, left-hand side of the equation above is positive. But right-hand side
of the equation is negative because ϕ12(c) < c < ϕ21(c) for all c ∈ (0, ¯c), leading to a contradiction.
The same argument can be applied when I1 < I2. Therefore, there does not exist asymmetric
equilibrium in the unobservable investment case.

In the symmetric equilibrium, I1 = ··· = In = I∗ and P1(c) = ··· = Pn(c),∀c ∈ [0, ¯c], implying

φ1 = ··· = φn. Then, Equation (B.19) becomes:

Φ(cid:48)(I∗) = −

H(z)nI∗

ln H(z)dz = −

Þ ¯c

0

Þ ¯c

(cid:16)

0

(cid:17)nI∗

(cid:16)

(cid:17)

1− z
¯c

ln

1− z
¯c

dz =

¯c

(nI∗ + 1)2

(B.20)

It implies that the symmetric equilibrium of R&D investment in the unobservable investment

case is identical to the one in the complete information case.

By using standard methods in Myerson (1981), the symmetric price equililbrium can be derived

as:

P1(c) = P2(c) = P∗(c) = c +

1

H(c)(n−1)I∗
1

=

(n− 1)I∗ + 1

¯c +

H(t)(n−1)I∗
c
(n− 1)I∗
(n− 1)I∗ + 1c

Þ ¯c

dt

(B.21)

for all c ∈ [0, ¯c]. In the equilibrium, the expected price at which the product is sold is:

EP∗

UI =

1

(n− 1)I∗ + 1

(n− 1)I∗
(n− 1)I∗ + 1 E(c) =

(2n− 1)I∗ + 1

((n− 1)I∗ + 1)(nI∗ + 1) ¯c

¯c +

(B.22)

Compared to the complete information case, the expected price is the same in the unobservable

investment case.

Derviation of Equilibrium in the Baseline Model: Observable Investment Case

In the observable investment case, only the rival’s investment level is commonly known, while
firms cannot observe the rival’s realized cost. The observed investment level by the rival can directly
affect firms’ pricing strategies in the second stage. Since firms can observe rival’s investment level,
it can directly affect firms’ pricing strategies in the second stage. Given the pricing equilibrium

89

(cid:0)P1(c;I), ···, Pn(c;I)(cid:1), firm i’s ex ante expected payoff is given by:

E πi(Pi, c;I) =

Þ ¯c
Þ ¯c

0

0

=

k(cid:44)i

(cid:0)Pi(c;I(cid:1) − c)· Pr(cid:2)ck > ϕki(c;I), ∀k (cid:44) i(cid:3)dGi(c)− Φ(Ii)
(cid:0)Pi(c;I)− c(cid:1)

(cid:2)1− Gk(ϕki(c;I))(cid:3)dGi(c)− Φ(Ii)
Þ ¯c
(cid:2)1− H(z)Ii(cid:3)

Þ ¯c
(cid:2)− H(z)Ii

+

(cid:2)1− H(z)Ii(cid:3) ∂H(ϕji(z;I))Ij

H(ϕki(z;I))Ik(cid:3)dz

H(ϕki(z;I))Ik dz− Φ(Ii)

H(ϕki(z;I))Ik ln H(z)

k(cid:44)i

E πi =

0

k(cid:44)i

Φ(cid:48)(Ii) =

0

j(cid:44)i




k(cid:44)i, j

∂Ii

(B.23)

By using the same method in the unobservable investment case, we obtain the following

equation:

We have the first order condition for firm i in R&D investment stage as:

Equation (B.23) implicitly defines firm i’s reaction function Ii = Θi(Ij). The first term in right-
hand side of Equation (B.23) shows the benefit from the R&D investment. The investment gives
the firm i a higher probability of having the cost advantage over rivals so that it can win the market.
The second term is an additional effect compared to the unobservable investment case.

Since the firms are ex ante homogeneous, there exists a unique symmetric Nash equilibrium
in the R&D competition stage if the stability condition is satisfied. Throughout the analysis, we
assume that the stability condition is satisfied and focus on the symmetric equilibrium. In fact, we
cannot rule out the possibility of the existence of asymmetric Nash equilibria in the first stage due to
the additional effect mentioned above. The stability condition is assumed for analytical tractability
but since there exists a unique symmetric investment equilibrium in the unobservable investment
case, the direct comparison between the two cases can be achieved under this assumption.

We cannot apply the standard theorems regarding differentiability with respect to the parameters
of the solution of a differential system because, in general, explicit forms of pricing functions and
inverse pricing functions cannot be derived. We can circumvent this difficulty by defining functions

90

1 (q; Ii);I) and βji(q;I) = G j(φ j(Pi(G−1

αi(q;I) = Pi(G−1
(q; Ii);I);I); Ij), that is, αi(q;I) is firm i’s
price as a function of its cost quantile and βji(q;I) is firm j’s cost quantile when firm i and firm j
set the same price given firm i’s cost quantile.

i

From the first order condition in the price competition stage, we have the following equations:




k(cid:44)i

dlog(1− Gk(φk(p;I); Ik))

dp

=

1

p− φi(p;I)

Summing up all equations except the one for firm i and subtracting the equation for firm i, we

find:

i(p)

−G(cid:48)

i(φi(p); Ii)· φ(cid:48)
1− Gi(φi(p))

1
n− 1
Simliarly, we obtain the equation for firm j.
1
n− 1

j(φ j(p); Ij)· φ(cid:48)
1− G j(φ j(p))

−G(cid:48)

j(p)

=

=

(cid:27)
(cid:27)

1

p− φk(p;I)

1

p− φk(p;I)

k(cid:44)i

(cid:26) −(n− 2)
p− φi(p;I) +

(cid:26) −(n− 2)
p− φ j(p;I) +

(q;Ii) +
k(cid:44)i, j
(βji(q;I);Ij) +
k(cid:44)i, j

k(cid:44)j

(B.24)

(B.25)

(B.26)

By multiplying Equation (B.25) by the inverse of Equation (B.24), we obtain:
d
dq βji(q;I) =
1− βji(q;I)

1
α(q;I)−G−1
i

−(n−2)
α(q;I)−G−1
j
−(n−2)
α(q;I)−G−1
i

(βji(q;I);Ij) +
(q;Ii) +

1
α(q;I)−G−1
j

1
α(q;I)−G−1
(βki(q;I);Ik)
k
1
α(q;I)−G−1
(βki(q;I);Ik)
k

1− q

The initial conditions are βji(0;I) = 0 and βji(1;I) = 1 and we can use the former condition to

|Ik =I∗ ∀k. Differentiating Equation (B.26)

(cid:21)

j (q; I∗)
∂q

+

1
1− q

ˆβji(q) =

(B.27)

solve the differential equation.

(cid:48)
ji(q) +
ˆβ

Let ˆαi(q) = ∂αi(q;I)
∂Ii

|Ik =I∗ ∀k and ˆβji(q) =
(cid:20)

∂ βji(q;I)
∂Ii
with respect to Ii, setting all Ik equal to I∗, we have:
(q; I∗) · ∂G−1
(q; I∗)
∂Ii

n− 1
αi(q;I∗)− G−1
i
(q; I∗) · ∂G−1
n− 1
αi(q;I∗)− G−1
i
Þ cH

i

In the symmetric investment level, each firm’s pricing function αk(q;I∗) is given by:

αk(q;I∗) = G−1

k (q; I∗) +

G−1
k

(q;I∗)(1− Gk(t; I∗)n−1)dt

(1− q)n−1

91

Substituting this expression into Equation (B.27), we have:

(cid:20)

Þ cH

G−1
k

(cid:48)
ji(q) +
ˆβ

(cid:21)

ˆβji(q)

(n− 1)(1− q)n−1
(q;I∗)(1− Gk(t; I∗)n−1)dt

· ∂G−1

j (q; I∗)
∂q

+

1
1− q

=

Þ cH

G−1
k

(n− 1)(1− q)n−1
(q;I∗)(1− Gk(t; I∗)n−1)dt

· ∂G−1

i

(q; I∗)
∂Ii

(B.28)

From Equation (B.28) and the initial condition ˆβji(0) = 0, we obtain:

ˆβji(q) = −A·(1− q)(((n−1)2 +2)I∗ + n−1)log(1− q)− I∗)− A· I∗(1− q)
where A = (n−1)((n−1)I∗+n−1)
∂G−1

I∗(((n−1)2+2)I∗+n−1)2 . By the definition of ϕji(c;I) and βji(q : I), we have:

j (Gi(c; Ii); Ij)

∂ϕji(c;I)

((n−1)2+1)I∗+n−1

I∗

(B.29)

|Ik =I∗ ∀k =

∂Ii

(cid:18) ∂ βji(Gi(c; Ii);I)

∂q

∂q

|Ik =I∗ ∀k
· ∂Gi(c; Ii)

+

∂Ii

(cid:19)

|Ik =I∗ ∀k

(B.30)

∂ βji(Gi(c; Ii);I)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂Ii
= ˆβji(q)

Subtituting the expression for ˆβji(q) in Equation (B.29) into Equation (B.30), ∂ϕji(c;I)/∂Ii|Ik =I∗ ∀k

can be rewritten as:

∂ϕji(c;I)

∂Ii

(cid:16)

1− c
¯c

(cid:17)1−I∗
(cid:19) I∗(cid:0)((n− 1)2 + 2)I∗ + n− 1(cid:1) log
(cid:16)
(cid:17)((n−1)2+1)I∗+n−1(cid:111) −(cid:16)
−(cid:16)
(cid:17) I∗

1− c
¯c

(cid:17)

1− c
¯c

1− c
¯c

1− c
¯c

¯c
I∗

|Ik =I∗ ∀k =
(cid:18)
(cid:20)
+ B·(cid:110)(cid:16)

− B·

(B.31)

(cid:17) I∗

(cid:16)

log

1− c
¯c

(cid:17)(cid:21)

1− c
¯c

where B = A· I∗.
This implies that ∂ϕji(c;I)/∂Ii|Ik =I∗ ∀k > 0, ∀c ∈ [0, ¯c]. Coupled with ∂H(c)Ij/∂c < 0,
∂H(ϕji(c;I))Ij/∂Ii|Ik =I∗ ∀k < 0, ∀c ∈ [0, ¯c]. We know that left-hand side and the first term in
right-hand side in Equation (B.23) are the same as the one in unobservable investment case but the

92

second term in right-hand side is negative for all c ∈ [0, ¯c]. Given the convexity of the R&D cost
function, Φ(·), symmetric level of R&D investment in the observable investment case cannot be
greater than the one in the unobservable investment case.

Welfare Analysis in Baseline Model

Under the box demand, the results above are ready to be applied to the welfare analysis. First,

the R&D investment level in the equilibrium in each case can be compared:

I∗
CI = I∗

UI > I∗

OI

If all agents have the same valuation of the product, social welfare is a function of minimum
actual cost, corresponding to the investment level. As shown above, whereas the investment level
in complete information and unobservable investment case is the same, the observable investment
case induces underinvestment. This finding implies that social welfare decreases in the observable
investment case, compared to the complete information and unobservable investment case.

The symmetric investment equilibrium gives rise to the standard pricing function (analogous to

Þ ¯c

c

the bidding function in auction theory):1

P∗(c; I) = c +

1

H(c)(n−1)I∗

H(t)(n−1)I∗

dt =

1

(n− 1)I∗ + 1

(n− 1)I∗
(n− 1)I∗ + 1c

¯c +

Thus, the expected price of the product is given by:

EP∗(I) =

1

(n− 1)I∗
(n− 1)I∗ + 1 E(c) =
UI > I∗

¯c +
(n− 1)I∗ + 1
= − n(n−1)I((2n−1)I+2) ¯c
(((n−1)I+1)(nI+1))2 < 0 and I∗
UI. We already
= EP∗
CI. Therefore, the expected price of the product under each regime is

((n− 1)I + 1)(nI + 1) ¯c
OI > EP∗

OI, we obtain EP∗

(2n− 1)I + 1

∂I

Since ∂EP∗(I)
showed that EP∗
UI
ranked as:

EP∗

OI > EP∗

UI = EP∗

CI

The comparison of consumer surplus under each regime follows as:

1See Myerson (1981) for the detailed derivation.

CSCI = CSUI > CSOI

93

Derviation of Equilibrium in the Oligopoly Model: Complete Information Case

In complete information case, given that each firm’s investment level, firm i’s expected payoff

is EΠi(ci) = E(cj − ci|ci < cj < ck, ∀k (cid:44) i, j)· Pr[ci < cj < ck, ∀k (cid:44) i, j]− Φ(Ii).
E(cj − ci|ci < cj < ck, ∀k (cid:44) i, j) = E( ¯c− ∆j −( ¯c− ∆i)| ¯c− ∆i < ¯c− ∆j < ¯c− ∆k, ∀k (cid:44) i, j)

(B.32)

= E(∆i − ∆j|∆k < ∆j < ∆i, ∀k (cid:44) i, j)
Þ ¯c
Þ ¯c

∆i · gi(∆i|∆k < ∆j < ∆i, ∀k (cid:44) i, j)d∆i

=

0

0

k(cid:44)i

−

0

gk(x)dxgi(z)dz

Pr[∆k < ∆i, ∀k (cid:44) i] =

∆j · gj(∆j|∆k < ∆j < ∆i, ∀k (cid:44) i, j)d∆j
gi(x)· Pr[∆k < x, ∀k (cid:44) i]
gi(x|∆k < ∆j < ∆i, ∀k (cid:44) i, j) =
Pr[∆k < ∆i, ∀k (cid:44) i]
Þ ¯c
Þ z


Þ ¯c


Þ ¯c
(cid:17)

(cid:16) z
Ii
n

n
Þ ¯c
¯c(
n
k=1 Ik

n
Ii
k=1 Ik)
k=1 Ik + 1

E(∆i|∆k < ∆i, ∀k (cid:44) i) =

· Ii ·(cid:16) z

Gk(z)· gi(z)dz

(cid:16) z

(cid:17) Ii−1

k(cid:44)i Ik · Ii
¯c

(cid:17)
n

k=1 Ik dz

0

0

0

=

=

=

k(cid:44)i

¯c

k=1 Ik

0

=

dz

¯c

¯c

Simliarly, we have:

E(∆j|∆k < ∆j < ∆i, ∀k (cid:44) i, j) =

94


k(cid:44)i Ik

k(cid:44)i Ik + 1

· ¯c(
n

n

k=1 Ik)
k=1 Ik + 1

Therefore, equation (B.32) can be rewritten as:
E(cj − ci|ci < cj < ck, ∀k (cid:44) i, j) =

¯c(
n
(
k(cid:44)i Ik + 1)(
n

k=1 Ik)

k=1 Ik + 1)

Firm i’s expected payoff is given by:

and the first order condition for firm i is:
Φ(cid:48)(Ii) =

¯c· Ii

EΠi(ci) =

(
k(cid:44)i Ik + 1)(
n
(
n
= −2 ¯c/{(
k(cid:44)i Ik +1)(
n

¯c

k=1 Ik + 1)2

k=1 Ik + 1) − Φ(Ii)

(B.33)

(B.34)

(B.35)

(B.38)

From ∂2EΠi/∂I2

k=1 Ik +1)3}− Φ(cid:48)(cid:48)(Ii) < 0, we can see that the second
order conditions for the firm’s maximization problem are satisfied. In the symmetric equilibrium,
firms’ optimal R&D investment level is:

i

Φ(cid:48)(I∗) =

¯c

(nI∗ + 1)2

In the complete information case, the expected price at which the product is sold is:

EP∗

CI = E(cj|ci < cj < ck, ∀k (cid:44) i, j) = E( ¯c− ∆j|∆k < ∆j < ∆i, ∀k (cid:44) i, j)

(2n− 1)I∗ + 1

((n− 1)I∗ + 1)(nI∗ + 1) ¯c

=

Derivation of Equilibrium in Oligopoly Model: Unobservable Investment Case

(B.36)

(B.37)

Next, we consider the situation where firms’ investment and costs are unobservable. Given that
firm i’s marginal cost is ci, suppose that firm i’s pricing strategy is Pi(ci) and let φi be the inverse
pricing function. Suppose further that H(c) ≡ 1− c/ ¯c. Firm i’s profit is given by:

πi =(cid:0)Pi(ci)− ci
=

=


k(cid:44)i

k(cid:44)i

(cid:1) · Pr[ck > φk(Pi(ci)), ∀k (cid:44) i]− Φ(Ii)

(cid:2)1− Gk(ϕki(ci))(cid:3)(cid:0)Pi(ci)− ci
(cid:1) − Φ(Ii)
(cid:1) − Φ(Ii)
H(ϕki(ci))Ik(cid:0)Pi(ci)− ci

95

Then, the ex ante expected profit for firm i is:

H(ϕki(0))Ik(cid:0)Pi(0)− 0(cid:1)

(B.39)

(B.40)

E πi =

0

Þ ¯c


= G( ¯c; Ii)

Þ ¯c

k(cid:44)i

k(cid:44)i
G(c; Ii) d
dc

−

0

= 0

k(cid:44)i

k(cid:44)i

−G(0; Ii)
= 0

(cid:1)dG(ci; Ii)− Φ(Ii)
H(ϕki(ci))Ik(cid:0)Pi(cI)− ci


H(ϕki( ¯c))Ik(cid:0)Pi( ¯c)− ¯c(cid:1)
(cid:124)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)


H(ϕki(c))Ik(cid:0)Pi(c)− c(cid:1)dc− Φ(Ii)
H(ϕki(c))Ik(cid:0)Pi(c)− c(cid:1) = −

Þ ¯c
(cid:2)1− H(z)Ii(cid:3)

Þ ¯c
H(z)Ii





Φ(cid:48)(Ii) = −

k(cid:44)i

k(cid:44)i

k(cid:44)i

E πi =

0

d
dc

H(ϕki(z))Ik ln H(z)dz

0

k(cid:44)i

H(ϕki(z))Ik dz− Φ(Ii)

H(ϕki(c))Ik

By using the Envelope Theorem, we have:

So, firm i’s expect profit can be rewritten as:

The first order condition for firm i in R&D investment stage follows as:

Þ ¯c

Let’s consider two firms case. Assume that I1 > I2. Then, from Maskin and Riley(2000b)
(Proposition 3.5. (ii)), we have φ1(p) < φ2(p) for all p ∈ (p, ¯c), where p = Pi(0),i = {1,2}. From
(B.40), we obtain the following equation:

0

H(z)I1H(ϕ21(z))I2 ln H(z)dz

H(z)I2H(ϕ12(z))I1 ln H(z)dz−

Φ(cid:48)(I1)− Φ(cid:48)(I2) =
Since I1 > I2 and Φ(cid:48)(cid:48)(·) > 0, left-hand side of the equation above is positive. But right-hand side
of the equation is negative because ϕ12(c) < c < ϕ21(c) for all c ∈ (0, ¯c), leading to a contradiction.
The same argument can be applied when I1 < I2. Therefore, there dose not exist asymmetric
equilibrium in the unobservable investment case.

0

Þ ¯c

96

In the symmetric equilibrium, I1 = ··· = In = I∗ and P1(c) = ··· = Pn(c),∀c ∈ [0, ¯c], implying

φ1 = ··· = φn. Then, (B.40) becomes:

Þ ¯c
Þ ¯c

0

0

Φ(cid:48)(I∗) = −

= −

H(z)nI∗
(cid:17)nI∗
(cid:16)

1− z
¯c

(cid:16)

ln H(z)dz

ln

1− z
¯c

(cid:17)

dz

(B.41)

=

¯c

(nI∗ + 1)2

It implies that the symmetric equilibrium of R&D investment in the unobservable investment

case is identical to the one in the complete information case.

By using standard methods, the symmetric price equililbrium can be derived as:

Þ ¯c

P1(c) = P2(c) = P∗(c) = c +

1

H(c)(n−1)I∗
1

=

(n− 1)I∗ + 1

¯c +

H(t)(n−1)I∗
c
(n− 1)I∗
(n− 1)I∗ + 1c

dt

(B.42)

for all c ∈ [0, ¯c]. In the equilibrium, the expected price at which the product is sold is:

(n− 1)I∗
(n− 1)I∗ + 1 E(c)
(n− 1)I∗
¯c
(n− 1)I∗ + 1

·

nI∗ + 1

EP∗

UI =

1

(n− 1)I∗ + 1

¯c +

1

=

=

(n− 1)I∗ + 1

¯c +
(2n− 1)I∗ + 1

((n− 1)I∗ + 1)(nI∗ + 1) ¯c

(B.43)

Compared to the complete information case, the expected price is the same in the unobservable

investment case.

Derivation of Equilibrium in Oligopoly Model: Observable Investment Case

Under the observable investment case, only rival’s investment level is commonly known, while
firms cannot observe rival’s realized cost. Since firms can observe rival’s investment level, it
can directly affect firms’ pricing strategies in the second stage. Since firms can observe rival’s
investment level, it can directly affect firms’ pricing strategies in the second stage. Given the pricing

97

By using the same method in the unobservable investment case, we obtain the following

=

k(cid:44)i

E πi(Pi, c;I) =

Þ ¯c
Þ ¯c

0

0

equilibrium(cid:0)P1(c;I), ···, Pn(c;I)(cid:1), firm i’s ex ante expected payoff is given by:
(cid:0)Pi(c;I(cid:1) − c)· Pr(cid:2)ck > ϕki(c;I), ∀k (cid:44) i(cid:3)dGi(c)− Φ(Ii)
(cid:0)Pi(c;I)− c(cid:1)

(cid:2)1− Gk(ϕki(c;I))(cid:3)dGi(c)− Φ(Ii)
Þ ¯c
(cid:2)1− H(z)Ii(cid:3)

Þ ¯c
(cid:2)− H(z)Ii

+

(cid:2)1− H(z)Ii(cid:3) ∂H(ϕji(z;I))Ij

We have the first order condition for firm i in R&D investment stage as:

H(ϕki(z;I))Ik(cid:3)dz

H(ϕki(z;I))Ik dz− Φ(Ii)

H(ϕki(z;I))Ik ln H(z)

E πi =

0

Φ(cid:48)(Ii) =

equation:

k(cid:44)i

∂Ii

k(cid:44)i




k(cid:44)i, j

0

j(cid:44)i

(B.44)

The equation (B.44) implicitly defines firm i’s reaction function Ii = Ri(Ij). The first term in
right-hand side of (B.44) shows the benefit from the R&D investment. The investment gives the
firm i higher probability of having the cost advantage over rivals so that it can win the market. The
second term is an additional effect compared to the complete information case.

Since the firms are ex ante homogeneous, there exists a unique symmetric Nash equilibrium in
R&D competition stage if the stability condition is satisfied. Throughout the analysis, we assume
that the stability condition is satisfied and focus on the symmetric equilibrium. In fact, we cannot
rule out the possibility of the existence of asymmetric Nash equilibria in the first stage due to the
additional effect mentioned above. The stability condition is assumed for analytical tractability but
since there exists a unique symmetric investment equilibrium in the unobservable investment case,
the direct comparison between two cases can be achieved under this assumption.

We cannot apply the standard theorems regarding differentiability with respect to parameters
of the solution of a differential system because in general explicit forms of pricing functions and
inverse pricing functions cannot be derived. We can circumvent this difficulty by defining functions

98

1 (q; Ii);I) and βji(q;I) = G j(φ j(Pi(G−1

αi(q;I) = Pi(G−1
(q; Ii));I);I); Ij), that is, αi(q;I) is firm i’s
price as a function of its cost quantile and βji(q;I) is firm j’s cost quantile when firm i and firm j
set the same price given that firm i’s cost quantile.

i

From the first order condition in the price competition stage, we have the following equations:




k(cid:44)i

dlog(1− Gk(φk(p;I); Ik))

dp

=

1

p− φi(p;I)

Summing up all equations except the one for firm i and subtracting the equation for firm i, we

find:

i(p)

−G(cid:48)

i(φi(p); Ii)· φ(cid:48)
1− Gi(φi(p))

1
n− 1
Simliarly, we obtain the equation for firm j:
1
n− 1

j(φ j(p); Ij)· φ(cid:48)
1− G j(φ j(p))

−G(cid:48)

j(p)

=

=

(cid:27)
(cid:27)

1

p− φk(p;I)

1

p− φk(p;I)

k(cid:44)i

(cid:26) −(n− 2)
p− φi(p;I) +

(cid:26) −(n− 2)
p− φ j(p;I) +

(q;Ii) +
k(cid:44)i, j
(βji(q;I);Ij) +
k(cid:44)i, j

k(cid:44)j

(B.45)

(B.46)

(B.47)

By multiplying (B.46) by the inverse of (B.45), we obtain:
d
dq βji(q;I) =
1− βji(q;I)

1
α(q;I)−G−1
i

−(n−2)
α(q;I)−G−1
j
−(n−2)
α(q;I)−G−1
i

(βji(q;I);Ij) +
(q;Ii) +

1
α(q;I)−G−1
j

1− q

1
α(q;I)−G−1
(βki(q;I);Ik)
k
1
α(q;I)−G−1
(βki(q;I);Ik)
k

The initial conditions are βji(0;I) = 0 and βji(1;I) = 1 and we can use the former condition to

|Ik =I∗ ∀k. Differentiating (B.47) with respect

(cid:21)

j (q; I∗)
∂q

+

1
1− q

ˆβji(q) =

(B.48)

solve the differential equation.

∂Ii

∂ βji(q;I)

(cid:48)
ji(q) +
ˆβ

Let ˆαi(q) = ∂αi(q;I)
∂Ii

|Ik =I∗ ∀k and ˆβji(q) =
(cid:20)

to Ii, setting all Ik equal to I∗, we have:
n− 1
αi(q;I∗)− G−1
i
(q; I∗) · ∂G−1
n− 1
αi(q;I∗)− G−1
i
Þ cH

(q; I∗) · ∂G−1
(q; I∗)
∂Ii

i

In the symmetric investment level, each firm’s pricing function αk(q;I∗) is given by:

αk(q;I∗) = G−1

k (q; I∗) +

G−1
k

(q;I∗)(1− Gk(t; I∗)n−1)dt

(1− q)n−1

99

(cid:20)

Substituting this expression into the equation (B.48), we have:
j (q; I∗)
∂q

(n− 1)(1− q)n−1
(q;I∗)(1− Gk(t; I∗)n−1)dt

· ∂G−1

Þ cH

(cid:48)
ji(q) +
ˆβ

G−1
k

(cid:21)

ˆβji(q)

+

1
1− q

=

Þ cH

G−1
k

(n− 1)(1− q)n−1
(q;I∗)(1− Gk(t; I∗)n−1)dt

· ∂G−1

i

(q; I∗)
∂Ii

From the equation (B.49) and the initial condition ˆβji(0) = 0, we obtain:
ˆβji(q) =
− A·(1− q)(((n− 1)2 + 2)I∗ + n− 1)log(1− q)− I∗)− A· I∗(1− q)
where A = (n−1)((n−1)I∗+n−1)
∂G−1

j (Gi(c; Ii); Ij)

∂ϕji(c;I)

I∗(((n−1)2+2)I∗+n−1)2 . By the definition of ϕji(c;I) and βji(q : I), we have:

((n−1)2+1)I∗+n−1

I∗

|Ik =I∗ ∀k =

∂Ii

(cid:18) ∂ βji(Gi(c; Ii);I)

∂q

∂q

|Ik =I∗ ∀k
· ∂Gi(c; Ii)

+

∂Ii

(cid:19)

∂ βji(Gi(c; Ii);I)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂Ii
= ˆβji(q)

(B.49)

(B.50)

|Ik =I∗ ∀k

(B.51)

∂Ii

(cid:16)

¯c
I∗

− B·

1− c
¯c

|Ik =I∗ ∀k can be rewritten as:
(cid:17)

Subtituting (B.50) into (B.51), ∂ϕji(c;I)
(cid:17)1−I∗
∂Ii
∂ϕji(c;I)
(cid:19) I∗(cid:0)((n− 1)2 + 2)I∗ + n− 1(cid:1) log
(cid:16)
(cid:17)(cid:21)
(cid:17)((n−1)2+1)I∗+n−1(cid:111) −(cid:16)
−(cid:16)
(cid:17) I∗
1− c
¯c
where B = A· I∗. It implies that ∂ϕji(c;I)
|Ik =I∗ ∀k > 0, ∀c ∈ [0, ¯c]. Coupled with ∂H(c)Ij
∂Ii

|Ik =I∗ ∀k =
(cid:18)
(cid:20)
+ B·(cid:110)(cid:16)

< 0,
|Ik =I∗ ∀k < 0, ∀c ∈ [0, ¯c]. We know that left-hand side and the first term in right-hand
side in the equation (B.44) are the same as the one in the unobservable investment case but the

∂H(ϕji(c;I))Ij

(cid:17) I∗

1− c
¯c

1− c
¯c

1− c
¯c

1− c
¯c

1− c
¯c

(B.52)

log

(cid:16)

∂Ii

∂c

100

second term in right-hand side is negative for all c ∈ [0, ¯c]. Given the convexity of R&D cost
function, Φ(·), symmetric level of R&D investment in the observable investment case cannot be
greater than the one in the unobservable investment case.

Welfare Analysis in Oligopoly Model

Under the box demand, the results above are ready to be applied to welfare analysis. First, R&D

investment level in the equilibrium under each regime can be compared.

I∗
CI = I∗

UI > I∗

OI

(B.53)

If all agent have the same valuation on the product, social welfare is a function of minimum
actual cost, corresponding to the investment level. As shown above, whereas the investment level in
the complete information case and the unobservable investment case are the same, the observable
investment case induces underinvestment. It implies that social welfare decreases in the observable
investment case.

The symmetric investment equilibrium gives rise to standard pricing function (analogous to

Þ ¯c

1

H(c)(n−1)I∗
1

¯c +

dt

H(t)(n−1)I∗
c
(n− 1)I∗
(n− 1)I∗ + 1c

bidding function in auction theory):

P∗(c; I) = c +

(n− 1)I∗ + 1
Thus, expected price of the product is given by:

=

EP∗(I) =

1

(n− 1)I∗ + 1

(n− 1)I∗
(n− 1)I∗ + 1 E(c)

¯c +

∂I

Since ∂EP∗(I)
showed that EP∗
UI
ranked as:

= − n(n−1)I((2n−1)I+2) ¯c
(((n−1)I+1)(nI+1))2 < 0 and I∗
UI. We already
= EP∗
CI. Therefore, the expected price of the product under each regime is

OI, we obtain EP∗

OI > EP∗

=

(2n− 1)I + 1

((n− 1)I + 1)(nI + 1) ¯c
UI > I∗

EP∗

OI > EP∗

UI = EP∗

CI

101

The comparison of consumer surplus under each regime follows as:

CSCI = CSUI > CSOI

Derivation of Equilibrium in Discrete Cost Model: Complete Information Case

As mentioned in the footnote in section 2.6.1, when possible realizations of the R&D result are
simplified to two discrete types, all results except social welfare are sustained in the broad class of
demand function. Therefore, in this section, we analyze the model under the assumption of general
downward-sloping demand. D(p) is a market demand with D(cid:48)(p) < 0 and define profits of a firm
with low cost, excluding R&D cost, as π(p) = (p− c)D(p). We assume that π(p) is strictly concave
in price. In order to simplify the result, denote the monopoly price for the firm with low cost as
p∗ = argmax
π(p) and define ˆp = min{ ¯c, p∗}. Let’s further denote the maximized profit with low
p
cost as ˆπ = π( ˆp). The timing of the game is the same as the baseline model.

Given firms’ R&D investment level, firm i can earn positive profit, ˆπ, only if firm i succeeds
in reducing its cost whereas the rival fails in full disclosed case. Therefore, firm i’s expected
payoff is given by EΠi = Ii(1− Ij) ˆπ − Φ(Ii). Then, the first-order condition in the R&D stage is
(1− Ij) ˆπ− kIi = 0 and, by symmetry, the equilibrium of R&D investment is given by:

I∗
CI =

ˆπ

ˆπ + k

Derivation of Equilibrium in Discrete Type Model: Unobservable Investment Case

When cost information is private, a two stage Bayesian game is considered. We first show
strategies of firms in equilibrium and then show that firms have no incentive to deviate from the
strategy. In the second stage, high cost firms would set price at the marginal cost. For privately
informed low cost firms, however, there does not exist a pure strategy equilibrium in the second
stage subgame because it has two motivations under cost uncertainty. On the one hand, the firm
is tempted to enjoy market power given that the rival remains at high cost. On the other hand, the
firm try to undercut and capture the market if it believes the rival also succeeds in R&D. Regardless
of which strategy the rival uses in the second stage, the low cost firm i for sure expects to earn

102

(1− Ij) ˆπ so two moves of the low cost firm should be balanced at the expected profit. Let Ki(p) be
the distribution function of firm i which describes the mixed strategy the low cost firm plays. Then,
it requires:

(1− Ij) ˆπ = (1− Ij)(p− c)D(p) + Ij · Pr(p < pj)(p− c)D(p)

= (1− Ij)(p− c)D(p) + Ij(1− Kj(p))(p− c)D(p)

(B.54)

in the support of [p, ˆp], where p satisfies πi(p) = (1− Ij) ˆπ. From Equation (B.54), we obtain:

Kj(p) =

− (1− Ij) ˆπ
Ij(p− c)

1
Ij

(B.55)

Suppose that (I∗

= ˆπ

i and invests by I∗

i , I∗

j ) is a Nash equilibrium in the first stage. If firm i deviates from I∗

still believes that the rival plays I∗
in Equatioh (B.55) over [p∗
price in the second stage, its expected profit is given by EΠi = Ii(1− I∗
same shape of best response in the complete information case if replacing I∗
that I∗
UI

i , firm j
j . Then, firm j plays the mixed strategy describes
j ) ˆπ. No matter how firm i sets
j ) ˆπ − Φ(Ii) which gives the
j with Ij. It implies

ˆπ+k is a Nash equilibrium in R&D stage.

j, ˆp] where p∗

j satisfies π(p∗

j) = (1− I∗

Derivation of Equilibrium in Discrete Type Model: Observable Investment Case

In the observable investment case, there does not exist pure strategy pricing equilibrium because
firms compete with prices under cost uncertainty. The nature of mixed strategy in the second stage
is similar to the one in the unobservable investment case. However, we should consider the case
where firms invest asymmetrically as well because of the observability of investment level in this
case. Let us assume that firm 1 invests more than firm 2 in the first stage. Suppose that both firms
use the same pricing strategy as in the unobservable investment case. Then, it turns out that firm
1’s minimum price level in the support of pricing strategy would be less than the rival’s minimum
price level and obviously firm 1 does not have any incentive to set price less than rival’s minimum
price. So this equilibrium cannot be sustained anymore.

Rather, firm 1 would set prices at higher level as it can expect that it is more likely to have the
market power. And for firm 2, it would set prices aggressively relative to the rival because it knows

103

its investment level is lower than the rival. Thus in equilibrium, the expected payoff to the low cost
firm is equalized for both firms even when firms invest asymmetrically. Given that I1 > I2, the
distribution used in mixed strategy for firm 1 is:
K1(p) =

− (1− I2) ˆπ
I1(p− c)
and the distribution describing the strategy for firm 2 is:
− (1− I2) ˆπ
I2(p− c)

K2(p) =

1
I1

1
I2

The distributions show that firm 1 sets price less aggressively compared to the rival and note
that K1(p) has a mass point on ˆp. In the observable investment case, therefore, firms have two types
of investment strategy: aggressive investment and passive investment. Firms’ expected profit for
each investment strategy is given by:



Πi =

Ii(1− Ii) ˆπ− Φ(Ii)

if Ii ≤ Ij

Ii(1− Ij) ˆπ− Φ(Ii)

if Ii ≥ Ij





Ii =

Ij

if Ij ≤ ˆπ
2 ˆπ+k

ˆπ

2 ˆπ+k

if Ij ≥ ˆπ
2 ˆπ+k

Ii =

Ij

if Ij ≥ ˆπ
ˆπ+k
k(1− Ij) ˆπ if Ij ≤ ˆπ
1
ˆπ+k

If a firm choose the aggressive investment strategy, the optimal investment level is:

And if a firm choose the passive investment strategy, the optimal investment level is:

Denote Π A
the expected profit of firm i when it chooses the aggressive investment strategy and
i
the expected profit when firm i chooses the passive investment strategy. Lemma A.1 says how

ΠP
i

firms use these two investment strategy given the rival’s R&D level.
Lemma 1. There exists ˜I ∈ (

ˆπ+k) such that Π A
ˆπ

i > ΠP

2 ˆπ+k ,

ˆπ

i , ∀Ii < ˜I and Π A

i < ΠP

i , ∀Ii > ˜I.

104

If the rival’s R&D level is very low, then a firm would be willing to invest aggressively so that
it can effectively increase the chance to hold a dominant position in the market in terms of cost.
But as the rival increases its investment level, switching the strategy would be profitable because as
shown above, once the rival observes that its own investment level is greater than the other, the rival
tends to set price at p hat. Then, this firm may want to invest passively and enjoy the undercutting
chance.

The fact that ˜I < ˆπ

ˆπ+k implies that there is no symmetric equilibrium in R&D stage in the
observable investment case. The equilibrium of investment in the observable investment case is:

1 OI, I∗
(I∗

2 OI) = (

ˆπ

2 ˆπ + k

(1 +

),

ˆπ
k

ˆπ

2 ˆπ + k

) or (

ˆπ

2 ˆπ + k ,

ˆπ

2 ˆπ + k

(1 +

))

ˆπ
k

Comparing the investment equilibrium with the one in previous cases, the aggregate level of
investment in the observable investment case is less(greater) than the one under other cases if
ˆπ < (>) k. Interestingly, regardless of value of parameters, both firms earn more profits in the
observable investment case. This is because unlike the baseline model, one firm’s additional
investment does not affect the rival’s pricing strategy. So the firm with passive investment strategy
does not price aggressively and, in turn, price competition becomes softened.

If we assume that demand is inelastic and the valuation of all consumer on the product is v, we
can easily see that I∗
UI achieves socially desirable level and the observable investment case
CI
induces over- or under-investment, depending on the value of parameters. Moreover, considering
that firms capture more surplus in the observable investment case, it hurts consumers as well.

= I∗

Derivation of Equilibrium in Asymmetric Firms Model

I first consider the complete information case. The lower bound of firm i’s cost distribution
is reduced by Ii after the investment, i.e, the support of firm i’s cost distribution is changed to
c ∼ U[ ¯c− δi − Ii, ¯c], or U[ ¯c− ˜δi, ¯c]. Let ˜Fi(c) = 1− ¯c−c
be the c.d.f of firm i’s cost after R&D
˜δi
activities.

In price competition stage, if ci < cj, firm i sets price at rival’s marginal cost and its profit
is cj − ci. Otherwise, firm i would charge its marginal cost and earn zero profit. Thus, firm i’s

105

expected payoff E πi is equal to E[cj − ci|ci < cj]Pr[ci < cj]. If ˜δ1 > ˜δ2, firm 1’s expected payoff
can be written as:

Þ ¯c
Þ ¯c− ˜δ2

¯c− ˜δ1

¯c− ˜δ1

c[1− ˜F2(c)]d ˜F2(c)−

c[1− ˜F2(c)]d ˜F2(c)−

c[1− ˜F2(c)]d ˜F1(c)− Φ(Ii)

cd ˜F1(c|c < ¯c− ˜δ2)

c[1− ˜F2(c)]d ˜F1(c|c > ¯c− ˜δ2)− Φ(Ii)

E π1(c;I) =

=

−

¯c− ˜δ2

Þ ¯c
Þ ¯c
Þ cH

¯c− ˜δ2

¯c− ˜δ2

Þ ¯c

(B.56)

(B.57)

c[1− ˜F2(c)]d ˜F2(c)−

Þ ¯c

¯c− ˜δ1

c[1− ˜F2(c)]d ˜F1(c)− Φ(Ii)

− Φ(Ii)
+ δi) = 1/3. Then, firm i’s best response in R&D competition stage is

˜δ2
2
6 ˜δ1
If ˜δ1 < ˜δ2, firm 1’s expected payoff is:

( ˜δ1 − ˜δ2) +

1
2

=

− Φ(Ii)

E π1(c;I) =

¯c− ˜δ2
˜δ2
1
6 ˜δ2
i such that Φ(cid:48)(I∗

=

i

Define I∗

given by:

Φ(cid:48)(Ii) =



1

2 − ˜δ2
j
6 ˜δ2
i
˜δi
3 ˜δj

if Ii < I∗

i

if Ii ≥ I∗

i

(B.58)

which gives the equilibrium in R&D competition stage at the intersection of best responses. By
using the same method in the section 2.4, the expected price in the complete information case is
given by:

E( ˜pCI) =

˜F2(c) + c ˜F(cid:48)

2(c)− ˜F1(c) ˜F2(c)dc

and consumer surplus is derived by CS = v − E( ˜pCI), followed by total welfare, W = CS +
 E πi.

¯c− ˜δ2

(B.59)

Þ ¯c

106

In the observable investment case, by applying the same method in section 2.3.3, the pricing

(by using integration by parts)

(B.60)

(B.61)

(B.62)

strategy for firm i can be derived as:

Pi(c;I) = ¯c−

and the inverse pricing function is:

1 +(cid:114)1−( 1

¯c− c
− 1
˜δ2
j

˜δ2
i

)( ¯c− c)2

φi(p;I) = ¯c−

2( ¯c− p)
− 1
1 +( 1
˜δ2
˜δ2
j
i

)( ¯c− p)2

E π1(P1, c;I) =

=

¯c− ˜δ1

Þ ¯c
Þ ¯c
Þ ¯c

¯c− ˜δ1

= −

¯c− ˜δ1

By using Envelope Theorem, we obtain:

Given the pricing equilibrium (P1(c;I), P2(c;I)), firm 1’s ex ante expected payoff is given by:

(P1(c;I)− c)· Pr[c2 > ϕ21(c;I)]d ˜F1(c)− Φ(I1)

(P1(c;I)− c)(cid:8)1− ˜F2(ϕ21(c;I))(cid:9)d ˜F1(c)− Φ(I1)

˜F1(c) d
dc

(P1(c;I)− c)(cid:8)1− ˜F2(ϕ21(c;I))(cid:9)dc− Φ(I1)
(P1(c;I)− c)(cid:8)1− ˜F2(ϕ21(c;I))(cid:9) = −(cid:8)1− ˜F2(ϕ21(c;I))(cid:9)
˜F1(c)(cid:8)1− ˜F2(ϕ21(c;I))(cid:9)dc− Φ(I1)

d
dc

Substituting this expression into the integral, we then have:

¯c− ˜δ1

E π1(P1, c;I) =

Þ ¯c
∂ ˜F1(c)(cid:8)1− ˜F2(ϕ21(c;I))(cid:9)
(cid:20) ∂ ˜F1(c)

∂I1

Þ ¯c
Þ ¯c

¯c− ˜δ1

The first order condition with respect to I1 is:
Φ(cid:48)(I1) =

dc−(−1) ˜F1( ¯c− ˜δ1)

(cid:8)1− ˜F2(ϕ21( ¯c− ˜δ1;I))(cid:9)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:8)1− ˜F2(ϕ21(c;I))(cid:9)(cid:21)

dc

= 0

(cid:8)1− ˜F2(ϕ21(c;I))(cid:9) + ˜F1(c) ∂
(cid:8)1− ˜F2(ϕ21(c;I))(cid:9) < 0 for all c ∈ [ ¯c − ˜δ1, ¯c], cost-reducing investment

(B.63)

∂I1

=

¯c− ˜δ1

∂I1

From the fact that ∂
∂I1

has a negative strategic effect as in the symmetric model. Applying the same method in the section

107

2.4, the expected price in the observable investment case is:

Þ ¯c

¯c− ˜δ2

E( ˜pPD) =

˜F2(c) + c ˜F(cid:48)

2(c) + ˜F1(ϕ12(c;I)){ϕ12(c;I) ˜F(cid:48)

2(c)− ˜F2(c)− c ˜F(cid:48)

2(c)}dc

(B.64)

The equation (B.64) and the corresponding equation for firm 2 give the industry profits and

consumer surplus is derived by CS = v − E( ˜pPD). Total welfare follows as W = CS +
 πi.

108

APPENDIX C

PROOFS IN CHAPTER 3

Proof of Proposition 3.2.

Proof. This proof proceeds in three steps. First, we show that there exists some positive value of ν
for which firm 1 would want to offer a grandfather clause. Second, we show that if firm 1 wants to
offer a grandfather clause for some value ν, it must also want to offer a grandfather clause for any
average quality improvement lower than that value. Conversely, we show that if firm 1 does not
want to offer a grandfather clause for some value of ν, it will not want to offer one for any larger
average quality improvement. Finally, we show that there exists a value of ν for which firm 1 does
not want to offer a grandfather clause.
Step 1. To show existence of a ν for which firm 1 would want to offer a grandfather clause, we
must first show that prices and consequently, firm 1’s profit, are continuous in ν. This requires a
continuous implicit function that gives the equilibrium prices in terms of ν. First, suppose that
firm 1’s equilibrium price is increasing in ν when firm 1 does not offer a grandfather clause. This
assumption will be confirmed in Step 2. Let

(cid:2)p2(1− F(ν + p2 − p1))(cid:3)(cid:17)

(C.1)

(cid:16) ∂

∂p1

(cid:2)p1F(ν + p2 − p1)(cid:3),

∂
∂p2

H(ν, p1, p2) :=

Note that under Assumption 2, p1F(ν + p2 − p1) represents either firm 1’s expected profit under
uniform pricing, or double the non-constant component of expected profit when it offers a grand-
father clause. Similarly, p2(1− F(ν + p2 − p1)) represents either firm 2’s expected profit when it
believes that firm 1 will set a uniform price, or double that amount when firm 1 is expected to offer
a grandfather clause.1

1Thus, the price that satisfies firm 2’s first order condition is invariant to its beliefs regarding

firm 1’s decision to grandfather.

109

Let J(ν, p1, p2) represent the matrix of the partials of H(ν, p1, p2):

−2 f(γ) + p1 f (cid:48)(γ)

f(γ) + p2 f(γ)



∂H(ν, p1, p2)
∂(p1, p2)

:= J(ν, p1, p2) =

f(γ)− p1 f (cid:48)(γ)
−2 f(γ)− p2 f(γ)

(C.2)

where we define γ ≡ ν + p2 − p1. In equilibrium, it must be that H(ν, p1, p2) = 0. Thus, we need
to find a continuous implicit function g(γ) = (p1, p2) such that H(ν, g(ν)) = 0. Such a g exists
if H(ν, p1, p2) is continuously differentiable and J(ν, p1, p2) is invertible. The first condition is
satisfied by assumption. The second follows if the determinant of J(ν, p1, p2),

|J(ν, p1, p2)| = 3 f(γ)2 +(p2 − p1) f(γ) f (cid:48)(γ)

(C.3)

is non-zero. The first term on the right-hand side of Equation (C.3) is clearly positive. Our
assumption that idiosyncratic preferences are independently and identically distributed in the initial
and mature market periods together with Assumption 2 imply that Lemma 1 applies. Therefore,
p2 − p1 < 0 and γ > 0. Then, single-peakedness and symmetry imply that f (cid:48)(γ) < 0, such that
the second term on the right-hand side of Equation (C.3) is likewise positive, as is |J(ν, p1, p2)|.
Thus, J(ν, p1, p2) is invertible and the implicit function theorem guarantees the existence of a
continuously differentiable g. Therefore, the function that gives firm 1’s equilibrium expected
profit in terms of ν is likewise continuous. Note that if firm 1 chooses to set a uniform price in the
mature market period, its equilibrium expected profit is given by π∗
1(ν) ≡ g1(ν)F(ν + g2(ν)− g1(ν))
where g(ν) = (g1(ν), g2(ν)).

Recall that when ν = 0, firm 1’s expected profit in the mature market period under uniform
pricing is 1/(4 f(0)) (this was also its realized profit in the initial period). Under Assumption 2, if
firm 1 chooses to offer a grandfather clause at some positive ν, its expected profit is 1/(4 f(0)) +
π∗
1(ν)/2. This will exceed firm 1’s non-grandfathered expected profit if and only if π∗
1(ν) <
1/(2 f(0)). Continuity of π∗
1 implies that for any 
, there exists ν such that if |ν| < ν, |π∗
1(ν)−
1/(4 f(0))| < 
. By choosing 
 < 1/(4 f(0)), we can ensure that for any ν < ν, firm 1’s profits will
be higher with a grandfather clause than without one.

110

Step 2. For the proof of this step, it suffices to show that π∗
immediately if p1 and γ are increasing in ν.

1(ν) is increasing in ν. This follows

According to the implicit function theorem the derivative of prices with respect to ν is
−(J(ν, p1, p2))−1DνH(ν, p1, p2), where DνH(ν, p1, p2) represents the derivative of H(ν, p1, p2)
with respect to ν. After some algebraic manipulation, we find that:

dp1
dν

=

f(γ)( f(γ)− p1 f (cid:48)(γ))

3 f(γ)2 +(p2 − p1) f(γ) f (cid:48)(γ)

(C.4)

dγ
dν

f(γ)2

=

3 f(γ)2 +(p2 − p1) f(γ) f (cid:48)(γ)

(C.5)
Note that the denominator in Equations (C.4) and (C.5) equals |J(ν, p1, p2)|, which is already
established to be positive. Likewise, both numerators are positive. In particular, with regard to
Equation (C.4), as established in the previous step, f (cid:48)(γ) is negative because of Lemma 1, single-
peakedness, and symmetry. Therefore, because price is increasing and quantity is non-decreasing
in ν, π∗

1(ν) is increasing in ν.

Because π∗

1(ν) is increasing, π∗

1( ˆν) < 1/(2 f(0)) implies that π∗

1(ν) < 1/(2 f(0)) whenever ν < ˆν.
That is, if firm 1 wants to offer a grandfather clause for some ˆν, it will want to offer a grandfather
clause for all ν < ˆν. Conversely, whenever π∗
1( ˆν) > 1/(2 f(0)), this will also be the case for all ν > ˆν.
Thus, if firm 1 does not want to offer a grandfather clause at ˆν, it will not want to offer one for any
ν > ˆν.
Step 3. To complete the proof we must show that there exists some ¯ν such that π∗
1( ¯ν) > 1/(2 f(0)).
Because quantity is bounded above by 1, to guarantee this, we need to show that price is unbounded
in ν—or alternatively, that its derivative provided in Equation (C.4) is bounded away from zero.
Using the first order conditions that f(γ) = 1/(p1 + p2), Equation (C.4) can be rewritten as:

dp1
dν

=

3−(p2

1
1 − p2

2) f (cid:48)(γ) +

−p1 f (cid:48)(γ)

3 f(γ)−(p1 − p2) f (cid:48)(γ)

(C.6)

Lemma 1, single-peakedness, and symmetry imply that both terms on the right-hand side are
positive. Thus, it suffices to show that the first term is unbounded. Because single- peakedness
implies that f (cid:48)(γ) goes to zero as ν approaches infinity, the first term could only go to zero if

111

(p2
1 − p2
2) were unbounded. But if that were true, p1 would be unbounded. Thus, p1 is unbounded
in ν and it must be the case that there exists some ¯ν such that π1∗( ¯ν) > 1/(2 f(0)). Finally, because
π∗
1(ν) is strictly increasing, there exists ˆν such that π∗
1( ˆν) = 1/(2 f(0)) and for any average quality
improvement ν < ˆν, firm 1 will wish to offer a grandfather clause, whereas for any ν > ˆν it will
(cid:3)
not.

Proof of Proposition 3.3.

Proof. We first show that prices and firm 1’s profits are continuous in ν and α. However, because
we fix ν for the duration of this proof, going forward, we will suppress ν as an argument in
functions. Suppose that the distribution of the difference in idiosyncratic preferences is given by
Gα. Additionally, suppose that firm 1’s equilibrium price is increasing in ν when firm 1 does
not offer a grandfather clause.
It is possible to confirm that this assumption does indeed hold
in equilibrium following the same methodology used in the proof of Proposition 2 (see Equation
(C.4)). Define the vector
H(α, p1, p2) :=

(C.7)

(cid:20)

∂
∂p1

∂
∂p2

(cid:2)p1F(α(ν + p2 − p1))(cid:3),
−2α f(αγ) + p1α2 f (cid:48)(αγ),

α f(αγ) + p2α2 f (cid:48)(αγ),

(cid:2)p2(1− F(α(ν + p2 − p1)))(cid:3)(cid:21)


α f(αγ)− p1α2 f (cid:48)(αγ)
−2α f(αγ)− p2α2 f (cid:48)(αγ)

and let J(α, p1, p2) represent the matrix of the partials of H(α, p1, p2):

As in the proof of Proposition 3.2 we need to find a continuous implicit function g(α) =
(p1, p2) such that H(α, g(α)) = 0. Such a g exists if H(α, g(α)) is continuously differentiable and
J(α, p1, p2) is invertible. The first condition is satisfied by assumption. The second follows if the
determinant of J(α, p1, p2),

|J(α, p1, p2)| = 3α2 f(αγ)2 + α3(p2 − p1) f(αγ) f (cid:48)(αγ)

(C.9)

is non-zero. As in Proposition 3.2, this follows from Assumption 2, Lemma 1, as well as our single-
peakedness and symmetry assumptions. Therefore, the function that gives firm 1’s equilibrium

112

J(α, p1, p2) =

where we again define γ ≡ ν + p2 − p1.

(C.8)

expected profit in terms of α and ν is likewise continuous. Note that if firm 1 chooses to set a
uniform price in the mature market period, its equilibrium expected profit is given by π∗
1(α) ≡
g1(α)F(α(ν + g2(α)− g1(α))) where g(α) = (g1(α), g2(α)).

Under this new specification, when ν = 0, it is readily shown that firm 1’s expected profit in the

mature market period under uniform pricing is:

¯π1(α) ≡

1

4α f(0)

(C.10)

Moreover, under Assumption 2, if firm 1 chooses to offer a grandfather clause at some positive
1(α)/2 so that its profit when it grandfathers exceeds its profit

ν, its expected profit is ¯π1(α) + π∗
when it doesn’t if and only if π∗

1(α) < 2 ¯π1(α). Define:
rπ ≡ π∗
1(α)
¯π1(α)

(C.11)

Thus, firm 1 wants (doesn’t want) to offer a grandfather clause if rπ(α) < 2 (rπ(α) > 2). Then
to complete our proof it suffices to show that (i) rπ(α) is increasing in α, (ii) limα→0 rπ(α) = 1, and
(iii) limα→∞ rπ(α) = ∞.

For (i), using the implicit function theorem together with the first order condition p1 f(αγ) =

F(αγ)/α, we find that the derivative of rπ(α) with respect to α is:

∂rπ(α)

∂α

=

4ν f(0)F(αγ)(2 f(αγ)− αp1 f (cid:48)(αγ))

3 f(αγ)− α(p1 − p2) f (cid:48)(αγ)

(C.12)

In the numerator, ν,

f(0), F(αγ), α, and p1 are all positive, whereas f (cid:48)(αγ) is negative
(Assumption 2, Lemma 1, single-peakedness and symmetry). Additionally, in the denominator,
f(αγ) and p1− p2 are positive (Lemma 1). Thus, both the numerator and denominator are positive,
so that rπ(α) is increasing in α.

In proving (ii), we note that as α approaches zero, ¯π1(α) increases to infinity, and because
π∗
1(α) is bounded below by ¯π1(α) (this follows because we assume that ν > 0 and because π∗
1(α)
is increasing in ν—see Step 2 in the proof of Proposition 2) π∗
1(α) likewise approaches infinity.
Therefore, we can use L’Hopital’s Rule to evaluate the limit of rπ(α) as it approaches zero. Again

113

using the implicit function theorem together with firm 1’s first order condition, we have:

(cid:46) ∂ ¯π1(α)

∂α

∂π∗
1(α)
∂α

4α f(0)F(αγ)[(3p1 − 2ν) f(αγ) + αp1γ f (cid:48)(αγ)]

3 f(αγ)− α(p1 − p2) f (cid:48)(αγ)

=

(C.13)

Using firm 1’s first order condition, the numerator in Equation (C.13) can be written:

12 f(0)F(αγ)2 − 8αν f(0) f(αγ)F(αγ) + 4αγ f(0) f (cid:48)(αγ) f(αγ)−1F(αγ)

(C.14)

Using the fact that p1 − p2 ≤ v (Lemma 1) and F(0) = 1/2, we see that the first term in Expression
(C.14) equals 3 f(0) in the limit whereas the remaining two terms equal zero. Similarly, the limit
of the first term in the denominator of Equation (C.13) equals 3 f(0) and that of the second term
equals zero, such that rπ(α) goes to 1 as α approaches zero.

Finally, for the proof of (iii), assuming that prices are restricted to being non-negative, it suffices
to show that firm 1 can always make a profit of at least ν/2. Because π∗
1(α) increases in p2, at
worst, when p2 = 0, firm 1 can name a price of ν to earn profit of νF(α × 0) = ν/2. Therefore,
rπ(α) ≥ 2αν f(0) and as α goes to infinity, so does rπ(α).
(cid:3)

Proof of Corollary 1.
Proof. Consider α∗(ν) for some ν > 0. From the proof of Proposition 3.3, we know that α∗(ν) is
the value of α that satisfies π∗
1(α)/ ¯π1(α) = 2. Next, consider some ν(cid:48) > ν. Using Step 2 in the proof
of Proposition 3.2, we know that holding α fixed, π∗
1(α) rises, but that ¯π1(α) remains unchanged.
Thus, rπ(α∗(ν)) > 2 at ν(cid:48). Because rπ(α) is increasing in α, α∗(ν(cid:48)) < α∗(ν).
(cid:3)

114

BIBLIOGRAPHY

115

BIBLIOGRAPHY

[1] Amelio, A., Julien, B., 2012. “Tying and Freebies in Two-Sided Markets.” International

Journal of Industrial Organization 30, 436-446.

[2] Amir, R., Jin, J.Y., Troege, M., 2010. “Robust Results on the Sharing of Firm-specific
Information: Incentives and Welfare Effects.” Journal of Mathematical Economics 46, 855-
866.

[3] Acquisti, A., Varian, H.R., 2005. “Conditioning Prices on Purchase History.” Marketing

Science 24, 367-381.

[4] Anderson, S.P., Renault, R., 1999. “Pricing, Product Diversity, and Search Costs: A Bertrand-

Chamberlin-Diamond Model.” The RAND Journal of Economics 30, 719-735.

[5] Armstrong, M., 2006. “Competition in Two-Sided Markets.” The RAND Journal of Economics

37, 668-691.

[6] Armstrong, M., Wright, J., 2007. “Two-Sided Markets, Competitive Bottlenecks and Exclusive

Contracts.” Economic Theory 32, 353-380.

[7] Arozamena, L., Cantillon, E., 2004. “Investment Incentives in Procurement Auctions.” Review

of Economic Studies 71, 1-18.

[8] Banerjee, A.V., Summers, L.H., 1987. On Frequent Flyer Programs and Other Loyalty-

Inducing Economic Arrangements. Harvard Institute of Economic Research.

[9] Bolton, L.E., Warlop, L., Alba, J.W., 2003. “Consumer Perceptions of Price (Un)Fairness.”

Journal of Consumer Research 29, 474-491.

[10] Bonanno, G., Vickers, J., 1988. “Vertical Separation.” The Journal of Industrial Economics

3, 257-265.

[11] Butters, G.R., 1977. “Equilibrium Distributions of Sales and Advertising Prices.” The Review

of Economic Studies 44, 465-491.

[12] Caminal, R., Matutes, C., 1990. “Endogenous Switching Costs in a Duopoly Model.” Inter-

national Journal of Industrial Organization 8, 353-373.

[13] Chen, Y., 1997. “Paying Customers to Switch.” Journal of Economics & Management Strategy

6, 877-897.

[14] Chen, Y., Riordan, M., 2008. “Price-Increasing Competition.” The RAND Journal of Eco-

nomics 39, 1042-1058.

[15] Choi, J., Stefanadis, C., 2001. “Tying, Investment, and the Dynamic Leverage Theory.” The

RAND Journal of Economics 32, 52-71.

116

[16] Choi, J., 2004. “Tying and Innovation: A Dynamic Analysis of Tying Arragements.” Economic

Journal 114, 83-101.

[17] Choi, J., 2010. “Tying in Two-Sided Markets with Multi-Homing.” The Journal of Industrial

Economics 58, 607-626.

[18] Choi, J., Jullien, B., Lefouili, Y., 2017. “Tying in Two-Sided Markets with Multi-Homing:

Corrigendum and Comment.” The Journal of Industrial Economics 65, 872-886.

[19] Coughlan, A.T., Wernerfelt, B. 1989. “On Credible Delegation by Oligopolists: A Discussion

of Distribution Channel Management.” Management Science 35, 226-239.

[20] Dolan, R., 1987. “Quantity Discounts: Managerial Issues and Research Opportunities.”

Marketing Science 6, 1-22.

[21] Ellison, G., 2005. “A Model of Add-On Pricing.” The Quarterly Journal of Economics 120,

585-637.

[22] European Commission, 1995. “Information Exchanges among Firms and Their Impact on

Competition.” Office for Official Publications of the European Communities.

[23] Farrell, J., Klemperer, P., 2007. “Coordination and Lock-in: Competition With Switching

Costs and Network Effects.” Handbook of Industrial Organization 3, 1967-2072.

[24] Fershtman, C., Judd, K.L., 1987. “Equilibrium Incentives in Oligopoly.” American Economic

Review 77, 927-940.

[25] Fudenberg, D., Tirole, J., 2000. “Customer Poaching and Brand Switching.” The RAND

Journal of Economics 31, 634-657.

[26] Fudenberg, D., Villas-Boas, J.M., 2006. “Behavior-Based Price Discrimination and Customer

Recognition.” Handbook of Economics and Information Systems 1, 377-436.

[27] Gal-Or, E., 1986. “Information Transmission- Cournot and Bertrand Equilibria.” Review of

Economic Studies 53, 85-92.

[28] Herrmann, A., Xia, L., Monroe, K.B., Huber, F., 2007. “The Influence of Price Fairness on
Customer Satisfaction: An Empirical Test in the Context of Automobile Purchases.” Journal
of Product & Brand Management 16, 49-58.

[29] Homburg, C., Hoyer, W.D., Koschate, N., 2005. “Customers’ Reactions to Price Increases:
Do Customer Satisfaction and Perceived Motive Fairness Matter?” Journal of the Academy
of Marketing Science 33, 36-49.

[30] Janssen, M., Non, M., 2008. “Advertising and Consumer Search in a Duopoly Model.”

International Journal of Industrial Organization 26, 354-371.

[31] Jeitschko, T., Tremblay, M., 2014. “Platform Competition with Endogenous Homing.” DICE

Discussion Paper No. 166.

117

[32] Klemperer, P., 1987a. “Markets With Consumer Switching Costs.” The Quarterly Journal of

Economics 102, 375-394.

[33] Klemperer, P., 1987b. “The Competitiveness of Markets With Switching Costs.” The RAND

Journal of Economics 18, 138-150.

[34] Klemperer, P., 1995. “Competition When Consumers Have Switching Costs: An Overview
With Applications to Industrial Organization, Macroeconomics, and International Trade.” The
Review of Economic Studies 62, 515-539.

[35] Klemperer, P., 1999. “Auction Theory: A Guide to the Literature.” Journal of Economic

Surveys 13, 227-286.

[36] Krishna, V., 2002. “Auction Theory.” Acamedic Press.
[37] Lagerlöf, J.N.M., 2016. “Bertrand under Uncertainty : Private and Common Costs.” Working

paper, University of Copenhagen.

[38] Lebrun, B., 2009. “Auctions with Almost Homogeneous Bidders.” Journal of Economic

Theory 144, 1341-1351.

[39] Li, T., 2009. “Tying in Two-Sided Markets.” Working paper, Fudan University.
[40] Maskin, E., Riley, J., 2000. “Asymmetric Auctions.” Review of Economic Studies 67, 413-438.
[41] McFadden, D., 1974. “Conditional Logit Analysis of Qualitative Choice Behavior.” in P.

Zarembka, ed., Frontiers in Econometrics. New York: Academic Press.

[42] Myerson, R.B., 1981. “Optimal Auction Design.” Mathematics of Operations Research 6,

58-73.

[43] Narasimhan, C., 1984. “A Price Discimination Theory of Coupons.” Marketing Science 3,

128-147.

[44] Padilla, A.J., 1992. “Mixed Pricing in Oligopoly With Consumer Switching Costs.” Interna-

tional Journal of Industrial Organization 10, 393-411.

[45] Perloff, J.M., Salop, S.C., 1985. “Equilibrium with Product Differentiation.” Review of Eco-

nomic Studies 52, 107-120.

[46] Piccione, M., Tan, G., 1996. “Cost-Reducing Investment, Optimal Procurement and Imple-

mentation by Auctions.” International Economic Review 37, 663-685.

[47] Plum, M., 1992. “Characterization and Computation of Nash-Equilibria for Auctions with

Incomplete Information.” International Journal of Game Theory 20, 393-418.

[48] Png, I.P.L., Hirshleifer, D., 1987. “Price Discrimination Through Offers to Match Price.”

Journal of Business 60, 365-383.

[49] Robert, J., Stahl, D.O., 1993. “Informative Price Advertising in a Sequential Search Model.”

Econometrica 61, 657-686.

118

[50] Rochet, J.C., Tirole, J., 2003. “Platform Competition in Two-Sided Markets.” Journal of the

European Economic Association 1, 990- 1029.

[51] Rochet, J.C., Tirole, J., 2006. “Two-Sided Markets: A Progress Report.” The RAND Journal

of Economics 37, 645-667.

[52] Sakai, Y., 1986. “Cournot and Bertrand Equilibria under Imperfect Information.” Journal of

Economics 46, 213-232.

[53] Sakai, Y., Yamato, T., 1989. “Oligopoly, Information and Welfare.” Journal of Economics 49,

3-24.

[54] Schmidt, B.C. Jr., 1982. “Principle and Prejudice: The Supreme Court and Race in the
Progressive Era. Part 3: Black Disfranchisement from the KKK to the Grandfather Clause.”
Columbia Law Review 82, 835-905.

[55] Shaffer, G., Zhang, Z.J., 2000. “Pay to Switch or Pay to Stay: Preference-Based Price
Discrimination in Markets With Switching Costs.” Journal of Economics & Management
Strategy 9, 397-424.

[56] Tan, G., 1992. “Entry and R&D in Procurement Contracting.” Journal of Economic Theory

58, 41-60.

[57] Taylor, C., 2003. “Supplier Surfing: Competition and Consumer Behavior in Subscription

Markets.” The RAND Journal of Economics 34, 223-246.

[58] Thomas, C.J., 1997. “Disincentives for Cost-reducing Investment.” Economics Letters 57,

359-363.

[59] Tirole, J., 2005. “The Analysis of Tying Cases: A Primer.” Competition Policy International

1, 1-25.

[60] United States of America, Federal Communications Commission, 2014. “Protecting and
Promoting the Open Internet.” Report and Order on Remand, Declaratory Ruling, and Order.
GN Docket No. 14-28. Washington, DC, March 12, 2015.

[61] Varian, H.R., 1980. “A Model of Sales.” American Economic Review 70, 651-659.
[62] Villas-Boas, J.M., 1999. “Dynamic Competition With Customer Recognition.” The RAND

Journal of Economics 30, 604-631.

[63] Vives, X., 1984. “Duopoly Information Equilibrium: Cournot and Bertrand.” Journal of

Economic Theory 34, 71-94.

[64] Williams, A.F., Zador, P.L., Harris, S.S., Karpf, R.S., 1983. “The Effect of Raising the Legal
Minimum Drinking Age on Involvement in Fatal Crashes.” The Journal of Legal Studies 12,
169-179.

[65] Winston, M., 1990. “Tying, Foreclosure, and Exclusion.” The American Economic Review

80, 837-859.

119

[66] Wolinsky, A., 1986. “True Monopolistic Competition as a Result of Imperfect Information.”

The Quarterly Journal of Economics 101, 493-511.

[67] Xia, L., Monroe, K.B., Cox, J.L. 2004. “The Price is Unfair! A Conceptual Framework of

Price Fairness Perceptions.” Journal of Marketing 68, 1-15.

120