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TWO ESSAYS ON PRODUCT BUNDLING AND ONE ESSAY ON VERTICAL
INTEGRATION

By

Kyonghwa Jeong

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

2006

ABSTRACT

TWO ESSAYS ON PRODUCT BUNDLING AND ONE ESSAY ON VERTICAL
INTEGRATION

By

Kyonghwa Jeong

Discounts are often offered to consumers attached to competitors in various in-
dustries where consumers make repeated purchases. Chapter 1 studies such a practice
in a vertically related industry, i.e. U.S. telecom industry, where ﬁrms offer a package
of handsets and phone service plan with long-term contracts of one or two years. I
show that the equilibrium incentives for bundling can be explained with the practice
of consumer poaching. The reason is that bundling acts as a commitment device
through which ﬁrms can lock in consumers by creating artiﬁcial switching costs and
ﬁrms can commit to future prices in combination with long-term contracts.

Chapter 2 studies the strategic use of mixed bundling of two independent goods
in duopoly. I show that mixed bundling may arise in equilibrium in markets with
heterogeneous consumers’ valuations. Mixed bundling dominates pure bundling in
duopoly when a number of customers with low valuations switch to rivals when of-
fered a bundle only. This result is consistent with that in a monopoly where mixed
bundling is preferred to pure bundling when consumers’ valuations of two goods are
not perfectly correlated.

In high tech industries, a competitive concern about a vertical merger is that
merged entity transfers proprietary information of a competitor’s innovation to its
downstream or upstream division. In several recent vertical merger cases, FTC/DOJ
requires Firewalls to prevent the passage of competitively-sensitive information within
an merged entity. Chapter 3 studies the welfare consequences of such a merger and re-
lated information transfers in markets with vertical R&D spillovers. The main result
is that a Firewalls provision does not necessarily improve social welfare. The reason

is that Firewalls may affect a nonintegrated ﬁrm’ decision for an input supplier.

To My Parents

iii

ACKNOWLEDGEMENTS

I am in debt of gratitude to Jay Pil Choi for his encouragements and helpful
comments on this dissertation. I also thank to Carl Davidson, Thomas Jeitschko,
and Steven Wildman for helpful advice and comments.

I am grateful to my family for their support and love. With their unconditional
love and support, I have completed this work.

Finally, I also owe a special debt to Yongjae Choi for his friendship throughout

the entire procass of writing this dissertation. He is responsible for making my life at

MSU rich and worthwhile.

iv

Contents

LIST OF TABLES viii
LIST OF FIGURES ix
1 Brand Switching and Product Bundling 1
1.1 Introduction ................................ 1
1.2 Model ................................... 5
1.3 Benchmark: No price discrimination ................... 8
1.3.1 No bundling ............................ 8

1.3.2 Pure bundling ........................... 9

1.3.3 No bundling vs. Pure bundling ................. 10

1.4 Model with consumer poaching ..................... 11
1.4.1 No bundling ............................ 11

1.4.2 Pure bundling ........................... 22

1.4.3 No bundling vs. Pure bundling ................. 38

1.5 Conclusions ................................ 39

2 Strategic Mixed Bundling in Duopoly ' 41
2.1 Introduction ................................ 41
2.2 Model with discrete valuations of good B ................ 45
2.2.1 No Bundling ............................ 46

2.2.2 Pure-Bundling ........................... 47

2.2.3 Mixed-Bundling .......................... 51

2.2.4 Sub-game perfect equilibrium .................. 56

2.3 Model with continuous valuations of good B .............. 57

2.3.1 No—Bundling ............................ 57

2.3.2 Pure bundling ........................... 57

2.3.3 Mixed bundling .......................... 59

2.3.4 Sub-game perfect equilibrium .................. 62
2.4 Conclusions ................................ 62
Vertical Integration and Firewalls with R&D Spillovers 64
3.1 Introduction ................................ 64
3.2 Model ................................... 68
3.3 Equilibrium outcomes ........................... 70

3.3.1 Equilibrium Outcomes with Information Transfer ....... 71

3.3.2 Equilibrium Outcomes with Firewalls .............. 74

3.3.3 Information Transfer vs. Firewalls ................ 78
3.4 Welfare .................................. 79
3.5 Conclusion ................................. 81
Appendix: Unilateral Bundling 83
A.1 Short-term contract ............................ 83
A.2 Long- and Short-term contracts ..................... 88
A3 Long-term contract ............................ 91
Appendix: Proof of chapter 1 94
Appendix: Model with homogenous valuations of good B 97
Cl Pure Bundling .................... g ........... 97
C2 Mixed-Bundling .............................. 98
C3 Sub-game perfect equilibrium ...................... 100
Appendix: Proof of Chapter 2 101
Appendix: Figures of Chapter 2 103

vi

F Appendix: Figures of Chapter 3 108

Bibliography 1 10

vii

List of Tables

1.1
1.2
1.3
1.4
1.5
1.6

2.1
2.2

3.1

Payoffs in No Bundling ((5 = 1) ..................... 21
Payoffs in short-term contract under (52%;15153) 03 < t (6 = 1) . . . . 25
Payoffs in short-term contract under CB < t S (52%) 63 (6 = 1) . 25
Payoffs in long-term contract under t S c B .............. 33
Payoffs in Pure Bundling under t S 63 (6 = 1) ............. 38
Payoffs in Pure Bundling under 63 < t (6 = 1) ............. 38
Equilibrium outcomes of Pure Bundling ................. 49
Equilibrium outcomes of Mixed Bundling ................ 53
Payoffs of nonintegrated ﬁrm D2 ..................... 78

viii

List of Figures

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11

2.1
2.2
2.3
2.4
2.5

3.1
3.2
3.3
3.4

E.1

No Bundling ................................ 5
Pure Bundling ............................... 6
No bundling in Benchmark ........................ 8
Pure bundling in Benchmark ....................... 9
No bundling and Short term contract .................. 12
No bundling and Long and Short term contracts ............ 15
No bundling and Long term contract .................. 18
Pure bundling and Short term contract ................. 22
Pure bundling and Long and Short term contracts ........... 27
Pure bundling and Long term contract ................. 32
Pure bundling and Long term contract with CB < t .......... 34
No bundling in discrete valuations of good B .............. 46
Pure bundling in discrete valuation of good B ............. 48
Mixed bundling in discrete valuations of good B ............ 52
Pure bundling in continuous valuations of good B ........... 58
Mixed bundling vs No bundling ..................... 61
Under Information Transfer ....................... 71
Under Firewalls .............................. 74
Firm D2’ R&D level Comparison with A = 25, c = 0.5 ......... 77
Welfare Comparison with A = 25, c = 0.5 ................ 80
Proﬁts of ﬁrms in Pure Bundling .................... 103

E2
E3
E4
E5

F.1
F2
F3
F4

Pure vs. No Bundling .......................... 104

Proﬁts of ﬁrms in Mixed Bundling .................... 105
Mixed vs. No Bundling .......................... 106
Mixed vs.Pure Bundling ......................... 107
Under Firewalls, Proﬁts Comparison .................. 108
Total R&D Level Comparison (Firewalls vs. Information Transfer) . . 108

Consumers Surplus Comparison (Firewalls vs. Information Transfer) . 109

Producers Comparison (Firewalls vs. Information Transfer) ...... 109

Chapter 1

Brand Switching and Product
Bundling

1 .1 Introduction

Firms often offer discounts to consumers attached to competitors in subscription
markets where consumers can be easily distinguished into groups, ﬁrst time and re-
peated buyers. Early termination fees are also fairly standard in a subscription-based
business. The whole point of early termination fees is that ﬁrms offer new customers
a discounted price if they agree to stay for a minimum period. Hence, early termi-
nation fee acts like switching costs for customers who may switch between brands in
the middle of a contract. In the US. wireless industry, the persistence of the pattern
of discounts of handsets suggests that discounts on handsets are used to lock-in con-
sumers. Such strategic behavior of loyalty inducing discounts can arise in order for
consumers to incur costs when they switch between suppliers. Customers must be
connected to one operator for at least one- or two-year term arrangements. If they
switch between operators and breach a contract, they are obligated to pay a release

fee.1 The phone itself does not work if used on a different network. SIM cards are

 

1According to FCC(http://ftp.fcc.gov/cgb/NumberPortability/#earlyfee), customers are oblig-
ated to pay any early termination fees they may have under an existing contract. In the US. wireless
industry, an early termination fee ranges from $150 to $240, depending on the company. As of Sep.
22, 2005, Verizon charges $175 per line as an early termination fee. Other examples of early ter-
mination fees are $200 per line of T-mobile, $240 or $150 of Cingular, $200 of Alltel, and $150 of

typically locked so that handsets only work on one network (hardware lock). So, if
consumers switch between providers, they also need to purchase another handsets
locked by a new provider.2

With consumers’ recognition, however, intertemporal price discrimination gen-
erates a more competitive price and even more in the presence of switching costs
(Klemperer (1987), Caminal and Claici (2005)). Hence bargaining and rip-off strat-
egy is not optimal for ﬁrms to pursue in a market with switching costs. This raeult
critically depends on an important assumption: whether ﬁrms can commit themselvae
to future behavior or not.

In the literature on intertemporal price discrimination based on consumers’ past
behavior, it is common to assume that ﬁrms cannot commit themselves to future be-
havior, not to discriminate between loyal customers and new customers. There might
be many reasons why ﬁrms may not be able to commit themselves to future behavior.
For instance, there may be uncertainty about demand, costs, or possible entry into the
market where customers make repeated purchases. Under such circumstances, ﬁrms
are in prisoners’ dilemma when ﬁrms engage in price discrimination purely based on
consumers’ preferences even in the absence of switching costs (Villas-Boas (1999);
Fudenberg and Tirole (2000)). Fudenberg and Tirole (2000) use a two-period model
where the ﬁrst period choices leaves ﬁrms with two different types of consumers in the
second period: its own customers and those of its competitor. Since the competitor’s
customers can be targeted with discounts in the second period, consumer poaching,
consumers are less sensitive to prices in the ﬁrst period. By offering a long-term con-
tract in addition to a short-term contract, ﬁrms are able to have smaller portion of
customers who switch between brands. Nevertheless, ﬁrms make less proﬁt than when
they offer a sequence of short-term contracts because each ﬁrm competes with itself
over its turf by offering both long-and short-term contracts. Consumer poaching with

an inﬁnite horizon model is also studied by Villas-Boas (1999). The general result

 

Sprint &. N extel.

2Some further switching costs relevant to mobile users can be the subscriber number (stationary
costs), and special links with other services. Since those switching costs do not affect our main
result, I ignore them in the present paper.

that each ﬁrm will offer discounts to the other ﬁrm’s customers remains unchanged.
On the other hand, Chen (1997) studies a two-period model of a duopoly where ﬁrms
offer discounts to customers attached to the other ﬁrm in the presence of switching
costs. He ﬁnds that both ﬁrms will offer discounts to customers with a history of
purchasing from the other ﬁrm. Shaffer and Zhang (2000) extend the work of Chen
(1997) by generalizing the demand side. They ﬁnd that if the loyalties of customer
groups differ, i.e. if its own customers are not very loyal, it may be optimal for a ﬁrm
to offer discounts to its own customers rather than to poach customers from competi-
tors. Taylor (2000) also extends Chen’s (1997) work in several ways by allowing for an
arbitrary number of periods and randomly varying switching costs. The modeling of
many periods enables Taylor to be explicit about strategic switching on the consumer
side. This occurs when a customer switches solely to establish a history of frequent
switching, thereby securing better future offers from his suppliers. In those papers,
consumer poaching results in prisoners’ dilemma. The reason is that intertemporal
price discrimination based on purchase histories intensiﬁes price competition either
when consumers are already locked-in or when they are not yet locked-in. The critical
assumption is that ﬁrms are unable to commit not to poach customers attached to
competitors

This paper extends the existing studies on consumer poaching based on consumers’
past behavior and connects two important streams of brand switching and product
bundling in a complementary market where a good/ service is perishable and the
other good is durable. Since consumers switch between brands as a result of con-
sumer poaching, ﬁrms may have strategic incentives to lock-in consumers. Bundling
can act as such a locking-in device because bundling creates artiﬁcial switching costs
in a complementary market.

Our main results are as follows. I show that bundling can be an effective commit-
ment device through which ﬁrms can lock-in consumers and ﬁrms will offer long-term
contracts in the presence of consumer poaching. That is, the equilibrium incentives
for bundling can be explained with the presence of consumer poaching. The reason is

that bundling creates an artiﬁcial switching cost that locks in customers. Such arti-

3

ﬁcial switching costs are different from learning (switching) costs to use new brands.
Learning costs can arise when customers get to know a new word processing system
or when customers receive an added beneﬁt if they stays with the previous word-
processing-system providers in the second period. Tying can be a proﬁtable strategy
in the presence of such learning-by-doing switching costs (Carlton and Waldman,
2005).3 On the other hand, artiﬁcial switching costs in the present paper arise as
results of ﬁrms’ actions, bundling. When customers are locked-in with such switching
costs after the ﬁrst purchase, ﬁrms may have an incentive to take advantage of locked-
in customers. However, this intensiﬁes competition for market share in the ﬁrst period
and results in lower proﬁts than in the absence of switching costs. Hence, ﬁrms have
a collective incentive to commit themselves to future prices by offering long-term con-
tracts in the presence of switching costs. On the other hand, when customers are not
locked-in with switching costs after the ﬁrst purchase, pure bundling with short-term
contracts may arise in equilibrium as a result of bundling game. When customers are
not locked-in with switching costs, ﬁrms have incentives to offer a sequence of short-
term contracts because long-term contracts intensify competition for market base in
the ﬁrst period. Nevertheless, ﬁrms make lower proﬁts in pure bundling than in no
bundling. The reason is that ﬁrms face more competitive market in both periods in
the presence of both consumer poaching and switching costs, while ﬁrms face less
elastic demand in the ﬁrst period, but more elastic market in the second period in
the absence of switching costs.

In the following section, I set up the benchmark model where ﬁrms are unable to
price discriminate between customers and across periods. In section 4, I analyze a
model where ﬁrms may have an incentive to offer discounts to customers attached to
competitors under alternative scenarios, no bundling, pure bundling. In appendix A,
I show the case where one ﬁrm bundles while the other ﬁrm does not. Conclusions

are at the end.

 

3An example of Learning-by—doing costs in Carlton and Waldman (2005) paper: If a consumer
purchased Word 2002 rather than Word Perfect 11 in the ﬁrst period, then he will prefer Word 2003
over Word Perfect 12 in the second period because of the similar ways in which Word 2002 and
Word 2003 operate.

1.2 Model

There are two ﬁrms, A1 and A2, in market A with a horizon of two periods. They
are located at extremes of an interval of [0, 1] in a traditional Hotelling model. Firm
A1 is at 0 and A2 is at 1. Product A is perishable and it must be repurchased in
each period. The ﬁrms are symmetric in the sense that they have the same marginal
cost, c A 2 0. Consumers are uniformly located on the interval of [0,1]. A consumer
at 9, where 9 E [0, 1], incurs the transportation cost of t6 in each period if good A
is purchased. To consume a product A, each customer needs to buy a product B
that lasts for two periods. There are n 2 2 ﬁrms in market B. Each ﬁrm produces
a homogeneous good 8,, where 2' = 1, 2, ...., n, at the same marginal cost of C3 > 0.
Each consumer enjoys a gross utility of U from consuming goods/services A and B
together per period. I assume U to be large enough that in equilibrium all consumers
in fact purchase one unit of both A, and Bj, where i, j = 1, 2, in each period.

I assume that ﬁrms in market A make the decision of whether‘ to bundle or

Figure 1.1: No Bundling

Consumers

W“

not and the bundling decisions are irreversible. In no bundling, ﬁrms in market A

 

 

 

 

 

sell their products A,, 2' = l, 2, separately from products 8-, j = 1,2, ....,n, in both

periods. Consumers independently purchase products B -, j = 1,2, ....,n, in market

Figure 1.2: Pure Bundling

man a

 

MaketA n

131

 

~—~ s

I Germ-mt:

 

 

 

 

B. Product B is durable and technically compatible with any product A, and hence
can be used with any A in the second period. On the other hand, in pure bundling, I
assume that ﬁrm Al purchases BI and ﬁrm A2 buys 32, at the price of CB and pack
it together with A1 and A2, respectively, making the bundle of AiBi, i = l, 2, at the
cost of CA + C3. Firms in market A sell only a package of AiBi, z" = 1,2, to all ﬁrst
time consumers. Hence consumers who may switch between providers need to buy a
package of AB, , 2' = 1, 2 , rather than A,, 2' = 1, 2. Product B,- purchased in the ﬁrst
period cannot be used with A -, i, j = 1, 2 , z' 74 j, in the second period. For second
time users, ﬁrms sell A1, 2' = 1, 2, because product B is durable.

Consumers are rational in the sense that they can correctly anticipate the conse-
quences of current actions on future decisions. Both ﬁrms and consumers discount
their second-period payoffs with the discount factor of 6 E (0, 1].

I consider three different types of contract arrangements in the present paper,
short-term contract, long-and short-term contracts, and long-term contract. In a
short-term contract, ﬁrms offer the ﬁrst period and second period spot prices sequen-
tially. When ﬁrms offer both long-and short-term contracts, ﬁrms offer a schedule of
prices for customers who purchase long-term contracts in the ﬁrst period. Poaching

prices to new customers and the second period spot prices for loyal customers who

purchase a sequence of short-term contracts are determined in the second period. In
a long-term contract, ﬁrms offer discounts of d,, where z' = 1, 2, in exchange for long
term contracts with a schedule of prices. Since discounts are given to customers in
return for a long-term contract, customers are obligated to pay a termination fee of
3i: where 2' = 1,2, if they breach the contract earlier than the contract term. The
early termination fee is a charge to compensate ﬁrms for customers’ failure to sat-
isfy the commitment on which the price of a service/ good is based, 3,- = 7d,, where
7 E (0, 00]. In other words, an early termination fee allows ﬁrms to recoup the costs of
providing customers with goods/services and discounts. In this sense, I assume that
a customer is subject to an early termination fee equal to the amounts of damages
caused by terminating the contract early. Klemperer (1995) points out that “Loyalty
Contracts” are implemented by offering customers discounts in exchange for long term
contracts committing the customers to pay damages if they terminate the contract.
Signing such a contract that speciﬁes damages of s is equivalent to paying 3 for a
discount coupon of present-value s that is offered in exchange for the next purchase.
In the present paper, it is equivalent to the sum of the ﬁxed amounts of discounts
and an interest of the discounts, s,- = (1 + r)d,- = 2154-, where 6 E (0,1], 1' = 1,2, r is
an interest rate.

There are two assumptions that hold throughout this paper: (i) Each consumer’s
preferences are constant over time. (ii) Firms in market A cannot commit not to
poach consumers from competitors.

The timing of the game is the following. In the ﬁrst stage, ﬁrms make bundling
decisions. In the second stage, ﬁrms choose whether to offer a sequence of short-term
contracts, both long— and short-term contracts, or a long—term contract. The proper

solution concept for this two-stage game is Subgame Perfect Nash equilibrium.

1.3 Benchmark: No price discrimination

1.3.1 N o bundling

This is the static model where ﬁrms do not price discriminate between customers
and over time. Since consumers’ preferences are constant over time, in each period,

a consumer at 6* is indifferent between A1 and A2 if

U"¢B"Pl41 —t6* =U—cB —piz—t(1—6*)
1 . .
=>6*=§(t+pj42-p:41), i=1,2. (1.1)
where pin and 123.42 are the prices of A1 and A2, respectively, in period 2' = 1,2.

Customers in [0, 0*] and [9*, 1] purchase A1 and A2, respectively, in each period.

Figure 1.3: No bundling in Benchmark

A, A

 

Period 1

 

Period 2 l l I

By maximizing the following proﬁt functions with respect to prices, pin, pin,

i=1,2,

7m = (pi. —- one" + 609,241 — cm. (1.2)

77.42 = (pig - CA)(1- 9*) + 507342 - CA)(1- 9"), (1~3)

The equilibrium prices and proﬁts in market A are as follows.

Pin =Pi2 =12?“ =P342 =t+CA, (1-4)

1! * t

In market B, since products B are durable, there is no sale in the second pe-
riod. The market price and proﬁts are the general Bertrand competition outcomes as

follows.

Phi = CB, (1-5)
“trig,- = 0, i=1,2,....,n. (1.7)

1.3.2 Pure bundling

Firms in market A offer a package of AiBi, i = 1,2, but not A,- separately,
2' = 1,2, in the ﬁrst period. In the second period, ﬁrms offer A5, 2' = 1,2, to their
own customers because products B are durable. Customers in [0, 6*] and [0*, 1]
purchase the bundle A181 and A282, respectively, in the ﬁrst period, and A1 and

A2, respectively, in the second period in the absence of price discrimiation.

Figure 1.4: Pure bundling in Benchmark
A 13 1 A 2 B 2 I

l

Perlod 1 l l l
o 6 * 1

1 l
l |

0 9* 1

Period 2

A customer at 0* is indifferent A181 and A232 in the ﬁrst period and A1 and

A2 in the second period if

B B

pA2 —pA1 (1.8)
2t ’

2 2

PA2 ‘M1 (19)
2t ’

* It 1
p§1+t6*=pﬁ2+t(1—0)=>0 =-2-+
,, 1
p§1+t6* =pi2+t(1—6*) =>9 = 5+
where p31 and p32 are the bundle prices of A181 and A282 of ﬁrm A1 and A2,
respectively, in the ﬁrst period and p?“ and p12“ are the prices of A1 and A2 of ﬁrm

A1 and A2, respectively, in the second period. By maximizing the following proﬁt

functions with respect to prices, {pﬁv pin} and {p32, 19312}, respectively,

1011 = (1331 - CA - CB)” + 5(P341 - CAW, (1-10)

7012 = (P32 - CA - CB)(1- 9*) + 50342 - CA)(1- 9*), (111)

The equilibrium prices and proﬁts in market A are as follows.

p§1=p32=t+cA+cB, (1.12)

1931:1342 =t+cA, (1.13)
t

at“: 7rj'42 = (1+ (5)5. (1.14)

In market B, ﬁrm A1 purchases Bl and ﬁrm A2 buys 32 at CB due to Bertrand

competition in market B. The market price and equilibrium proﬁts are as follows.

p231 =p*32 = 63. (1.15)

1.3.3 No bundling vs. Pure bundling

The equilibrium outcomes of pure bundling are the same as in no bundling when
ﬁrms engage in no price discrimination between consumers and over periods. The

reason is that no positive effects of bundling such as a production differentiation role

10

of bundling can take place in the present paper because products B are complementary

to goods/ services A in a Bertrand competition model.

1.4 Model with consumer poaching

Price discrimination between loyal and new consumers is well observed in vari-
ous industries where consumers make repeated purchases. I analyze such a practice
in a vertically related industry. Consumers correctly anticipate the second period
prices and maximize their total utilities when sellers are able to price discriminate
based on consumers’ past behavior. Firms can offer three different types of contract

arrangements: short-term, long-and short-term, and long-term contracts.

1.4.1 N o bundling

Firms in market A and B sell independently to consumers. Consumers purchase
A,- +B-, i,j = 1, 2, in the ﬁrst period and purchase 14,-, 2' == 1, 2, in the second period.
In market B, ﬁrms have no sale in the second period under any contract arrange-
ments because customers can use product B purchased in the ﬁrst period, even when
they switch between brands. Firms sell product B at the marginal cost of CB and

make zero proﬁts.

pBi = 63, (1.17)
”Bi = 0, i=1,2,.....,n (1.18)

(1) Short-term contract

Firms offer a short-term contract each period. Customers in [0, 8*] purchase A1
and customers in [6*, 1] purchase A2 in the ﬁrst period. After observing consumer
preferences in the ﬁrst period, ﬁrms in market A offer discounts to consumers attached
to competitors in the second period. That is, consumers in [61, 9*] and [0*, 02] are

targeted with discounts by competitors with A2 and A1, respectively, in the second

11

period.

Figure 1.5: No bundling and Short term contract

A. A.

Period 1

 

 

 

O
Q__
*

Period 2

I solve the problem by backward induction. In the second period, given 6*, a

consumer at 91 and 92 is indifferent between A1 and A2 if

. 1 .
pg“ +t01 =p312+t(1-91) =>91 = 2—t(t+p?42—p?41), (1.19)
. 1 .
p3,, + tag = p342 +t(1— 02) => 02 = 5,}(15‘1'1’312 -—pA1), (1.20) p

where pin and 19342 are the second period prices of A1 and A2 for loyal consumers of
ﬁrms A1 and A2, respectively. 1331 and 13312 are the second period poaching prices
of A1 and A2, respectively. Consumers in the intervals [0, 01] land [62, 1] are loyal
customers to ﬁrms, A1 and A2, respectively. They do not switch between suppliers
even in the presence of consumer poaching. On the other hand, consumers in the
intervals [01, 6*] and [0*, 02] switch between brands in the second period, from A1 to
A2 and from A2 to A1, respectively. The proﬁt functions of the ﬁrms'in the second

period are

nil”) =(p3u—cA)61+(ﬁil—cl)(62—0*). , (121)
«$59) = (p?42 — and - 02)+(13,242 — chm: — 61). ' (1.22)

By maximizing the proﬁt functions with respect to the second period prices, The

12

second period prices as a function of 0* are as follows.

29*+1 3—26”

.2 3—40* .2 40*—1
pA1=CA+ 3 t2pA2=CA+ 3 t,

In the ﬁrst period, a consumer at 6* is indifferent between A1 and A2 if

 

 

 

 

p)“ + w“ + 603?“, +t(1— 0*)) = p342 + t(1 - 0*) + 6(13311-i-t0")

1 1 ~2 ~2 1 1
1 — -6 — 3 —
=_+(P.42 9.41) (222 10.41) 21+ (1022 PA1)_(1_24)

=’ 6 2t 2(1 — a): 2 2(3 + 6)t

Firms maximize the following proﬁt functions with respect to pin and pin, respec-

 

 

tively.
NB 5 1 3(101 -p1 ) NB S A
7r1 ( )= (pin-6’4) (5+ 22“;+6)’t“ +6152 ( ), (1.25)
N S 1 3071 ‘Pl NB S
71'2 B( )= (pig—CA) (~2- — 2?32+6)’:1))+57r22 ( ). (1.26)
NB(S)

a . ._
Note that at 6* = 1/2, ligg—i—aiglf- = —% = 0, where 2' = 1,2. When ﬁrms
”.4:

have the same market share of 1 / 2, an increase in the ﬁrst period prices do not affect
the second period proﬁts. The reason is that in the presence of consumer poaching,
the gains from a market with loyal customers are exactly offset by the costs from the

other market with new buyers. In equilibrium, the market prices are as follows.

1 __1 _ é
pAl —pA2—t+cA+3t,
2
p311 =p342=§t+cA, (1.27)

~2 .2 1
PA1=pA2 = §t+CA -

Note that the ﬁrst period prices of pin and 1232 are higher than those in the benchmark

model. This reﬂects the fact that consumers are willing to pay more in the ﬁrst

13

period (less elastic demand) because they can take advantage of discounts offered by
competing ﬁrms in the second period (more elastic demand). However, the second
period prices are lower than those in the benchmark model.4

In equilibrium, each ﬁrm shares the market evenly in both periods. By substituting

the optimal prices into eq.( 1.19) and eq. (1.20), the followings are derived.

9,: ,9*= ,92=§. (1.28)

I 1
3 2

1 / 3 of consumers switch between brands as a result of consumer poaching. The
portion of loyal customers is 2/ 3. By offering discounts to consumers attached to
competitors, each ﬁrm is worse off than in the benchmark model where ﬁrms do not
price discriminate based on customers’ past behavior. The equilibrium proﬁts with

short-term contracts are as follows.

S 86 t
as 2.1128(8): (1+3) .,. (1.29)

(2) Long- and Short-term contracts

Firms offer both long-and short-term contracts. With long-term contracts, ﬁrms
offer a schedule of prices for the ﬁrst and second period. Firms do not commit to
second period prices for customers who purchase a short-term contract in the ﬁrst
period. Since customers who have high preferences for A1 will be more willing to com-
mit themselves to consuming A1, I assume that long-term contracts are purchased
by the consumers who most prefer that ﬁrm’s product. This tie-breaking assump-
tion is needed because our model is deterministic and hence a consumer who plans
to purchase from ﬁrm Al chooses between a long-term contract and a sequence of
short-term contracts based on his costs. Consumers in [0, 9*] and [9*, 1] purchase
A1 and A2, respectively, in the ﬁrst period. Consumers in the interval of [0, 9 Al
and [9B, 1] purchase long-term contracts at the price of 1')" A1 and 13,12 from ﬁrm A1

and A2, respectively. Hence, competition in the second period is over the interval

 

“These are the main results of Phdenberg and Tirole (2000)

14

Figure 1.6: No bundling and Long and Short term contracts

 

 

 

Al A2
Period1 l l l
l * I |
o 6.1 9 63 1
A1 A2 A1 A2
Period2 l l l l
I l I l* I l
0 9A 01 6 62 93 1

[9 A, 93] . Consumers in the intervals [9 A, 91] and [92, 93] buy a sequence of short-
term contracts from A1 and A2, respectively. Consumers in the intervals [91, 9*]
and [9*, 92] are targeted with discounts by rivals and switch between brands in the
second period, from A1 to A2, from A2 to A1, respectively.

In the second period, given 9*, a consumer at 91 and 92 is indifferent between A1

and/12 if
. 1 .
P341+t91=1z242+t(1—01) => 91=E(t+P,242-P?41). (1-30)
“2 2:0:2 t1-9 9=—1-t 2—‘2 131
PA1+ 2 PA2+( 2) => 2 2t( +PA2 R41): ( l

where 12241 and p342 are the second period prices of A1 and A2 for loyal customers of
ﬁrm A1 and A2, respectively. 131241 and 151242 are the second period poaching prices of

A1 and A2, respectively. Firms maximize the following proﬁt functions with respect

to second period prices, {121241, 12241} and {p32, 151242},

riff“) = on?“ — c.0091 - 6.4) + (2}... — CA)(92 — 6*). (1.32)
7423"“) = (p312 — c.0093 - 62) + (23.2 - char — 61). (1.33)

15

Then, we have

 

 

 

 

26*+1 4 26*+1 4
P31 = CA + t - ~02t. 10312 = CA — t + —03t, (1.34)
3 3 3 3
A 49* — 1 2 , 49* — 1 2
p311=CA—( 3 )HgOBt, p312=cA+( 3 )t—E9At. (1.35)

In the ﬁrst period, a consumer at 9* is indifferent between A1 and A2 if

pi] + M" + 6032,12 + t(1 — 9*)) = pf.” +t(1— 0*) + 6(13341+t6*)
1 1 -2 .
1 PA2 ‘PA1 _ 5(PA2 "PAD

:6 =2+ 2(1—6)t 2(1—6)t ’

1 1 2 2
10.42 -P.41 + 501.12 -PA1) (1.36)
2t 4t '

 

1
=>9*=—
2+

A consumer at 9 A and 9 B is indifferent between a long-term contract and a sequence

of short-term contracts if

15,11 + (1+ 6)t9A = ph1+t9A + 6(1)?“ + t9A)
=> 23211 = 1211+ 519311, (1-37)
2.12 + (1 + 6)t(1- 63) = p112 +t(1— 613) + 50.3,, +t(1— 63))

=> 2.12 = 12:12 + 6103.2. (1.38)

where 13,41 and 13 A2 are the prices of A1 and 31 for a long-term contract, respectively.

Firms are maximizing the following proﬁt functions with respect to the ﬁrst period

prices, p341 and p342.

NB L&S .1. - .1

7., l ’ = (loll—cm +6003“-c.1)61+6(p§u—c.1)(62—6). (1.39)
NB L&S at

«2 ‘ ’ (th‘CAX1-9)+5(P312—CA)(1-92)

+6922 — cw" - 01). (1.40)

In equilibrium, the ﬁrst and second period prices are lower than in a sequence of

short-term contracts. The reason is that each ﬁrm competes with itself over its turf

16

by offering both long-and short-term contracts simultaneously.

.. - 1

PA1= PA2 = (1+ 55) t+ (1 + (DC/1,

9341: P342 = t + CA, (1.41)
l

sz11== ,t +

- , 1

2311= 37342 = 4t + CA-

The fraction of consumer poaching is 1/4, which is less than in a sequence of
short-term contracts, where consumer poaching is 1 / 3. By substituting the optimal
prices into eq.(1.30) and eq. (1.31), I can ﬁnd {91, 92}. I can derive {9A, 93}
from the ﬁrst order conditions of eq. (1.39) and eq. (1.40) at £2: = 0, where

ﬁrst-period sales are unaffected if ﬁrms increase phi, but decrease 13A,- by the same

amount, 2' = 1,2.
3 5 7
,9=—,9*=—,9=—,9 =—. 1.42
1 8 2 8 B 8 ( )

The equilibrium proﬁts of ﬁrms are

7 t
”{VB(L&S)= 7réVB(L&S) = (1 + 1—66) 5, (1.43)

Proposition 1.1 When offering both long-and short-term contracts under no bundling,
ﬁrms have positive portions of consumers who buy long-term contracts, 1/4. There
are also consumers who switch between brands in the second period, 1/4, which is
smaller than in a sequence of short-term contracts. The equilibrium prices and proﬁts
are lower than in a sequence of short-term contracts.

Proof. See appendix B.

(3) Long-term contract

To prevent any consumer poaching, rather than simple long-term contracts where

ﬁrms offer a schedule of the ﬁrst and second period prices for repeated buyers, ﬁrms

17

may have an incentive to offer long-term contracts in which ﬁrms offer a schedule of
prices of (pin, pin} and {12142, 12312} with discounts of di, 2' = l, 2, in exchange for
long-term contracts. In this long-term contract, I allow customers who committed
to a long-term contract to switch between brands. But they are subject to early

termination fee of 3,, 2' = 1, 2, under the long-term contract that they signed up for.

Figure 1.7: No bundling and Long term contract

Al A2 I

Period 1 l ]
o 9 * 1

Period 2

 

o—r—

Customers in [0, 9*] and [9*, 1] purchase long term contracts of A1 and A2 ,
respectively, in the ﬁrst period. After observing consumer preferences in the ﬁrst
period, ﬁrms in market A may offer discounts to consumers committed to competitors.
Consumers in [91, 9*] and (9*, 92] are targeted with discounts by competitors with
A2 and A1, respectively, in the second period.

In the second period, given 9*, customers are indifferent between A1 and A2 at

91 and 92 if

. 1 .
19311 + t61=p2Az 'l' 81 +t(1— 01) => 01: .2—t-(t +1932 '1‘ 31—193“), (1.44)

. ‘ 1 .
[2341+ 32 + t92 = R242 +t(1— 92) => 62 = -2—t(t +1232 - p?41_ 82), (1.45)

where p341 and 12312 are the second period prices of A1 and A2 for consumers of ﬁrm
A1 and A2, respectively, who committed to long-term contracts. 151241 and 13312 are the
second period prices of A1 and A2 for consumers who switch between providers in

the second period. 3,, where 2' = 1, 2, is an early termination fee that customers are

18

obligated to pay when they breach a long-term contract. By maximizing the following

proﬁt functions with respect to the second period prices for new customers,

max (pal—ems —— 9*), (1.46)

max (pig—cm“ — 9,), (1.47)

I have the second period prices to new buyers as follows.

- 1 .
pin: — (t + cA — 32 — 220* +1312) , p312 =

2 (2t9* — t + CA — 31+P2Al) . (1.48)

NIH

In the ﬁrst period, a consumer at 9* is indifferent between A1 and A2 if

p), + 29*—5sl+5(133,2+sl +t(1—9*))

= P112 +t(1— 0") - 682 + 5053,, + 32 + 20*)
:> 9* : _1_ + 12342 _plt1 _ 6032,42 ‘93“)
2 2(1— 6)t 2(1 — 6)t

l I 2 2
PA2 —pA1 + 609,42 “ 17,41) __ 6(32 ‘ 31) (149)
2t 4t 4t ’ '

 

1
=> 9*:—
2-l-

where 12]“ and p142 are the ﬁrst period prices of A1 and A2 in long-term contracts,
respectively. Note that a customer receives a ﬁxed amount of discounts of d,- = 6.9,- in
return for a long-term contract. Hence, if terminating early, the customer is obligated
to pay back 32': 2' = 1, 2, as an early termination fee.

Firm Al and A2 maximize the following proﬁt functions with respect to (pin, p341}

and {p34}? p32}, respectively, for customers committed to long-term contracts.

max (9341'- CA — 631)9* + 6(1):“ — CA)61+ 6(9311- CA)(92 — 9*) + 631(9" — 91),

s. t. 91 = 9“. (1.50)

max (th‘CA’552X1—9U+5(P,242—CA)(1—92)
swig—chm—91)+632(02-0*). (1.51)
s. t. 92 = 9*.

19

By substituting the optimal prices and the condition, 91 = 92 = 9* , into eq. (1.44)
and eq. (1.45), I have the optimal amount of discounts of d" and early termination

fee of 3* as follows.

d1“: d3 = 6t, (1.52)
3’] = s; = t (1.53)
And the equilibrium prices are as follows.
1 _ 1 _. é
pAl —pA2 —t+cA+ 2t,
pi] =19?” = t + 6.4.
13311 = 2.212 = CA- (1.54)

The ﬁrst period price is higher than in the benchmark model. However, with dis-
counts, proﬁt margin, price—cost-discounts, is lower than in the benchmark model. At
the price of CA, no consumer switches between brands due to early termination fee
3* = t. With loyalty contracts, ﬁrms allow no consumer poaching. However, they
face intensiﬁed competition in the ﬁrst period and are worse off than in short-term
contracts. The equilibrium proﬁts of ﬁrms are

rim“): ”323(2) = (1+ $6) :- (1.55)
Proposition 1.2 In no bundling with long-term contracts, (i) The optimal discounts
are d‘i‘ = d; = 6t and early termination fees are 3’] = s; = t. (ii) The ﬁrst period
proﬁt margin, price-cost-discounts, is lower than in a sequence of short-term contracts
while the second period proﬁt margin is higher than in short-term contracts. (iii) By
offering discounts in exchange for long-term contracts, ﬁrms are worse off than in a
sequence of short-term contracts.

Proof: See appendix B.

Firms in market A may not allow any consumer poaching by offering loyalty

20

inducing discounts in return for long-term contracts. That is, by creating an artiﬁcial
switching cost to block consumer poaching, no customer switches between brands.
However, this intensiﬁes competition for market share in the ﬁrst period and hence
ﬁrms are worse off than in a sequence of short-term contracts while payoffs from
offering long-term contracts are higher than from offering both long-and short-term

contracts.

(4) Short-term contract vs. Long-term contract

In the presence of consumer poaching, ﬁrms may have an incentive to offer dis-
counts to customers in exchange for long-term contracts in order not to allow any
consumer poaching by competitors. However, this intensiﬁes competition for market
base in the ﬁrst period, which results in fewer proﬁts than in short-term contracts
where positive customers switch between brands.

In Table 1.1, I compare payoffs from three different types of contract arrange-

Table 1.1: Payoffs in No Bundling (6 = 1)

 

 

 

 

 

 

 

 

Firm A2 Short-term Long-term Long&Short
Firm A1
Short-term 0.944t, 0.944t 0.963t, 0.790t 0.880t, 0.816t
Long-term 0.790t, 0.963t 0.750t, 0.750t 0.667t, 0.800t
Long&Short 0.816t, 0.880t 0.800t, 0.667t 0.719t, 0.719t

 

 

ments at 6 = 1. Short-term contract arises in equilibrium in the presence of consumer
poaching. The reason is that by offering both long-and short-term contracts, each ﬁrm
faces intensiﬁed competition due to its own competition between long-and short-term
contracts. Long-term contract may not allow any consumer poaching, but intensi-
ﬁes competition for a market base in the ﬁrst period. In a short-term contract, while
facing intensiﬁed competition in the second period, ﬁrms are able to beneﬁt from soft-
ened competition in the ﬁrst period because customers become less elastic in the ﬁrst
period in the presence of consumer poaching. Hence, ﬁrms are better off when they
offer a sequence of short-term contracts and allow consumer poaching by competitors

than either long-term contracts or long-and short-term contracts.

21

1.4.2 Pure bundling

I consider a model where ﬁrms in market A offer a package of AiBi, where
2' = 1,2, to all ﬁrst time consumers. Consumers who switch between brands incur
switching costs because they need to buy a whole package of A281, rather than 24,,
where 2' = 1,2. Note that I assumed that ﬁrm A1 purchases 81 and ﬁrm A2 buys
32 at the price of CB and pack it together with A1 and A2, respectively, making the

bundle of AB), 2' = 1,2, at the cost of CA + C3.

(1) Short-term contract

I consider the case where ﬁrms offer a sequence of short-term contracts in the
presence of consumer poaching in market A. Consumers in [0, 9*] and [9*, 1] buy
the bundle A181 and A282, respectively, in the ﬁrst period. In the second period,
consumers in the intervals [91, 9*] and [9*, 92] are targeted with discounts by com-
petitors and switch between brands, purchasing the whole package of AiBi, rather
than A, separately, where 2' = 1, 2. On the other hand, consumers in the intervals

[0, 91] and [92, 1] do not switch suppliers. In the second period, given 9", a customer

Figure 1.8: Pure bundling and Short term contract
l A] B 1 A2 B 2
Period 1

|
I
o 9 * 1

 

 

_
H

Period 2

is indifferent between A1 and A282 at 91 and between A181 and A2 at 92 if

. 1 .
pfn+261 =p§2+t(1—01)=>61 = §(t+p§2—p§u), (1.56)

. 1 .
p§1+t92 = p3,, + t(1— 92) => 02 = 2%“ +113,2 — 102,), (1.57)

22

where p?“ and p32 are the second period prices of A1 and A2 for loyal consumers of
ﬁrm A1 and A2, respectively. 13% and 1322 are the second period prices of A181 and
A282 for consumers who switch between providers. The proﬁt functions of the ﬁrms

in the second period are as follows.

«$39) = (9311 —c.961 + (921 — .., — c1992 — 6*). (1.58)
“568(5) == (P1242 - CA)(1 - 92) + (P32 - CA - CB)(9* - 91)- (1-59)

By maximizing the proﬁt functions with respect touthe second period prices, I have

the second period prices as a function of 9* as follows. ‘

 

1 29*+1 1 3—20*
pil=CA+§cB+(——3—)t,pig=cA+§cB+( 3 )2, (1.60).

~B 2 3—49* AB 2 49*-I
19,41 =CA+§CB+ T t2 pA2=CA+§CB+ 3 t.

 

In the ﬁrst period, a consumer at 9* is indifferent between A181 and A232 if

p3, + 29* + 605,132 +2(1- 9*)) = p22 +2(1 — 9*) +663,“ + 29*)

=i+(p§2—pgl)+6(ﬁ§l-ﬁ§2)=l 3(p§2_p21) (161)
22 2(1— 6)2 2 2(3+6)t ' '

 

=>9*

Firms set the ﬁrst period prices, pin and p142, by maximizing the following proﬁt

functions,

 

1 3 B -
WfB(S) = (Pgl-CA- CB) (5 + (2232+ 551)) + 6233(5) , (1.62)

l-i

PB 5 3(PB -PB ) PB 5
7r2 ( )= (1022—94—63) (2 — 2332+ (5)211 )+6"22( )- (1'63)

PB(S)
_ ‘97" 99* _ 5 29*—1 c _ .-
Note that at 9* — 1/2, #ggt—gg— -— —[ 3(3+6 + 3+6)2] — - 3+“, where
‘l

2' = 1, 2, which implies that when there are switching costs of CB created by bundling,

ﬁrms compete more aggressively for market share in the ﬁrst period than in the

23

absence of switching costs. The equilibrium prices are as follows.

6
17/811:sz =t+CA +CB + 3a — 2cm,

2 1
p341 = p312 = 5t + CA + ch, (1.64)

. . 1 2
pg, = pﬁz = 52 + cA + 363.
By substituting the optimal prices into eq.(1.56) and eq. (1.57), I derive

1

1
9:-
1 3 6t’ 2 6t

(1.1) t > CB

Customers are not locked-in after the ﬁrst purchase if t > c3. Consumers who
switch between brands are located in the intervals [61, 0*] and [6*, 62]. The portion
of consumer poaching is 313(1 — 9?), which is less than in the absence of switching
costs. In no bundling, 1 / 3 of consumers switch between providers in the second
period. By creating switching costs, bundling locks-in some of consumers who may
otherwise switch between providers. However, the equilibrium proﬁts are lower than
in no bundling because ﬁrms face intensiﬁed competition in the ﬁrst period in the
presence of switching costs. The equilibrium proﬁts are

$522586): (1.2955);— $7892-63). (1.66)

In appendix A, I show the case where ﬁrm A1 offers the bundle A181 to new
customers while ﬁrm A2 does not bundle. Firm Al makes higher proﬁts in pure
bundling at the cost of ﬁrm A2. Hence, in equilibrium, both ﬁrms engage in bundling
as a result of bundling game and are worse off than in no bundling (See Table 1.2 &

1.3).

24

Table 1.2: Payoffs in short-term contract under (2274351455) CB < t (6 = 1)

 

 

NB(Firm A2) PB(Firm A2)
C2
NB 0.94442 —- 0.2803 + 0.15Jti,
2
(Firm A1) 0.94442, 0.94442 0.94442 + 0.06cB + 0.15512

 

2 22
PB 0.94442 + 0.0693 + 0155?, 0.94442 — 0.22c3 + 0.11%,

 

2 2
(Firm A1) 0.94442 — 0.28cB + 0.153)3 0.94442 - 0.22cB + 0.115?

 

 

 

 

Table 1.3: Payoffs in short-term contract under 63 < t _<_ (727773”13163) CB (6 = 1)
NB(A2) PB (A2)
NB(Al) 0.9444t, 0.9444t 0.7897t, 0.9633t

 

 

 

 

 

 

 

22 c2
PB(A2) 0.9633t, 0.7897t 0.944t—0.22cB+0.11—F, 0.9442—0.22cB+0.11—F

 

Proposition 1.3 In pure bundling with a sequence of short-term contracts, if c B < t,
(i) %(1 — SF) of consumers switch between brands (ii) The ﬁrst and second period
prices are given in eq.(1.64). (iii) If CB < t S 2cB, the ﬁrst and second period proﬁt
margins, price—marginal costs, are lower than in the benchmark model. If 2cB S t,
the ﬁrst period proﬁt margin is higher than in the benchmark model while the second
period proﬁt margin is lower than in the benchmark model. (iv) Pure bundling with
a sequence of short-term contracts arises in equilibrium as a result of bundling game
in the presence of consumer poaching, resulting in prisoners’ dilemma.
Proof: See appendix B.

In market B, ﬁrms B1 and B2 have a positive sale, %(1 - 9F), where cB < t, in the
second period because some customers switch between brands and need to purchase
a package of AB), 2' = 1, 2. The market prices of BI and 82 for both periods are the

marginal cost of CB. Each ﬁrm makes zero proﬁt.

(1.2) t S 03

No consumer switches between brands in the second period if t *3 CB. Knowing

25

that customers are locked-in after the ﬁrst purchase, ﬁrms may have an incentive to
charge locked-in customers as much as they can, U — t0" for A1 and U — t(1 - 0*)
for A2, where U is the consumers’ reservation value for AiBi, 2' = 1,2. This leads
to intensiﬁed competition in the ﬁrst period and hence rasults in less proﬁt in pure
bundling than in no bundling.

In the ﬁrst period, given t S 03, a consumer at 0* is indifferent between A181

and A282 if

122, + 29* + 6(U—29* + 29*) = pﬁz +2(1— 9*) + 6(U — 2(1 — 9*)+2(1— 9*))

+0“? ———”——A1). (1.67)

t
:0 :22" 22

Firms maximize the following proﬁt functions with respect to p31 and pi?

12f”) = (1921—614— CB)9* + 6(U — 29* — CA)9* , (1.68)

PB S :1: :1: t
122 ( )= (232—24— 23x1- 9 )+ 6(U — 2(1 — 9 )— 2A)(1— 9). (1.69)
The equilibrium market prices and proﬁts are as follows.

pgl =p§l=t+cA+cB—6(U—t—CA),

2
pin =piz = U '7 52 (1°70)

ﬂfB(S) = «58(5) = (1+ g) %.
Proposition 1.4 In pure bundling with a sequence of short-term contracts, if t S 63,
(i) No customer switchas between brands due to high switching costs. (ii) The ﬁrst
period price of the bundle is lower than in the benchmark model while the second
period price is higher than in the benchmark model. (iii) Firms make lower proﬁts
than in no bundling.
Proof: See appendix B.
The intuition behind proposition 1.4 is that ﬁrms face lessened competition in

the second period due to locking-in effects of switching costs while switching costs

26

intensify competition for market base in the ﬁrst period. The loss from intensiﬁed
competition in the ﬁrst period cannot be covered by the gains from softened compe-
tition in the second period.

Firms in market B have no sale in the second period while in the ﬁrst period ﬁrms

B1 and B2 have half of market share at the price of cB, making zero proﬁts.

(2) Long- and Short-term contracts

Firms offer long-term contracts as well as short-term contracts. Consumers in
[0, 0*] and [0*, 1] buy the bundle A131 and A282, respectively, in the ﬁrst period.
Consumers in [0, 6A] and [03, 1] commit to a long-term contract at the price of 19,11
and 2' A2 from ﬁrm A1 and A2, respectively. Consumers in [6,4, 0*] and [0*, 03] buy
a short-term contract at p31 and 1’22 from ﬁrm A1 and A2, respectively. In the
second period, some consumers who bought a short-term contract in the ﬁrst period,
[01, 6*] and [6*, 02], are targeted with discounts and switch between providers, and
pay the second period poaching price 1332 and 1531 for A282 and A181, respectively.
Consumers in [6,4, 01] and [92, 03] stay with the previous providers and pay pi] and

p342 for A1 and A2, respectively. Since customers in [0, 6A] and [63, 1] committed to

Figure 1.9: Pure bundling and Long and Short term contracts

 

 

A B A B
Period1 I I ' 1 I 2 2 I I
a:
0 6.4 0 63 1
A 9 A B A
Period2 I 1 IA2 2I ' 1I 2 I
I I I* I I I
0 6A 01 0 92 68 1

a long term contract at 2.41 and 15/12, respectively, competition in the second period is

over the interval [6A, 03]. Given 0*, a customer is indifferent between A1 and A282

27

at 91 and between A181 and A2 at 92 if

. 1 .
p.241+t61 =1)?” +t(1— 61) => 61: —t'(t +1322 —p?41), (1.71)

9§I+292=p§42+2(1-92) => 92=—(t+pA2- 13,121). (1.72)

Firms are maximizing the following proﬁt functions with respect to the second period

prices: {19311, 152]} and {p32, 1532}

W£B(L&S)= (P1241— CA)(61"’ 0A) + (ﬁﬁl-cA—cngBg - 9*) , (1.73)

QB‘L&S’= (piz—cmeg— 92)+(p§2— cA— 23x9 .9) (1.74)

Then, I have

 

 

 

29* 1

pA2=cA+ é-CB (26 3+ )t+:§ —9Bt, (1.75)
49*—1

PA1=CA+ g-CB 3 t+- $9,3t

 

“B —c +—2-c + 46*1t—-2-9t
In the ﬁrst period, a consumer at 9* is indifferent between A181 and A282 if

pg, + 29* + 609,122 +2(1— 9*)) = p32 + 2(1 — 9*) + 60521 + 29*)
g 9* =1 P22 ‘P31 _ M1532 ‘939
2 2(1 — 6)2 2(1— 6)t
2 2
1+ p32 -p31 + 60,112 _ 19/11) (1.76)
2 22 42 ‘

 

 

 

=>9*=

28

A consumer at 9 A and at 9 B is indifferent between a long-term contract and a sequence

of short-term contracts if

13,11 +(1 + 5)t9A = 1231 + WA + 5(1)?“ + tQA) => 13241: 1931+ dpil, (1.77)
9.42 +(1+ c020 — 93) = 2322 +t(1- 93) + 60222 +t(1— 93))

=> PA2 = P22 + 517312- (1'78)

Firm A1 and A2 maximize the following proﬁt functions with respect to p21 and 12%,

respectively.

PB L823 2:
«1 ‘ ’=(p§1—cA—c3)6 +5(P311—CA)91

+6931 - 2,, — c1292 — 9*) . (1.79)
“58““) = (P32 - CA - CB)(1 - 9*) + 509312 - CA)(1- 6’2)
+50% — CA — CB)(9* — 91). (1.80)

The equilibrium prices are

6
I512“: ﬁAg = (1 + E) t+(1+ (”CA +63,
6
p§1=p§2=t+cA+cB (1-5) , (1.81)

2 _ 2 1 1
9,41— 1222 = 513+ CA + 563,

. . l 3
p31: p22 = it + CA + 168'

By substituting the optimal prices into eq. (1.71) and eq. (1.72), I have {91, 92}.

I can also derive {9 A: 93} from the ﬁrst order conditions of eq. (1.79) and eq. (1.80)

at fig— = 0, where ﬁrst-period sales are unaffected if ﬁrms increase 17.1322 but decrease
Ai

9A2 by the same amount, 2' = 1, 2.

8 T 8 32
9*=%,92=§( —%‘ti),93=g(1+%) (1.82)

(2.1) t > CB

By offering both long-and short-term contracts, ﬁrms have a smaller portion of
customers who switch between brands in pure bundling than in no bundling. The
reason is that it costs more for ﬁrms to poach in the presence of switching costs of
63. It is also smaller than in a sequence of short-term contracts where :1; (1 — 3F) of
consumers switch. I can ﬁnd that a positive portion of consumers commits to long-
term contracts, % (1 — 5F), where t > c 3. The portion of consumers who purchases
a sequence of short-term contracts is 21, (1 — SP), where t > c 3. Nevertheless, ﬁrms
make less proﬁt than in short-term contracts.

The equilibrium proﬁts are

fan”): 7253““) = (1 + 126-6) é- — $3051} (22 - 323). (1.83)
Proposition 1.5 In pure bundling with both long-and short-term contracts, (i)
A positive number of consumers purchases long-term contracts, % (1 — SF), where
t > C3. The fraction of consumer poaching is i (1 -— SF) , where t > C3, which is less
than in a short-term contact. (ii) Proﬁts are lower than in no bundling if 3cB S t,
but higher than in no bundling if C3 < t S 303, while no ﬁrm necessarily makes
more proﬁts than in a short-term contract.

Proof. See appendix B.

Firms face more intensiﬁed competition when offering both long-and short-term
contracts than a sequence of short-term contracts because each ﬁrm competes with
itself on its own turf. Hence, payoffs from short-term contracts are higher than in
long-and short-term contracts

In market B, ﬁrms have a positive sale, §(1— 3P), where c 3 < t, in the second pe-
riod because some customers switch between brands and need to purchase a package
of A2812, 2' = 1, 2. The market prices of 81 and 32 for both periods are the marginal

cost of C3. Each ﬁrm makes zero proﬁt.

(2.2) t S CB

30

Long-and short-term contracts may not arise in equilibrium if t S 63. No con-
sumers switch between brands and purchase long-term contracts in equilibrium. All
consumers purchase a sequence of short-term contracts. Since customers are com-
pletely locked-in in the second period, ﬁrms severely compete for market share in the
ﬁrst period, resulting in the same equilibrium outcomes as in the previous case of

offering a sequence of short-term contracts.

(3) Long-term contract

Since ﬁrms offer a package of AiBi, where 2' = 1, 2, ﬁrms in market A offer dis-
counts on the purchase of A23), where i = 1, 2. I consider long term contracts in
which ﬁrms in market A offer a schedule of prices and a ﬁxed amount of discounts
of d* in return for a long term contract of good/service A, and establish a fee of
3* = %d*, where 0 < 6 S 1, when consumers terminate the contract earlier than the
contract term. Such strategic discounts is well observed in markets with repeated
purchase. In the US. telecom industry, ﬁrms offer discounts on the purchase of hand-
sets in exchange for one or two year long-term contract on a service plan.

In pure bundling, customers face switching costs of CB when they switch providers.
Hence, there are two cases that we need to consider when ﬁrms in market A have an
incentive to offer long-term contracts. From the previous case of short-term contracts,
we know that if t S c 3, no customer switches between brands due to high switching
costs of c 3, and if t > cB, customers are not yet locked-in and hence some customers

switch between brands in the presence of consumer poaching.

(3.1) t S CB
Customers are locked-in in the second period. Unlike a sequence of short-term con-
tracts, ﬁrms commit themselves to future prices in long-term contracts. Consumers
in [0, 9*] and [9* , 1] purchase a long-term contract of {A181, A1} and {A282, A2},
respectively. Let us assume that ﬁrms in market A offer a price for both periods, 13,41

for ﬁrm A1 and 13 A2 for ﬁrm A2. In the ﬁrst period, given t S cB, a customer at 9*

31

Figure 1.10: Pure bundling and Long term contract

A131 A232

Period 1

o—
%___
*-

Perlod 2

is indifferent between {A131, A1} and {A282, A2} if

16.41 + H?“ + 605,411+ t6") = m +t(1- 9*) + 603.42 + t(1 — 6*»

=9ﬁ=é+ﬂﬁﬁﬂ&. ass

Each ﬁrm maximizes the following proﬁt function with respect to 13 A1 and 5A2: re-

spectively.

PB L ._ at _ a.
«1 ‘ ’= (m —cA —cB>6 +602.“ —c.4>6, (1.85)

63‘” = (m - cA — CB)(1— 6*) + m2 — cm -— 0*). (1.86)

I have the equilibrium market price and proﬁts as follows.

 

- - CB
= p =t+ + , 137

{3‘19 = «53(L)=(1+5)%, (1.88)

where t S cB. Discounts on the purchase of product B emerge endogenously, d“ =
ﬁcB. However, these discounts are recouped in the second period. Remember that
I assumed that when a customer is offered discounts in return for a long-term con-
tract, then the customer needs to pay back the discounts plus interest as an early

termination fee if he terminates early. Since customers get discounts of ﬁe B on

32

the purchase of good B, they need to pay back T653 as an early termination fee if
switching between brands in the middle of contract term.

Let us check whether customers have any incentives to breach a long-term contract
and switch between brands in the presence of consumer poaching. In the second pe-
riod, consumers who may switch between brands need to pay the sum of a discounted
price of the bundle and the early termination fee, 6 A + c B + {$3, which is higher than
the price of t + c A + £63, where t S 63, and hence no customer is willing to switch
between brands.5 The idea is that in the presence of consumer poaching, bundling
can act as a commitment device through which ﬁrms are able to lock-in customers by
creating switching costs and ﬁrms offer long-term contracts in the presence of such
switching costs. That is, bundling permits neither customer switching nor consumer
poaching.

An example can be found in the US. telecom industry where ﬁrms oﬁer handsets
at a price lower than the marginal cost in exchange for a long-term contract of phone-
service plan for 1 or 2 years. When customers terminate a long-term contract, they
are obligated to pay an early termination fee, ranging from $150 to $240. The early
termination fee discourages customers from switching between brands. This practice
is observed through bundling of handsets and service plan.

Now let us check whether ﬁrms have collective incentives to bundle with long-term

contracts when t S cB.6 In Table 1.4, I show that long-term contracts may arise in

 

 

 

equilibrium.
Table 1.4: Payoffs in long-term contract under t S CB
No Bundling (Firm A2) Pure Bundling (Firm A2)
No Bundling (Firm Al) (1+ $3, (1+ 3); (1 + 3);, (1 + 3);
Pure Bundling (Firm Al) (1+ 3);, (1+ 3% (1 + (5)13, (1 + (5)5

 

 

 

 

 

Proposition 1.6 In pure bundling with long-term contacts, if t S (33, the equilib-

rium price is given in eq. (1.87). Pure bundling with long-term contracts arises in

 

5The poaching price of bundle to new customers is 132, = {222 = CA + Ca.
°In appendix A, I show the case where ﬁrm A1 offers the bundle with loyalty contracts while ﬁrm
A2 does not.

33

equilibrium. Firms are able to earn the same equilibrium proﬁts as in the benchmark
model.
Proof: See appendix B.

The intuition behind proposition 1.6 is that ﬁrms have incentives to bundle with
loyalty contracts in order to create artiﬁcial switching costs in the presence of con-

sumer poaching and to commit to future prices in the presence of such switching costs.

(3.2) CB < t

This is the case where ﬁrms offer a ﬁxed amount of discounts of d’“ in exchange
for a long-term contract of good/service A with a schedule of prices, {12%, p.311} and
{pgw p342} when customers are not yet locked-in after the ﬁrst purchase. Consumers
in [0, 6*] and [0*, 1] buy the bundle A181 and A282, respectively, in the ﬁrst periodf
In the second period, consumers in the intervals [91, 0*] and [6*, 02] are targeted with
discounts by competitors and may switch between brands, paying an early termina-
tion fee of s,- and purchasing the whole package of AiBi, rather than A,- separately,
where i = 1, 2. On the other hand, consumers in the intervals [0, 61] and [62, 1] stay

with their previous suppliers and purchase A1 and A2, respectively. In the second

Figure 1.11: Pure bundling and Long term contract with CE < t

 

 

A B A B
Perlod1 I l 1 ] 2 2 I
o 0* 1
A A B A B A
Perlod2 I 1 I 2 2 I 1 1 I 2 I
l | l I
0 91 0* 92 1

period, given 6* , a customer at 61 and at 02 is indifferent between A1 and A282 and

34

between A181 and A2, respectively, if

- 1 .
p2AI+t61=pﬁ2+sl+t(1—01) => 61=§E(t+p§2+31—p341), (1.89)

- 1 ..
1921+ 82 + t92 = 19342 + t(1 -- 92) => 02 = 520 +1232 —p§1 - 52), (1.90)

where p.241 and p32 are the second period prices of A1 and A2 for consumers of ﬁrm
A1 and A2, respectively, who committed to a long-term contract. 1331 and 1322 are the
second period prices of A181 and A232 for consumers who switch between providers,
paying an early termination fee of 3,, i = 1, 2. Firms maximize the following proﬁt

functions with respect to the their own poaching prices.

max (pﬁl—cA—chag —— 6*), (1.91)

max (7322—91—0wa — 61). (1.92)
Then, I have the second period prices for new customers as follows.

1521 = (t +1932 — 82 — 2t6" + CA + 63), (1.93)

HNIt—l

.. 2
1932 = -2-(Zt9* - t — 81+pA1+ CA + 63).

In the ﬁrst period, a consumer at 6* is indifferent between A181 and A282 if

1931+ t0* — 631+ 603,122 + t(1 — 6*) + 31)

= p32 + t(1 - 6*) — 632 + 6(ﬁ§1+t6* + 32)

_1+ M _ 605% ‘13.}:1)

2 2(1— 6)t 2(1— 6)t

= E+M+6(p2A2-pil) _6(32—31)
2 2t 4t 4t ’

 

=>6*—

(1.94)

 

=>0*

where p21 and p22 are the ﬁrst period prices of A181 and A282 in long-term con-
tracts, respectively. 63, is a ﬁxed amount of discounts of d, offered to customers in
return for a long-term contract, 2' = 1, 2. s, is an early termination fee imposed on

customers who terminate the contract early, 2' = l, 2.

35

Firm A1 maximizes the following proﬁt function with respect to {p31, pill

max (P21*CA*CB—531)9* ”(pal-c301
+6921 -cA—c3)(62—0*) + elm—61> ,

8. t. 01 = 9*. (1.95)
Firm A2 maximizes the following proﬁt function with respect to {pﬁw p32}.

max (pﬁz—cA—cg—6s2)(1-0*)+6<piz—ci>(1—62)
«59322— cA—c3)(e*—91)+6s2(62—o*> ,
8. t. 92 = 6* . (1.96)

By substituting the optimal prices and the condition, 61 = 02 = 0*, into eq.
(1.89) and eq. (1.90), I have the optimal discounts of d* and early termination fee of

3* as follows.

d’i‘ = d; =6(t—cB), (1.97)
31' = 3; = t — C13. (1.98)

I have a positive early termination fee, 3* > 0, where CB < t, high enough that ﬁrms
are unable to gain additional customers attached to competitors. In other words,
ﬁrms need to offer a positive amount of discounts of d* = 6 (t — 63) in order to lock-in
customers in the second period.

The equilibrium prices are

6
P§1=p§1=t+cA+cB+§t-6CB,
P311 =P§12 =t+CA, (1.99)

- «B
1931:1542 =CA+CB,
where {pﬁv 191241} and {p32, p342} are a schedule of prices of the ﬁrst and second

36

period in long-term contracts. Firms also include early termination fee 3* in long-
term contracts.

The equilibrium proﬁts are as follows.

«PM = «fHL)=(1+6t (1.100)

55 '

Firms offer discounts in the ﬁrst period in order to lock-in customers in the second
period. This intensiﬁes competition for market base in the ﬁrst period and hence pay-
offs are lower than in short-term contracts. Even though the discounts are smaller
than in no bundling, the equilibrium proﬁts are the same as in no bundling. The
reason is that since the ﬁrst period price reﬂects the amount of discounts offered to
customers and the difference between prices for loyal customers and new customers,
proﬁt margin in pure bundling, price-marginal costs-discounts, is the same in no

bundhng.

Proposition 1.7 In pure bundling with long-term contacts, if t > 63, the opti-
mal discount is d* = 6(t — CB) and early termination fee is 3* = t — 63. It then
follows that no consumer has an incentive to switch between brands. Each ﬁrm gains
the same equilibrium proﬁts as in no bundling, but lower than in a sequence of short-
term contracts.
Proof: See appendix B.

In market B, since no consumer switches between brands, there is no sale in the
second period. The market price to ﬁrms in market A is 03. Firms in market B make

zero proﬁts.

(4) Short-term contract vs. Long-term contract

With high switching costs, i.e. t S CB, customers are locked-in after the ﬁrst
purchase. Hence, ﬁrms have an incentive to commit themselves to future prices
through long-term contracts in the presence of such switching costs because otherwise

ﬁrms face more competitive market in the ﬁrst period and are worse off than in no

37

bundﬁng.

In Table 1.5, I compare payoffs under different types of contract arrangements

Table 1.5: Payoffs in Pure Bundling under t S cB (6 = 1)
Short-term (Firm A2) Long-term (Firm A2)
Short-term (Firm A1) 0.750t, 0.750t 0.826t, 0.893t
Long—term (Firm A1) 0.893t, 0.826t t, t

 

 

 

 

 

 

 

 

Table 1.6: Payoffs in Pure Bundling under 03 < t (5 = 1)
Short-term Long-term Long&Short
(Firm A2) (Firm A2) (Firm A2)

3 0.94t-0.22cB+0.11$, 0.96t—0.25cB+0.09-€2F, 0 88t- o 15cB+0. 09%,
(A1) 0. 94t— 0. 22cB+O. 11523 0.79t-0.11cB+0.10J,3 0. 82t- 0. 16cB+0. 12—?
L 0791— 0. 11cB+0. 1041, 0. 67t+0. 02cB+0. 0652?,
(A1) 0. 9613— 0 25634-0. 09—]i 0. 75:, 0. 75t 0.80t—0.14cB+0 09£

L&S 0.82t—0.1603+0.12—C:B , 0 80t— 0. 14cB +0 09%, 0.72t—0.06 3 +0.09%,

02 c2
(A1) 0.88t—0.1503+0.09-f3- 067t+0ﬂ02cB+0 065,3 0.72t—0.06cB+0.09-t3

 

 

 

 

 

 

 

 

 

 

at 6 = 1. When customers are locked-in by switching costs, ﬁrms have incentives to
offer long-term contracts. Nevertheless, when customers are not locked—in after the
ﬁrst purchase, CB < t, a sequence of short-term contracts is an optimal strateg' in
the presence of consumer poaching (See Table 1.6). The reason is that ﬁrms face
more competitive market in the ﬁrst period (or in both periods) than in short-term

contracts by offering long-term contracts (or long-and short-term contracts).

1.4.3 No bundling vs. Pure bundling

Pure bundling with long-term contracts may arise in equilibrium. The equilib-
rium outcomes are‘the same as in the benchmark model with no price discrimination.
The reason is that bundling creates an artiﬁcial switching cost. When customers are
locked-in with such switching costs, i.e. t S 63, long-term contacts permit ﬁrms to
commit to future prices in the presence of such switching costs. Hence, ﬁrms in a

vertically related industry may have an incentive to bundle in order to lock-in con-

38

sumers in the presence of consumer poaching and to offer long—term contracts in order
to commit to future prices.

On the other hand, pure bundling with short-term contracts may arise in equilib-
rium as a result of bundling game when bundling creates switching costs that cannot
lock-in customers in the presence of consumer poaching, i.e. c B < t. Firms are worse
off in pure bundling than in no bundling. The reason is that ﬁrms face intensiﬁed
competition for market base in the ﬁrst period in the presence of switching costs
created by bundling. The intuition is that ﬁrms face more competitive market in the
second period, but softened market in the ﬁrst period in the absence of switching
costs in markets with consumer poaching, while ﬁrms face more competitive market
in the ﬁrst period as well as in the second period in the presence of switching costs

in those markets.

1.5 Conclusions

This paper extends the existing studies on consumer poaching based on con-
sumers’ past behavior and connects two important streams of brand switching and
product bundling in a complementary market where a good / service is perishable and
the other good is durable. This study shows that ﬁrms in a market where consumers
make repeated purchases may have a collective incentive to bundle two complemen-
tary goods/services in combination with long-term contracts in the presence of con-
sumer poaching. The reason is bundling creates artiﬁcial switching costs and hence
ﬁrms are able to lock-in consumers. Long-term contracts permit ﬁrms to commit to
future prices in the presence of such switching costs. That is, bundling acts as a
commitment device through which ﬁrms are able to lock-in customers and ﬁrms offer
introductory discounts on a durable good in exchange for a long-term contract on the
other good that needs to be repurchased in each period. In the US. telecom industry
where ﬁrms offer a package of handsets and mobile service plan, discounts are given
on the purchase of handsets in return for a long-term contract of mobile service plan.

I also show that pure bundling with short-term contracts may arise in equilibrium as

39

a result of bundling game when customers face switching costs, but are able to switch
between brands when targeted with discounts by competitors. Firms are in prisoners’
dilemma in the sense that ﬁrms make less proﬁt than in no bundling because they
face more competitive markets in the presence of switching costs than in the absence
of switching costs.

Our paper has limitations. First, I assume that consumers have constant prefer-
ences over periods. I expect that the results would not be different from our paper
even when I consider the case where consumers change their preferences over peri-
ods. Second, I do not consider a model with overlapping generation consumers. New
customers may enter the market in the second period. In this case, ﬁrms may ﬁnd it
difficult to distinguish between customers that have just entered the market and cus-
tomers attached to competitors. Third, I consider only pure bundling, but not mixed
bundling, in the present paper. Models with the last two ideas would be extensions
of our paper. Another extension would be in the possibility of entry.

By the construction in the present paper, bundling may seem to result in positive
effect on welfare, eliminating inefﬁcient brand switching and yielding the same out-
comes as in the benchmark model. However, if I consider possibility of entry in the
second period, bundling may not necessarily improve social welfare in the presence
of consumer poaching. The reason is that it is hard for an entrant to gain a positive
market share, less competitive than it would be with new entry (preemption). This
suggests that the practice of bundling of phone service plan and handsets should be

regarded with suspicion.

40

Chapter 2

Strategic Mixed Bundling in
Duopoly

2. 1 Introduction

Bundling is the selling of two or more goods as a package, and there are two types
of this practice, pure and mixed bundling. In pure bundling, the individual goods are
sold only as a package. Mixed bundling refers to the practice of offering consumers
' the option of either buying two goods separately or buying a package of two different
goods, generally at a discounted price. Mixed bundling includes the case where a ﬁrm
offers good A or B as well as the bundle of AB. For example, local exchange carriers
in the US. tend to offer local service as well as a package of local service and lntemet
service, but do not offer Internet service alone.1

Most of the previous analysis of bundling has focused either on the ability of
bundling to achieve price discrimination or its ability to foreclose competition. A clas-
sic analysis in price discrimination by a monopolist is that bundling leads monopolists
to do better than selling independently when consumers’ valuations of two goods are

not perfectly correlated (Stigler, 1968; Adams and Yellen, 1976; Schmalensee, 1982;
McAfee, McMillan, and Whinston, 1989). The idea is that bundling reduces the vari-

 

lSBC Internet service (DSL) requires SBC local service. Qwest Choice DSL is available only to
Qwest local service customers for residential use. AT&T DSL Service is available only to residential
AT&T local service customers.

41

ations of customers’ valuations. The argument of the foreclosure is that a monopolist
of one product increases its proﬁts by earning monopoly proﬁts in the monopolized
tied market. For a long time, this was quite controversial because the monopolist
needs not monopolize the tied market to earn all the potential monopoly proﬁts
(Posner, 1976; Bork, 1978). However, Whinston (1990) shows that bundling can be
an effective foreclosure device under a monopoly in the ﬁrst market and duopoly in
the second market.2 Under the same market structure, Carbajo et al (1990) show
that a monopolist has a strategic incentive to bundle when valuations are perfectly
correlated. Their idea is that bundling can be used as a product differentiation device.
Chen (1997) also has the same idea in markets where there is duopoly in a primary
market and more than two ﬁrms in the other. Other explanations of bundling regard
cost savings and R&D. Choi (1996) studies the effects of bundling on research and
development and shows that bundling serves as a channel to monopolize the second
market when taking R&D investments into account. Salinger (1995) examines the
role of costs in explaining why products are bundled.

In monopoly models, mixed bundling is preferred to pure bundling because the
bundle is priced to extract all the surplus of consumers who value both goods high A
and the separate markets are used to extract the surplus of consumers who value
one high, but the other low. Adams and Yellen (1976) were the ﬁrst to show that if
the ﬁrm has a monopoly over both goods, proﬁts may be strictly higher under mixed
bundling than pure bundling. Schmalensee (1984) has shown, using the Adams-Yellen
(1976) framework, that when reservation prices for the two goods are independently
normally distributed, mixed bundling is preferred, by a multi-product monopolist,
to either pure bundling or no bundling. However, there are papers that show mixed
bundling is not optimal in a more competitive environment, unlike a monopoly. In
the Carbajo et al (1990) and Chen ( 1997) papers where bundling acts as a product
differentiation device, mixed bundling is dominated by pure bundling. They recognize
the role of bundling as a product differentiation device. Mixed bundling undermines

the role of product differentiation, which intensiﬁes price competition and leaves ﬁrms

 

2Choi (2003) and Nalebuff (1999) study on entry deterrence in different environments.

42

worse off. Symmetric duopolies in complement markets are considered in the Matutes
and Regibeau (1992) and Anderson and Leruth (1993) papers. They show that mixed
bundling is more likely to be associated with monopoly due to the beneﬁts obtained
through price discrimination. In a duopoly, however, mixed bundling is dominated
by pure bundling, unlike the monopoly situation since ﬁrms face more competitive
markets under mixed bundling. Economides (1993) shows that when goods are not
very close substitutes, using mixed bundling leads to lower proﬁts for both ﬁrms in
comparison to the proﬁts achieved when both ﬁrms avoid using mixed bundling. In
these papers, mixed bundling leads to a typical prisoners’ dilemma situation.

In contrast to the previous works on mixed bundling in more competitive markets,
I show that mixed bundling is a dominant strategy in duopoly like in the monopoly
situation where mixed bundling allows the monopolist to extract more consumers’
surplus than pure bundling due to the beneﬁts reaped via price discrimination. In
this paper, I show that an integrated ﬁrm (ﬁrm 1) strategically uses mixed bundling in
markets with consumers’ heterogenous valuations. I consider a model where there are
two independent markets, i.e. Internet access and local / long-distance phone providers
(or cable TV providers). In each market, there are a horizontally integrated ﬁrm and
an independent ﬁrm: ﬁrm 1 and A2 competing in a Hotelling model in market A
and ﬁrm 1 and B2 competing in a Bertrand model in market B53 In Matutes and
Regibeau (1992), Anderson and Leruth (1993), and Economides (1993), two multi-
product ﬁrms produce complementary products. Their result is that mixed bundling
can decrease the equilibrium proﬁts for a duopolist. However, Choi (2003) shows
that mixed bundling is better than pure bundling for a vertically integrated ﬁrm in
a system market because the vertically integrated ﬁrm has a strategic incentive to
sell inputs to its rival in the downstream market. I consider a partial mixed bundling

where ﬁrm 1 offers product A1 as well as A181 . It is important to consider this type

 

3Kovac (2004) explores tying in a market where a multi-product ﬁrm in independent markets
competes against several single-product ﬁrms. He shows that a multi—product firm has an incentive
to use pure bundling. However, he only analyzes a specific partial mixed bundling where a multi-
product ﬁrm offers a bundle of A181 and a homogeneous good of Bl in a market where consumer’s
valuations of good B are heterogeneous. In my model, I analyze a partial mixed bundling where an
integrated ﬁrm offers a bundle of A181 and a differentiated good of A1.

43

of mixed bundling because the integrated ﬁrm might have an incentive to foreclose a
market by practicing the mixed bundling when pure bundling is not permitted.4

My main result is as follows. With heterogenous valuations of good B, bundling
has two effects: substitution effect and competition softening effect. Substitution
effect of bundling arises from a situation where consumers of ﬁrm 1 who value good
B less than the bundle price minus the price of good A1 switch to its rival, A2, when
ﬁrm 1 offers A1 only in the bundle of A131. On the other hand, competition soft-
ening effects occur in both markets A and B, resulting from interdependence of two
goods in price and product differentiation role of bundling. The bundle A181 is dif-
ferentiated from good 32. Bundling also interrelates two independent goods in price
in market A. Pure bundling is proﬁtable when products A are highly differentiated
so that competition softening effect of bundling in market B dominates substitution
effect of bundling occurred in market A. Firm 1 is better off in pure bundling than
in no bundling at the costs of ﬁrm A2 because ﬁrm A2 needs to decrease its price
to compete against the bundle as bundling lessens competition in market B. On the
other hand, mixed bundling is proﬁtable even when products A are close substitutes.
The reason is that mixed bundling dampens the substitution effect of bundling. In
addition, when products A are highly differentiated, mixed bundling is preferred, by
ﬁrm 1, to either pure bundling or no bundling because with high magnitude of prod-
uct differentiation degree, t, competition softening effect of bundling is signiﬁcant
compared the substitution effect of bundling. As in pure bundling, ﬁrm A2 is also
worse off in mixed bundling than in no bundling. Suppose «£23 < F < «2’28 and
«£428 < F < «X23 where F is an entry cost. Then, mixed bundling can act as an
entry barrier where pure bundling is not allowed by an anti-trust authority. In the
previous example, local exchange carriers might be interested in offering local phone
service separately as well as a package of Internet access service and local phone ser-

vice, but not Internet service alone, to deter an entry in local service. Therefore,

mixed bundling in duopoly should be regarded with suspicion, like pure bundling.

 

4In addition, ﬁrm 1 has no incentive to offer a mixed bundling of {Bl , A131} and {A1, Bl, A181}
because products B are homogeneous in my model.

In the following sections, I set up a model with discrete consumer’s valuations of
good B in section 2. As a special case, I analyze a model with homogeneous valuations
of good B in appendix A. For the robustness of our results, a model with continuous

valuations of good B is analyzed in section 3. Conclusions are at the end.

2.2 Model with discrete valuations of good B

There are two independent markets for indivisible products A and B. In market
A, there are two ﬁrms: ﬁrm 1 and A2. Product A is horizontally differentiated in a
simple Hotelling model. It is assumed that ﬁrm 1 is located at point 0 and ﬁrm A2 at
1. Each ﬁrm produces goods at the same constant marginal costs of CA. A continuum
of consumers is normalized its total mass to 1. Each consumer is described by her
location on an interval, :12 E [0, 1]. The consumers are uniformly distributed on the
interval. Each consumer enjoys a gross utility of u from good / service A. I assume the
value, 11, of the good to be large enough that in equilibrium, every consumer has a unit
demand for product A. A consumer at x, where a: E [0, 1], incurs the transportation
cost of tx, or disutility associated with the difference between the purchased product
and the consumer’s most preferred product. If a consumer located at a: purchases from
ﬁrm 1, then that consumer gains net utility of u -— p A1 — tx. If that same consumer
purchases from ﬁrm A2 then that consumer gains net utility u — p A2 — t(1 — x).
The marginal utility from any additional unit of good A is zero. This ensures that a
consumer will buy one unit of product A. I
In market B, ﬁrm 1 and 82 produce homogeneous goods at the constant marginal
cost of CB and compete in price. There are two types of consumers in market B,
high(17) and low(y) valuations. That is, the utility from the ﬁrst unit of good B is
heterogeneous. I assume that y < CB < 17 and there are /\ consumers with valuation 17
and 1 — A with y. For tractability and simplicity, I set 9 = 0 and 17 = 1, with /\ = 1/2.
As a special case, I show in appendix A the case where consumers have homogeneous
valuations of good B, 17 = 1 with A = 1. For robustness of our results, I will consider

continuous consumers’ valuations of good B in section 3.

45

There are assumptions that hold throughout the paper. If ﬁrm 1 is indifferent
between bundling and no bundling, ﬁrm 1 chooses to bundle. If it is indifferent
between mixed bundling and pure bundling, ﬁrm 1 chooses mixed bundling.

The timing of the game is the following. In the ﬁrst stage, ﬁrms make bundling
decisions: No-bundling, Pure-bundling, and Mixed-bundling and these decisions are
irreversible. In the second stage, ﬁrms compete in price. The proper solution concept

for this two-stage game is Subgame Perfect Nash equilibrium.

2.2.1 No Bundling

Under No Bundling, ﬁrm 1 sells A1 separately from Bl. Hence, there are products
A1, A2, BI, 32 availableoin the markets. In market A, ﬁrms compete in a' Hotelling

model. A consumer at :L'* is indifferent between A1 and A2 if

u—pAi—tx*=u—pA2—t(1—x*)

1
=59" = §Z(t+PA2"PA1)- (2-1)

where p A1 and p A2 are the prices of A1 and A2, respectively.

Figure 2.1: No bundling in discrete valuations of good B

Market A I | I

Firms 1 and A2 maximize the following proﬁts function with respect to 1) A1 and

46

p A2 respectively.

1
7n = -2-t(PA1 - CA)(t +PA2 - p.41), (2-2)

1
7rA2 = 5-;(PA2 - CA)(t - PA2 + PAil- (2-3)

In equilibrium, each ﬁrm has % of market share. The equilibrium price and proﬁts

for ﬁrm 1 and A2 are as follows.

p311 = P22 = t + CA, (2'4)
(2.5)

Since ﬁrms in market B compete in price with homogeneous products, the market

price of good B is the marginal cost and each ﬁrm makes zero proﬁt.

193* 2 63, (2.6)

2.2.2 Pure-Bundling

Firm 1 offers only the bundle of A181, but not A1 and BI separately. In the
markets, the bundle A181 and separate goods A2 and 32 are available. Since ﬁrm 1
offers only the bundle of A131, ﬁrm B2 is the only one producing good B. However,
it cannot enjoy a monopoly proﬁt since there exists the bundle of A181. In market
A, ﬁrm A2 competes against the bundle A181. Since a positive number of consumers
who have low valuations of good B switches to A2 when A1 is offered to them only
as a package of A181, ﬁrm A2 can get beneﬁts from pure bundling.

A consumer at $3 is indifferent between A181 and A2 + 32 when p + t:r3 =
19,42 + p32 + t(1 — :53). i5 is the price of the bundle of A131. 12,12 and p32 are the

prices of A2 and 82 respectively.

47

Figure 2.2: Pure bundling in discrete valuation of good B

 

Market A ] A'B' ] A2 (+32) ]
0 x3 1

Considering heterogeneous consumer’s valuations of good B: y = 0, 17 = 1, with
A = %, the demand for A181 is 6' = 213(t +pA2 + §p32 — 15), the demand for A2 is
9A2 = 2120-12212 - §P32 +13), and the demand for 32 is 932 = 3120 -PA2 -p32 +15)-
The proﬁt functions are

1 ~ 1 -

7r1 = 5; - CA - CB)(t + PA2 + 51232 - 10), (2-8)
I l ..

7rA2 = 2709.42 — CA)(t — PA2 - 517132 +10), (2-9)
I -

7r32 = 470032 - CB)(t - PA2 - P32 +P)- (33-10)

There are three possible pricing policies for ﬁrm B2 to consider.

CB
1782 = 63 < p32 < 1
1

I assume that if a ﬁrm cannot make a positive proﬁt, its best pricing policy is to set

the price equal to the marginal cost. In Table 2.1, I show the equilibrium outcomes

of pure bundling.

48

Table 2.1: Equilibrium outcomes of Pure Bundling

 

 

 

 

 

 

 

 

 

 

 

 

 

1’82ch CB<p32<I p82=1

P32 113(3t+ 403)

ii t+cA+ch ~]-(1,t+cA+§cB t+cA+é+§cB
10,42 t+cA+écB 190t+cA+§cB t+cA—%+§CB
«f’B 21,0 —,1;cB)2 2010,03 — 2a,?)2 21,0 + ,1, — 11103)?
«£23 3,0 + gm)? 20111391! + 2c,,)2 21,0 — ,1, + gag)?
«Si? 0 swat -— c3)2 31,0 — cBXt— :1 +33)

(1) 19*32 = CB

This would be the case if t < 11c 3.5 When goods /services A are close substitutes
such as t < %03, the competition between the bundle of A131 and A2 + 32 is so
severe that the best strategy for ﬁrm B2 is to set 12*32 = 03. The integrated ﬁrm 1 has
no incentive to offer bundling because when ﬁrm 1 only offers the bundle of A131,
consumers who value product B less than 15 — p A1 switch from ﬁrm 1 to ﬁrm A2.
That is, the substitution effect of bundling arises in market A, but no competition

softening effect of bundling occurs in market B.

(2) CB <1);32 < 1

If 11303 S t < £5333, ﬁrm B2 has an incentive to set C); < p332 < 1, which results
in a positive competition softening effect of bundling in market B.6 When t 2 2cB,
ﬁrm 1 has an incentive to bundle its products. The condition ensures for ﬁrm 1 that
competition softening effect of bundling dominates substitution effect of bundling that
occurs in market A. Combining the two conditions, I can derive 2cB S t < £4353,
where 63 < %, in which ﬁrm 1 offers the bundle of A181 and ﬁrm B2 sets its price
at a price higher than the marginal cost. On the other hand, ﬁrm A2 is worse off
because it competes against the bundle of A181, rather than a separate good A1.

As p*‘32 is priced more than the marginal cost, ﬁrm A2 needs to lower p A2 in order

 

5min{§cB, 2:791} = Sf, where 0 < cB < 1
6pzn<1,ift< 5—1391

49

to compete against A181. That is, with high magnitude of product differentiation
degree, t, positive substitution effect of bundling, from a view point of ﬁrm A2, is
dominated by negative competition softening effect of bundling. Note that the bundle
price is lower than the sum of prices of two separate products: 13 < p223 + 12ng , and
that if t Z 203, the bundle price is higher than the sum of prices of two separate
products in no bundling: {1ng + 10ng S 13, but the price of ﬁrm A2 is less than that
in no bundling: pig S pﬁzB. The following proposition summarizes the equilibrium

outcomes of pure bundling.

Proposition 2.1 If 2cB S t < 5—-§fB-, where 03 < é, ﬁrm B2 has an incentive
to set its price at %(3t + 4cB). Proﬁts of ﬁrm 1 and B2 are higher in pure bundling
than no bundling. However, proﬁt of ﬁrm A2 is lower in pure bundling than in no
bundling.

Proof: See appendix D.

The intuition behind proposition 2.1 is that bundling is proﬁtable because the
condition of 2cB S t < 32:23, where 63 < %, ensures for ﬁrm 1 that bundling yields
a signiﬁcant competition softening effect in comparison to a substitution effect. On
the other hand, if $63 S t S 2cB, the opposite is true: bundling is not proﬁtable
because the substitution effect dominates the competition softening effect even when

ﬁrm 82 sets its price higher than the marginal cost.

(3) P*32 = 1

Firm B2 has an incentive to set its price at 1 if 2—tgcﬂ S t. The necessary con-
dition for ﬁrm 1 to bundle is CB S %. The condition of CB S a], ensures for ﬁrm 1
that in market B, the positive competition softening effect of bundling dominates the
negative substitution effect of bundling. Both ﬁrm 1 and B2 are better off in pure
bundling than in no bundling whereas ﬁrm A2 is worse off because when 2:3C—B S t,

from a view point of ﬁrm A2, there is little positive effect of bundling (substitution

effect), but signiﬁcant negative effect of bundling(competition softening effect).

50

Preposition 2.2 If 2:3;3 S t, where CB S é’, ﬁrm B2 has an incentive to set its
price at 1. Proﬁts of ﬁrm 1 and B2 in pure bundling are higher than in no bundling
while that of ﬁrm A2 is lower than in no bundling.
Proof: See appendix D.

The idea behind proposition 2.2 is the same as in the previous case of CB < p*32 <
1. Pure bundling is optimal only when products A are somewhat differentiated so
that bundling lessens competition in market B signiﬁcantly enough to cover the costs

that result from heterogeneous consumers’ valuations of good B.

2.2.3 Mixed-Bundling

There are three types of mixed bundling strategies for ﬁrm 1 to choose: {A1, A131},
{Bl,AlBl}, and {A1,Bl,AlBl}. If ﬁrm 1 offers {BL/1181}, or {A1,Bl,AlBl},
there is no competition softening effect of bundling in market B. The results of these .
mixed bundling are the same as the case of no bundling. Therefore, our interests
are in the partial mixed bundling of {Ah/1181}. Firm 1 offers A1 and A131 in this
subgame. By offering .41 as well as A181, ﬁrm 1 holds consumers with valuations of
good B less than 1'5 — p .41 who otherwise may switch to A2 in pure bundling. However,
ﬁrm 1 competes against itself: A181 competes against A1 + 32. Firm 1 has to satisfy
the condition of ii S p A1 + 1232 in order to have positive sales of A131. A consumer

at 2:3 is indifferent between A131 and A2 + 82 when
P+t$3 =PA2 +P32+t(1 -$3) (2-11)
And a consumer at 2:4 is indifferent between A1 + 82 and A2 + B2 when
PA1 + P32 +t1’4 = PA2 +P32 + t(1 - 1C4) (2-12)

From (2) and (3) with discrete consumer’s valuations of good B, I can derive 5:3 =

‘212{t +pA2 +1];sz — 1‘5} and x4 = 2%{t +pA2 — PAll- In addition to market share in

51

pure bundling, ﬁrm 1 can gain additional market share of 3:4 — 5:3 by setting prices
such as p A1 + %p32 S p” S PM + p32. The idea is that mixed bundling permits ﬁrm
1 to keep those who may otherwise switch to A2 when A1 is offered to them only as

a package of A181.

Figure 2.3: Mixed bundling in discrete valuations of good B

 

MarketA ] A131 ] Al(+BZ) ] A2(+Bz) ‘ I
0 x3 x4 1

Firm 1 has positive demands for both A181 and A1, (1' = 213(t + 13,42 + épgg - 15)
and qu = 21365 — PAl -— ész), respectively. Demands for ﬁrm A2 and B2 are
9A2 = 2120 - PA2 + PA1) and 932 = 1120 - PA2 - P32 + P), reSPectively. The Proﬁt

functions are

7r1 = %” - CA — C3)(t +PA2 + $P32 -P)
+5120,“ - one - p.41 - gm), (2.13)
7rA2 = $(PA2 - CA)(t - PA2 +PA1): (2-14)
«32 = 41,032 - C3)(t - p.12 — p32 + :3). (2.15)

As in pure bundling, there are three possible pricing policies for ﬁrm B2 to

consider. Table 2.2 shows equilibrium outcomes.

52

Table 2.2: Equilibrium outcomes of Mixed Bundling

 

 

 

 

 

P32=C3 C3<P32<1 P32=1
P32 3t+ 3C3
p” §t+cA+I%cB §t+cA+§cB gt+cA+110+§cB
pm gt+cA+%cB 3t+cA+§cB §t+cA—§+§cB
pAg §t+CA+110CB gt+CA+%CB gt+CA-I!O+%CB

 

5%;(‘2ft-C3X3t-C3)

+21?(3t+ gag)?

«(”3 Amt-amass

+21,(§t+ §c3)2

5%?(6t-CB + §)(t—c3 + i)
+21,(§t + gag — 3;)?

 

 

 

 

 

 

 

 

0.15423 21%? + 1150192 21263” 11193)2 212$ - r10 +113C3)2
4
«9’23 0 he — lea)? ha — chm}: — 5 + gas)
(1) P232 = CB

If t < %cB, ﬁrm 82 would set p32 = 63.7 When t S ﬁcB or §cB S t, ﬁrm 1
gains more in mixed bundling than in no bundling even in the absence of competition
softening effect, 12232 = 63, in market B. The reason is that mixed bundling mini-
mizes the substitution effect and the bundle option softens competition in market A.
When t S -17cB, ﬁrm A2 also makes a higher proﬁt than in no bundling due to inter-
dependence in prices of two independent goods A and B while there is no negative
competition softening effect of bundling in market B.8 I can check that in equilibrium,
the bundle price is less than the sum of prices of separate goods: ii S 12%3 + 1215423,

where t < ACE. It also holds that pm + £sz < )5.

Pr0position 2.3 If t < .17cB, (i) Firm B2 sets its price at the marginal cost 63.
Both ﬁrms 1 and A2 gain more in mixed bundling than in no bundling. (ii) Firm 1
is better off in mixed bundling than in pure bundling. Firm A2 makes less in mixed
bundling than in pure bundling.

Proof: See appendix D.

When products A are close substitutes, i.e. t < $03, mixed bundling permits

 

7min{£ﬂ,4——%ﬂ-}=%CB,0<CB<I B

81ft 5 —c3,pg13 £1)le (ﬁt-Iraq S gt+CA+%CB ipﬁz

2 Spﬁ1284=>t+CAS §t+CA+$CB.

53

little substitution effect and hence is preferred to either no bundling or pure bundling
by ﬁrm 1. Firm A2 is also better off in mixed bundling than no bundling due to the

competition softening effect of bundling in market A.

If 303 S t < 9;;‘23, ﬁrm 82 has an incentive to set its price at 19"‘32 = $(7t+803)
and gains a positive proﬁt. Firm 1 can gain higher proﬁts in mixed bundling than
in no bundling if t S €03 or ‘ECB S t. Firm A2 also gains a higher proﬁt in mixed
bundling than in no bundling if t S go 3 due to interdependence of two goods in price
(competition softening effect), pg? S p963 and pIXZB S pﬁ’zB if t S §03. However,
if t Z #03, ﬁrm A2 is worse off in mixed bundling than in no bundling due to the
competition softening effect of bundling in market B. In order to have positive sales
of the bundle, it must hold that 15 S p A1 + p 32, where t S 403. Therefore, We need

$03 S t < £03 or #03 S t < min{403, £395} = t1, where 03 < 3%.

Proposition 2.4 If $03 S t S §03, or If g03 S t < min{403, 9—_7835}, where
03 < %, ﬁrm 82 sets its price at P32 = 3(7t + 803), making positive proﬁts and
ﬁrm 1 makes higher proﬁts in mixed bundling than in no bundling. Firm A2 gains
less proﬁts in mixed bundling than in no bundling if #03 S t < min{403, 2:353},
where 03 < %, but higher proﬁts if %03 S t S €03.

Proof: See appendix D.

The idea behind Proposition 2.4 is that both ﬁrms gain more in mixed bundling
than no bundling when products A are close substitutes. The reason is that both
ﬁrms can enjoy positive competition softening effect of bundling in market A. On the
other hand, when products A are sufficiently differentiated, i.e. {$03 S t < t1, ﬁrm
A2 is worse off in mixed bundling than in no bundling because it needs to decrease
its price far more than what it can beneﬁt from the competition softening effect of
bundling in market A.

There are intervals where mixed bundling is more proﬁtable than no bundling,

while pure bundling is not. For example, ﬁrm 1 has no incentive to offer pure bundling

54

in the interval of $03 S t < §03 or %03 S t < 203 where mixed bundling is pre-

ferred, by ﬁrm 1, to no bundling.

Proposition 2.5 If 303 S t S 12303 or %%03 S t < t1, where t1 = min{403, 9:393},
03 < $175, ﬁrm 1 is better off in mixed bundling than in pure bundling while ﬁrm
A2 gains less proﬁt in mixed bundling than in pure bundling. Firm B2 makes
higher proﬁts in mixed bundling than in pure bundling if 37103 S t S €03 or if
~}§cB S t < min{403, s£333}, where 03 < é-g, while it makes more proﬁts in pure
bundling than mixed bundling if $03 3 t g {It-B.

Proof: See appendix D.

(3) P232 = 1

When 2:393 S t, ﬁrm B2 sets 19:92 = 1. Firm 1 has an incentive to offer mixed
bundling if 03 S %. Unlike in pure bundling, ﬁrm 1 also prefers mixed bundling
to no bundling if t S 03 - % or 33(03 — %) S t, where 03 > r], due to signiﬁcant
competition softening effect of bundling. Firm 1 has a positive demand for the bun-
dle, 13 S PA] +1932, if t S 21:23. The necessary condition for 12,41 + §P32 S 15 is

t 2 max{1—_§EB, 0}. This holds under 4—:36—5 S t S 11:95.

Proposition 2.6 (i) (a) If 4:393 S t S 21:33, where 03 S %, (b) If t2 S t S 2:293,
where t2 = max{4—-37SB,1(E§——l)}, % < 03 S 170, or (c) If £353 S t S 03 — %,
where 2:] S 03, ﬁrm B2 has an incentive to set its price at 1, making positive proﬁts in
mixed bundling. Firm 1 makes higher proﬁts in mixed bundling than in no bundling
while ﬁrm A2 gains less in mixed bundling than in no bundling except the case of
(0). (ii) If t3 S t S £355, where t3 = max{4—_%c-B,ﬂl—;2-2£ﬂ}, {15$ 03 < 1%, or If
4:395 S t S 7—7403, where 312 S 03, ﬁrm 1 is better off in mixed bundling than in
pure bundling. Firm A2 gains less in mixed bundling than in pure bundling, while
ﬁrm B2 makes higher proﬁts in mixed bundling than in pure bundling.

Proof: See appendix D.

Proposition 2.6 implies that mixed bundling has business stealing effect in the

55

sense that ﬁrm 1 gains higher proﬁts in mixed bundling than in either no bundling or
pure bundling at the expenses of ﬁrm A2. Mixed bundling is preferred, by ﬁrm 1, to
pure bundling even when the marginal cost of good B is signiﬁcant, i.e. é S 03, unlike
in pure bundling. The reason is that mixed bundling allows little substitution effect.
High competition softening effect in comparison to substitution effect of bundling,
fromea view point of ﬁrm 1, occurs when products A are highly differentiated. This
can also occur when products A are close substitutes if the marginal cost of good B

is high.

2.2.4 Sub-game perfect equilibrium

Mixed bundling may arise as a sub-game perfect equilibrium. When products
A are close substitutes, i.e. t < $03 in which ﬁrm B2 sets its price at the marginal
cost, ﬁrm 1 is better off in mixed bundling than in pure bundling. The reason is
that mixed bundling dampens substitution effect of bundling. When $03 < t < 13:03
or $$03 S t < min {403,333} = t1, where 03 S $%, in which ﬁrm B2 has an
incentive to set its price at 03 < 17*82 < 1, ﬁrm 1 also makes higher proﬁts in mixed
bundling than in pure bundling due to competition softening effects in both markets
and reduced substitution effect in market A. When £395 S t S 7-2325, where
$ S 03 or when t3 S t S 1:;513, where t3 =max{4——$SB,1—9(—l;22—cB—)}, (15S 03 < $,
in which ﬁrm B2 has an incentive to set its price at p32 = 1, mixed bundling is
preferred, by ﬁrm 1, to pure bundling.

With mixed bundling, ﬁrm B2 gains a higher proﬁt than in pure bundling except
in the interval of $03 S t S £03. Note also that ﬁrm A2 is better in pure bundling
than mixed bundling. In addition, it prefers no bundling to mixed bundling unless
t S 32503 or £355 S t S 03 — é, where 2:] S 03. This implies that ﬁrm 1 may
have an incentive to offer mixed bundling in order to deter entry. In the previous
example, local exchange carriers might be interested in offering local phone service
separately as well as a package of Internet access service and local phone service, but

not Internet service alone, to deter an entry into local service.

56

2.3 Model with continuous valuations of good B

In this section, I consider a case in which the consumer’s valuation of good B is
uniformly distributed, 1) ~ U m' f [0, 1]. I follow the assumptions made in the previous
model where consumers have discrete valuations of good B in exception of continuous

valuations of good B.

2.3.1 N o-Bundling

This is the case where ﬁrm 1 sells goods A1 and Bl separately. Hence, equilibrium
outcomes of ﬁrms in market A and B are the same as in the model with discrete

valuations of good B.

2.3.2 Pure bundling

Firm 1 offers only the bundle of A131. A consumer at 1:5 is indifferent between
A181 and A2 +82 when 15+tx5 = 1))” +1332 +t(1 —- 2:5). ii is the price of A181. 19,42
and p32 are the prices of A2 and 32, respectively. Considering consumers’ valuations

of good B: 12 ~ U m’ f [0, 1], the demand functions for the bundle, A2 and 82, are

1 1 1
9 =/0 13300619 = 5; (t +PA2 - P+P32 - 517322) , (2-16)
.. 1 ~ 1
9A2 =1- 9 = 2—t(t -PA2 +P -P31+ 5P322), (2-17)

932 = (1 - $§(P32))(1 - 1932): 51;“ - PA2 +P - P32)(1- P32) - (2-18)

where xg(v) = 21:0 + pAg + min{p32,v} - 13). Clearly, ﬁrm B2 has no incentive to

set p232 = 1. The proﬁt functions are as follows.

I ~ ~ 1

7r1 = 52(1) - CA - C3) (t +PA2 - P+P32 - 517322) , (2-19)
1 ~ 1 2

7rA2 = 2—t(PA2 - CA) t- PA2 + P - P32 + 51332 , (2-20)
1 ..

7r32 = 52(P32 - C3)(t - PA2 +P - P32)(1 — P32) - (2-21)

57

I do simulation analysis for these outcomes since they are rather messy in expression.
In appendix C, I show proﬁts of ﬁrms when ﬁrm 1 offers only the bundle of A131.
Even though I do not have closed forms of the equilibrium outcomes, I can show
conditions where ﬁrm B2 sets its price 03 < p)” < 1 and ﬁrm 1 offers pure bundling

(see Figure 2.4).
0
Figure 2.4: Pure bundling in continuous valuations of good B

’3

1.75
1.50
1.25
1.00
0.75
0.50

0.25

 

 

 

Proposition 2.7 If tp < t < tp, optimal price of ﬁrm BZ is at 03 < p22 < 1. Firm
1 and 82 make higher proﬁts in pure bundling than in no bundling whereas ﬁrm A2

gains less proﬁts in pure bundling than in no bundling.

_ 803 +4(1 —cB),ﬂ—T— CB —4-c23

 

 

t i
p 2x/I-QCB-(l-CB)

- 2- 1

11,, = 368, where 03 < 5.

Proof: From the proﬁt functions of ﬁrm 1, 0'53 2 niVB, if a 2 03, where a =
p*32 -— $(p232)2, 12:32 is the optimal price of good B: p332 — $(p232)2 2 03. The
equality of p332 — $(p*82)2 Z 03 holds at 1 — m =13, where 03 < $9 So,
the condition of 13 < 11:32 is needed to hold in order for ﬁrm 1 to bundle. By

substituting 13 into the F.O.C. of 1132, I can ﬁnd a condition for 19:32 > 13 : (i)
t > 803+4(1~03)W3—4—023 __

2\/1—203—(1-03) —
its price at 03 < p332 < 1 to make positive proﬁts. Firm B2 has a positive proﬁt

tp . Now let’s ﬁnd conditions for ﬁrm B2 to set

 

9Note that 03 < 13.

58

margin, 1’,sz —- 03 > 0, if 13 < 9*82 because 03 < 13. (ii) In order for ﬁrm B2 to sell

a positive amount of 82, qu = (1 — a:* (19232)“ — 13*82) > 0, it needs to hold that

 

19*32 < 13, where 13 = min{$(\/1 + 4(3t + 03) -— 1), 1}. Hence, combining the two
conditions for pure bundling equilibrium, it needs to hold that 13 < 12*32 < 13. This
holds if tp < t < 2:395 = fp, where 03 < $.

Pure bundling vs No bundling The intuition behind proposition 2.7 is the same
as in the previous case of discrete valuations of good B. Bundling has a substitution

effect in market A and competition softening effects in both markets A and B. When

8CB+4(I—CB)‘/I-2CB-4—032 .
2 \/1—2CB‘ (1‘68) , competi-

tion between A131 and A2 + 32 is so severe that ﬁrm BZ is less willing to increase its

products A are close substitutes: i.e. t < tp =

price while pure bundling causes substitution effect due to heterogenous customers’
valuations of good B. In this case, ﬁrm 1 has no incentive to offer pure bundling.
On the other hand, if products A are sufﬁciently differentiated, i.e. t > 51,, in which
substitution effect is dominated by the competition softening effects of bundling, ﬁrm
1 has an incentive to offer pure bundling. I compare proﬁts of ﬁrms in pure bundling

with those in no bundling in Appendix E.

2.3.3 Mixed bundling

Firm 1 offers A1 as well as the bundle of A181. There are A], A2, 32, and A131

in the market. A consumer at 2:5 is indifferent between A181 and A2 + 82 when
- 1 -
P+t$5 =PA2 +P32 +t(1- 335) => $5 = é—tU‘I‘PAZ +P32 -P),

And a consumer at 3:5 is indifferent between A1 + Bg and A2 + 32 when

1
PA1+P32 +t16 = PA2 +P32 +t(1- $6) => 5’76 = E“ +PA2 - PA1),

And, to sell a positive amount of A181, the bundle price is less than the sum of

the two separate products: 13 S 1) A1 + p32. With continuous valuations of good

59

- 1 -
B, 135 = f0 x5(v)dv = 21105 + PA2 - P + P32 - éPfgg). where 935(9) = 2110 + PA2 +
min{p32,v} - 13). By offering A1 as well as A181, ﬁrm 1 can gain an additional
market share of 1:5 — 5:5 with PA] +1232 - $12232 S 3' S 12,11 +1232 . The demand

. - 1 1

for the bundle is q = [0 x5(v)dv = 21; (t +pA2 —p +1332 — $12232), demand for A1
. 1 - .
1S qu = 2:5 - f0 x5(v)dv = 213 (1)—11,41 —p32 + 317282), demand for ﬁrm A2 1s
qu = 1 — (3 -— qu = 213(t — 19,12 + PAI): ﬁrm B2 has a market share of q32 =
(1 - $5(P32))(1 - P32) = 2110 - PA2 + P - P32)(1 - P32) Clearly, ﬁrm B? has no
incentive to set p32 = 1 in order to sell positive amounts of 32. The proﬁt functions

are as follows.

1 - 1 1 2
7r1 = 5 (P—CA-CB) t+PA2“P+PB2"§P32
1 - 1 2
+5; (PAl-CA) P-PAl-P32+§PB2 , (2-22)
1
7r212 = §;(PA2—CA)(t-PA2+PA1): (2-23)
1 ~ ,
932 = 2—t(P32-C3)(t-PA2+P-P32)(1-P32)- (2-24)

In Appendix E, I do simulation analysis when ﬁrm 1 offers A1 and A181.

Example 1. (Mixed bundling vs No bundling) Simulation analysis shows us
that ﬁrm 1 has an incentive to offer mixed bundling, i.e. when products A are suf-
ﬁciently differentiated with small marginal cost of good B. It is also preferred to no
bundling even when products A are close substitutes if the marginal cost of good B
is high. Let’s suppose that ﬁrm BZ sets 1);” = 03. If t S $023 or £02}? S t, mixed
bundling is preferred, by ﬁrm 1, to no bundling even when pf32 = 03. From pure
bundling case, I know that bundling is proﬁtable if 13,, < t. Hence, if t S tp, pure
bundling cannot arise in equilibrium while this example suggests that mixed bundling
may arise in equilibrium, i.e. t S 115628 (see Figure 2.5).

Proof: We know that 11’le —- 77le Z 0 if t + gm — 03) + 211$ —- 03)2 Z 0, where

B =p32— 1192 . Suppose that 19* = c . Then it becomes t—9 03 2t-I- 7 CB)4 2 0-
2 32 32 B 21 8?

60

So, ift S $023, or 11-0?3 S t, 3])“; 2 «(VB even when 19*32 = 03.

”- 9t+c +40 +16 - Ea-t-I-c +20 _23 -:1-t+0 +10 -—16
1
7n = ﬁ((6t+(ﬁ- C3))(3t+3(ﬂ-C3)) + (3t—2w— c302).
MB _ _Lv

7rA2 - 50, (4t- (ﬁ—C3))2-

where [3 = 12232 - $02232? , 1);” is the optimal price of good Bg.

Figure 2.5: Mixed bundling vs No bundling
1‘ tr

OOOHHH
.01ch b

 

 

 

Example 2. (Mixed bundling vs Pure bundling) Now, let’s illustrate an ex-
ample where mixed bundling may arise in equilibrium while pure bundling may not.
If 12:32 2 1 — (ﬁt-2'03 = 13, where 03 < $, ﬁrm 1 has an incentive to offer mixed
bundling.10 When 19:32 = 13, then ,3 — 03 = 0, where 6 = 12*32 -— $(p’l‘32)2, 19*32 is the
optimal price of good 32. Note also that the F.O.C. of «£3428 increases at 13, 13*32 > 13,

, 8c3+4(1—cB),/T1— 03—4—02 1
1ft 2 tM, Where g ( 2\/l—QC3-(l—03) B = 9t], = tM, where 03 < 2. Now

let’s ﬁnd some intervals where ﬁrm B2 sets its price at 03 < 3'32 < 1 to make posi-

 

tive proﬁts. Firm B2 has a positive proﬁt margin, 192'32 — 03 > 0, if 13 < p332 because

 

”90m example 1, «1” — «1“: 2 o if t + 99 — .,.) + 9.0 — new 2 o. where 3 = p... - 93..

Hence, if 1252 S l — \/1 — 203 + 2t or 1 — 4M7 — 17403 +Zt S p232, ﬁrm 1 makes higher proﬁts
in mixed bundling than in no bundling. The conditions for mixed bundling equilibrium are rather
messy in expression. Hence, I show examples where mixed bundling is preferred to either pure

bundling or no bundling. Note that if 122,2 2 1 — \/1 — 203 = 13, ﬁrm 1 has an incentive to offer
mixed bundling because (1 — 4M7 — 1403 + 4t) < 13.

61

03 < 13. In order for ﬁrm B2 to sell a positive amount of .82, it needs to hold that

 

12*32 < 13, where 13 = min{$(\/9 + 28t + 1203 - 3), 1}. Hence, combining the two
conditions for mixed bundling equilibrium, it needs to hold that 3 < 19"1‘32 < 13. This
holds iftM < t < :39- = {M , where 03 < $. Therefore, iftM S t < t-M, where
03 < $, ﬁrm B2 has an incentive to set 13 < 1);” < 13 and ﬁrm 1 gains more in mixed
bundling than in no bundling. Remember that the necessary condition for ﬁrm 1
to offer pure bundling is tp < t, where 03 < $. This example suggests that there
are some intervals where mixed bundling is proﬁtable, but not pure bundling, due
to a large substitution effect of bundling compared to competition softening effects
of bundling: i.e. tM S t < tp. I compare proﬁts of ﬁrms in mixed bundling with
those in pure bundling in appendix C. Mixed bundling is preferred to pure bundling
when the marginal cost 03 is high, resulting in both signiﬁcant substitution effect
and competition softening effect of bundling. Firm 82 is better off in mixed bundling
when marginal cost is small while it prefers pure bundling to mixed bundling when

marginal cost is high. Firm A2 is worse off in mixed bundling than in pure bundling.

2.3.4 Sub-game perfect equilibrium

If tp < t < 51,, where 03 < é, pure bundling may arise in equilibrium. I
illustrated that mixed bundling is a proﬁtable strategy while pure bundling is not
in some intervals such as tM < t < tp, where 03 < 3. I also ﬁnd that ﬁrm 1 has
an incentive to offer mixed bundling even when ﬁrm B2 sets its price at p232 = 03.
Simulation analysis shows that mixed bundling is preferred to pure bundling when
there exists a competition softening effect, i.e. high 03, and a signiﬁcant substitution

effect of bundling, i.e. high marginal cost of good B and small degree of product

differentiation.

2.4 Conclusions

This paper studies strategic use of mixed bundling in duopoly where there is a

multi-product ﬁrm and an independent ﬁrm in each market. I show that in duopoly,

62

mixed bundling may arise in equilibrium like in a monopoly model where mixed
bundling dominates pure bundling when consumers’ valuations of two goods are not
perfectly correlated. In monopoly, the bundle is priced to extract all the surplus of
consumers who value both goods highly, while the separate markets are used to ex-
tract the surplus of consumers who value one high, but the other low. On the other
hand, in the previous duopoly models, pure bundling dominates mixed bundling be-
cause ﬁrms fear the extra degree of competition inherent in mixed bundling.

In this paper, I show that an integrated ﬁrm strategically uses the mixed bundling
strategy in a market where bundling has two effects: substitution effect and compe-
tition softening effect. Substitution effect of bundling arises from a situation where
consumers in market B have heterogeneous valuations of good B. If ﬁrm 1 only sells
the bundle of A181, consumers of ﬁrm 1 who value good B less than the bundle price
minus the price of A1 switch to ﬁrm A2. This effect is signiﬁcant when products A
are close substitutes and/ or the marginal cost of good B is high. On the other hand,
competition softening effects of bundling occur in both markets A and B, resulting
from interdependence in price of two independent goods and production differenti-
ation role of bundling. I show that mixed bundling is preferred, by the integrated
ﬁrm 1, to pure bundling not only when products A are close substitutes, but also
when products A are sufficiently differentiated due to competition softening effects
of bundling. Firm 1 gains more proﬁts in either mixed bundling or pure bundling
at the expenseof ﬁrm A2. Firm A2 is worse off in either mixed bundling or pure
bundling than in no bundling due to the interdependence of two goods in price. Our
result has an important implication on anti-trust policies. When «£23 < F < «X23
and «£423 < F < 32/23, where F is an entry cost, mixed bundling can be used as an
entry barrier in a market where pure bundling is not allowed by an anti-trust author-
ity. For instance, local exchange carriers might be interested in offering a package of
Internet access service and local service, but not Internet service alone, to deter an

entry into local service. Therefore, mixed bundling in duopoly should be regarded

with suspicion, like pure bundling.

63

Chapter 3

Vertical Integration and Firewalls

with R&D Spillovers

3.1 Introduction

Vertical integrations are mergers of noncompeting companies where one’s good
is a complementary good to the other’s. Such mergers can achieve procompetitive
efficiency beneﬁts in the sense that vertical integration can reduce costs or enhance
innovation efforts through several ways. When both the input and output markets
are imperfectly competitive, a vertical merger of a ﬁrm with an input supplier can get
efficiency by eliminating one of the two markups and hence reducing output prices.
Vertical mergers also facilitate better coordination between input suppliers and output
producers with respect to product design and make it possible to share technologi-
cal information common to separate stages within an industry. Design coordination
can lead to lower costs and higher product quality. There may be positive R&D
externalities, vertical R&D spillovers, through the sharing proprietary technological
information between two different production stages within a vertically integrated
ﬁrm engaging in innovation. Even though a vertical merger can increase the costs
to nonintegrated competitors (rasing rival’s cost), it does not necessarily represent
inefﬁciency of vertical merger. Vertical merger efficiency can arise from the fact that

inefficient output ﬁrm serves smaller output market share or is out of the market after

64

the merger.

Vertical acquisitions can be also anticompetitive under certain circumstances. By
engaging in exclusionary conducts in its pricing and purchasing decisions that create
or raise entry barriers at both upstream and downstream levels, vertical mergers can
lead to higher prices or lower innovation effort levels. In upstream market, this may
involve the upstream division raising the input price it charges to competitors of the
downstream division or refusing to supply them. In downstream market, this may in-
volve refusing to purchase from rival input suppliers. And also, a vertical merger raises
competitive concerns regarding competitively sensitive information transfers within
an entity. Since vertical acquisitions involve companies making products one of which
is a necessary complement to the other’s, vertical integrations can give the vertically
integrated entity the ability to obtain rival’ proprietary information in either market.
The information can involve nonpublic pricing or demand information, in which case
the competitive concern is that collusion in one of the markets is facilitated as a result
of the merger. For this matter, a settlement involves ﬁrewalls, a provision of Depart-
ment of Justice (DOJ) or Federal ’Ifade Commission (FTC) to prevent the sharing
of competitively sensitive information of competitor’s between different divisions of
a vertically combined entity. For example, in Lilly’s acquisition of PCS, a pharmacy
beneﬁt management company, the Commission’s settlement requires that the merged
ﬁrm constructs a ﬁrewalls provision to prevent information transfer to Lilly such as
other drug producers’ prices, bids, or other conditions of sale. In high tech indus-
tries where upstream and downstream companies work closely, a vertically integrated
ﬁrm’s division in one market can pass on competitors’ innovation information in the
other market to its division in that other market. A competitive concern here is
that the merged ﬁrm can free ride off its competitors’ innovation effort, potentially
reducing the rivals’ R&D effort levels. For this matter, in the the Silicon Graphics,
Inc. (SGI) acquisition of Alias & Wavefront, one of conditions in the settlement with
Silicon Graphics from the Commission in order to maintain fair competition at both
upstream and downstream levels while at the same time allowing the achievement

of potential vertical merger efﬁciencies is a ﬁrewalls provision preventing information

65

from passing nonpublic information from SCI (upstream supplier) to its Alias (down-
stream producer).l For the concern of vertical foreclosure, the Commission requires
as one of the conditions on approval of the merger that SGI is to refrain from discrim-
inating between its downstream division (Alias and Wavefront) and rivals, and also
to maintain an open architecture and provide its application programming interfaces
for its workstations.

There is, however, little economic research regarding the competitive effects of
ﬁrewalls in imperfect competition markets following a vertical integration. Riordan
and Salop (1995) study that vertical integration may allow downstream units to act
as conduits for the exchange of pricing, output demand, and other competitively sen-
sitive information by upstream suppliers. They offer a perspective that a vertical
merger can raise competitive concerns regarding competitively sensitive information
exchange under three conditions: projectable, nonpublic, and appropriate upstream
market structure for successful pricing coordination. Hughes and Kao (2001) con-
sider the consequences of information transfer within an entity on competition and
organizational structure, and assess the impact of a ﬁrewalls provision on social wel-
fare. In their analysis, they consider a market structure with competition in both
input and output markets and refer to the downstream competitors private output
demand information. Their idea is that there are costs of obtaining private demand
information for a vertically integrated ﬁrm, i.e. input price concession in exchange
for a private output demand information to its competitor in downstream market.
The price concession results in lower marginal costs of downstream competitors that
leads to consumers’ surplus and social welfare gains. This is the reason that Fire-
walls decrease both consumer surplus and total social welfare while they increase the
producer surplus. In a more recent paper, Milliou (2004) has investigated the impact
of Firewalls on innovation incentives and on social welfare in a partially vertically
integrated industry, in which a vertically integrated ﬁrm provides a nonintegrated

downstream ﬁrm with an input in upstream market and at the same time competes

 

lSGI is the dominant graphics workstation provider with a 90% market base and Alias and
Wavefront are two of the three dominant Unix-based software developers that operate on those
workstations.

66

against the nonintegrated ﬁrm in the ﬁnal output market. The upstream division of
the integrated ﬁrm pass on private information such as the nonintegrated ﬁrm’ R&D
efforts obtained in the process of trade to its downstream division. That is, the inte-
grated upstream division acts as a conduit for horizontal R&D spillovers. Due to the
elimination of double marginalization, a vertically integrated ﬁrm is more efficient in
the sense of smaller marginal cost of output than its rival. When allowed information
ﬂow within a vertically integrated ﬁrm, horizontal R&D spillovers, the more efficient
vertically integrated ﬁrm is willing to invest more in R&D and can serve more market
share and vice versa for less eﬁicient nonintegrated ﬁrm, which leads to consumers’
surplus and social welfare gain. This is the idea behind his paper to conclude that
Firewalls do not necessarily improve R&D investments and total social welfare.

This paper is different from the previous works in several ways. First, there are
symmetric ﬁrms in the upstream market competing in a Bertrand model rather than a
monopoly as in Million (2004). Second, unlike Hughes and Kao (2001) where informa-
tion transfer refers to demand side, I consider proprietary information of competitor’
R861) efforts, horizontal R&D spillovers. Third, ﬁrms in the downstream market
engage in R&D investments and there are vertical R&D spillovers once ﬁrms are ver~
tically integrated with one of upstream ﬁrms. There is lots of evidence of signiﬁcant
inter-industry spillovers such as vertical R&D spillovers. Technological links among
different stages within a vertically related industy are important features of many in—
dustries. Information about technology, design, or speciﬁc characteristics of products
can be shared between upstream and downstream divisions within a ﬁrm. This seems
to be most of the R&D intensive industries, where the coordination of the upstream
and downstream divisions is necessary in order for products to be compatible and
to avoid extra adjustment costs. Examples are electric utility, defense, telecommuni-
cations, and energy industries. However, this perspective of vertical R&D spillovers
with a Firewalls provision on social welfare has not been considered in the previous
works.

The main result is that such ﬁrewalls do not necessarily improve social welfare

while it may increases R&D incentives. The result here is partially consistant with

67

the general idea of FCC/DOJ that Firewalls may improve R&D incentives (Pitofsky,
2001). However, the result regarding social welfare is different from the general idea
that Firewalls improve overall social welfare. Under Firewalls, the nonintegrated ﬁrm
has a strategic incentive to purchase from the upstream division of the integrated ﬁrm
in order to soften competition in the ﬁnal output market because the integrated ﬁrm
faces the nonintegrated in both upstream and downstream markets and hence com-
petes less aggressively in downstream market.2 However, industry proﬁt (producers
surplus) is smaller than under Information 'Ifansfer because ﬁrms in downstream mar-
ket compete in R&D as well as in price. They severely compete each other in R&D
game and as a result, engage in a highly inefficient R&D investment. This occurs
when outputs are close substitutes. Consumers surplus is larger than under Informa-
tion Flow due to severe R&D competition. Hence, total social welfare is smaller than
under Information Transfer under certain curcumstances, i.e. high '7 and B. When
ﬁnal products are close substitutes, the nonintegrated ﬁrm is willing to take high risk
R&D investment because competition softening effect obtained by purchasing from
the upstream division of the integrated ﬁrm is signiﬁcantly high. Without the pro-
vision, ﬁrm D2 is not willing to trade with the integrated ﬁrm because information
ﬂow within the different stages of the integrated ﬁrm do harm itself.

The remainder of the paper is organized as follows. Section 2 describes the basic
model of the present paper. In the following sections, equilibrium outcomes under
Information 'Ifansfer (N o Firewalls) and under Firewalls are analyzed. I conclude

with welfare analysis at the end.

3.2 Model

Consumers purchase an output in a market where one of two downstream ﬁrms,
D1, is vertically integrated with an upstream ﬁrm, U1, while ﬁrm D2 is independent

from upstream ﬁrms. There is a ﬁxed-coefﬁcient technology such that each unit of

 

2Chen (2001) studies that a nonintegrated ﬁrm may strategically choose the integrated ﬁrm as
an input supplier due to competition softening effect.

68

output in downstream market requires one unit of input from upstream market. De-
mand functions are as follows: q,- = 1 — p,- + 7(pj — 19;), where 7 > O, i,j = 1,2,
and i 74 j. The parameter '7 represents the measure of product differentiation. The
products are highly differentiated if 7 is close to 0. The products are almost ho-
mogeneous as 7 goes to 00. There are upstream producers (U 1, U 2, ...., U n), where
n 2 2, producing a homogeneous input for the downstream industry at a constant
marginal cost of production, cu > 0 and competing in a Bertrand model. I assume
that upstream ﬁrms set linear prices for their products. Two-part tariffs are typically
set in a model where a single upstream ﬁrm supplies each downstream ﬁrm because
downstream ﬁrms are assumed to have no bargaining power. But in my model, I want
to model cases where there are upstream ﬁrms competing in the Bertrand framework
(112* = wl = mg = = 10,, = cu) to diminish the role of foreclosure and hence
to focus on the consequences of information transfer on R&D incentives and social
welfare.

Each downstream ﬁrm makes an investment in process innovating R&D that is
cost reducing. There is no uncertainty. Two downstream ﬁrms produce differentiated
products at the cost of 0,1,: = Cd — :01, where x,- < 0d, 2' = 1, 2. The cost of R&D is
given by I,- = A323, 2' = 1, 2, reﬂecting the existence of diminishing returns to R&D
expenditures. The parameter A is related to the marginal cost of R&D. 2:,- is the
amount of R&D undertaken by a downstream ﬁrm, Di, 2' = 1, 2. There is vertical
R&D spillovers of B, where 0 < 6 < 1, from the integrated downstream division to
its upstream division. Hence, the cost of upstream division of the integrated ﬁrm is
Cul = cu — 62:1, where 5:01 < cu.

The timing of the game is the following. In the ﬁrst stage, D1 and D2 simultane-
ously choose their R&D effort levels, 231 and 3:2. In the second stage, the integrated
ﬁrm sets the input price for ﬁrm D2 and ﬁrm D2 chooses its input supplier. I assume
that if ﬁrm D2 decides to purchase from one of other alternative suppliers, U 2, ...., U n,
ﬁrm D2 chooses ﬁrm U2. Finally, D1 and D2 compete in prices in the ﬁnal output
market. To make sure that the second order conditions are satisﬁed and that R&D

effort levels are positive, I make the following assumption.

69

Assumption 1: given 7 and 3, the degree of A in R&D satisﬁes the following condi-

tion.3

A > A = max{E1,E2} (3.1)

where,

 

2 4 2 2 4 2 1
E1 = 2(1 + 7)(1+ m2 [4019);]sz I + 4f1++77)::(72+7)I ’
= ﬁv(l+7+ﬂ)+{2l297(1+7)+(1+ﬂ)(2+47+72)+7(1+7+ﬁ)l}x
4(1+7)2 —72 [4(1+7)2 “1212
{(1 + 7 — Bv)(2ﬁ7(1 + 7) + (1 + B)(2 + 47 + 72”}
[4(1 +7)2 - 7212 '

E2

 

Note that E1 and E2 increase in 7 and 5. That is, this assumption assures that ﬁrm
D2 is willing to take R&D investments with high marginal costs when outputs are

close substitutes in the presence of high vertical R&D spillovers.

Assumption 2: The upstream division of the integrated ﬁrm is unable to foreclose
the nonintegrated downstream ﬁrm.

This assumption is reasonable because FCC/DOJ generally concerns about ver-
tical foreclosure and requires as a condition on approval of a vertical merger that
the integrated ﬁrm must maintain fair competition in both markets as in the SGI

acquisition of Allias & Wavefront example.

3.3 Equilibrium outcomes

The nonintegrated ﬁrm D2 chooses its input supplier under two different regimes,
Firewalls and Information Transfer. Under Information Transfer, ﬁrm D2 would face
cost disadvantage due to information ﬂow (horizontal R&D spillovers) between two
different production stages within the integrated ﬁrm if it purchases an input from

the upstream division of the integrated ﬁrm (U1) and hence has an incentive to buy

 

3E1 is for 2;” > 0 under Information Transfer and E2 is for x; > 0 under Firewalls.

70

an input from other than U1. On the other hand, under Firewalls, the integrated ﬁrm
U1 is no longer able to transfer ﬁrm D2’ proprietary information to its downstream
division. Firm D2 may have an incentive to purchase an input from the integrated
ﬁrm in order to soften competition in the ﬁnal output market. The integrated ﬁrm U1
cannot foreclose ﬁrm D2 by assumption 2 and carmot charge more than the marginal

cost cu due to alternative input suppliers.

3.3.1 Equilibrium Outcomes with Information Transfer

Firm D2 can purchase an input from the integrated ﬁrm U1 or from other than
ﬁrm U1. Under no Firewalls provision, ﬁrm D2 would purchase an input from ﬁrm
U2, rather than the integrated ﬁrm U1 as in Figure 3.1 because the upstream division
of the integrated ﬁrm may pass on ﬁrm D2’ nonpublic information to its downstream

division once ﬁrm D2 purchases from ﬁrm U1.

Figure 3.1: Under Information 'Ifansfer

 

 

\
X
mm
mm

 

 

 

I solve the game by Backward Induction. In the third stage, the best price response

functions are chosen to maximize the following proﬁt functions.

2 2

.. 35 _ I
7rD1 : (p1 "' Cl)Ql " A?’ ”02 = (p2 — 62)q2 - A—2g, (3.2)

where 01 = on + Cd — (1 + @131, 02 = 0., + Cd - :02. The best price response functions

71

are
_1+’7Pj+(1+7)Ci
2(1 +7)

 

szPj) (3.3)

where i = 1, 2, z' sé j. As we can see in eq. (3.3), price is a strategic complement in
the sense that 19,- increases in pj, i, j = 1,2, 1' 75 j. From the ﬁrst order condition, I
can derive demand for each ﬁrm, q,- = (1 + 7)(p,- — 0;), z' = 1, 2.

In the second stage, R&D levels are chosen to maximize the following proﬁt func-

tion.

3;?

«n.- = <1 + 00».- — a.)2 — Aj- (3.4)

where i = 1, 2. By differentiating the proﬁt functions, the best R&D response func-

tions are as follows.

= (1+ ﬂ)v(A — 111:2)

= v(A — (I + @1131)
AI — (1+ mvz .

1’ _ v2 (3.5)

 

 

R1(£L‘2) 7 R2($1)

’ 2+37 1—0 2+4 + 2 7 1+7 . .
h A: A ,A= ,= = .Unlke ,
W 2 1+7 W ” $33237, “ 35.32%? ‘ ......
R&D is a strategic substitute in the sense that 2:,- decreases in xj, i, j = 1, 2, i aé j.

The simultaneous Nash R&D levels are as follows.

I

xn f _ (1 + 6)le 311 f 9T2 (3.6)

1 -A’—(1+ﬂ)2vSl, 2 AI—ng.

2 2
whereT1=A(1—ji”—,).Si=v(1+73‘—,).T;=A(1 W),s£=
—v

— A -(1+ﬂ)21)2

 

1+3 2112
1 + —,£—L— .
v( A -(1+m2v2)
The R&D effort levels are positive under assumption 1. The integrated ﬁrm makes
higher R&D effort than the nonintegrate ﬁrm. The reason is that ﬁrm D1 has positive

vertical R&D spillovers ﬂ and hence has higher incentives to invest in R&D than ﬁrm

72

D2. The proﬁt for each ﬁrm is as follows.

n , ,\’
an1= (1 +7» (W1) ($162, (3.7)
“0,2: (1 +7» ’(’)— — 1) (33f)? (3.8)

The proﬁts are quadratic pay-offs of R&D effort levels. A ﬁrm with higher R&D
investments makes higher proﬁts. In the presence of a vertical spillover )6, ﬁrm D1 is

more willing to invest in R&D and hence makes higher proﬁt than ﬁrm DZ.

Proposition 3.1 Under Information 'Ifansfer, the amount of R&D effort level of
the integrated ﬁrm D1 are larger than that of nonintegrated ﬁrm D2. If A > A, the
proﬁt of the integrated ﬁrm D1 is higher than that of nonintegrated ﬁrm D2.

where,

2(1+ 7)(2 + 47 + 72)(2 + 57 + 272)[(1 + mz + 1]

’ = (40 + 712 — 702(2 + 67 + 372)

 

(3.9)

Proof: xlf — xzf > 0, because (1+ @le > UT, and AI — (1+ @2031 < A, — 115;.

sz'gn.[7r"fl — erf2] = sign[( (7_A_7_ (1 + )8)zv 2) (7L) 7—)14-‘7 —v(v+'u))2 — (X377) —
v2) (KIA?) —- (1 + ﬁ)2v(v + 11))2 ] = 3ign[A (A (v + 211) — 0(1) + u)2[(1 + ﬁ)2+ +1]) +

2(1+7)(2+47+72)(2+57+272)1(1+ﬂ)2+1}__
(4(1+'7)2-72)2(2+67+3*12 )
Note that A increases in 7 and ﬂ and that A > E1 is the condition for ﬁrm D2 to

(1+6)% 3(12+u)2] (i) 7r)D1-7rD£> 0 ifA >

make positive amounts of R&D effort under Information 'Ifansfer, T; > 0, in which
the integrated ﬁrm D1 also makes positive R&D effort levels. By comparison, we
know A > E1. (ii) Hence, it may be possible that the nonintegrated ﬁrm D2 makes
higher proﬁts than the integrated ﬁrm if A > A > E1.

73

3.3.2 Equilibrium Outcomes with Firewalls

This is the case where the upstream division of the integrated ﬁrm U1 cannot
pass on the proprietary information of ﬁrm D2 to its downstream division D1 and is
not allowed to foreclose ﬁrm D2. Hence, ﬁrm D2 may have an incentive to purchase
an input from the integrated ﬁrm U1 since it softens competition in the downstream
market because price is a strategic complement.

Under Firewalls, let’s check whether ﬁrm D2 has an incentive to purchase an input
from the integrated ﬁrm. When ﬁrm D2 purchases an input from U2, the problem
under Firewalls is the same as that in the previous case of Information Flow. Hence,
I solve by backward induction the case where ﬁrm D2 purchases an input from the

integrated ﬁrm as in Figure 3.2.

Figure 3.2: Under Firewalls

 

 

 

 

 

 

Upstream
market ' ' " ‘ U"
Downstream
market

 

 

 

 

 

 

As in the previous scenario, the cost function for the integrated ﬁrm is ('31 =
on + Cd — (1 + ,B)a:1, where 0 < ﬂ < 1, while that of ﬁrm D2 is 52 = w* + Cd — 2:2.
The proﬁt functions of the integrated and of the nonintegrated ﬁrms are respectively

given by
- at?
7'01 = (P1 - C1)Ql + (101 - Cul)Q2 - /\-2-, (3-10)
1‘3
7’02 = (P2 — 52m - A—Z", (3-11)

where cu] = cu — ﬁxl. The integrated ﬁrm sets the input price at cu due to alternative
suppliers and cannot foreclose ﬁrm D2 by assumption 2. Notice that the proﬁts of
the vertically integrated ﬁrm comes from both upstream and downstream markets,
in which the ﬁrm operates. Solving the production stage, demand for each ﬁrm is
(11 = (1+ ’Y)(P1 - 51) - 7(w1- C111) and (12 = (1 + 7)(P2 - 59.)- The integrated ﬁrm
losas a portion of market share in the downstream market. This is because it serves
the upstream market and makes positive profits, and hence is less aggressive in the

downstream market. The best price reaction functions are as follows.

 

 

R1(P2) = 1 + 7P2 + (1 :(171517)7(w1 - Cal), (312)
122(1),) = 1+ 72212517; ”52. (3.13)

Note that the price reaction function of the integrated ﬁrm has an extra term that
does not appear in eq. (3.3). The extra term will not be zero unless the market
price of the integrated ﬁrm is more than the competitive market price, wl = cu.
This implies that ﬁrm D2 may have an incentive to purchase from the integrated ﬁrm
because the integrated ﬁrm is less aggressive in the ﬁnal product market in the sense
that R1092) in eq. (3.12) > R1(p2) in eq. (3.3). Hence, when ﬁrm D2 buys from the
upstream division of the integrated ﬁrm, the integrated ﬁrm maximize the following

proﬁt function.

2
$1

7‘01 = (1 + 7)(P1- 51)2 + (101 - C111)[(1 + ’7)(P2 - 52) - 7(P1— 51)] - 3,
subject to wl = cu. (3.14)

and ﬁrm D2 maximizes the following proﬁt function.

«02 = (1+7)<p2 —52)2— %. (3.15)

By differentiating the proﬁt functions with respect to 33,-, i = 1,2, the best R&D

75

response functions are as follows.

[2(1+ 'y)vl + ﬂ]A— [2(1+ 'y)vfu — ﬁ'yu — B(1+ 'y)v]:1:2

 

121(19): A + 25721] + 2,6(1 + ﬂu; —- 2(1 + 7)(vf)2 ’ (3°16)
()<__> (3.17)

where vi = (1 + ﬁ)v + 2311, u; = (1 + 11—45;?) u. By substituting, I can derive simul-

taneous Nash R&D levels as follows.

[2(1+7)vf+ﬁlA- [2(1+7)vfu—ﬁ7u- ﬁ(1+7)vl(-r—2-)

 

 

x! = —” ”S2 ,(3.18)
1 A + 2,6722] + 2130+ ’Yluz ‘ 2(1 '1‘ 7)(v1)2
35 = 7172—. (3.19)
A — 1235‘
where,
T, = A(1_ u;[2(1+ 7):); + m >
2 A + 2ﬂ7vf + 2B(1+ ﬂu; - 2(1 + “1)(vi)2
... _ u§[2(1 + '7)vi‘u - ﬂvu - 5(1 + ”(M

 

2 ‘ v A + 2572;; + 2M1 + m; — 2(1+ 7)(v;)2'

Note that each ﬁrm has positive R&D effort level, xif > 0, i = 1,2, under the
assumption 1. For all 3 and '7, the integrated ﬁrm D1 has higher R&D effort level
than the nonintegrated ﬁrm D2. The total proﬁt functions of the integrated ﬁrm and

of the non-integrated ﬁrm are as follows.

 

’ _ 2
"D1: (1+ 7)(A— ﬁ2) (A021; -1) (x{)2, (3.20)
«£2: (1 + 7)). (32- — 1) ($2 f)2. (3.21)

When ﬁrm D2 purchases an input from U1 under Firewalls, for moderate ﬂ and '7,
the integrated ﬁrm makes high proﬁt than the nonintegrated ﬁrm and vice versa for
sufﬁciently high ﬂ and 7 even when ﬁrm D1 has higher R&D effort level. 4 The

 

4For proﬁt comparison between the integrated ﬁrm and nonintegrated ﬁrm, see appendix F. The

76

reason is that for sufficiently high )6 and 'y, in which outputs are close substitues in
the presence of high vertical R&D spillovers, ﬁrms in downstream market engage in
inefﬁcient R&D game in the sense of high value of A.

Now, let’s check when ﬁrm D2 would like to purchase an input from ﬁrm U1 rather
than ﬁrm U2. When ﬁrm D2 purchases from ﬁrm U2, the analysis is the same as that
in the previous case of Information Transfer. Since the expressions here are rather
messy, I do simulation analysis. Figure 3.3 shows ﬁrm D2’ R&D level comparison
between purchasing from U1 and from U2. Since proﬁt is a quadratic function of
R&D effort level, ﬁrm D2 makes higher proﬁt by purchasing from ﬁrm U1 in the are
of X(U2) < :c(Ul) in Figure 3.3.

Figure 3.3: Firm D2’ R&D level Comparison with A = 25, c = 0.5
10

8 X(uz)>xw1)

X(U2) < X(U1)

 

 

0 0.2 0.4 0.6 0.8 1

Proposition 3.2 Under Firewalls, there exists a A such that for A S A where A(T§‘ —
7g) — 2(1+ r)v(S§T; — $ng) 2 0, the nonintegrated ﬁrm D2 has an incentive to buy
an input from the integrated ﬁrm U1. The R&D level and the proﬁt of ﬁrm D2 are
higher when it buys from the integrated ﬁrm than from other alternative suppliers.

 

simulations are made with A = 25, c = 0.5.

77

Sketch of Proof: if MT; — T2,) — 2(1 + r)v(S§T£ — SgTé") Z 0, in which firm D2
purchases an input from ﬁrm U1 and makes higher R&D effort levels, ﬁrm D2 makes
higher proﬁts. As we can see from Figure 3.3, for sufﬁciently high 0 and '7 given
A = 25 and c = 0.5, ﬁrm D2 makes higher R&D effort levels when it purchases an
input from the integrated ﬁrm U1 than from ﬁrm U2. The intuition here is that the
extra term in the price response function eq. (3.12) is a increasing function of '7 and
ﬂ. That is, when outputs are close substitutes (high 7) and vertical R&D spillovers
are high, competition softening effect is signiﬁcant and hence ﬁrm D2 can get beneﬁts
from purchasing an input from ﬁrm U1 under no pressure of information ﬂow within

the integrated ﬁrm.

3.3.3 Information Transfer vs. Firewalls

In a vertically related market where a vertically integrated ﬁrm competes with
other input suppliers in upstream market for nonintegrated downstream ﬁrm D2 while
competing with nonintegrated ﬁrm D2 in downstream market, the nonintegrated
downstream ﬁrm D2 make a decision whether to buy an input from the integrated
ﬁrm U1 or ﬁrm U2 under two different policy regimes, Firewalls vs. Information
Transfer. As we can see from Table 3.1, payoffs of ﬁrm D2 depend on its R&D
effort levels. The higher R&D investments, the higher payoffs it can get. Under
Information Transfer, ﬁrm D2 buys an input from ﬁrm D2 rather than ﬁrm U1 due
to cost disadvantage resulted from horizontal R&D spillovers. Under Firewalls, ﬁrm
D2 may have an incentive to purchase an input from the integrated ﬁrm due to

competition softening effect.

Table 3.1: Payoffs of nonintegrated ﬁrm D2
Buy from U1 . Buy from U2

N F' alls 1+ )1’ Al-1 "“2 1 A, 2,—1 "f2)2
ouew (7 33 (23) (+7) 53 (x2

r I
Firewalls (1 + ’7)A’ (32 - I) (3:51)2 (1 + 'y)A’ (:7 - 1) ($552

 

 

 

 

 

 

. nf2__ f2
note. x2 —:r2 .

78

When A S A, ﬁrm D2 makes higher proﬁt as well as higher R&D effort levels when
purchasing an input from the integrated ﬁrm than from ﬁrm U2, 3:51 > 1752 =
2:3”. Hence, in equilibrium, the nonintegrated ﬁrm D2 purchases an input from the
integrated ﬁrm U1 under Firewalls.

On the other hand, as we can see in Appendix F, when A S A, in which ﬁrm D2
makes higher R&D effort levels and proﬁts under Firewalls than under Information
'Dansfer, for low 7 and 6, ﬁrm D1 makes higher proﬁts under Firewalls than under
Information Transfer and vice versa for high 7 and B. And also, ﬁrm D1 makes higher

proﬁts than ﬁrm D2 for low 7 and ﬂ and vice versa for high 7 and B.

3.4 Welfare

Total social welfare is deﬁned as the sum of the consumers’ surplus and the

producers’ surplus, W = CS + PS.

mm = 1—“g1 ((1 - plxpl — 5)+(1-p2)(p2 - 6))
2 2
+(1+ 7) ((1)1 - 502 - 2% + (P2 - 52)2 - 252-2) 1 (3-22)
mm = ”T1 ((1 —p1)<p1 — e>+(1—p2><p2 - a»

$2 $2
+(1+ 7) ((131 - 51)2 + (@1302 - 2‘31 + (P2 - 52)2 - A32) (3-23)

In order to compare total social welfare under two different policy regimes, Informa-
tion Transfer (W(NF)) vs. Firewalls W(F), I do simulation analysis. Figure 3.4 shows
that for sufficiently high ﬂ and 7, social welfare under Information Transfer is higher
than under Firewalls and vice versa for moderate ﬂ and 7. When ﬁnal products are
close substitues (high 7), ﬁrms engages in inefﬁcient R&D game, which leads to wel-
fare losses. If A S A, in equilibrium, ﬁrm D2 purchases an input from the integrated
ﬁrm, in which consumers surplus and R&D levels are higher under Firewalls than
under Information Transfer. This result is consistant with that a Firewalls provision

can retain the integrated ﬁrm’ incentives to free ride on competitors’ hard work and

79

hence improve ﬁnal output competition as well as R&D incentivas. However, my
result regarding producers’ surplus and overall social welfare tells a different story.
Final output producers suffer from severe R&D competition with sufficiently high
value of A. A Firewalls provision may lead to inefﬁcient R&D game by affecting ﬁrm

D2’ decision for an input supplier and hence producers’ surplus losses and overall

social welfare losses.5

Figure 3.4: Welfare Comparison with A = 25, c = 0.5

10

I
O
8 ,1 mum
> m1);
0
e ,o'
1 ,o'
4 .v WWW)

 

 

0 0.2 0.4 0.6 0.8 1
,0

 

 

 

Pr0position 3.3 For all ﬂ and 7, consumers surplus is larger under Firewalls than
under Information Transfer. However, for sufficiently high 6 and 7, producers surplus
is smaller under Firewalls than under Information Transfer. As a result, for sufﬁciently
high H and 7, total social welfare is smaller under Firealls than under Information
'D'ansfer.

Sketch of Proof: As downstream ﬁrms engage in severe R&D game, consumers get

beneﬁts while ﬁrms suffer from high costs of R&D. From a social point of view,

 

5see the appendix for consumers surplus, producers surplus, and R&D comparisons.

80

Firewalls may lead to inefﬁcient R&D game and as a result not necessarily improve

total social welfare.

3.5 Conclusion

I analyze the consequence of a Firewalls provision on social welfare and on R&D
incentives in a market where a vertically integrated ﬁrm and a nonintegrated ﬁrm
compete in price and engage in R&D in downstream market. Firms in upstream
markets are symmetric, except the integrated ﬁrm has a cost advantage in the sense
that there exists vertical R&D spillovers. In such circumstances, the nonintegrated
ﬁrm needs to make a decision whether to buy an input from the upstream division
of the integrated ﬁrm or other alternative input suppliers under two different pol-
icy regimas, Information Transfer vs. Firewalls. When authority allows Information
'Ikansfer, i.e. horizontal R&D spillovers from ﬁrm D2 to ﬁrm D1 through the up-
stream division of the integrated ﬁrm, ﬁrm D2 has no incentive to purchase an input
from the integrated ﬁrm due to cost disadvantage of horizontal R&D spillovers. On
the other hand, when authority does not allow Information Transfer by establish-
ing a Firewalls provision, in equilibrium, ﬁrm D2 may have an incentive to purchase
an input from the upstream division of the integrated ﬁrm. The reason is that the
integrated ﬁrm competes less aggressively in downstream market because in both up-
stream and downstream market, it operates and faces ﬁrm D2 as an input provider
as well as a ﬁnal output competitor. The integrated ﬁrm can make positive proﬁts in
both upstream and downstream markets.

FCC and DOJ generally allows a vertical merger under conditions of Firewalls and
Nonforeclosure in a high tech industry where upstream and downstream ﬁrms work
closely. My study shows that for high 3 and 7, a Firewalls provision with nonforeclo—
sure condition alters the nonintegrated ﬁrm’ decision for an input supplier and leads
to total social welfare smaller under Firewalls than under Information Transfer due
to inefficient R&D investment (sufﬁciently large value of A). Nevertheless consumers

surplus is larger than under Information Transfer. The result regarding ﬁnal output

81

market competition in the present paper is consistant with the competition concerns
suggested by FOC/DOJ. The commission suggests that Firewalls along with non-
foreclosure provision may improve both upstream and downstream competition and
hence increase consumers’ surplus and social welfare. However, my study regarding
overall social welfare indicates that under certain curcumstances Firewalls may not
necessarily improve social welfare due to produecers’ surplus losses.

My result is based on two critical assumptions. First, with signiﬁcantly high value
of A, I assumed that each downstream ﬁrm makae positive R&D investments in order
to make the anyalsis simple. Second, the integrated ﬁrm is unable to foreclose the
nontegrated downstream ﬁrm. Given high 7 and B, in which case the integrated ﬁrm
makes less proﬁt by supplying an input to the nonintegrated ﬁrm, the integrated ﬁrm
simply stop supplying an input to the nonintegrated ﬁrm if it may. However, under
the assumption, the integrated ﬁrm must supply an input to the nonintegrated ﬁrm
if the nonintegrated ﬁrm make an order to the integrated ﬁrm. Nevertheless, my
result suggests that when the Commission evaluates vertical merger efficiency in line
with competition concerns in both markets, overall social welfare should be carefully

considered in high tech industries with signiﬁcantly high R&D investments.

82

Appendix A

Appendix: Unilateral Bundling

In this section, I consider the case where one ﬁrm offers the bundle 1443,, i = 1, 2
while the other ﬁrm does not bundle. I assume that ﬁrm A1 offers the bundle of

A181 to new customers, while ﬁrm A2 does not practice bundling.

A.1 Short-term contract

Consumers in [0, 6*] buy the bundle A131 and customers in [0*, 1] purchase
A2 and 8,, where z' = 2, ....,n, in the ﬁrst period. In the second period, consumers
in the intervals [61, 0*] and [6*, 02] are targeted with discounts by competitors and
switch between brands, purchasing A2 and A181, respectively. On the other hand,

consumers in the intervals [0, 01] and [62, 1] do not switch their suppliers.

A131 A2+ Bi

Period 1 l

O

Q.___
*

—L

Period 2

 

In the second period, given (9*, a customer is indifferent between A1 and A2 at 61

83

and between A181 and A2 at 02 if

. 1 .
p3“ +t61 =pi2+t(1—91)=>01 .—. 5i(t+p§,2—p§,1), (A.1)

. 1 .
p3, + as = p242 +t(1— 62) a 02 = 27c +1132 - p21), (A2)

where p?“ and 192112 are the second period prices of A1 and A2 for loyal customers of
ﬁrm A1 and A2, respectively. 1531 and 15342 are the second period prices of A131 and
A2 for customers who switch between providers. The proﬁt functions of ﬁrms in the

second period are as follows.

«533(3) = (P311 -CA)91 + (1331 — CA - CB)(92 - 9") , (A3)
71"NB(S)""-(px12'_ CA)(1 _ 92) + (15312 _ CA)(6* _. 01) . (AA)

By maximizing the proﬁt functions with respect to the second period prices, I have

the second period prices as follows.

 

 

 

 

26*+1 1 3—20"

2 _ 2 _

pAl—CA+( 3 )t,pA2—CA+§CB+( 3 )t, (A.5)
, 2 3—40" 46*-1
p§1=cA+§cB+( 3 )t, P312=CA+( 3 )t. (A.6)

In the ﬁrst period, a consumer at 0* is indifferent between A181 and A2 + B,,
2': 2, ....,n, if

p2, + t0* + 6033,, + t(1— 0*)) = p1,, + CB +t(1— 6*) + 5052, + w")
_1_+ (17:12 '1' CB - 1031)- 603342 “1331)

 

=>0* =

 

2 2(1 - 6)t
_ 1 3(1)), - p52) 3 + 26
‘ E + 2(3 + 6)t 2(3 + (5)363 (M)

Firms set the ﬁrst period prices, 1’21 and p342, by maximizing the following proﬁt

84

functions,

”3(5) (11.3)

PB S 1.1
7V1 ( )=(PA1-CA"CB)9 +6012 ,

39’3“): (p3,, M,— -cA)(1-0*)+67rPB(S) (11.9)

By substituting the optimal prices into eq. (A.1) and eq. (A2), we derive

+ 4663 0,, = 1 4663 (27 — 195)CB

1 2
3 3(27 — 116)t ’ 2 + (27 —116)t ’ 2 _ 3 - 6(27 — 116)t '

 

01 = (A.10)
We need the condition of t > (2%) 63 for 02 > 0*. Otherwise, ﬁrm A1 is unable
to gain customers attached to ﬁrm A2 1n the second period while ﬁrm A2 15 able to
gain a positive portion of customers attached to ﬁrm A1. That is, customers who
purchased from ﬁrm A2 in the ﬁrst period are locked-in if t S (7272:5155) CB. Firm
A2 becomes a monopoly over the interval [0* , 1] and charge 12342 = U — t(1 - 0*) to

its locked-in customers in the second period. Hence, we need to consider two cases,

27-1-55 27 55
t) (m) CB and t S (W) CB.

27
(1) t > (273%) 63
Both ﬁrms are able to gain a positive portion of customers attached to competi-

tors. The equilibrium proﬁts are as follows.

PB(S)_ 85 _3 _§_ 2- é

7r1 _ (1+ 9)2+ (2g 965 96)cB +(2g—4 —6§ 2+1—186) t ,(A.11)

NM)- 85 .1 _1 2 is:

7r, _(1+9)2— (2g 93»: 95)CB 11—(254-265 + 1186)t ,(A.12)
46

where 62m.

(2) t s (327%)c3
Customers of ﬁrm A2 are locked-in after the ﬁrst purchase, 02 = 0*. Consumers
in [0, 0*] buy the bundle A131 and customers in [0*, 1] purchase A2 and 3,, where

85

i = 2, ....,n, in the ﬁrst period. After observing customers’ behavior, consumers in
the intervals [01, 0*] are targeted with discounts by ﬁrm A2 and switch to A2. On

the other hand, consumers in the intervals [0, 01] and [0*, 1] do not switch their

 

 

suppliers.
A1 B1 A2 + B i
Period 1 l ] ]
o 0 * 1
A A A
Period 2 I 1 I 2 I 2 I
| | l
0 91 9 * 1

In the second period, given 0*, a customer is indifferent between A1 and A2 at 01
if

. 1 .

where p?“ is the second period price of A1 for loyal customers of ﬁrm Al and 13342
is the second period price of A2 for ﬁrm Al’ consumers who switch to ﬁrm A2. The

proﬁt functions of ﬁrms in the second period are as follows.

«53(5) = (pin «.061. (A14)

«2123(5) = (p32 — chl — 9*)+(13?42 — cA>(6* - 01). (A15)

where 12312 = U - t(1 - 0*). By maximizing the proﬁt functions with raspect to the

second period prices, 12241, 15312, I have the second period prices as follows.

26*+1 , 40*-—1
P2A1=CA+( 3 )t.pﬁ2=c,4+( 3 )t. (A.16)

 

 

In the ﬁrst period, a consumer at 0* is indifferent between A131 and A2 + B,- ,

86

p§1+ t6* + 6931242 + t(1 — (9*)) = phz + c3 +t(1— 6*) + 6(U — t(1 — 0*)+t(1— 9*))

_ 3 1 _ B _
3 26+ (pA2+cB Pm) M, (A.17)

:0 = 6+6 (6+6)t (6+6)t

 

 

Firms set the ﬁrst period prices, pgl and pin, by maximizing the following proﬁt

functions,

{WE (:08 _CA“ cBw + 6(1),.“ — anal, (A.18)
§B(S)= (p Az-CA)(1 - 9*) + 6(U - t(1 - 6*)— —c,4)(1 — 9*)
+5(13A1"CA)(9* - 91)- (A.19)

By substituting the optimal pricm into eq. (A.13), I derive

+ 27+136 ,, _ 27+136

1
_ __ _ _, A2
6 3(54 + 76) ’ 54 + 76 I 0)

01:

The equilibrium prices are as follows.

B_ _2 18-6 274-136
pAl—CA+CB 96t+(———9 )(—54+76 t,

17 18+56 27+136
1 — _ —— — —— _—
pA2—2t+cA+96t 6U+6CA ( 9 )(54+75)t, (A21)

2 1 2 27+136 2 27—66
”Al “4+ +3(54+76 t’ ”A? U 54+76 ’

3
' 1 4 27+136
19.42:“ 23% m‘

The equilibrium proﬁts are as follows.

 

 

 

 

2
PB(S) _ i 18 + 6 27 + 136
7'1 _ 18t+ ( 9 54 + 76 t’ (A22)
7rNB(S) = t 36 +176 36 + 86 27 + 136
2 18 54 + 76
18 + 46 27 + 136
+t( 9 )(t—‘___521+76)2 ‘ (A23)

87

A.2 Long— and Short-term contracts

 

 

 

 

 

 

A B A B.
Period1 I I l 1 ] 2 + ' I ]
0 *
0A 6 0B 1
A A A B A
Period2 1 2 1 1I 2
1 * I 1
o 6.4 0l 0 02 0B 1

Consumers in [0, 0*] and [0*, 1] buy the bundle A181 and A2 + 8,, respectively,
where i = 2,....,n, in the ﬁrst period. Consumers in [0, 0A] and [0B, 1] commit
to a long-term contract of {A131, A1} and {A2, A2}, at the price of 16,41 and 5A2,
respectively. Consumers in [0 A: 0*] and [0*, 0 Bl buy a short-term contract at p21 and
p312, from ﬁrm A1 and A2, respectively. In the second period, some consumers who
bought a short-term contract in the ﬁrst period, [01, 0*] and [0*, 02], are targeted with
discounts and switch between providers and pay the second period poaching prices
15342 and ﬁﬁl for A2 and A181, respectively. Consumers in [0 A: 01] and [02, 0 B] stay
with the previous providers and pay 122/“ and 121242 for A1 and A2, respectively. Since
customers in [0, 0 AI and [0B, 1] committed to a long term contract at 15 A1 and 15 A21
respectively, competition in the second period is over the interval [0 A, 0 B]-

In the second period, given 0*, a customer is indifferent between A1 and A2 at 01

and between A131 and A2 at 02 if

.. 1 .
pf“ +101 =p312+t(1—01) => 01: E-t-(t+p?42—p3u), (A24)
- 1 .
1121+ 102 = pig +t(1— 02) => 02 = Eur-+16312 - p31) . (A25)

Firms are maximizing the following proﬁt functions with respect to the second period

88

prices, {12241, 1331} and {12342, 151242}.

PB L863 .. :1:
7:12 ( ) = (Pin—CAXQI - 9A)+(P§1-CA-CB)(92 - 9 ) , (A26)
NB L&S ..
«22 ‘ ’ = (Aroma — 62) + (pig-cw? — 61) . (A27)
Then, I have
1 4 2 1 1 4 2
2 It 2 :1:
pA1=CA+§t_'3—t6.4+§t6 , pA2=CA— §t+§CB+§th—§t6 , (A28)
.3 2 1 2 4 * .2 1 2 4 *
= — -t — - —t0 = — —t - -t0 -t0 .
pAl cA+3cB+3 +3t03 3 , PA2 CA 3 3 A+3

In the ﬁrst period, a consumer at 0*is indifferent between A181 and A2 + 3,,

2': 2, ....,n, if

p§1+t6* + 603,242 +t(1— 0*)) = pin + c3 + t(1 — 0*) + 6(13A1+t0*)
1 0,142 + CB —p§1 _ 505312 ’031)

 

 

 

 

: 0* = —
2 + 2(1 — 6)t 2(1 — (5)t
)1 1 19112 + CB #231 60232 - 2311) 563
= _ — . A.2
=> 6 2 + 2t + 4t + 4t ( 9)

A consumer at 0 A and at 0 B is indifferent between a long-term contract and a sequence

of short-term contracts if

1241 +(1+ «MA = p21 + 29.4 + 5003,, H61) => 6.41 = p31 + 620?... (630)
m +(1 + 6)t(1— 03) = (0142 + t(1 — 63) + 623.2 + t(1 - 63))

=> ﬁA2 = P312 + 519312- (A31)

89

Firms maximize the following proﬁt functions with respect to the ﬁrst period prices.

«fB‘L&S’ = (pi?1 -— CA — cBw“ + 602,241 — 0001

+5033, — cA — cB)(02 -— 9*), (A32)
«5’3“:3‘5) = (p22 — owl — 0*) + 5003,, — cm — 62>

+5053, - cm“ - 01). (A33)

The equilibrium prices are

1
ﬁA1=cA+cB—-6cB+t+2v,ch— 6¢CB+6(- t+cA),
1 l
pA2=cA——6cB+t—21/)CB+6z,/JCB+6 -t+cA+2cB ,
PA1=CA+CB-Z<SCB+t+21/)CB’5¢CBa
1
R142: CA—chB+t—2¢CB+6¢CB, (A34)
2 —1t+c 2 -lt+c +1c
19,41-2 A: PA2-2 A 23:

, 1 3 . 1 36
p231: Z1: + cA + 108 - wag, 19,242 = at + CA + 1068’ Where ¢ = 24 — 86'

 

By substituting the optimal prices into eq. (A24) and eq. (A. 25), I can derive
{01, 62}. I can also get {6A, 63} from the ﬁrst order conditions of eq. (A32) and eq.

(A. 33) at 5;— = 0, 2': 1, 2, where ﬁrst-period sales are unaffected if ﬁrm Al and A2

increase p Al and p A2 , but decrease p A1 and 1) A2 by the same amount, respectively.

_1 @0013 _3 $63 *_1 $03

0’4-8 2t ’ 01_8+ 2t ’0 ‘24.“ t ’ (A35)
_5 03 11163 _7 CB 'dJCB __ 36

02—8 8t+ 2t ’03‘8+8t+ 2t ’wherw 24— 8—7

90

The equilibrium profits are

 

{EMS = (1 + 1766) %+ (2w — $61!) —1%6)CB
.1 2-1 25%
+ (3254.210 25¢ ) t , (A36)
ﬂaws) = (1 + 1766) g — (w — $6111 — $5) CB
1 2_1 2fi
+ (166+22/J 26¢) t , (A.37)
where w = 24?:586'

A.3 Long-term contract

Consumers in [0, 6*] buy the bundle A181 and customers in [6*, 1] purchase
A2 and 3,, where i = 2, ....,n, in the ﬁrst period. In the second period, consumers
in the intervals [61, 6*] and [6*, 92] are targeted with discounts by competitors and
switch between brands, purchasing A2 and A181, respectively. On the other hand,

consumers in the intervals [0, 01] and [62, 1] do not switch their suppliers.

 

 

A1 B1 A2 + Bi
Period1 l : {
o 6* 1
A A A B A
Period2 l ' I 2 l ‘ 1 I 2 I
I I I l
0 91 6* 62 1

In the second period, given 0*, a customer is indifferent between A1 and A2 at 61

and between A181 and A2 at 62 if

. 1 -
131241 + t91 =p2AZ + 81+t(1 — 01) => 91: 2—t’(t +1932 -p?41+ 31), (A.38)
. 1 .
pi, + 32 + tog = p32 + t(1 — 02) => 02 = -2—t(t +1232 -p§, — 52), (A39)

91

where p31 and p32 are the second period prices of A1 and A2 for loyal customers
of ﬁrm A1 and A2, respectively. 15,13, and 1332 are the second period prices of A181
and A2 for customers who switch between providers. 3,- , i = 1, 2 , is an early termi-
nation fee that customers who breach a long-term contract are obligated to pay. By

maximizing the following proﬁt functions with respect to poaching prices,

max (1531 - CA - CB)(92 - 9*) , («440)

max (ﬁiz-CA)(0*-61),
I have [3&1 and 13312 as follows.
‘3 —1 9* ' *2 1 0* A4
pA1—§(t+ﬁ—32—2t +cA+cB), pA2=-2-(2t -t+a+cA—sl). ( .1)

In the ﬁrst period, a consumer at 0" is indifferent between A131 and A2 + B,- ,

i: 2, ....,n, if

pgl — 531 + ta,“ + (505312 + 81+t(1— 0*»

=19},12 —632 +c3 +t(1 —0*) +6032, + 32 + t6*),

4 _ 1 (P342 + CB -1021)- 505342 _ﬁgl)
=> 0 —— ,
2(1 —6)t

1 B 2 2
(PA2 + CB " PA2) + 5(1’A2 ' pAl) _ 6(32 _ 31) + 663 (A.42)

2t 4t 4t 4t

[0

1
=> 0* = -
2 +
By maximizing the following proﬁts functions with respect to {19%, pin} and {19142, p312},

max (pgl— cA— CB — (531)6“
+6003“ - «2.001 + 60531 - cA — chez — 6*) + 681W -— 01)
st. 01 = 9* , (A43)
max (13342— CA- 632)(1- 9*) + 50032 - CA)(1" 92)
+5033, — cha'“ — a) + 582(92 — 0*)
s.t. 92 = 0*. (A44)

92'

the equilibrium prices are as follows.

1 1
p§1=t+cA+cB+§6t, ph2=t+cA+§6t—JCB,

17,241=t+CA, piz=t+cA,

.B «2
pA1=CA+CBa PA2=CA-

(.445)

By substituting the optimal prices and the condition 01 = 0* = 02 into eq.(A.38)

and eq. (A39), the optimal amount of discounts and early termination fees are

obtained as follows.

The equilibrium proﬁts are as follows.

”53(3) = #9309) = (1+g)

NIH-

(A46)
(A47)

(A48)

In market B, ﬁrms have no sale in the second period. The market prices of Bl

and 32 for both periods are the marginal cost of c B . Each ﬁrm makes zero proﬁt.

93

Appendix B

Appendix: Proof of chapter 1

1 _ I 5 2 _ 2
ProofofProposition 1.1: 0* = é+£A2212A1+—(mzﬁffﬂ—) => £3411- : —21—{—35.£:—:f1

max(p},, — CA)9* + 6003, — CA)91+ 9033,, - CA)(92 — 9*) (13.1)

F.O.C. of eq.(B.1) : 0* +(11— CA)_6_91*_ + 6(pA12cA)a_:§L- + 59,25,241. - 503311-

‘3‘1’A61p2 A1 A1
CA)ap——1——Al = 0. 1 From (i), we know that ﬁ=~ ©551— — -,0 and an increase in

pA1 has no effect on 62. Hence, F..O C.: 0* +6(p:1 -cA) )gigl— +661(— §)= 0. By
apAl
substituting the second period prices of eq. (1.34) and eq. (1. 35)into eq. (1 30), I have

91: (+9*+9A=>) =— —+—
ap—AI 3 319.141 5P1“

Gilt—I
colt—1

691 1(50" 80A)_l6614

 

3 2 - . 3 _ 2 _
M=(20;+1-49A)=9A=204+1- (pAlt CA),91=Z+Q;--p CA.
26;}— : -;,1;@141= 2%? Hence F.O.C. : 6* + 6(pA1- 9053-; + 661(—%) = 0
6PA126PA1

=> pin — CA = (201 — 0*)2t. By substituting p?“ — CA into 9A and 91, I have
9,, = §+ g9 — 391 and 91=§(21,+ 9*) = ,1, + 9;. Hence, at 9* =1/2,91 = 3/8,
9A = 1/8. Since ﬁrms in market A are symmetry, I can derive 62 = 5/8, 9A = 7/8.

In order to calculate pA1 and p.421 suppose that ﬁrm Al considers increasing

both p A1 and p A by the same amount, which will leave :—:fu= 0 and hence
A1

 

l 59?— = 0 by envelope theorem.
pAl

94

80* — 1 . .. 1 66* 2 09
552—1 — —?Z' F.O.C. of eq.(B.1) . 6 + (pA1 — CA)6—p_IA; + 6(pA1— CA)57:1AL1 _
6(pA1 - CA)%1—=.O Since —-IL = 262T6*’ F. O. C. becomes 0*+ +(a — cA)a%9;— =

A1 A1

((p A1 — c A)- 1209.41 —cA))6p66 6721:; The right side is zero in a symmetric equilib-

rium and hence pA1=—pA2 = t + cA.
By substituting cutoff points into eq.( 1.34) and eq.( 1.35), I can ﬁnd the second

period prices.
Proof of Proposition 1.2: (i) (1) F.O.C. {191141}: 6* — 212(1)}41— cA —631)— 553(1)?“ —
CA) + 21620531 - CA) = 0
(2) mo. {pal}: 6101— 31,923,, —c.4) — $003.1 -c,4) — 11292341 —cA —6sl)+ £793,241—
CAll = 0-
(3) F-O-C- {31}: él-9*+2112(P,141’CA -581)+ 311(P311”CA)+362(P,241—CA)— 35205311-
cAll = 0-
By (1) and (2), in a symmetric equilibrium, p3,, = t+ cA and 12312 = t+ cA. By
substituting the equilibrium price of p341 and 1&2 into eq.(1.44) and eq. (1.45) with
the condition 01 = 62 = 9*, I have 31‘ = 8; = t.
Proof of Proposition 1.3: (i) Customers in the interval [61,6*] and [0*,02] switch
between brands. 0* - 61 = 315— a? = 02 - 0*. Hence, consumer poaching: 313(1 — 9?),
where t > C3.
Proof of PrOposition 1.4: (i) 01 = :1; + 96% S 0* = 1/2, if t 5 cB. In other words,

customers are locked-in after the ﬁrst purchase if Bt < CB.

_ _ 2 ...
Proof of Proposition 1.5: (i) )0* = 1 + €427,441+ {91134731512 => —a—B—9

apAl
2 2
1 6 ‘9? . 6P 2 99*
—ﬁ—B—§1:(I)If =—5,then——B—= .
apAl a—PAl 6pm

mpg. - cA — c3)6* + 6923.1 — c.9611 + 69331 — cA - choz — 6*) (B2)

. 3p? -
F.O.C. ofeq.(B.2)z 0 +(pA1—CA— cﬂi’b— +6(pA1-cA)a:_og—+5gla_p§l_5(p§1_
612A, A1 A1

CA - cﬂﬁ- —— 0. From (i), we know that —§1=—3 4» (7:;— = 0. Hence, F.O.C.
A1 A1

. 0* + 6(p A1 — cA)5—é— + 601( —3)— — 0. By substituting the second period prices of
1!’A1

95

eq.(1.75) into eq.(1.71), I have

 

1 1 1 ,, 661 1 66* 66A 1 66A
01=—+—cB+—(6 +6A)=> ——=- + =-——-. (8.3)
6 6t 3 0p21 3 317.131 01021 3 6921

 

3 2 _ 2 _ 2
M =(29*+1-49A)+9{3- => 0.4 = 292+1-3mirc’4)+%§ => 2% = 3:397:91,
apAl A1

01=i+ Q2: - 22% + 93?. Hence, I have 5%?— = 1,. F.O.C.: 0* +<$(p,24l —
010%; + 601(—§) = 0 => 192,41 — cA = (201 — 0*)2tf4113y substituting p311 — cA into 9A
and 91, I have, at 0* = %, 01:3(1 + 93?), 0A = §(1 — 3F). Since ﬁrms in market A
are symmetry, I can derive 02 = 3(1 + Pg), 03 = §(1 + 5%).

In order to calculate p31 and p32, suppose that ﬁrm A1 considers increasing both

2
pg, and 5.41 by the same amount, which will leave 6p = 0 and hence 5%;— = -21-£.
60* Al 80 A1
F.O.C. of eq.(B.2) : 9* +(1»le — CA — CB)? + 6(1)?“ — CA)?— — 50513,— CA —
86* 2 59 1 60* A] B Al 39*
63) = 0. Since = 2 , F.O.C. becomes 9* + (pAl - CA — CB) =
6 ((1331 - cA — CB) - 9(1’311 - cA)) (ii—03- = ——023%—. In a symmetric equilibrium,
Al A1

Ihavepg1 =1)?2 =t+cA+cB — gag.

Proof of Proposition 1.6: Each ﬁrm has an incentive to bundle and ends up being
in {bundling, bundling} situation (see Table 1.4).

Proof of Proposition 1.7: (ii) (1) F.O.C. {p31}: 0* — afﬁne, — cA -— CB —- 6.91) —
£29321 — cA) + £93, — cA — ca) = o. (2) me. {pin}: 6191 — 31,913,, — cA) —
$003,, — CA) — 31,013, - CA — CB — 631) + 3520531 — CA — CB” = 0. (3) F.O.C. {3,}:
5l—9*+2112(P§1 —CA"CB"531)+2112(P?41"CA)+36?(P,241 -CA)-§Sz(13§1-CA-CB)J = 0-
By (1) and (2), p2A1 = t+ cA and in a symmetric equilibrium, 192A2 = t + cA. By
substituting the optimal prices into eq.(1.89) and eq. (1.90) with 61 = 02 = 6*, I

haves’i'=s§=t—CB. Cl

 

2 5—9?- = O by envelope theorem.
P.“

96

Appendix C

Appendix: Model with homogenous valuations of good B

I consider that consumers have homogenous valuations of good B, U > 03. For
simplicity, I assume that each consumer in market A purchases at most one unit of

good B. That is, consumers buy one unit of good/service A.

C.1 Pure Bundling

The integrated ﬁrm 1 bundles its products A1 and 81 in this sub-game. There
are A1B1, A2, and 32 available in the markets. Since ﬁrm 1 offers only the bundle of
A181, ﬁrm B2 is only one producing good B. However, it cannot enjoy a monopoly
proﬁt since there exists the bundle of A181. On the other hand, ﬁrm A2 competes
against A131, rather than A1 . Hence, pricing decision of ﬁrm A2 depends on ﬁrm
Bl as well as ﬁrm A1 in pure bundling.

A consumer is indifferent between A181 and A2 + 32 at 1:1 when 13 + txl =
pAz +1932 + t(1 - x1) . p" is the price of A131 . pAz and p32 are the prices of A2 and

82 , respectively. The proﬁt functions are

1

7r1 = E ~-CA-CB)(t+PA2+1732-i5), (Cl)
1 -

7rA2 = 2700.42 - CA)(t - PA2 - 10132 + p) . (C2)
1 ~

7r132 = $0732 — CB)(t -PA2 - P32 + P) , (C3)

97

The equilibrium prices and proﬁts in pure bundling are

- 5 3 3
P =Zt+CA+CBa PA2=Zt+CAa PBZ=Zt+CB, (C4)
25 9 9
= —t = —t = _ , 0.5

Note that in equilibrium, 12ng + pg? > 13. Firm B2 charges a price higher than the
marginal cost. The proﬁt margin for ﬁrm 1 is higher in pure bundling than in no
bundling: pg 18 +19ng < 13 . These results can be explained by competition softening
effect in market B resulting from a product differentiation role played by bundling.
However, the proﬁt margin for ﬁrm A2 is lower in pure bundling than in no bundling.
Since ﬁrm A2 competes against A181 rather than A1, ﬁrm A2 has to reduce the price
of A2 as ﬁrm B2 sets the price of B2 at a price higher than the marginal cost of good
B. N o substitution effect of bundling occurs because consumer’s valuation of good B
is homogeneous and is high enough that all consumers in market A buy one unit of
good B.
Proposition C.1 When ﬁrm 1 offers only a package of A181, the proﬁts of ﬁrms
1 and B2 are higher than in no bundling, while that of ﬁrm A2 is lower than in no
bundling.
Proof: It is clear from direct comparisons of proﬁts and prices.

The intuition behind proposition A.1 is that bundling has a business stealing effect

as a result of a product differentiation role of bundling in market B.

C.2 Mixed-Bundling

Firm 1 offers A1 as well as AlBl. A consumer at 3:1 is indifferent between the

bundle A131 and A2 + 32 when

.. 1 ~
P +1331: PA2 +P32 + t(1 — 171) => Jr1= 27At + PA2 + P32 -P)- (06)

98

A consumer at 2:2 is indifferent between A1 + 82 and A2 + 82 when

1
PA1+PBZ + t332 = PA2 + P32 +t(1— 332) => 132 = E“ + PA2 —PA1)- (07)

And also, ﬁrm 1 can sell positive amounts of A181 if p A1 + 1232 2 13. Otherwise,
consumers would buy A1 + 32 rather than the bundle of A181. Rom eq.(C.6) and
eq.(C.7), I have 1:1 2 x2 under p A1 + 1232 2 15. That is, by offering A1 separately to
consumers who value product A highly and the bundle of A181 at a discounted price
to the consumers who have less preference over A1, ﬁrm 1 can gain an additional
market share of $1 — 2:2. The demand of the bundle is ('1' = 212(pA1 + p32 — 13), the
demand of A1 is qu = 217505 +pA2 — pm), the market share of ﬁrmA2 is qu =
21203 -10A2 -p32 +13). and that of ﬁlm 132 is (132 = 1 - ('1' = 217m _PAI -p32 +13)-

Proﬁt functions of ﬁrms are as follows:

1 ,, - 1

”ii” = $0? - CA - CB)(PA1+ P82 - P) + £00141 - CA)(t + PA2 - pA1)(C-8)
1 a-

“£28 = $00212 - CA)(t - PA2 - P32 + P), (0.9)
1 ~

7’11‘342B = $0132 - CB)(2t - PAl - P32 + p), (0.10)

The equilibrium prices and proﬁts are

9 11 5 7
p =-t+cA+cB, pA1=—t+cA, pA2=§t+cA, p32=§t+cB,

8 8
103 25 49
MB MB MB
_ t —_— -_ —’t. 011
”1 128 ’ "A2 128 ’ "32 128 (C )

Note that the bundle price is less than p A1 + p32 and p A2 + p32. The price of A2 is
less than in pure bundling, while the price of 82 is higher than in pure bundling. The
reason is that the integrated ﬁrm 1 becomes aggressive in price by offering discounts
for the bundle option to consumers who have less preference over A1, while the bundle
option softens competition in market B. Note also that with homogeneous valuations

of good B, no substitution effect of bundling arises in market A.

99

Proposition C.2 By offering mixed bundling, ﬁrm 1 makes higher proﬁts than in
either no bundling or pure bundling. Firm A2 is worse off. Firm BZ is better off
with mixed bundling. (i) «£43 > «f3 > nil/B , (ii) {1:423 < W523 < WXZB , (iii)
MB > «528 > «228

Proof: It is clear from direct comparisons of equilibrium proﬁts under alternative

scenarios.

The idea behind proposition A2 is that mixed bundling acts as a price discrimi-

natory device based on consumers’ preferences.

C.3 Sub-game perfect equilibrium

With homogeneous valuations of both goods, the outcomes of (partial) mixed

bundling arise in equilibrium.

100

Appendix D

Appendix: Proof of Chapter 2

Proof of proposition 2.1: 1r1 = malt - 2cB)2 = %%t — $08 + ﬁne"); > é, iff
t 2 2cB . «£423: m(9t+2cB)2 <2 4:» (9t+203)2_ < 100t2=(10t)2 e 9t+2cB_ <
10t, iﬂ't 2 2cB. «$423 = ﬁat—CB)? > 0, ifft 2 303. t 2 2a,; > 363 -+ «32 > 0.

t<5—-.§£ﬁ,lhavepg.2<l=.*»If2cBSt<5;;SB ,wherecB<%,ﬁrmB2would

set its price such as CB < p32 < 1 and ﬁrm 1 makes higher proﬁts in pure bundling
than no bundling.

Proof of proposition 2.2: m = 21f(t+%—%CB)2 2 5 and “A2 = 21?(t—%+%CB)2
5,11ch < .1, «32 = ,}(1 —cB)(t- g + :lch)_ > o, ifft> §(2—cB).

Proof of proposition 2. 3. (i) 7rBM2B - ﬁng = 317(9t — ch)2 — O > 0, ift > 7cB.
ﬂle-ﬂ'ivB = 2%?(t2—9 3th+ 3032) Z 0 , ifft S 2C3 or t Z 303. (1:423 —7er23 =
,1,(§t + foam? — g = 2130: + 110(63 — 2t))2 — .3, 2 0, ift 3 $03.

(ii) «PB —1rNB < 0,ift<:1;cB. «MB —7r{VB Z 0, ift S £03 or t 2 E63. Hence,

ift<17CB, 7TPB <7TNB<7rMB.1T§§3-7rﬁ/IZB= 213(t+%CB)2—217(gt+1;063)2 >0,

for all CB and t .
Proof of proposition 2.4: wggB- «£23: Ell-{(gt — ﬁcB)2 - 0 2 0,ift 2 $03.
MB

4 - l4
1r1 - niVB = 6%?“2 — 1305-th4-28 3503 2) 2 O, 1ft S gag or t 2 "CB. And,

(1)423 —7er2B = 213(t — 158t+ «[2863? —§ _>_ 0 if t S ch. Now, let’s check the condition
ofﬁ Sp,“ +1932 4:) 13t+cA+ ch S gt+cA+ 9cB+ gt+ ch ift S 463. Hence,
when 7cB S t S 503, or 1'7ch t S min{4cB,9—7—B}, where 63 S 2'61 mixed
bundling arises in equilibrium.

2
c 1
MB ”rim = (6%)th " $3203” 1132‘?) 4200101“

Proof of proposition 2.5: 7r1

101

2cB)2) 2 O , for all t and CB. #2123 — 7rA2B = 7(f8 3t+ 2cB)2 - 212(190t + IUCB)2 < 0,

for all t and cB. #2423 — «£23 = z—(gt - 2cB)2 — m(3t - CB)2 2 0 if t Z 3.3-CB.
Hence, if 203 St S grog, «$423 —1r§23 S 0. If 1?ch S t S ch or If 2%CB S t <

min{4cB, 2:395}, where 03 < 2%, «23423 — «1523 > 0
_ NB_
Proof of proposition 2. 6. (i) For p_ < PAI +1932, I needt S ;gcj-Jr 2323 1r 32 =

2117(1‘ CB)(5t ‘ 508 - 5) > 0 if 4—7—3 < t- “MB - ”iVB= 252(t2+ (12833)”
(2623- -263 + 3)) > 0 if 03_ < 2. Firm 1 also makae higher proﬁts in mixed bundling

than in no bundling ift 2 Keg—J or t S CB — 2, where CB > 2. Hence if

3c

maic,{thBK2—C§L12}StS tggﬂ,where2<cBS¥U,orif4- StS
min{LéEB, cB — 2} = cB— 2, where3 Z S 03, then ﬂiMB — «{VB_ > 0.

Firm A2 makes more proﬁts 1n mixed bundling than in no bundling, «£423 —7rQI2B =

211(§t+§CB-1i3)2-§= 212(t+%(CB-t—%))2—é20.iffc3-g 2t2 4_-7_a
where 63 2 3. Otherwise, ﬁrm A2 is strictly better off in pure bundling than in
Z
9—18c
MB —7rf’3 = 215:“? + (73)t+ (2623 ——
23 l—2c .
208 + 3)) - 2120 + 6' — chlz = 22202 " ($72.13))“. 199(CB — %)2) 2 (3,1ch 2 2
Hence, combining the conditions for ﬁrm 1 and B2, (*) if £393 S t S 7:52-93, where
(23 2 2, then, «MB—#131 > 0. Andalso,1rMB—1r2DB Z 0, ift S 2(1-203) or

1
t 2 $22332, where cB < 2. Combining these conditions with the condition for ﬁrm

mixed bundling. (ii) It is clear that 7r1

82 to set p*B2 = 1, I derive conditions with which ﬁrm 1 is better off in mixed bundling
than in pure bundling, (**) t3 S t S £293, where t3 = max{4——$EB, £01722—632},
CB <2. (On the other hand, £355 S t S min{7———g—CB, 2(1 — 203)}, where CB 2 2,
cannot hold because $393- > 2(1— 2cB)).

Firm A2 makes less in mixed bundling than in pure bundling. «£23 — #2423 =
212((t—2+2CB)2—(gt+2cB—110)2)) > O. sign[7rA2B ""142 M—B] - sign[3t+203- 1] > 0,
under (*) and (**). Firm 82 makes higher proﬁts 1n mixed bundling than in pure

bundling, 2242B — wgg- -— 302(1 - CB)(3t + 2cB — 1) > 0, under (*) and (**).

102

Appendix E

Appendix: Figures of Chapter 2

Figure E.1: Proﬁts of ﬁrms in Pure Bundling

F1 (P)

FBZ(P)

 

Note: F1(P), F2(P), and FB2(P) are proﬁts of ﬁrms 1, 2, and 82 respectively.

103

Figure E.2: Pure- vs. No Bundling

 

0.1
F1(P)-F1(N) 0
-o.1

F2(P)-F2(N)

0.075
0.05
FBZ(P)-FBZ(N) 0.025

 

Note: F1(P)—F1(N), F2(P)-F2(N), and FB2(P)-FB2(N) are ﬁrm 1’s, ﬁrm 2’s, and

ﬁrm B2’s proﬁt difference between pure bundling and no bundling respectively.

104

Figure E.3: Proﬁts of ﬁrms in Mixed Bundling

F1(M)

 

0.10
F82(M) 0.05

 

Note: F1(M), F2(M), and FB2(M) are proﬁts of ﬁrm 1, ﬁrm 2, and ﬁrm B2 respec-

tively.

105

Figure EA: Mixed vs. No Bundling

F1 (M)-F1 (N)

 

0.1

o
F2(M)—F2(N) '9612
o

 

0.1 o
FBZ(M)—F82(N) 0.05

0

 

Note: F1(M)-F1(N), F2(M)—F2(N), and FBZ(M)-FBZ(N) are ﬁrm 1’s, ﬁrm 2’s, and

ﬁrm B2’s proﬁt difference between mixed bundling and no bundling respectively.

106

Figure E.5: Mixed vs.Pure Bundling

0.2
F1 (M)-F1 (P) 0.1

 

-0.1
-0.2

F2(M)-F2(P) 0 3
' 0

 

F82(M)-FBZ(P)

 

Note: F1(M)—F1(P), F2(M)—F2(P), and FB2(M)-FB2(P) are ﬁrm 1’s, ﬁrm 2’s, and

ﬁrm B2’s proﬁt difference between mixed bundling and pure bundling respectively.

107

Appendix F

Appendix: Figures of Chapter 3

Figure F.1: Under Firewalls, Proﬁts Comparison

 

 

 

 

 

 

 

Note: P1(F) and P2(F) are proﬁts of ﬁrm 1 and D2 under Firewalls, respectively.
P1(NF) and P2(NF) are proﬁts of ﬁrm 1 and D2 under Information Flow, respectively.

Figure F2: Total R&D Level Comparison (Firewalls vs. Information Transfer)

 

Note: DX is R&D(Firewalls)-R&D(Information Flow)

108

Figure F3: Consumers Surplus Comparison (Firewalls vs. Information Transfer)

0.10

 

Note: DCS is CS(Firewalls)-CS(Information Flow)

Figure FA: Producers Comparison (Firewalls vs. Information Transfer)

 

 

 

10

a

‘5 m’m
1

4

2

mums»)
o 0.2 0.4 0.6 0.8 1
p

 

 

 

Note: PS(F) and PS(NF) are producer’ surplus under Firewalls and under Information

Flow, respectively.

109

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