GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 369

Multiproduct competition in vertically related industries

Shohei Yoshida

May 2015

GCOE Secretariat
Graduate School of Economics
OSAKA UNIVERSITY
1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Multiproduct competition in vertically related industries∗
Shohei Yoshida†
Graduate School of Economics, Osaka University
May 10, 2015

Abstract
The paper investigates how competition between two multiproduct downstream firms in vertical
relationships aﬀects horizontal relationships: competitor collaboration and performance diﬀerence.
When the upstream market consists of exclusive suppliers, the eﬃcient firm may have incentive fortechnology transfer without any payment to its less eﬃcient rival, which can be a credible device ofthe eﬃcient firm to enlarge its more profitable product. Moreover, such technology transfer enhances
both consumer surplus and social welfare. The ineﬃcient downstream firm may earn more than theeﬃcient firm under upstream markets with exclusive suppliers and with discriminatory monopolist.
Keywords: Multiproduct firm, Technology transfer, Vertical relationship, Competitor collaboration,
Firm performance
JEL Classification Numbers: L14, L24, L41∗

I am especially grateful to Noriaki Matsushima for his valuable advice. I would also like to thank Makoto Hanazono,
Shinji Hasegawa, Takeshi Ikeda, Ikuo Ishibashi, Junichiro Ishida, Akio Kawasaki and Takao Ohkawa, the conference
participants at the Japanese Economic Association (Doshisha University), and the seminar participants at Kobe University,
Kwansei Gakuin University, Nagoya University and Osaka University for their very useful comments. All remaining errors
are my own.
†
Address: 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail: pge031ys@student.econ.osaka-u.ac.jp.

11Introduction

In the modern economy, the overwhelming majority of large enterprises has two features: they are
multiproduct firms (MPFs) (e.g. Bailey and Friedlaender, 1982) and they have vertical business relations.1 Needless to say, automobile manufactures produce several varieties of automobile, mini car,passenger car, sports car, and so on. Moreover, almost all of them also sell several varieties of passenger car. To do so, automobile manufacturers purchase steel, tires, and a number of parts produced bytheir suppliers. Buyer-supplier relationships are crucial for firm’s performance especially in automobile industry. Previous researches in management have found a positive relationship between cooperative inter-firm relations and the performance of manufactures and suppliers (Cusumano and Takeishi,
1991; Helper and Sako, 1994; Dyer, 1996).2 This implies that vertical relationships significantly
aﬀect many strategic decisions of downstream firms, which determine these firms’ performances.3
As well as vertical relationships, horizontal relationships such as competitor collaboration amongautomobile manufacturers are widely observed and also would have a great influence on the firms’
performances. Cooperative relationships among manufactures take various forms such as partial equity ownership, joint research/production, OEM, license, etc.4 Such competitor collaboration is still
controversial. Economists and policymakers dealing with antitrust and regulatory issues have recognized that conventional wisdom on single-product firms does not always apply to MPFs.5 We
therefore investigate how competition between MPFs in vertical relationships aﬀects horizontal relationships such as competitor collaborations or firms’ performances.
1

Bernard et al. (2010) show good evidence for the importance of MPFs. Investigating firms in the U.S., they showthat MPFs are present in all industries and produce 87% of total output. They think of firms producing a single-product as
those whose range of products falls within a single five-digit SIC category. Multiproduct firms, on the other hand, are those
whose product range is wide enough to span several five-digit SIC categories. In fact, some firms would produce several
varieties of product which are classified into the same category. In a sense, almost all firms must be MPFs.
2
Some studies show that cooperative relationships in Japan outperform their U.S. counterparts (Nishiguchi, 1994; Dyer,
1996) and the buyer-supplier relationship in Japan has several distinctive features (Morita, 2001). First, a large manufacture
typically owns partial shareholding of their suppliers. Second, suppliers make investments that are customized to theirpurchaser. Third, the relationship is long-term and stable.
3
It is well-known in economics literature that vertical structure has a great influence on behaviors of firms (e.g., DeGraba,
1990; Mukherjee and Pennings, 2011).
4
For example, partial ownership between Renault and Nissan, joint research between GM and Honda, Toyota and Ford,joint production between GM and Toyota, Mitsubishi and Nissan, OEM among Nissan, Mazda, Suzuki and Mitsubishi.
5
Theoretical studies point out the importance of studies on MPFs for a long time (e.g. Bailey and Friedlaender, 1982;
Brander and Eaton, 1984). For a decade, a number of studies have been made not only in industrial organization (e.g.,
Johnson and Myatt, 2003, 2006) but also in other fields of economics such as international trade (e.g., Eckel and Neary,
2010; Bernard et al., 2011).

2

We consider several types of vertically related industry where duopolists produce two varietiesof final product. The product that can be produced at a lower cost is called core product, and theother is called non-core product. Downstream firms require common input (or labor) in order toproduce both of products. They have diﬀerent technologies for one of the products with respect toeﬃciency. The present setup yields several new results. First, when an upstream market consists of
exclusive suppliers, an eﬃcient firm may have incentive for technology transfer, which improves anineﬃcient firm’s technology for producing the non-core product, without any payment. Second, suchtechnology transfer enhances both consumer surplus and social welfare. Finally, an ineﬃcient firm
may earn more than an eﬃcient firm under some types of upstream structures.6
The eﬀect of the technology transfer on profits of downstream MPFs can be decomposed into thetwo eﬀects: the production shift eﬀect and input price eﬀect. The first eﬀect makes MPFs modify their
output portfolios of the two products.7 A reduction in a firm’s marginal cost of a product increases theoutput of the product, but decreases its competitor’s output of the same product. This is the standard
business-stealing eﬀect. Furthermore it also makes the firm decrease its output of the substitute product to avoid cannibalization. This allows its competitor to expand the other product. To summarize,
when a firm improves eﬃciency of a product, the business-stealing eﬀect directly hurts its competitor
whereas the latter eﬀect benefits it. Thus the technology transfer has direct and indirect eﬀects on
profits of downstream firms, with both eﬀects working in the opposite directions.
The input price eﬀect works through the upstream market. The technology transfer also aﬀects
the pricing of upstream firms. A firm-specific supplier utilizes its input price for extracting rents fromits trading downstream firm. The technology transfer reduces the total output and input demand of theeﬃcient firm, while it increases those of the ineﬃcient firm. It thus decreases the input price of theeﬃcient firm and increases that of the ineﬃcient one. The technology transfer benefits the eﬃcient
firm through the lower input price. From the view point of the ineﬃcient firm, the technology transfer
reduces ex post marginal cost of the product but increases that of the other product due to the increaseof the input price. Consequently, it accelerates the production shift between the two products.
One may wonder why we consider whether firms transfer its knowledge or technical know-how
6

Sen and Stamatopoulos (2015) consider a Cournot duopoly with strategic delegation, where quantities of firms arechosen by their managers and shows a similar result.
7
Lin and Zhou (2013) point out this eﬀect.

3

without any payment. Generally, explicit knowledge can be transferred through licensing and contracting since it is verifiable. Contrary to such transfer, codification of tacit knowledge is diﬃcult andsuch knowledge is unverifiable. Hence, licenses and contracts play, at best, the limited role in thetransfer of tacit knowledge (Mowery, 1983, p354).8 Moreover, this allows us to isolate the strategic
eﬀects of technology transfer from other incentives by licensing payments.
In some industries, technology transfer without direct compensation indeed occurs through strategic alliances or joint production ventures.9 Morita et al. (2010) investigate into collaborations amongfirms. They take a survey, and observe 88 cases in which a firm transfers its technology without direct compensation. Especially, 59% of the technology transfers are performed between firms without
partial equity ownerships. In the automobile industry, we can find some cases: the joint production
ventures between Nissan and Mitsubishi (NMKV), GM and Toyota (NUMMI) and the strategic alliance between Ford and Mazda.10 We here give the particulars of the first case. In 2013, Nissan and
Mitsubishi began the joint production of mini cars at the NMKV in Japan. Nissan had not produced
mini car in person before, and therefore Nissan purchased it from Mitsubishi, Suzuki and Mazda
by OEM. Since Mitsubishi has superior technology for mini car, the joint production would involve
technology transfers.
Our results suggest some implications for both policymakers and managers. The result on thetechnology transfer tells us the eﬀect of competitor collaborations on social welfare. Firms often use
complex collaborations to achieve a goal (e.g., profit maximization). Antitrust agencies had viewed
competitor collaborations as only collusive behaviors. However, in 1993, the National Cooperative
Research Act of 1984 was amended to the National Cooperative Research and Production Act of1993 (NCRPA) so that firms can engage in joint production. Furthermore, the Antitrust Guidelines
for Collaborations Among Competitors in 2000 proclaims that “the Federal Trade Commission (FTC)and the U.S. Department of Justice (DOJ) regard such collaborations as often being not only benign

8

A joint venture is viewed as a mechanism to transfer knowledge that cannot be done by licensing (Kogut, 1988). For
further details of contractual diﬃculties in transferring tacit knowledge, see also Ghosh and Morita (2012).
9
Employees frequently give technical information to colleagues in other firms, including direct competitors (Schrader,1991). In the literature on international economics, technology spillovers through FDI have been explored extensively. See
Lin and Saggi (1999) and Ishikawa and Horiuchi (2012).
10
Creane and Konishi (2009) argue that the joint production between GM and Toyota involves technology transfers. They
provide the large cases of joint production. Ghosh and Morita (2012) also provide the rich cases for knowledge transfers.

4

but procompetitive.”11 In the last three decades, the federal antitrust agencies indeed have come toadmit competitor collaborations. However, the welfare eﬀect of such collaboration is still controversial. Several recent studies obtain diﬀerent implications on joint production with technology transfer
or know-how disclosure. For instance, joint production as a device to predate non-recipient firms
harms consumer surplus but enhances social welfare (Creane and Konishi, 2009).12 It is true that
economists usually define social welfare as the object function of the social planner, but policy makers often regard consumer surplus as a proper measure. In contrast to previous studies, as the joint
production certainly raises not only consumer surplus but also social welfare in our paper, our result
would promote such joint production.
Patent protection is one of the most important problems for managers. Our first result may imply,
however, that some technology spillovers are welcome.13 In other words, mangers do not have tostrive to protect their technological information for non-core products. Furthermore, the result on the
profit of MPFs also provides some implications for managers. In the real economy, MPFs face theconstraint on selecting the distribution of the management resource, such as R&D, advertisement andhuman resources. Our result implies that there exist some situations when firms should not invest
their resource in less profitable segment.
Several recent studies have raised instances where technology transfer or know-how disclosure
without direct compensation can increase profit of technologically superior firm. Creane and Konishi (2009) consider an asymmetric oligopoly model with entry or exit, and show technology transfer
without any compensation can be profitable when it works as a type of predation or deterrence. Matsushima and Ogawa (2012) examine incentive of knowledge disclosure for MPFs. They provide theview of know-how disclosure which induces some firms to change their plans for location or specialized products. In those papers, technology transfer drives some rivals out of market or aﬀects
decisions for entrants. The primary diﬀerence with our model is that we restrict a situation where
neither entry nor exit occurs. Milliou and Petrakis (2012) show that a vertically integrated firm can
11

Potential eﬃciencies from the collaborations may be attained through a variety of contractual arrangements, including
joint ventures, trade or professional associations, licensing arrangements, or strategic alliances, etc.
12
The eﬀect of know-how disclosure on consumer surplus is positive while that on social welfare is ambiguous (Matsushima and Ogawa, 2012).
13
Suzuki (1993) analyzes the eﬀects of technological diﬀusion among and within vertical keiretsu groups in the Japaneseelectrical machinery industry, and find that positive R&D spillovers, which stem from the R&D activities of other keiretsu
groups. It is remarkable that there are the spillovers between core firms of competing keiretsu groups.

5choose to fully disclose its knowledge to its downstream rival. In their paper, the technology transfer
intensifies downstream competition but expands downstream market size, which increases the wholesale revenue of the vertically integrated firm. Ghosh and Morita (2012) examine technology transfer
among competitors with a partial equity ownership, and show an equity alliance may induce the technology transfer. In contrast, our model does not include any direct elements such as PEO or wholesalerevenue, which increase the profit of technologically advanced firm. Matsushima and Zhao (2015)
examine a bilateral duopoly market in which each downstream firm has outside options and upstream
firms can engage in cost reduction and generate technological spillover. They find that each upstream
firm has incentive to voluntarily generate the spillover to its upstream rival. As well as our paper,
they also point out the importance of linkage between vertical relationship and technology transfer.
However, the mechanism of their paper is clearly diﬀerent from ours since it stems from the existenceof outside options.
Our study is also in line with the vast literature on technology licensing. However, few studies
examine the impacts of technology licensing on MPFs. Except for analyzing single-product firms,
one of the related works is Mukherjee and Pennings (2011), who examine the eﬀects of the unionization structure (viz., decentralized and centralized unions) on incentive for licensing. They show
the incentive for licensing is stronger under decentralized unions than centralized. A similar input
price eﬀect of licensing in their paper also works in our setting. A notable diﬀerence is that in ourmodel technology transfer without payment can indeed occur, while such license never occurs in theirmodel.
The rest of this paper is organized as follows. Section 2 sets up our model and obtains a preliminary result on a property of multiproduct firms. Section 3 analyzes incentive for technology transfer
under vertical relationship. Section 4 analyzes alternative scenarios in upstream market. Section 5oﬀers some concluding remarks. All proofs of the results and derivations of the equilibrium outcomesin the extension models are provided in the Appendices.

62Model and downstream competition: Preliminary result

2.1

Model

We first characterize downstream market, and then introduce the timing of the game. We follow themodel of Lin and Zhou (2013) without R&D investment.14 The downstream market is composed oftwo multiproduct firms, named D1 and D2, which produce two types of final product, goods A and
B. The inverse demand functions are specified as
pk = v − (q1k + q2k ) − γ(q1l + q2l )k, l ∈ {A, B}, k , l,(1)where pk is the price of good k, qik shows the quantity of good k by downstream firm i ∈ {1, 2}, andv is a positive demand parameter. The parameter γ ∈ [0, 1) reflects the substitution between goods A
and B.
Downstream firms require one unit of common input, e.g., raw material, which is commonlyused for goods A and B, to produce one unit of final goods. Each firm has diﬀerent transformation
technologies for each good. Therefore the per unit production cost of good k of downstream firm i,cik , is denoted by
cik = wi + zik ,(2)where wi is the input price for Di, which is determined in an upstream market. zik is the eﬃciency
measure of a downstream firm and shows the technological level for good k of Di.15 We can alsointerpret the production technology of the downstream firms as requiring two inputs, one is producedin the upstream market and the other is supplied by a competitive sector. They have an uneventechnological level of input from the competitive sector. One product that the firms can produce at alower cost is called core product and the other one is called non-core product. We assume that good
A is the core product, and technological gap between firms is present only in the technology for the
non-core product.16 We refer to D1 (D2) as the eﬃcient (ineﬃcient) firm. To summarize, we assume
z2B ≥ z1B > ziA and ziA is normalized to zero for simplicity.
14

Lin and Zhou (2013) investigate the R&D portfolio choices of multiproduct firms without vertical relationships.
We obtain qualitatively similar results under the assumption that cik = zik wi , where zik > 0, which is employed in
Mukherjee and Pennings (2011).
16
All of the results we present are equally true in the case where D1 is eﬃcient in both technologies. Moreover, we
can also show that a quality advantage, which is reflected by demand parameters, rather than a cost advantage, will lead toessentially the same result. See the discussion in Lin and Zhou (2013, p.88).
15

7

The timing of the game is as follows. The model has two stages: in the first stage, input prices
are determined in the upstream market. We specify how the input prices are determined in Section 3.
In the second stage, observing the input prices, the downstream firms engage in Cournot competition.
In what follows, we are looking for subgame perfect equilibria. Note that the decision of technology
transfer is not included in the game. We analyze whether the firms have incentives for technology
transfer by comparative statics.

2.2

Downstream competition

In this subsection, we examine an important property of MPFs. In the second stage, given the input
price(s), each downstream firm chooses its quantity pair (qiA , qiB ) in the final good market. Eachfirm’s problem in this stage can be written as
max πDi = (pA − ciA )qiA + (pB − ciB )qiB .

(qiA ,qiB )(3)

The first-order conditions for profit maximization of Di are as follows.
∂πDi= (pk − cik ) − qik
∂qik

−γqil
|{z}

= 0.

(4)

negative eﬀects

The last term captures cannibalization eﬀects, which are present only for MPFs. That is, an increasein quantity of one good reduces not only the price of the good but also that of the other one, so that it
discourages itself from producing the other one.
It is evident that a decrease in a firm’s marginal cost of a product increases its output of that
product, but decreases its competitor’s output of the same product. The standard business-stealing
eﬀect also works as well as for single-product firms. What is unique about MPFs is that the firm also
reduces its output of the substitute product to mitigate within-firm cannibalization, which allows its
competitor to expand the substitute product. Note that the above property is deeply related to our
results.
Assuming interior solutions exist, the equilibrium quantities are obtained asqik =(1 − γ)v − 2cik + c jk + 2γcil − γc jl.
3(1 − γ2 )8(5)From (5), we certainly confirm the property mentioned above. This leads to the equilibrium profit ofeach downstream firm,(v − 2ciA + c jA )2 − 2γ(v − 2ciA + c jA )(v − 2ciB + c jB ) + (v − 2ciB + c jB )2
πDi =.
9(1 − γ2 )3How does upstream structure aﬀect technology transfer?

In this section, we examine how upstream structure aﬀects technology transfer in a vertically relatedindustry. The structure of the upstream market is the important element in this paper. We consider
five types of vertical structure: the upstream market consists of (I) competitive suppliers, (II) two upstream firms, named U1 and U2, which exclusively supply its input for D1 and D2 respectively, (III)a monopolist employing uniform pricing, (IV) upstream market consists of a monopolist employing
discriminatory pricing, and (V) exclusive suppliers for D1 and D2, respectively, and common suppliers, which supply its input for both D1 and D2. For simplicity, upstream firm(s) produces at nocosts. We consider the relationship between input supplier(s) and final goods producers, although thismodel is suitable for much more general application. For example, the model can be applied to therelationship between labor unions and firms.
Figure 1 summarizes the structure of the model (the last case is introduced in Section 4). We first
analyze the case (I).
[Figure 1 about here]

3.1

Competitive suppliers: Benchmark

Consider a competitive upstream market where the input price is zero. We can interpret this case asa multiproduct duopoly model without vertical structure. We then introduce vertical structures andshow our main results in the next subsection.
In the benchmark case, the equilibrium quantity of each firm and price are solved as (substituteziA = 0 when k = A)
qCik =(1 − γ)v − 2zik + z jk + 2γzil − γz jl,
3(1 − γ2 )

(6)

and
1
pCk = (v + zik + z jk ).
3
9

(7)

We then obtain the equilibrium profit of each firm:
πCDi =

v2 − 2γv(v − 2ziB + z jB ) + (v − 2ziB + z jB )2.
9(1 − γ2 )(8)

The conditions that the interior solutions indeed exist are
qCik > 0 ⇔ γ <v − 2zik + z jk
≡ γ̄Cik .
v − 2zil + z jl

(9)Hereafter, we restrict our attention to the case where both downstream firms provide both products.
We examine the incentive for the eﬃcient firm to transfer its technology. The technology transfer
improves the D2’s eﬃciency for the non-core product, but does not aﬀect the technology for thecore product. In words, the technology is valuable for the non-core product but can not be applied
to the core one.17 Consider a situation that the technology transfer yields no direct compensation
for D1. Therefore the firms agree to the technology transfer if a technology improvement for the
non-core product of D2 increases not only the profit of D2 but also that of D1. From (8), we obtaindπD1 /dz2B > 0, which implies,
Proposition 1 When the upstream market is competitive, the eﬃcient firm has no incentive for thetechnology transfer to the ineﬃcient firm.
The technology transfer has two eﬀects, which work oppositely. A decrease in the D2’s marginal costof good B discourages D1 from producing good B. On the other hand, it causes that D2 produces
good A less whereas good B more. Since D2 shifts its production from goods A to B, D1 can produce
good A aggressively. In this case, the former eﬀect dominates the latter, and hence the technology
transfer is not profitable for D1.
We are also interested in the relationship between the profits of the downstream firms. Comparing
them, we have
Proposition 2 When the upstream market is competitive, the eﬃcient firm always earns more profit
than the ineﬃcient firm.

17This is an important assumption for the technology transfer. However, we would obtain qualitatively similar results if
the technology is available for the core product but the availability is suﬃciently low.

10

From (6), D2 produces good A more than D1 and good B less. D2 benefits from the higher marginal
cost of good B, because it works as a commitment device to produce the core product aggressively.
However this force is not strong enough to compensate for the disadvantage in eﬃciency.

3.2

Exclusive suppliers

In this subsection, we examine the case where the upstream market is composed of two firms which
supply its input to the downstream firm exclusively.18
Each upstream firm simultaneously decides its input price so as to maximize its own profit.19
Each upstream firm’s problem can be written as
max πUi = wi Qi (w1 , w2 ),wiwhere Qi ≡ qiA + qiB .20 The first-order condition for the upstream firm i is
∂πUi
−4
= wi
+ Qi (w1 , w2 ) = 0.
∂wi3(1 + γ)(10)

The reaction function of upstream firm i is1
wi = (2v − 2ziB + z jB + 2w j ).
8(11)Equation (11) implies that the input prices are strategic complements. The equilibrium input prices
are obtained as
wiE =1(10v − 7ziB + 2z jB ).
30

(12)

We get that dwiE /dziB < 0, and dwEj /dziB > 0. This implies
Lemma 1 When the upstream market consists of exclusive suppliers, as the eﬃciency of downstream
firm i increases, the input price of upstream firm i ( j) increases (decreases).
18
According to Nobeoka (1996), Nissan bought fuel filters from Tsuchiya Seisakusho, while Mitsubishi bought it from
Nippon Denso and Tokyoroki. With respect to fuel filters, Tsuchiya Seisakusho was the exclusive supplier for Nissan andNippon Denso and Tokyoroki were those for Mitsubishi. In 2014, Nissan buys some engine cooling and air-conditioning
system products from MAHLE berh Japan while Mitsubishi does not, which is another example for the exclusive supplier.
19
This setting means the contract is take-it-or-leave-it oﬀer. Some readers would think that the bargaining power ofsuppliers is relatively weak in reality. However, Ahmadjian and Oxley (2013) indicate that the profitabilities of Japanese
auto suppliers are not lower than those of Japanese auto assemblers in terms of ROA (Table 2 in their paper), which implies
that auto suppliers are not always relatively weak to their trading assemblers.
20
We assume that the upstream firms determine their input prices such that the downstream firms can produce both of
the final goods. Since the downstream firms are MPFs, their input demand functions are kinked. Therefore, we must check
not only the local optimality (i.e., first-order condition) but also the global optimality (i.e., deviations “beyond” the kinked
point). The calculations are described in the appendix.

11An eﬃciency improvement of Di’s non-core product raises the profitability of the firm. This leads toa higher input price by its exclusive supplier. On the other hand, D j decreases its total quantity. The
downward shift of input demand decreases the input price.
The equilibrium quantity and price are obtained asE
qik=20(1 − γ)v − 4(11 + 4γ)zik + (19 + 11γ)z jk + 4(4 + 11γ)zil − (11 + 19γ)z jl,
90(1 − γ2 )(13)

1(10v + 5z1k + 5z2k − z1l − z2l ).
18

(14)

pkE =

This leads to the equilibrium profits of the firms:
πEDi =(20v + 16ziB − 11z jB )2 − 2γ(20v + 16ziB − 11z jB )(20v − 44ziB + 19z jB ) + (20v − 44ziB + 19z jB )2(15)
9 · 302 (1 − γ2 )

and
E
πUi
=

(10v − 7ziB + 2z jB )2.
15 · 45(1 + γ)

(16)

E > 0, areThe conditions for the interior solutions, qik

γ<

20v − 44zik + 19z jk + 16zil − 11z jl≡ γ̄ikE .
20v + 16zik − 11z jk − 44zil + 19z jl(17)

We also have to check whether the upstream firms have no incentive to deviate from the their input
prices. Let γ̄dE be the upperbound of γ, and we can show that the condition γ < γ̄dE assures the
upstream firms do not deviate.21 Hence, the condition for the equilibrium is γ < min{γ̄ikE , γ̄dE }.
We now turn to the central issue of this section, namely the incentive for the technology transfer
in the vertical relationship.
Proposition 3 When the upstream market consists of exclusive suppliers, the eﬃcient firm may havethe incentive for the technology transfer to the ineﬃcient firm and the ineﬃcient firm may accept theE ) where γ E and γ̄ E areoﬀer. Formally for v > v̂E , πEDi is decreasing in z2B if and only if γ ∈ (γ E , γ̄π2
π2
π1

π1

the threshold values of γ such that dπEDi /dz2B = 0 respectively.22

We derive the threshold values γ̄d for each upstream structure in Appendix B.
It might be possible to define the condition for the technology transfer as when there exists z′2B ∈ [z1B , z2B ) such that
πDi (v, z1B , z′2B , γ) > πDi (v, z1B , z2B , γ) for i = 1, 2. Our definition is suﬃcient for the other. Moreover, that is suitable when
the cost reductions of the technology transfer are uncertain in advance.
21

22

12

The technology transfer has the two notable eﬀects. An eﬃciency improvement for good B of D2
increases (decreases) the quantity of good B (A) from D2, whereas decreases (increases) that of good
B (A) from D1 - what we call the “production shift eﬀect.” In addition, the technological improvement
also induces a lower (higher) input price for D1 (D2) - what we call the “input price eﬀect.” Therefore,
D1 can produce the core product aggressively, and purchase cheaper inputs from its exclusive supplier.
In turn, although D2 can not produce good A aggressively, and besides must purchase more expensive
inputs, it can produce good B more owing to the lower ex post marginal cost of good B. When thecondition in Proposition 3 is satisfied, the benefit can be large enough to compensate for the loss
for both firms, and consequently the technology transfer can indeed occur. The firms can utilize thetechnology transfer as a credible device for the “output specialization.”
Figure 2 illustrates an example for the proposition above. When z2B is suﬃciently low, a decreasein z2B increases both firms’ profits. The detail of the conditions is shown in Figure 3. For the technology transfer, γ plays an important role for the condition. That is, γ deeply aﬀects the degree ofthe production shift eﬀect. Equation (13) shows how the technology transfer modifies their production portfolios. Clearly, the diﬀerence between z1B and z2B is also important. When the diﬀerence
between z1B and z2B is relatively large, the D1’s quantity of good A is small while that of good B islarge. Suppose that the technology transfer decreases z2B , it yields an decrease in the price of good B
and the D1’s quantity of good B, resulting in the large loss for D1.23[Figures 2 and 3 about here]
When competing firms intend to perform cooperative behaviors such as technology transfer, antitrust authority must beware that it does not have anti-competitive eﬀects.
Proposition 4 If the technology transfer benefits both downstream firms, then it always enhances
both consumer surplus and social welfare.
How does eﬃciency of firms aﬀect consumer surplus or social welfare? It is well know that making
an ineﬃcient firm more eﬃcient can reduce welfare (Lahiri and Ono, 1988). The intuition of their
23

Marjit (1990) also shows that technology licensing with fixed fees is likely to occur between firms which are reasonably
close in terms of their initial technologies.

13

result is as follows. A cost reduction in a relatively less eﬃcient firm results in an increase in the total
output, which clearly enhances consumer surplus. The cost reduction, however, shifts production
from the more eﬃcient firms to the less eﬃcient one, so that producer surplus may fall. Although thelatter negative eﬀect also exists in our model, the technology transfer benefits both downstream firms.
Hence, the eﬀect to producer surplus is also positive. The technology transfer indeed occurs only if itenhances both consumer surplus and social welfare.
We turn to another interest: which firm can outperform? Comparing the profits of the downstream
firms, we get the following:
Proposition 5 When the upstream market consists of exclusive suppliers, the ineﬃcient firm may earnmore profit than that of the eﬃcient firm. Formally for v > vE , πED2 > πED1 if and only if γ ∈ (γ E , γ̄dE )where γ E is the threshold value of γ such that πED2 = πED1 .
The intuition behind the result is as follows. There exist two positive eﬀects of the ineﬃciency on thetechnology for good B in this case. First, by Lemma 1, thanks to its ineﬃciency, the ineﬃcient firm
can purchase its input cheaper than the eﬃcient rival. Since the input is used for both good A and B
commonly, the non-core segment of D2 serves to indirectly subsidize its core segment.24 In addition,
the ineﬃcient technology for the non-core product works as a commitment device to expand the coreone. For the reason above, this counterintuitive result may occur in our model.
Figures 2 and 4 illustrate examples for the proposition. Note that when z2B or γ are suﬃciently
high, the profit of the ineﬃcient firm exceeds that of the eﬃcient firm. The detail of the conditions
is shown in Figure 5. The figure shows that when γ is large, in other words, when the production
shift eﬀect works strongly, the ineﬃcient firm outperforms the eﬃcient rival. The diﬀerence between
z1B and z2B is also important. When γ is relatively small (e.g., γ = 0.6), the condition requires
the diﬀerence between z1B and z2B is relatively large. This means when the production shift eﬀect isweak, D2 needs less ineﬃcient technology to outperform its rival because both the commitment eﬀect
to expand the core product and the input price eﬀect must work strongly.
[Figures 4 and 5 about here]
24

This eﬀect is also emphasized in the accounting literature. Due to the latent cross-subsidization eﬀect, firms which sellmultiple markets can understate the value added by less profitability market (Arya and Mittendorf, 2011).

14

4

Extensions of the model

So far, we have shown that the technology transfer may occur under the upstream market with theexclusive suppliers. As shown in this section, the technology transfer does not occur when an upstream market is composed of a monopolist, but occurs when many suppliers, which are introducedin Section 4. The result on the profit is also obtained under an upstream market with discriminatory
monopolist and many suppliers. Table 1 summarizes the results of this paper.

Table 1. Summary of the Results
Upstream structures
Competitive
Suppliers
Technology transfer×
Ineﬃcient firm outperforms
×

Exclusive
Suppliers⃝⃝

Uniform Pricing

Discriminatory

Many

Monopolist
××

Monopolist
×⃝Suppliers⃝⃝

Consider now the upstream market consists of a monopolist. The following two subsections investigate two kinds of upstream market, that with the monopolist employing uniform pricing and thatwith the monopolist employing discriminatory pricing. The timing structure of the game is the samein the previous section. Main purpose in the following two subsections is to show the technology
transfer does not occur in the markets with upstream monopolists. Subsection 4.3 examines an extended model, named “many suppliers”, which is composed of both exclusive and common suppliers.

4.1

Monopolist employing uniform pricing

In this subsection, we consider the case where the upstream monopolist, M, sets a same input priceto both downstream firms. The outcome in the second stage is the same as that in the last section. In
the first stage the upstream monopolist faces the problem:
max π M = w

∑ ∑

qik .

(18)

1
w MU = (4v − z1B − z2B ).
8

(19)

w

i∈{1,2} k∈{A,B}

The equilibrium input price is obtained as

We thus get the following lemma.
15Lemma 2 When the upstream monopolist employs uniform pricing, as the eﬃciency in good B increases, the input price increases.
An eﬃciency improvement in a good increases the quantity of the good and decreases that of theother. Since the increase in quantity of the good exceeds the decrease in quantity of the others, thetotal input demand shifts upward so that the input price goes up.
We obtain the equilibrium profits of the firms:
MU
πDi
=

(4v + ziB + z jB )2 − 2γ(4v + ziB + z jB )(4v − 15ziB + 9z jB ) + (4v − 15ziB + 9z jB )2,
9 · 82 (1 − γ2 )
π MU
M =

(4v − z1B − z2B )2,48(1 + γ)

(20)

(21)

where the condition for the equilibrium is γ < min{γ̄ikMU , γ̄dMU }.
We are interested in the incentive for the technology transfer and conclude that it never occurs inthis case.
Proposition 6 When the upstream monopolist employs uniform pricing, the eﬃcient firm has no incentive for the technology transfer to the ineﬃcient firm.
The intuition behind this proposition is similar to that in Proposition 1. There is an additional negativeeﬀect for the eﬃcient firm through input price. As Lemma 2 shows, the technology transfer yields
a higher input price for both downstream firms. This interferes with an expansion in good A of theeﬃcient firm directly. The negative eﬀect therefore dominates the positive eﬀect from an expansion
in the core product.
We turn to the next question: which firm will get a higher profit. Comparing the profits of thedownstream firms, we get
Proposition 7 When the upstream monopolist employs uniform pricing, the eﬃcient firm always
earns a higher profit than that of the ineﬃcient firm.
Proposition 7 is parallel to Proposition 2. Under uniform pricing, the diﬀerence in eﬃciency aﬀects
only competition in the final good market. Hence, the intuition in Proposition 2 can be also applied
to this case.
16

4.2

Monopolist employing discriminatory pricing

We next analyze the case where the upstream market consists of the monopolist employing discriminatory pricing.
In the first stage the upstream monopolist faces the problem:
max π M = w1 Q1 (w1 , w2 ) + w2 Q2 (w1 , w2 ),(w1 ,w2 )where Qi ≡

∑

(22)

k∈{A,B} qik . The equilibrium input prices are

1
wiMD = (2v − ziB ).
4(23)

Lemma 3 When the upstream monopolist employs discriminatory pricing, an eﬃciency improvementof a downstream firm increases its input price but has no eﬀect on the input price of the other downstream firm.
An eﬃciency improvement of Di shifts the input demand of Di (D j) upward (downward). Therefore the input price for Di (D j) increases (decreases). However, since the pricing for downstream
firms is complement, there is another eﬀect which works oppositely toward the input demand eﬀects
above. For the input price of D j, two eﬀects are canceled out, and consequently the Di’s eﬃciency
improvement has no impact on the D j’s input price.
We obtain the equilibrium profits of the firms:
MD
πDi
=

(2v + 2ziB − z jB )2 − 2γ(2v + 2ziB − z jB )(2v − 6ziB + 3z jB ) + (2v − 6ziB + 3z jB )2,
9 · 42 (1 − γ2 )

(24)

and
π MD
M =

(2v − z1B )2 + (2v − z2B )2 + (z2B − z1B )2,24(1 + γ)(25)

where the condition for the equilibrium is γ < min{γ̄ikMD , γ̄dMD }.
We first compare the input prices under uniform pricing and under discriminatory pricing.
Lemma 4 When the upstream monopolist employs discriminatory pricing, it sets a higher (lower)
input price to the (in)eﬃcient firm than when it employs uniform pricing. That is w1MD > w MU > w2MD .

17

Since the ex ante profitability of the eﬃcient firm is higher than that of the less eﬃcient one, themonopolist sets a higher input price for the eﬃcient firm in order to exploit it when using pricediscrimination.
We check whether the eﬃcient firm has incentive for the technology transfer or not.
Proposition 8 When the upstream monopolist employs discriminatory pricing, the eﬃcient firm has
no incentive for the technology transfer to the ineﬃcient firm.
In this case, the technology transfer also raises the input price for the ineﬃcient firm. However itdoes not decrease input price for the eﬃcient firm. The positive eﬀects can not be large enough tocompensate for the loss from giving up the dominance in ex ante eﬃciency.
We next examine which firm can outperform its rival. Comparing the profits of the downstream
firms, we then have
Proposition 9 When the upstream monopolist employs discriminatory pricing, the ineﬃcient firm
MD > π MD if and only
may earn a higher profit than that of the eﬃcient firm. Formally for v > v MD , πD2
D1
MD = π MD .
if γ ∈ (γ MD , γ̄dMD ) where γ MD is the threshold value of γ such that πD2
D1

We can also apply the intuition behind Proposition 5. Figures 6 and 7 illustrate the numerical examples of Proposition 9 and the detail of the conditions is shown in Figure 8. Figure 6 also shows thatthe more ineﬃcient the ineﬃcient firm becomes, the more profit both firms get. This observation issimilar to Lemma 1 in Arya and Mittendorf (2010), which shows a multi-market firm may lose by anincrease in a market size.
[Figures 6 and 7 and 8 about here]

4.3 Many suppliers
We complete this section by discussing what happens if the model is extended to an upstream market
with many suppliers. Consider an industry where both D1 and D2 trade with nE exclusive suppliers
respectively, and nC firms supply to both D1 and D2. Hereafter, we call the latter “common suppliers.”
Figure 9 summarizes the structure of the extended model. Note that when nE = 1 and nC = 0, this18

model is reduced to the model in the subsection 3.2 and when nE = 0 and nC = 1, it is also reduced
to the model with the uniform pricing monopolist.
[Figure 9 about here]
The technologies of the downstream firms are modified as follows. The downstream firms require
one unit of nC + nE kinds of input, which is commonly used for goods A and B, to produce one unitof final goods. Therefore the per unit production cost of good k of downstream firm i, cik , is denoted
by
cik =

nC
∑

wcs +

s=1

nE
∑

wih + ziks ∈ {1, ..., nC }, h ∈ {1, ..., nE },

(26)

h=1

where wcs is the input price of common supplier s, wih is that of downstream firm i’s exclusive supplierh and zik is an eﬃciency measure.
The timing of the game is as follows. In the first stage, the common suppliers and the exclusive
suppliers simultaneously decide their input prices. In the second stage, observing the input prices, thedownstream firms engage in Cournot competition.
The profits of the common supplier s and the exclusive supplier ih are respectively
πUcs = wcs (Q1 + Q2 ),πUih = wih Qi .

(27)

In the first stage, each upstream firm maximizes its own profit above. After some calculations, theequilibrium input prices are obtained as
4v − z1B − z2B,2(2nC + nE + 2)(28)

2(3nE + 2)v − (3nC + 3nE + 4)ziB + (3nC + 2)z jB.
2(3nE + 2)(2nC + nE + 2)

(29)

wcs =
and
wih =

Substituting the input prices into the outcome in the second stage, we can obtain the equilibrium profits of the downstream firms. Figures 10 and 11 provide some numerical examples of the relationshipbetween the profits of the downstream firms and z2B . As the figures indicate, when z2B is suﬃciently
low, the technology transfer benefits both firms. When z2B is suﬃciently high, the profit of the inefficient firm exceeds that of the eﬃcient one. We conclude that the results in the exclusive supplierscase also hold in the industry with more complex vertical relationship.
[Figures 10 and 11 about here]
19

5Concluding remarks

This paper explores the impact of competition between multiproduct firms in vertical relationships
on horizontal relationships: technology transfers between competitors and a diﬀerence in firms’ performances. To this end, we develop a multiproduct Cournot model with a vertical structure andexamine incentive for the technology transfer without any payment. We find that when the upstream
market consists of exclusive suppliers or many suppliers, the technology transfer may benefit bothdownstream firms. Such technology transfer enhances both consumer surplus and social welfare. In
addition, an ineﬃcient downstream firm may outperform its eﬃcient rival.
Since cost structures of MPFs influence their output portfolios, an MPF with advanced technology
will have incentive to utilize the technology transfer so as to modify the output portfolio in its favor.
For the recipient MPF, although such technology transfer shifts its output from more profitable good
to less profitable one, the profit may indeed increase. This is because the positive eﬀect from the eﬃciency improvement of the recipient product can dominate the negative eﬀects from the modificationon the output portfolio and the increase in its input price.
The policy implications of our result are as follows. FTC or DOJ recognize that consumers may
benefit from competitor collaborations in various ways. However, the welfare eﬀect of competitor
collaborations such as joint production is still controversial. Creane and Konishi (2009), for example,
show that the joint production, which includes technology transfer, also has a positive eﬀect on social
welfare while the market price rises up. In general, cooperative agreements that tend to raise prices
or to reduce output are challenged by the agencies as per se illegal, while agreements not challenged
as per se illegal are analyzed as the rule of reason to determine their overall competitive eﬀects.
Therefore, based on Creane and Konishi’s (2009) result, we can not conclude joint productions should
be promoted. In contrast to their research, the joint production in our model certainly raises not only
consumer surplus but also social welfare. Our result may justify some approvals for joint production
agreements by antitrust agencies.
The overall point that we want to emphasize is that making an analysis of MPFs suggests a numberof possibly important and certainly interesting economic consequences. We believe that this paper
provides a new insight into competition among MPFs.

20

Appendix A: The equilibrium when upstream market with monopolist
A.1 The equilibrium when monopolist employing uniform pricing
The first-order conditions is
∑ ∑
dπ M
−4
=w
+
qik = 0.
dw
3(1 + γ) i∈{1,2} k∈{A,B}(30)

The equilibrium input price is obtained as
1
w MU = (4v − z1B − z2B ).
8(31)

The equilibrium quantity and price are obtained as
qikMU =

4(1 − γ)v − (15 + γ)zik + (9 − γ)z jk + (1 + 15γ)zil − (9γ − 1)z jl
24(1 − γ2 )(32)

1(8v + 3z1k + 3z2k − z1l − z2l ).
12

(33)

and
pkMU =

This leads to the equilibrium profits:
MU
πDi
=

(4v + ziB + z jB )2 − 2γ(4v + ziB + z jB )(4v − 15ziB + 9z jB ) + (4v − 15ziB + 9z jB )2,
9 · 82 (1 − γ2 )
π MU
M =

(4v − z1B − z2B )2.
48(1 + γ)

(34)

(35)

The conditions that the interior solutions indeed exist, qikMU > 0, areγ<

4v − 15zik + 9z jk + zil + z jl
≡ γ̄ikMU .
4v − 15zil + 9z jl + zik + z jk

(36)

We also have to derive the condition the upstream firm has no incentive for changing its equilibrium
input price, and the condition is γ < γ̄dMU (see Appendix B). Hence, the condition for the equilibriumis γ < min{γ̄ikMU , γ̄dMU }.

A.2 The equilibrium when monopolist employing discriminatory pricing
The first-order conditions are
∂π M
−4
2
= wi
+ Qi (wi , w j ) + w j= 0.
∂wi
3(1 + γ)3(1 + γ)21(37)

The equilibrium input prices are
1
wiMD = (2v − ziB ).
4(38)

The equilibrium quantities and prices are
qikMD =

2(1 − γ)v − 2(3 + γ)zik + (3 + γ)z jk + 2(1 + 3γ)zil − (1 + 3γ)z jl
12(1 − γ2 )(39)

1(8v + 3z1k + 3z2k − z1l − z2l ).
12

(40)

and
pkMD =

This leads to the equilibrium profits:
MD
πDi
=

(2v + 2ziB − z jB )2 − 2γ(2v + 2ziB − z jB )(2v − 6ziB + 3z jB ) + (2v − 6ziB + 3z jB )2,
9 · 42 (1 − γ2 )

(41)

and
π MD
M =

(2v − z1B )2 + (2v − z2B )2 + (z2B − z1B )2.
24(1 + γ)(42)

The conditions that the interior solutions indeed exist, qikMD > 0, areγ<

2v − 6zik + 3z jk + 2zil − z jl
≡ γ̄ikMD .
2v − 6zil + 3z jl + 2zik − z jk

(43)

We also have to derive the condition that the upstream firm has no incentive for deviation to the otherprice (see Appendix B). Hence, the condition for the equilibrium is γ < min{γ̄ikMD , γ̄dMD }.

Appendix B: Conditions that upstream firms do not deviate
So far, we assume that the upstream firms determine their input prices such that the downstream
firm can produce both of the final goods. Since the downstream firms are MPFs, their input demand
functions are kinked. The upstream firms under the kinked input demand may find the deviation
beyond the kinked point profitable. In this section, we derive the conditions that such deviations are
unprofitable for them.
In the case of exclusive suppliers, given the input price of U1, U2 faces the following problem,
max w2 q2A |w1 =wE s.t q2B |w1 =wE ≤ 0.25
w2

25

1

1

We check whether U2 deviate or not, because U2 tends to deviate than U1.

22

The problem reflects the possibility that U2 may find another price profitable by giving up the inputE
demand from the non-core product. Define πU2d
as the solution above, and therefore the no deviation
E > πE , orcondition for the exclusive supplier is πU2
U2d

γ<

√
320v(5v + 2z1B − 7z2B ) + 289z21B − 1348z1B z2B + 1684z22B − 120 2(2z2B − z1B )(10v + 2z1B − 7z2B )80v(20v − 7z1B + 2z2B ) − 401z21B + 1772z1B z2B − 1796z22B≡ γ̄dE .
(44)

Similarly, the no deviation conditions for monopolist employing uniform pricing and discriminating
pricing are, respectively,√
12v(v + z1B − 2z2B ) + (7z2B − 5z1B )(z1B + z2B ) − 2 3(2z2B − z1B )(4v − z1B − z2B )γ<≡ γ̄dMU (45)12v2 − (z1B + z2B )2
and
γ<

√
v(20v − 34z2B + 14z1B ) + (27z22B − 21z1B z2B + 3z21B ) − 2 69(2v − z2B )(2z2B − z1B )v(20v + 22z2B − 14z1B ) − (25z22B − 13z1B z2B + z21B )≡ γ̄dMD .
(46)

Appendix C: Proofs
Proof of Proposition 1:
Diﬀerentiating πCD1 with respect to z2B , we havedπCD1
v − 2z1B + z2B
<0⇔γ>
.
dz2B
v
This violates the interior solution condition, q1B > 0.
Proof of Proposition 2:
We first show min{γ̄C1A , γ̄C1B , γ̄C2A , γ̄C2B } = γ̄C2B to identify the condition for interior solution. Define
NiB ≡ v − 2ziB + z jB . Note that N1B > N2B , and then γ̄C2A > γ̄C1A and > γ̄C1B > γ̄C2B . Comparing γ̄C1A andγ̄C2B , we have γ̄C1A > γ̄C2B . Therefore the interior solution conditions are satisfied if and only if γ < γ̄C2B
holds.
The diﬀerence between the profits of the downstream firms is as follows.
πCD1 − πCD2 =(z2B − z1B )(N1B + N2B − 2vγ).
3(1 − γ2 )23

We can thus prove the proposition if N1B + N2B − 2vγ > 0. Since the expression is decreasing in γ,
it suﬃces to show the above inequality at γ = γ̄C2B . Substituting γ = γ̄C2B , we have 3(z2B − z1B ) > 0.
This shows that for any γ < γ̄C2B , πCD1 > πCD2 .
Proof of Proposition 3:
E , γ̄ E , γ̄ E , γ̄ E } = γ̄ E . Define M ≡ 20v + 16z − 11z , M ≡ 20v − 44z +We first show min{γ̄1A
iA
iB
jB
iB
iB
1B 2A 2B
2B
E > γ̄ E and γ̄ E > γ̄ E . Comparing γ̄ E
19z jB . Note that M2A > M1A and M1B > M2B , and then γ̄2A
1A
1B
2B
1A
E , we also have γ̄ E > γ̄ E . Therefore the interior solution conditions are satisfied if and only ifand γ̄2B
1A
2B
E holds. A few lines of computations establish that γ̄ E > γ̄ E . Thus it is suﬃcient to show theγ < γ̄2B
2B
dproposition for γ < γ̄dE .
Diﬀerentiating πED1 with respect to z2B , we havedπED180v − 506z1B + 241z2B<0⇔ γ>≡ γE .
π1dz2B
80v + 394z1B − 209z2B
Diﬀerentiating πED2 with respect to z2B , we havedπED2140v + 253z1B − 548z2BE<0⇔ γ<≡ γ̄π2.
dz2B
140v − 197z1B + 352z2B
E . A few lines of computations establish that
We compare γ E and γ̄π2
π1

E
γ̄π2
> γE ⇔ v >π1

z2B (16z1B − 11z2B )≡ v̂E .
10(6z1B − 5z2B )E , and thus v > v̂E is the condition
After some calculations, we can show that γ̄dE is greater than γ̄π2

for the proposition.
Proof of Proposition 4:
We first show that if the technology transfer occurs, then the consumer surplus increases. The condition that the technology transfer enhances the consumer surplus is
dCS E
16v − 13(z1B + z2B )<0⇔γ<≡ γ̄CS .
dz2B
16v + 5(z1B + z2B )E ). Hence, we
Proposition 3 shows that the technology transfer can occur if and only if γ ∈ (γ E , γ̄π2
π1E . Define T ≡ 140v + 253z
will show γ̄CS > γ̄π2
1
1B − 548z2B and T 2 ≡ 140v − 197z1B + 352z2B . We

24

have
E
γ̄CS − γ̄π2

=
=
>
=

108(110vz2B − 90vz1B − 17z22B − 5z1B z2B + 12z21B )[16v + 5(z1B + z2B )]T 2
108[90v(z2B − z1B ) + 17z2B (v − z2B ) + 3vz2B − 5z1B z2B + 12z21B ][16v + 5(z1B + z2B )]T 2
108[90v(z2B − z1B ) + 17z2B (v − z2B ) + 3z22B − 5z1B z2B + 12z21B ][16v + 5(z1B + z2B )]T 2
2
108[90v(z2B − z1B ) + 17z2B (v − z2B ) + 3(z2B − 65 z1B )2 + 119
12 z1B ]> 0.
[16v + 5(z1B + z2B )]T 2

We next show that if the technology transfer occurs, then social welfare increases. The condition
that the technology transfer enhances social welfare is
280v + 407z1B − 727z2B
dW E
<0⇔γ<
≡ γ̄W .
dz2B
280v − 223z1B + 263z2B
E .
Hence we will show γ̄W > γ̄π2E
γ̄W − γ̄π2==(280v + 407z1B − 727z2B )T 2 − (280v − 223z1B + 263z2B )T 1
(280v − 223z1B + 263z2B )T 2
540[(z2B − z1B )(70v − 67z2B ) + 140z2B (v − z2B ) + z1B (114z2B − 44z1B )]> 0,(280v − 223z1B + 263z2B )T 2

which proves the proposition.
Proof of Proposition 5:
Comparing the profits of the downstream firms, we have
πED2 > πED1 ⇔ γ >16v − 19(z1B + z2B )≡ γE .
16v + 11(z1B + z2B )Now we show that there exists γ satisfying both the above condition and the interior solution condition.
√√(z1B + z2B )[(43 − 24 2)z2B − (11 − 9γ̄dE > γ E ⇔ v >√82z1B − (60 2 − 46)z2B
This yields the condition for the proposition.

252)z1B ]

≡ vE .

Proof of Proposition 6:
The proof follows the lines of the proof of Proposition 1. Define E A ≡ 4v + z1B + z2B , EiB ≡MU , γ̄ MU , γ̄ MU , γ̄ MU } = γ̄ MU . Note that E
4v − 15ziB + 9z jB . We first show min{γ̄1A
1B > E 2B , and
1B
2A
2B
2B
MU > γ̄ MU and γ̄ MU > γ̄ MU . We then compare γ̄ MU and γ̄ MU ,therefore γ̄2A
1A
1B
2B
1A
2B
MU
MU
γ̄1A
− γ̄2B=(E A − E2B )E A − (E1B − E A )E2B.
E A E1B

MU > γ̄ MU . Therefore the interior solution conditions
Since E A > E2B and E A −E2B > E1B −E A , thus γ̄1A
2B
MU holds. (since we can confirm γ̄ MU > γ̄ MU , it is suﬃcient to show
are satisfied if and only if γ < γ̄2B
2B
d
MU .)
the proposition for γ < γ̄2B
MU with respect to z , we haveDiﬀerentiating πD1
2B
MU
dπD1
E A + 9E1B
<0⇔γ>
≡ γ MU .
π1
dz2B
E1B + 9E A

We now show that the above condition violates the interior solution condition.
MU
γ MU − γ̄2B=
π1

9E A (E1B − E2B ) + E A2 − E1B E2B> 0,
E A (E1B + 9E A )MU > γ̄ MU implies E 2 − E E
which proves the proposition. (γ̄1A
1B 2B > 0)A
2B

Proof of Proposition 7:
The proof follows the lines of the proof of Proposition 2. Comparing the profits of the downstream
firms, we have
MU
MU
πD1
− πD2
=

(E1B − E2B )(E1B + E2B − 2γE A ).
9 · 82 (1 − γ2 )We can prove the proposition if E1B + E2B − 2γE A > 0. Since the expression is decreasing in γ, it
MU . Substituting γ = γ̄ MU , we have E
suﬃces to show this at γ = γ̄2B
1B − E 2B > 0. This shows that for
2B
MU , π MU > π MU .
any γ < γ̄2B
D1
D2

Proof of Proposition 8:
MD with respect to z , we
The proof follows the lines of the proof of Proposition 1. Diﬀerentiating πD1
2B

have
MD
dπD1
2v − 10z1B + 5z2B
<0⇔γ>
≡ γ MD .
π1
dz2B
2v + 6z1B − 3z2B

26

MD with respect to z , we haveDiﬀerentiating πD2
2B
MD
dπD2
2v + 5z1B − 10z2BMD<0⇔γ<≡ γ̄π2.
dz2B
2v − 3z1B + 6z2B

Now we show there does not exist γ satisfying the above conditions. Since 2v − 10z1B + 5z2B >MD , which proves the
2v + 5z1B − 10z2B and 2v − 3z1B + 6z2B > 2v + 6z1B − 3z2B , thus γ MD > γ̄π2
π1proposition.
Proof of Proposition 9:
The proof follows the lines of the proof of Proposition 5. Define HiA ≡ 2v + 2ziB − z jB , HiB ≡MD , γ̄ MD , γ̄ MD , γ̄ MD } = γ̄ MD . Note that H
2v − 6ziB + 3z jB . We first show min{γ̄1A
2A > H1A and
1B
2A
2B
2B
MD > γ̄ MD and γ̄ MD > γ̄ MD . We then compare γ̄ MD and γ̄ MD ,H1B > H2B , and thus γ̄2A
1A
1B
2B
1A
2B
MD
MD
γ̄1A
− γ̄2B=(H1A − H2B )H2A − (H1B − H2A )H2B.
H2A H1B

MD > γ̄ MD . Therefore the interior solution conditions
Since H2A > H2B and H1A − H2B > H1B − H2A , γ̄1A
2B
MD holds.
are satisfied if and only if γ < γ̄2B
MD ≥γ̄ MD , which implies the no deviation condition assures the interior solution.
We next show γ̄2Bd√√√2(z2B − z1B )(2v − z2B )[2( 69 − 3)v + (9 − 69)z1B − 2(12 − 69)z2B ]
MD
MD
> 0.
γ̄2B − γ̄d =H2A [v(20v + 22z2B − 14z1B ) − (25z22B − 13z1B z2B + z21B )]Comparing the profits of the downstream firms, we have
MD
MD
πD2
> πD1
⇔γ>

4v − 5z1B − 5z2B
≡ γ MD4v + 3z1B + 3z2B

Finally, we show there exists γ satisfying the above condition and the no deviation condition. With
some algebra, we obtain the condition for the proposition,√√[2(17 − 69)z2B − (5 + 69)z1B ](z1B + z2B )MDMD≡ v MD .
γ̄d > γ⇔v>√
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30

Upstream

Upstream

Firm, U1

Firm, U2

W₁

W₂

Dowmstream

Dowmstream

Firm, D1

Firm, D2

q1A q1Bq2A q2B

Consumer
Upstream
Firm, M

(W₁)

W

(W₂)

Dowmstream

Dowmstream

Firm, D1

Firm, D2

q1A

q1B

q2A q2B

Consumer
Figure 1: The structure of the model.
Top: exclusive suppliers
Bottom: monopolist employing uniform (discriminatory) pricing
31

ΠDi
0.0564

Inefficient firm

0.05630.0562
0.0561

Efficient firm
0.05600.05590.0558

0.10

0.11

0.12

0.13

0.14

0.15

z2 B

Figure 2: The relations between profits and z2B (Exclusive suppliers case).
Solid (dotted) line shows the profit of the (in)eﬃcient firm
[ v = 1, z1B = 0.1, γ = 0.64]
Γ
0.75

0.70

0.65

0.60

0.55

0.100

0.105

0.110

0.115

0.120

z2 B

Figure 3: The condition under which the technology transfer raises the downstream firms’ profits(Exclusive suppliers case).
[ v = 1, z1B = 0.1]
32

ΠDi

0.060

0.058

0.056

0.054

Γ
0.55

0.60

0.65

0.70

0.75

Figure 4: The relations between profits and γ (Exclusive suppliers case).
[ v = 1, z1B = 0.1, z2B = 0.12]

Γ

0.80

0.75

0.70

0.65

0.60

0.55
0.12

0.14

0.16

0.18

0.20

z2 B

Figure 5: The condition under which the ineﬃcient firm earns more profit (Exclusive suppliers case).
[ v = 1, z1B = 0.1]

33

ΠDi
0.032220.032200.032180.032160.032140.03212

0.100

0.105

0.110

0.115

z2 B

Figure 6: The relations between profits and z2B (Discriminatory monopolist case).
[ v = 1, z1B = 0.1, γ = 0.64]

ΠDi
0.03280.03260.03240.03220.03200.0318
Γ
0.60

0.61

0.62

0.63

0.64

0.65

0.66

0.67

Figure 7: The relations between profits and γ (Discriminatory monopolist case).
[ v = 1, z1B = 0.1, z2B = 0.11]

34

Γ
0.72

0.70

0.68

0.66

0.64

0.62
0.105

0.110

0.115

0.120

z2 B

Figure 8: The condition under which the ineﬃcient firm earns more profit (Discriminatory monopolistcase).
[ v = 1, z1B = 0.1]

U1nE

U11

UCnC

UC1

wC1

U21

U2nE

wCnC

w11

w2nE

D1
q1A

D2

q1B

q2A

q2B

Consumer
Figure 9: The structure of the extended model (Many suppliers case).
35

ΠDi
0.01930

0.019290.019280.019270.01926

0.01925
0.105

0.110

0.115

0.120

0.125

z2 B

Figure 10: The relations between profits and z2B (Many suppliers case).
[ v = 1, z1B = 0.1, γ = 0.2, nC = 1, nE = 2]

ΠDi
3.2134
3.2132
3.21303.21283.2126
3.2124

0.12

0.14

0.16

0.18

Figure 11: The relations between profits and z2B (Many suppliers case).
[ v = 100, z1B = 0.1, γ = 0.2, nC = 8, nE = 30]

36

0.20

z2 B