EFFECTS OF INTEGRATION,
TYING AND UPSTREAM MARKET STRUCTURES(Revised Edition)

Chuan Yangand

Yasuo Kawashima

Faculty of Economics
Chuo University
November, 2010

Abstract
This paper shows that vertical integration enables an upstream monopolist to exploit
leverage in a market and to exclude a rival from a market, but ﬁrms other than a monopolist
can not foreclose a rival from a market. It will be shown when integration is permissible
and when it is not. Moreover, the model is applicable to the analysis of tying and shows
the reason why tying MS Windows to Internet Explorer browser drives rival products out of
the Japanese market and why introduction of Face Screen in Europe has reduced the market
share of Internet Explorer browser.

Acknowledgement
Financial support from Chuo University (Research Grant, 0382) and the Japan Ministry of
Education, Culture, Sports, Science and Technology (Grand-in-Aid for Scientiﬁc Research(C),20604001) is highly appreciated. We would like to thank Professors Akio Matsumoto,
Masahiro Yabuta of Chuo University for their suggestions and Ferenc Szidarowszky of
University of Arizona for his kind advices.
JEL classiﬁcation: L2, L4, L5
Keywords: vertical integration, tying, cost eﬀect, eﬃciency eﬀect, market foreclosure.
Address: 742-1, Higashinakano, Hachioji-shi, Tokyo 192-0393,Japan
fax:+81 426 74 3425
e-mail: yasuokaw@tamacc.chuo-u.ac.jp

1

Introduction

Market foreclosure has been a controversial issue in U.S. antitrust law and industrial organization. Vertical integration is one of typical means to achieve market foreclosure. There is the fearthat vertical integration makes a market ineﬃcient by serving to reduce competition throughmarket foreclosure. Thus, it is crucially important and controversial not only for the ﬁrm’s
management strategies, but also for courts and regulators to determine whether the merger iscompetition-neutral or not. On the other hand, if it is competitive-neutral or improves market
eﬃciencies, the integration is likely to lower market prices through eﬃciency gains of the merger.
Thus, there have been two strands of thought about eﬀects of the vertical integration; oneis that it is competitively neutral or promotes market eﬃciencies through its eﬃciency gains;for example, see Bork (1978) and Chen (2001). Especially, Chen (2001) makes an important
contribution on the subject; he develops an equilibrium theory of vertical integration and showsthat the merger lowers downstream prices and that it can even if market foreclosure arises.
The other stand is that it has anticompetitive eﬀect on a market; for example, see Salinger(1988,1991), Ordover, Saloner and Salop (1990), Hart and Tirole (1990), Ma (1997), Riordan(1998), Choi and Yi (2000), Church and Gandale (2000). However, Chen and Riordan (2007)
show that the vertical integration and the exclusive contract lead to market foreclosure. If
the integration has anticompetitive eﬀect, market price is enhanced because it confers moremonopolistic power on the integrated ﬁrm.
In this paper, the upstream market structures and the productivity of downstream ﬁrms
which have been ignored will play crucial roles in our analysis of vertical integration. We
shall argue that vertical integration beneﬁts consumers and that the foreclosure eﬀects of theintegration depend crucially upon upstream market structures; an upstream monopolist can
exploit leverage through the integration in a second market, but an upstream duopolist can not.
It is interesting to note that our model can apply to the analysis of tying and that tying by amonopolist can exclude a rival from a market. Moreover, the model can explain why “keiretsu”
emerges in games in which there are multiple upstream ﬁrms.
The model presented in this paper will mainly examine eﬀects of the vertical integrationin vertically related markets. We model these markets as a two-stage game: the ﬁrst stage
corresponds to the upstream market and the second stage to the downstream market. Given the
total demand for their outputs, the downstream ﬁrms compete to supply it. Once equilibrium
outputs of downstream ﬁrms are determined, demand for inputs of the upstream ﬁrms is derived

1

from equilibrium outputs of the downstream ﬁrms. With known total demand for inputs by thedownstream ﬁrms, the upstream ﬁrms compete to supply it. Note that the present model does
not consider incentives for the merger.
The characteristics of the integration eﬀect are diﬀerent from others because the integration
has two distinct kinds of eﬀects. One is cost-reducing eﬀect (hereafter, cost eﬀect) which reduces
the price because the integrated ﬁrm can get access to inputs (or throughputs) at the marginal
costs. The other is eﬃciency eﬀect, which is due to the fact that the downstream division of
the integrated ﬁrm may have superior or inferior production facilities relative to a rival. For
example, if the downstream division holds superior facilities, eﬃciency gains have a tendency
to lower market price, but inferior eﬃciency has an opposite eﬀect. Chen (2001) also identiﬁes
two eﬀects of the integration: the eﬃciency and the collusive eﬀects. The former is our costeﬀect and the latter, which is diﬀerent from our eﬃciency eﬀect, comes soley from the fact that
downstream products are diﬀerentiated.
The model will examine two distinct kinds of games. The ﬁrst is called the upstream
monopoly game (hereafter, monopoly game) in which there is an upstream monopolist and two
downstream ﬁrms. The other is called the upstream duopoly game (hereafter, duopoly game) inwhich there are two ﬁrms both operating in the upstream and the downstream market. It will
be shown that an upstream monopolist can foreclose a rival ﬁrm by the vertical integration per
se even if there are no eﬃciency gains through the integration. In other words, the cost eﬀect
is so strong in the monopoly game that a rival has to exit from a market even if the eﬃciency
eﬀect does not work against the rival.
On the other hand, when we consider the duopoly game, the integrated ﬁrm can not exclude arival from the downstream market even if an eﬃcient upstream duopolist merges with an eﬃcient
downstream ﬁrm; the cost eﬀect is moderate enough for a competitor to stay in the downstream
market even if the eﬃciency eﬀect works to the advantage of the integrated ﬁrm. Although
the vertical integration enables the upstream monopolist to exploit leverage too eﬀectively forthe rival to stay in the downstream market, the duopolist fails. Thus, the strength of the costeﬀect depends upon upstream market structure. These results are in stark contrast with thosein Chen (2001) and Chen and Riordan (2007) in which the presence of an upstream rival does
not make any diﬀerence to their analysis.
Moreover, our model shows that there are no trade-oﬀs between competitiveness and market
eﬃciency in both games. Even if the merger is competitively neutral in the sense that it does not
result in foreclosing rival ﬁrms, market eﬃciencies are enhanced because post-merger equilibrium2

price goes down.1 Finally, it is of some interest to note that the merged duopoly maximizes
proﬁts by not supplying inputs to the upstream market.2 This may correspond to “keiretu” inJapanese industries.
As pointed out by Scherer (1980), the analysis of the vertical integration can be applied
to that of tied sales because the tied sales is a special type of the vertical merger in verticallyrelated markets. It will be shown that tying a product of an upstream monopolist with productsof a downstream ﬁrm excludes a downstream rival from the market.3 Considering a real case,the tying of MS Windows to Internet Explorer browser (henceforth, IE) has enabled Microsoft,which is a de facto upstream monopolist of computer OS, to foreclose Netscape’s Navigator web
browser. Tied sales have been allowed in Japan until now, and hence IE has been able to enjoy
an almost 100 percent market share. Recently, the introduction of a Choice Screen in Europe
has caused the market share of IE to shrink, and other major browsers to enjoy gains of market
shares in 2009. 4 However, it will be shown that tying a product of one of upstream duopolists
with products of a downstream ﬁrm does not foreclose a rival from the market. For example,
tying a Toyota car with an air-conditioner, which is produced by its subcontractor, has not
excluded rival ﬁrms from this Japanese auto market. Although the details of these examples
diﬀer from our model, they illustrate the empirical relevance of our arguments.
Our paper is organized as follows. Section 2 sets forth the eﬀects of the integration by an
upstream monopolist. The integration by the monopolist causes market foreclosure even if themonopolist has the same production facilities. This in turn enables us to show that tied sales
result in excluding un-tied products. This explains why Microsoft excludes Netscape Navigator
web browser and the IE has enjoyed a 100% share in the Japanese market. Section 3 explores
the duopoly game and shows that the integration does not exclude a rival ﬁrm from the market.
It will be shown that the upstream division does not supply inputs to the upstream market.
This may explain “keiretu” . We end with our conclusions and a discussion of limitations of themodel in Section 4.

1

These observations are consistent with what Hortacsu and Syverson (2007) observed in their empirical analysis

of cements and ready-mixed industries in the US.
2
Salinger (1988) also reached the same conclusions under some assumptions.
3
A contribution to the leverage theory of tying was made in Whinston (1990). His model, however, does notenable us to draw up appropriate guidelines for antitrust and regulatory rules.
4
For detail, see Net Applications’ Annual Industry Report.

3

2

The basic Ingredients

Consider a vertically related market in which two ﬁrms 1 and 2 supply a consumer good. The
consumer good is produced by using inputs which are provided by two ﬁrms, ﬁrms 3 and 4. The
former market, supplied by ﬁrms 1 and 2, is called the downstream market and the latter the
upstream market. This game is called a duopoly game and is a benchmark game against which
other games are evaluated. The demand for the consumer good is given by
P = a − x = a − (x1 + x2 ),(1)

where P represents the price of consumer good, xi the quantity of output by ﬁrm i, i = 1, 2,
and a a ﬁxed constant.

To simplify our analysis, assume that αi units of inputs are translated into a unit of output(or a consumer good) by ﬁrm i, i = 1, 2. It follows thatxi =

1yi , i = 1, 2,αi(2)where yi is the quantity of inputs for ﬁrm i. This is the production function of ﬁrms 1 and 2.
To proceed with our analysis, it will be assumed that productivity of ﬁrm 2 is higher than thatof ﬁrm 1, and that the diﬀerence between them is not very large. Formally, this assumption canbe expressed asAssumption 1. 1 < α = α1 /α2 ≤ 2,
Together with this, it is also assumed thatci = αi p,

i = 1, 2,(3)where ci stands for the unit cost of the consumer good produced by ﬁrms 1 and 2, and p for the
price of inputs. This means that ﬁrm i using αi units of inputs produces one unit of a consumer
good at a cost αi p.
To simplify our analysis and to ensure positive outputs of ﬁrms in various types of games,
assume that the marginal costs (β3 and β4 ) of ﬁrms 3 and 4 are positive and that0 < β3 ≤ β4 .
To proceed with our analysis, the following assumption is made:4(4)

Assumption 2. a ≥ 4α1 β4 .
Several interesting integrated market structures emerge depending upon the number of independent ﬁrms in upstream and downstream markets. In what follows, consider ﬁrst a gamein which there is a monopolist in the upstream market and two independent downstream ﬁrms.
Assume that outputs, which is henceforth called inputs, produced by the monopolist, is an essential input without which downstream ﬁrms can not produce outputs. Words “inputs” and“outputs” are named from the viewpoint of downstream ﬁrms. Thus, this game is the monopoly
game, which is depicted in Fig. 1. When the upstream monopolist mergers with one of the two
downstream ﬁrms, one independent ﬁrm is left in the downstream market. We will now analyze
when the independent ﬁrm can be viable and when it exits.
Second, consider a one-sided vertical integration in a duopoly game in which there are two
independent ﬁrms in the upstream and also in the downstream market. One of the upstreamﬁrms merges with one of the downstream ﬁrms, and the other ﬁrms remain independent. The
problem examined will be whether the independent ﬁrms can stay in the market or not in the
post-merger game. Comparison of the pre- and the post-merger game enables us to explore
eﬀects of vertical integration in vertically related markets. It is interesting to know whether
the integration in these games forecloses an independent ﬁrm in the downstream market or not.
These comparisons reveal the diﬀerences in the eﬀects of the integration in diﬀerent market
structures. These games will be examined in the next section.
In this section, we study ﬁrst the monopoly game. The upstream monopolist (or ﬁrm 3)
merges with downstream ﬁrm 1 and downstream ﬁrm 2 remains independent; see Fig. 1. Notealso that the focus is mainly on the eﬀects of vertical integration and that the motivations forthe integration are not studied in this paper.
Note also that ﬁrms 1 and 3 form the integrated ﬁrm and the marginal cost of inputs for thedownstream division of the integrated ﬁrm is β3 . Thus, the proﬁts of the two ﬁrms, integrated
ﬁrm I and independent ﬁrm 2, are given byπI= πd + πu = (P − α1 β3 )xI + α2 (p − β3 )x2 = (a − α1 β3 − (xI + x2 ))xI + α2 (p − β3 )x2 ,

π2 = (P − α2 p)x2 = (a − α2 p − (xI + x2 ))x2 ,where πd and πu stand for proﬁts in the downstream and upstream markets by the integrated
monopolist. The ﬁrst order conditions for the maximization of proﬁts are
∂πI
∂xI

=

∂πd
= a − α1 β3 − 2xI − x2 = 0,∂xI
5

Upstream monopoly ﬁrm?

Upstream market
Q
Q

Q
Q

?

Q
Q
s
Q

Cournot competition

1
Q

Q
Q

Q
Q


Q
+
s
Q







2


Consumers

Figure 1: The upstream monopoly game

∂π2
∂x2

= a − α2 p − xI − 2x2 = 0,

where xI stands for output of the monopolist and x2 for that of the independent competitor.
The Nash equilibrium outputs of these ﬁrms are expressed as
x∗I =
x∗2 =

a − 2α1 β3 + α2 p,
3a + α1 β3 − 2α2 p
.
3

(5)
(6)

Firm 2 has to purchase inputs from the integrated ﬁrm and its demand is given by α2 x∗2 . Demandfor inputs of ﬁrm 2 isα2 x∗2 = Y =α2 (a + α1 β3 − 2α2 p).
3Solving this equation for p, we have the inverse demand p for inputs:p=−3Y + α2 (a + α1 β3 ).
2α22Firm I faces demand for its products derived above. Its proﬁt is
πI= πd + πu = (P − α1 β3 )x∗I + (p − β3 )Y = (P − α1 β3 )x∗I + (=−5Y 2 + 4Y (α1 − α2 )α2 β3 + α22 (a − α1 β3 )2
.
4α22
6

−3Y + α2 (a + α1 β3 )− β3 )Y
2α22

It follows that the condition for optimality is
5Y
α1
dπI
= − 2 + (−1 +)β3 = 0.
dY
2α2
α2
The monopolist supplies inputs to ﬁrm 2, which are given byY∗ =2(α1 − α2 )α2 β3.
5Noting the inverse demand for input, price charged for this input by the supplier isp∗bm =−3Y ∗ + α2 (a + α1 β3 )5a − α1 β3 + 6α2 β3
=
.
2
10α2
2α2

(7)

It would be of some interest to check whether the equilibrium price in the upstream market ishigher than the provider’s marginal cost β3 . Subtracting β3 from p∗bm , we havep∗bm − β3 =5a − α1 β3 + 6α2 β35a − α1 β3 − 4α2 β3− β3 =
> 0,
10α2
10α2

which will be positive under our Assumptions 1 and 2. Thus, the integrated monopolist has
cost advantage over its competitor in the downstream market.
Thus, we can now summarize our analysis above as:
Lemma 1. Equilibrium prices in the upstream and downstream markets where there is an upstream monopolist are given byp∗bm =PBM=

5a − α1 β3 + 6α2 β3.
10α2
5a + β3 (3α1 + 2α2 ),10where PBM is higher than α2 p∗bm . Moreover, output of the independent ﬁrm is given byx∗2 =2(α1 − α2 )β3> 0.
5Proof. Substituting eq. (7) into eq. (5) and eq. (6) yields,
x∗I

=
=

a − 2α1 β3 + α2
a − 2α1 β3 + α2 p∗bm=
335a − 7α1 β3 + 2α2 β3,105a−α1 β3 +6α2 β3
10α 2

and

x∗2 =

a + α1 β3 − 2α2a + α1 β3 − 2α2 p∗bm=
3375a−α1 β3 +6α2 β3
10α 2

=

2(α1 − α2 )β3> 0,5where the inequality comes from Assumption 1 and x∗2 is equal to zero at α = α1 /α2 = 1. Then,it follows from eq. (1), x∗I and x∗2 that we have
PBM = a − x∗I − x∗2 =5a + β3 (3α1 + 2α2 )
.
10

(8)

As noted before, ﬁrm I has cost advantage over its downstream competitor. However, it wasstated in Assumption 1 that the competitor, ﬁrm 2, has more eﬃcient technology. Then, consider
whether this cost advantage is enough for ﬁrm I to drive ﬁrm 2 out of the market. Using PBM
and p∗bm derived above, we get:
PBM − α2 p∗bm =

2β3 (α1 − α2 )> 0.
5where the sign comes from Assumption 1 and the equality holds at α = 1. Thus, both the
integrated ﬁrm and the independent ﬁrm can make positive proﬁts and they can be viable.
Noting these, the model shows that the vertical merger has two distinct eﬀects on market
price: cost eﬀect and eﬃciency eﬀect. The former comes from the fact that the merged ﬁrm can
get access to inputs at lower costs, which in turn causes lower price in a downstream market. Thisis the cost eﬀect, which is due to the cancellation of the double marginalization. The eﬃciency
eﬀect is due to the fact that the merger causes market prices to go down if the downstream
division is more productive than ﬁrm 1 and causes the prices to go up otherwise. This eﬀect
works for the merged ﬁrm in the former case (α ≤ 1). If α > 1 (the merged ﬁrm has less eﬃcient
production technologies relative to ﬁrm 1), the eﬃciency eﬀect works against the merged ﬁrm.
Lemma 1 states that although the integrated ﬁrm has cost advantages over the independent,
the latter can be viable after integration in the monopoly game if the latter has superior production technologies. Consequently, the output of the independent ﬁrm is positive and the ﬁrm
can make positive proﬁts as long as the independent has more eﬃcient technologies. It follows
from Lemma 1 that the downstream rival can not be foreclosed from a market if the eﬃciency
eﬀect works against the merged ﬁrm, but that the competitor can be excluded from a market if
the eﬃciency eﬀect is inactive or works for the integrated ﬁrm.
Our conclusions are mainly dependent on Assumptions 1 and 2. If either of these two
assumptions fails, market structure changes into pure monopoly. If Assumption 1 does not hold,there does not exist competitors in the downstream market. It is interesting to note that the
integrated ﬁrm can foreclose the rival from the market, even if its downstream division has thesame productivity as the rival does.

8

If α1 = α2 = 1, one unit of essential inputs produces one unit of outputs. For example, MS
Windows and IE enable us to enjoy internet services, and the combination of MS Windows andNetscape Navigator provides us with the same services. Thus, when α1 = α2 = 1, the integration
causes the monopoly game to change into a game of tying by products of an upstream monopolist
and ﬁrm 1. Generally, tied sales emerge when α1 = α2 . In fact, Scherer (1980) pointed out thatties have similar eﬀects as vertical integration by a ﬁrm to exercise power over price of the tiedmarket. Following Scherer (1980) and the arguments above, the analysis of tie-in sales can bemade in our model in which α1 = α2 . In our model, it will be assumed away that the upstream
monopolist has incentives to engage in tying. The focus here is on what eﬀect tying has.
Proposition 1. Tying enables the integrated upstream monopolist to foreclose a competitor froma downstream market.
Proof. Noting that tying is equivalent to the condition that α1 = α2 , our proof goes as follows.
In the proof of Lemma 1, the output of ﬁrm 2 is given byx∗2 =2(α1 − α2 )β3.
5

In the present model, tie-in sales are considered as α1 = α2 . This in turn causes x∗2 to become
zero.
These results can reveal some of the important aspects of tying by an upstream monopolist
such as MS. From the viewpoint of our model, companies oﬀering application software whichenables us to enjoy internet service are sure to be foreclosed. In fact, Netscape’s Navigator web
browser was foreclosed in the U.S. and Ichitaro has been foreclosed by MS Word in Japan. As
mentioned in the Introduction, the presence of a Choice Screen that displays a list of 12 diﬀerent
web browsers has resulted in the shrinkage of MS IE dominance in 2009. More freedom of choiceof browsers has caused MS IE to lose market share. These observations are consistent with our
Proposition 1.

93

One-sided Vertical Integration in Duopoly Game

There is a long history of whether the vertical integration matters because it may cause market
foreclosure of rival ﬁrms. Two controversial issues have been examined by those concerned withantitrust proceedings and with regulation. One is whether the vertical integration is anticompetitive or not, and the other is whether it promotes market eﬃciencies. Eﬀects of one-sided
vertical integration are examined in this section.
To simplify our analysis, in what follows, consider the duopoly game. In particular, it will
be considered whether tie-in sales result in foreclosing a competitor from a market. The point
to note is that there is no upstream monopolist in the duopoly game. When we examine eﬀects
of vertical integration in the monopoly and the duopoly games, the eﬀects of vertical integrationand tie-in in a market will be examined in diﬀerent market structures.
In the duopoly game, there are two competitors in each of the two markets as shown in Fig.
2. This game is analyzed as a two-stage game with perfect and complete information. In theﬁrst stage, upstream ﬁrms supply their products to downstream ﬁrms. Demand for upstream
products is derived from outputs by downstream ﬁrms. In the second stage, downstream ﬁrms
sell their products given the demand function eq. (1) of their product. To solve our two-stage
game, we rely on backward induction. It follows that ﬁrms in a downstream market play asimultaneous-move (or a static) game given the demand function eq. (1), and maximize theirrespective proﬁts.
Using eqs. (2) and (3), proﬁts of ﬁrm i are given by
πi = (P − ci )xi = (a − αi p − (x1 + x2 ))xi,

i = 1, 2,

where ﬁxed costs are assumed away because they will not play any role in the analysis to follow.
The objective of each ﬁrm is to maximize πi , which requires∂π1= a − α1 p − 2x1 − x2 = 0
∂x1
and
∂π2
= a − α2 p − x1 − 2x2 = 0.
∂x2
Solving these equations for equilibrium outputs yields
x∗1 =
x∗2 =

a − (2α1 − α2 )p,3a + (α1 − 2α2 )p
,
3
10

3

4

?

Upstream market?

?
Cournot competition

2
Q

Q
Q

Q
Q


Q
+
s
Q







1


Consumers

Figure 2: The duopoly gamewhere the outputs of both ﬁrms are assumed positive. Total equilibrium output X ∗ isX ∗ = x∗1 + x∗2 =2a − (α1 + α2 )p.
3Demand for inputs is derived from outputs in the downstream market. In view of the production
function of ﬁrms in this market, total equilibrium demand Y for inputs is given by
Y = α1 x∗1 + α2 x∗2 =a(α1 + α2 ) + 2p(α1 α2 − α1 2 − α2 2 ).
3Solving this for p yields inverse demand for inputs, which is given byp=

3Y − a(α1 + α2 ).
2(α1 α2 − α1 2 − α2 2 )(9)Given the demand for inputs, the two ﬁrms play a simultaneous-move game in the upstream
market. Under these assumptions, upstream ﬁrms supply inputs to an upstream market. Then,Y = y3 + y4 .
It follows from this and eq. (9) that proﬁt of ﬁrm 3 isπ3 = (p − β3 )y3 = (3Y − a(α1 + α2 )3(y3 + y4 ) − a(α1 + α2 )− β3 )y3 = (− β3 )y3 .
222(α1 α2 − α1 − α2 )2(α1 α2 − α1 2 − α2 2 )11where β3 is the constant marginal cost of ﬁrm 3 and yi is the inputs of ﬁrm i, i = 3, 4. Diﬀerentiating proﬁts with respect to y3 yields∂π3
6y3 + 3y4 − a(α1 + α2 )− β3 = 0.
=∂y32(α1 α2 − α21 − α22 )Similarly, we have
π4 = (p − β4 )y4 = (3Y − a(α1 + α2 )3(y3 + y4 ) − a(α1 + α2 )− β4 )y4 = (− β4 )y4 ,222(α1 α2 − α1 − α2 )2(α1 α2 − α1 2 − α2 2 )where β4 is the constant marginal cost of ﬁrm 4. Diﬀerentiating π4 with respect to y4 andequating the derivative to zero gives
3y3 + 6y4 − a(α1 + α2 )∂π4
− β4 = 0.
=∂y42(α1 α2 − α21 − α22 )Solving these equations for y3 and y4 , we get the equilibrium values of inputs supplied by
upstream ﬁrms:
y3∗ =
y4∗ =

a(α1 + α2 ) + 2(β4 − 2β3 )(α21 − α1 α2 + α22 )9a(α1 + α2 ) + 2(β3 − 2β4 )(α21 − α1 α2 + α22 ),9which are both positive in view of Assumption 1, eq. (4) and Assumption 2. In fact, it follows
from Assumption 1, eq. (4) and Assumption 2 that the numerator of y3∗ is reduced to2(α21 − α1 α2 + α22 )(β4 − 2β3 )) + a(α1 + α2 )≥ 2(α21 − α1 α2 + α22 )(β4 − 2β3 )) + 4α1 β3 (α1 + α2 )> 2(α21 − α1 α2 + α22 )(−2β3 ) + 4α1 β3 (α1 + α2 )= β3 (−4α21 + 4α1 α2 − 4α22 + 4α21 + 4α1 α2 )= 4α22 β3 (2α − 1) > 0,
where the inequality is due to Assumption 1.
The numerator of y4∗ is reduced to2(α21 − α1 α2 + α22 )(β3 − β4 − β4 ) + a(α1 + α2 )> 2(α21 − α1 α2 + α22 )(−2β4 ) + 4α1 β4 (α1 + α2 )= 4β4 (−α21 + α1 α2 − α22 + α21 + α1 α2 )= 4β4 (2α1 α2 − α22 )= 4α22 β4 (2α − 1) > 0,12

where again the inequality is due to Assumption 1.
Then, the total demand for inputs is
Y ∗ = y3∗ + y4∗ =2(α1 α2 − α21 − α22 )(β3 + β4 ) + 2a(α1 + α2 ).
9The equilibrium price p∗N of inputs is given by substituting Y ∗ into eq. (9), and it isp∗N =a(α1 + α2 ) + 2(β3 + β4 )(α21 − α1 α2 + α22 ).
6(α21 − α1 α2 + α22 )It is easy to show that price p∗N is higher than β4 . In fact,p∗N − β4 =a(α1 + α2 ) + 2(β3 − 2β4 )(α21 − α1 α2 + α22 ),6(α21 − α1 α2 + α22 )where the denominator is positive under Assumption 1 and the numerator is positive because itis the same as that of y4∗ in the duopoly game.
Next, consider if x∗1 is positive or not at p = p∗N . Substituting p∗N into x∗1 yieldsx∗1 ==a − (2α1 − α2 )p∗Na − (2α1 − α2 )a(α1+α2 )+2(β3 +β4 )(α21 −α1 α2 +α22 )6(α21 −α1 α2 +α22 )

=
3
3
2
2
6a(α1 − α1 α2 + α2 ) − (2α1 − α2 )(a(α1 + α2 ) + 2(β3 + β4 )(α21 − α1 α2 + α22 ).
18(α21 − α1 α2 + α2 )To simplify our calculations, consider the following:
6a(α21 − α1 α2 + α22 ) − (2α1 − α2 )(a(α1 + α2 ) + 2(β3 + β4 )(α21 − α1 α2 + α22 ))= a(4α21 − 7α1 α2 + 7α22 ) − 2(2α1 − α2 )(α21 − α1 α2 + α22 )(β3 + β4 )> 4α1 β4 (4α21 − 7α1 α2 + 7α22 ) − 8α1 (α21 − α1 α2 + α22 )β4 = 4α1 β4 (2α21 − 5α1 α2 + 5α22 )= 4α1 β4 (2α2 − 5α + 5)α22 > 0,
where the last inequality is due to Assumption 1. It then follows from these calculations thatx∗1 is positive. Similar calculations will yield the result that x∗2 is positive.
In view of the demand function for products in the downstream market, it is easy to showthat the equilibrium price PN in the downstream market is
PN = a − x∗1 − x∗2 =a(7α21 − 4α1 α2 + 7α22 ) + 2(α31 + α32 )(β3 + β4 )a + (α1 − α2 )p∗N,=318(α21 − α1 α2 + α22 )which is larger than α1 p∗N . In fact, the diﬀerence is reduced toPN − α1 p∗N==a(4α21 − 7α1 α2 + 7α22 ) − 2(2α31 − 3α21 α2 + 3α1 α22 − α32 )(β3 + β4 )18(α21 − α1 α2 + α22 )a(4α2 − 7α + 7) − 2α2 (2α3 − 3α2 + 3α − 1)(β3 + β4 ).
18(α2 − α + 1)
13

(10)

It is easy to show that the denominator of the diﬀerence is positive, but the sign of the numerator
is not obvious. However, it will be shown that it is also positive. From Assumptions 1, eq. (4)and Assumption 2, the numerator is given bya(4α2 − 7α + 7) − 2α2 (2α3 − 3α2 + 3α − 1)(β3 + β4 )≥ 4α1 β4 (4α2 − 7α + 7) − 4α2 β4 (2α3 − 3α2 + 3α − 1)
≥ 4α2 β4 (4α2 − 7α + 7) − 4α2 β4 (2α3 − 3α2 + 3α − 1)
= 4α2 β4 ((4α2 − 7α + 7) − (2α3 − 3α2 + 3α − 1))= 4α2 β4 (−2α3 + 7α2 − 10α + 8) > 0,
which is positive in view of Assumption 1. Then, PN is larger than α1 p∗N . It follows from thesearguments that the independent ﬁrms can make positive proﬁts and can stay in the up- anddown-stream markets.
We can now establish:
Lemma 2. The equilibrium prices in the upstream and downstream markets are given by
p∗N

=

PN

=

a(α1 + α2 ) + 2(β3 + β4 )(α21 − α1 α2 + α22 ),6(α21 − α1 α2 + α22 )a(7α21 − 4α1 α2 + 7α22 ) + 2(α31 + α32 )(β3 + β4 ).
18(α21 − α1 α2 + α22 )Next consider the following duopoly game: If eﬃcient ﬁrm 2 integrates with upstream ﬁrm
3 as shown in Fig.2, the downstream division of the integrated ﬁrm is superior in technologies
to a downstream rival, and it can get access to inputs at lower costs than independent ﬁrm 4.
This game is a benchmark against which eﬀects of the integration are evaluated. In the previous
section, eﬀects of vertical integration were examined in the monopoly game. Now, we will focus
on eﬀects of the vertical integration on a market in the duopoly game. To examine the featuresof the integration, consider a game in which eﬃcient ﬁrm 2 merges with the more productive
ﬁrm 3. Thus, the integrated ﬁrm is the strongest both in costs and eﬃciency.
The integrated ﬁrm may be able to make proﬁts by supplying outputs produced by using
inputs not only from the upstream division, but also from an upstream market. Thus, proﬁts of
the integrated ﬁrm and the un-integrated ﬁrm are expressed asπI= (P − α2 β3 )xI + (P − α2 p)x21 + (p − β3 )y3= (a − α2 β3 − xI − x21 − x1 )xI + (a − α2 p − xI − x21 − x1 )x21 + (p − β3 )y3 ,π1 = (P − α1 p)x1 = (a − α1 p − xI − x21 − x1 )x1 ,14

where xI stands for output of the integrated ﬁrm produced by through-puts , x21 for output of
the ﬁrm by using inputs purchased in the upstream market and p for input price.
The ﬁrst order conditions for proﬁt maximization of two ﬁrms yield
∂πI
∂xI
∂πI
∂x21
∂π1
∂x1

= a − α2 β3 − 2xI − x21 − x1 − x21 = a − α2 β3 − 2xI − 2x21 − x1 = 0,
= a − α2 p − xI − 2x21 − x1 − xI = a − α2 p − 2xI − 2x21 − x1 = 0,
= a − α1 p − xI − x21 − 2x1 = 0.

Generally, β3 is not equal to p. This means that simultaneous equations above are unsolvable.
Moreover, if β3 is equal to p, the integrated ﬁrm can not decide how much inputs are to bepurchased from the upstream market and how much is to be provided by its upstream division.
This in turn means that demand for inputs is indeterminate and that vertical merger does notplay a role. Thus, it is not possible to explore games in which the integrated ﬁrm purchases
inputs in the upstream market. These games will not be considered in what follows.
Noting arguments above, the demand for inputs comes solely from ﬁrm 1. When the integrated ﬁrm and ﬁrm 1 compete in the downstream market, their proﬁts areπI= (a − α2 β3 − xI − x1 )xI + (p − β3 )y3 ,π1 = (a − α1 p − xI − x1 )x1 .
The ﬁrst order conditions for maximum proﬁts are
∂πI
∂xI
∂π1
∂x1

= a − α2 β3 − 2xI − x1 = 0,
= a − α1 p − xI − 2x1 = 0.

Solving these equations for xI and x1 , we have
x∗I =
x∗1 =

a − 2α2 β3 + α1 p,
3a + α2 β3 − 2α1 p,3which are analogous to eqs. (5) and (6). As noted above, derived demand d for inputs by ﬁrm
1 is given byd = α1 x∗1 = α1

a + α2 β3 − 2α1 p.
3Solving this for p yields inverse demand for the inputs, and is given byp=(a + α2 β3 )α1 − 3d
.
2α21
15

(11)

Given this demand for the inputs, the upstream division and independent ﬁrm 4 supply the
inputs to the downstream ﬁrms, and their proﬁts areπI= (a − α2 β3 − x∗I − x∗2 )x∗I + (p − β3 )y3 = (a − α2 β3 − x∗I − x∗2 )x∗I + ((a + α2 β3 )α1 − 3d
− β3 )y3
2α21(a + α2 β3 )α1 − 3(y3 + y4 )− β3 )y3 ,2α21(a + α2 β3 )α1 − 3d(a + α2 β3 )α1 − 3(y3 + y4 )− β4 )y4 = (− β4 )y4 ,π4 = (p − β4 )y4 = (
2
2α1
2α21
d = y3 + y4 .
= (a − α2 β3 − x∗I − x∗2 )x∗I + (Substituting x∗I , x∗2 , and p derived above into πI and π4 , the ﬁrst order condition for
maximum proﬁt yields
∂πI
∂y3

= −

5y3 + 2(y4 + α1 (α1 − α2 )β3 )≤ 0, f or y3 , y4 ≥ 0,
2α21

where the inequality is due to Assumption 1. It follows from the Kuhn-Tucker conditions that
optimal supply ŷ3 of inputs to ﬁrm 3 is equal to 0. Thus, the integrated ﬁrm alone produces
throughputs and does not supply inputs to the upstream market. It follows from these argumentsthat ﬁrm 4 is the sole supplier of inputs, which are demanded by the downstream independentﬁrm. Firm 4 maximizes its proﬁts given the derived demand for inputs. Then, we have
∂π4
∂y4

=

−6y4 + α1 (a + α2 β3 − 2α1 β4 )
= 0,
2α21

where (a + α2 β3 − 2α1 β4 ) is positive because of Assumption 2. Solving this equation for y4 , itis given byŷ4 =α1 (a + α2 β3 − 2α1 β4 )> 0,
6where the inequality is due to Assumption 2. Substituting this into eq. (11) and noting thatd = y4 , equilibrium price p̂ of inputs is determined and then substituting this p̂ into x∗I , x∗1 yields
equilibrium outputs x̂I , x̂1 of the two ﬁrms.
Now we can summarize our arguments above as:
Lemma 3. If the most eﬃcient ﬁrm 2 merges with the most productive upstream ﬁrm 3, theone-sided vertically merged ﬁrm can maximize proﬁts by not supplying inputs to an upstream
market. Equilibrium outputs, prices and inputs of ﬁrms in the post-merger game are given byŷ3 = 0,α1 (a + α2 β3 − 2α1 β4 )
,
ŷ4 =
6
16

a + α2 β3 + 2α1 β4,4α15a − 7α2 β3 + 2α1 β4,
12a + α2 β3 − 2α1 β4,65a + 5α2 β3 + 2α1 β4
.
12

p̂ =
=

x̂I

x̂1 =
P̂

=

Proof. In the post-merger game, ﬁrm 4 maximizes proﬁts by supplying products for the given
demand eq. (11) for inputs. Note also that d = y4 because the upstream division does not
supply inputs to the upstream market. Then, proﬁts of ﬁrm 4 are given by
π4 = (p − β4 )y4 = (α1 (a + α2 β3 ) − 3d
− β4 )y4 ,2α21Solving the ﬁrst order condition for y4 yieldsŷ4 =

α1 (a + α2 β3 − 2α1 β4 ).
2α21Substituting this into eq. (11), we havep̂ =a + α2 β3 + 2α1 β4.
4α1It is easy to check that p̂ > β4 under Assumption 2. In fact,p̂ − β4 =a + α2 β3 + 2α1 β4
a + α2 β3 − 2α1 β4− β4 =
> 0.
4α1
4α1

As ŷ4 is positive, proﬁts of ﬁrm 4 are positive and ﬁrm 4 can supply inputs to the upstream
market. Substituting p̂ into eqs. (5) and (6) yields
x̂I

=

x̂1 =

5a − 7α2 β3 + 2α1 β4
a − 2α2 β3 + α1 p̂
=
> 0,
3
12
a + α2 β3 − 2α1 β4
a + α2 β3 − 2α1 p̂
=
> 0.
3
6

Finally, together with eq. (1), equilibrium price in the downstream market isP̂ = a − x̂I − x̂2 =5a + 5α2 β3 + 2α1 β4.
12It is easy to show that P̂ is larger than α1 p̂. In fact, per-units proﬁts (P̂ − α1 p̂) areP̂ − α1 p̂ =5a + 5α2 β3 + 2α1 β4
a + α2 β3 − 2α1 β4
a + α2 β3 + 2α1 β4
− α1
> 0,
=
12
4α1
6

where the sign comes from Assumption 2. This means that ﬁrm 1 can reap positive proﬁts andsell outputs in the downstream market because equilibrium output x̂1 of ﬁrm 1 is positive.
17

It is of some interest to note that the vertical integration does not exclude a rival ﬁrm
from a market in spite of the fact that the integrated ﬁrm is the most competitive because thedownstream division (ﬁrm 2) is more productive in view of Assumption 1 and the upstream
division (ﬁrm 3) has advantages in costs in view of (4). Thus, these results are in a striking
contrast with the eﬀects of the vertical integration by the upstream monopolist, who can foreclosea rival from a market even if the downstream division does not have advantages in production
technologies (see, for example, Lemma 1). In other words, even if the cost eﬀect and theeﬃciency eﬀect work for the integrated ﬁrm, a downstream rival can be viable in a duopoly
game. On the other hand, the cost eﬀect is so strong in the monopoly game that a rival has toexit from the market even if the eﬃciency eﬀect is inactive. However, as was shown above, evenif both the cost and eﬃciency eﬀects do work for the duopolist, the integration does not enable
the upstream duopolist to exclude a rival from the market. Thus, the strength of the cost eﬀect
depends upon upstream market structure.
Moreover, the ﬁrm can reap the maximum proﬁts by not supplying inputs to a market becauseŷ3 = 0 in Lemma 3. As the downstream division does not purchase inputs in the upstream
market, this result is similar to the Salinger (1988) model, where the integrated ﬁrms do notparticipate in an upstream market.5 This reminds us of “ keiretu ” in Japanese manufacturingindustries. For example, Toyota has purchased inputs (or parts) solely from its subcontractors
and so has Nissan. The subcontractors of Toyota have not supplied their products to Nissan
and vice versa, though the situations have recently been changing. Our results above may besimilar to these situations. The model explains situations of “ keiretu ”, in which parties to thecontract are not allowed to trade with ﬁrms not concerned with the contract. If ﬁrm 1 integrates
with upstream ﬁrm 3, it will be shown that the integrated ﬁrm may maximize proﬁts by notsupplying inputs to the upstream market.6
There are two important controversial questions on eﬀects of the vertical merger on a market:
is it anti-competitive and does it promote market eﬃciency? The present model can answer thesequestions. We ﬁrst examine if it is anti-competitive or not. Our Lemma 3 shows that the merger
is not anti-competitive in the duopoly game because rival ﬁrms are not excluded by the merger.
However, it follows from Lemma 1 that the vertical merger enables the upstream monopolist to
5

Chen and Riordan (2007) observed that vertical integration enables the integrated firm to preempt an up-

stream independent firm.
6 ∂πI> 0 for positive y3 , y4 . If the best response functions of two firms intersect in the first quadrant, then∂y3ŷ3 > 0 and the upstream division can maximize profits by supplying inputs to the upstream market. However,calculations similar to previous ones will show that ŷ3 = 0.

18

drive a rival out of a downstream market when an integrated upstream monopoly has equally
productive facilities (i.e., α = 1). Thus, the answer to the ﬁrst issue depends crucially upon
upstream market structure. However, Chen and Riordan (2007) concluded in Prop. 2 that theanswer is independent of upstream market structure.
Second, it is possible to examine whether the one-sided vertical merger improves market
eﬃciency or not. If the vertical merger improves eﬃciency in a market, post-merger market
prices decline. Usually, it has been considered that the merger confers more monopoly power
to the integrated ﬁrm and enables the ﬁrm to control a market. Then, the merger results inaggravating market eﬃciency. However, it is not easy to show which is correct. We can be more
precise than this:
Proposition 2. One-sided vertical integration of ﬁrm 2 and 3 is neutral in competitiveness, but
it does promote market eﬃciency.
Proof. As shown in Lemma 3 and the argument above, ﬁrm 4 can reap positive proﬁts bysupplying inputs in the upstream market. Moreover, it will be shown that ﬁrm 1 can be viablein the downstream market. Proﬁts of ﬁrm 1 are given by x̂1 × (P̂ − α1 p̂), where x̂1 is positivein view of Lemma 3. Per-units proﬁts (P̂ − α1 p̂) were shown in Lemma 3 to be positive.
Thus, the independent ﬁrm can be viable in the post-merger game. The one-sided merger isneutral from the viewpoint of the competitiveness of a market.
It follows from Lemmas 2 and 3 that we can compare equilibrium prices in pre- and postmerger game:
PN − P̂

=
=
=
=
≥
≥
≥
=

a(7α21 − 4α1 α2 + 7α22 ) + 2(α31 + α32 )(β3 + β4 ) 5a + 5α2 β3 + 2α1 β4−18(α21 − α1 α2 + α22 )
12
2
2
2
2
−(α1 − 7α1 α2 + α2 )a + (α1 − α1 α2 + α2 )(−6α1 β4 − 15α2 β3 + 4(1 + α)α2 (β3 + β4 ))36(1 − α + α2 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(−6α1 β4 − 15α2 β3 + 4α2 (β3 + β4 ) + 4α1 (β3 + β4 ))36(1 − α + α2 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(−11α2 β3 + 4α1 β3 + 4α2 β4 − 2α1 β4 )36(α21 − α1 α2 + α22 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(4α1 β3 − 4α1 β3 − 7α2 β3 + 2β4 (2α2 − α1 ))36(α21 − α1 α2 + α22 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(4α1 β3 − 4α1 β3 − 7α1 β4 + 2β4 (2α2 − α1 ))36(α21 − α1 α2 + α22 )−4α1 β4 (α21 − 7α1 α2 + α22 ) − 7α2 β3 (α21 − α1 α2 + α22 )36(α21 − α1 α2 + α22 )α1 β4 (28α1 α2 − 4α21 − 4α22 − 7α21 + 7α1 α2 − 7α22 )36(α21 − α1 α2 + α22 )
19

=
=

α1 β4 (35α1 α2 − 11α21 − 11α22 )36(α21 − α1 α2 + α22 )α1 β4 (35α − 11α2 − 11)> 0,36(1 − α + α2 )where the inequality comes from Assumptions 1 and 2. The calculations above show thatequilibrium price is higher in the pre-merger game than in the post-merger game: PN > P̂ .
Equilibrium price goes down after the merger. This means that the merger promotes market
eﬃciency.
It was shown in Yang and Kawashima (2008) that in the monopoly and the duopoly games,
the vertical integration causes equilibrium downstream prices to go down. Together with this
proposition, whether the upstream market is a monopoly or not, the answer to the second
question is that the integration results in higher market eﬃciency.
Noting that tying is equivalent to α1 = α2 in this model, we can examine eﬀects of tying byusing our model. When α1 is equal to α2 , x̂2 , (p̂ − β4 ) and (P̂ − α1 p̂) are positive, and then ﬁrm
2 can reap positive proﬁts. In fact, it follows from Lemma 3 that for α1 = α2 ,x̂2 =p̂ − β4 =P̂ − α1 p̂ =a + α1 (β3 − β4 )a + α2 β3 − 2α2 β4
=
> 0,
6
6
a + α2 β3 + 2α1 β4
a + α1 β3 − 2α1 β4− β4 => 0,
4α1
4α1
a + α1 β3 − 2α1 β4> 0,6where inequalities are due to Assumption 2. Thus, tying is not anti-competitive. Our results arein a stark contrast with those for the merger in the monopoly game. In fact, as P roposition 1
has shown, the merger enables the integrated upstream monopolist to drive a rival from themarket whose downstream division has the same productivity as the independent ﬁrm. On theother hand, tied sales by ﬁrms in the duopoly game do not drive a rival from a market even if
the merged ﬁrm has an eﬃcient downstream division. Eﬀects of the merger depend upon market
structures in the upstream market.
Now we can establish:
Proposition 3. Tied sales are anti-competitive in the monopoly game, but not in the duopoly
game.
Tie-in sales are a ubiquitous business practice. For example, in the auto industry a car is soldwith an air-conditioner which is usually supplied by subcontracting ﬁrms. Thus, this may be
regarded as one example of tying. This market is featured by the fact that there is no upstream
20monopolist. As with the case of the Japanese auto industry, P roposition 2 implies that tying a
car with an air-conditioner does not foreclose a rival ﬁrm from a market. Although the details of
the two examples are diﬀerent from our theoretical model, they illustrate the empirical relevanceof our arguments.

21

4Conclusions

The present model has been applied to examine conditions under which the vertical integration
per se causes market foreclosure and when it does not. The integration has two distinct eﬀects on
a market: (1) the cost eﬀect which comes from the fact that the integration enables the integrated
ﬁrm to get access to inputs at lower costs, and (2) the eﬃciency eﬀect which comes from thefact that the integrated ﬁrm owns a downstream division. Both the cost and eﬃciency eﬀects
exert inﬂuences on market prices. If an upstream monopolist merges with an equally eﬃcient
downstream ﬁrm, the monopolist can exploit leverage in the downstream market and exclude
the rival from the market. The cost eﬀect (or leverage) is strong enough for the integrated ﬁrm
to exclude the rival from a market. On the other hand, when the merged ﬁrm has advantages incosts and production relative to an independent ﬁrm in the duopoly game, market foreclosure
does not take place because the cost eﬀect (or leverage) of the duopolist is not strong and hence
the integration does not exclude the independent from the market.
Our analysis can also shed some light on tied sales. Tied sales are a special form of the vertical
integration. Tying by the upstream monopolist excludes a rival from the downstream market. In
fact, MS is a de facto upstream monopolist in computer OS market and MS Windows has beensold with IE installed. Consequently, tying by MS has resulted in the foreclosure of Netscape’s
Navigator web browser. From the viewpoint of welfare, the vertical integration results in lower
downstream prices even if the foreclosure occurs. Although it is anti-competitive because thenumber of rivals is reduced, it nevertheless promotes market eﬃciency by lowering prices. Thus,
there is no trade-oﬀ between anti-competitiveness and market eﬃciency.
In a one-sided vertical integration game, ﬁrm 2 and 3 are merged and the upstream division does provide throughputs only to the downstream division. This is similar to “keiretu” ofJapanese industries. This is a game in which an eﬃcient upstream ﬁrm and an eﬃcient downstream ﬁrm merge. Moreover, as mentioned in Footnote 5, our result above is independent of thepair of ﬁrms which merge. Our future work will explore games in which the upstream divisionof the merged ﬁrm supplies inputs to the upstream market.
There are several limitations to our model. For example, it was assumed that the upstream
and the downstream divisions of the integrated ﬁrm have constant returns to scale. If the
upstream division exhibits increasing returns to scale, it is probable that the division can maximize proﬁts by supplying inputs to the upstream market. The model assumes a linear demand
function. This enabled us to make our model tractable. However, when we try other demand22

functions, it may be very diﬃcult to solve the two-stage games even with complete and perfect
information because it is necessary to derive an upstream demand from downstream equilibrium
outputs. It is easy to solve for downstream equilibrium, but it is hard to solve for upstream
equilibrium. Until now we have not found other demand functions which enable us to explicitly
solve the two-stage games.

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24

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25