EFFECTS OF INTEGRATION AND UPSTREAM MARKETSTRUCTURES
Chuan Yangand

Yasuo Kawashima
Faculty of Economics
Chuo University
May, 2011
Abstract
This paper shows that vertical integration can lower equilibrium prices in a downstream
market, whether an upstream market is monopolized or not. However, there is not necessarily
a trade-oﬀ between competitiveness and market eﬃciency. In fact, the integration enables
an upstream monopolist to exploit leverage in a market and to exclude a rival from thatmarket, but ﬁrms other than a monopolist can not drive a rival from a market. Thus,
upstream market structure matters. It will be shown when integration promotes eﬃciency
and when it does not, and also that situations of “keiretsu”, found in Japanese industries,
emerge.

Acknowledgement
Financial support from Chuo University (Research Grant, 0382) and the Japan Ministry of
Education, Culture, Sports, Science and Technology (Grant-in-Aid for Scientiﬁc Research(C),20604001) is highly appreciated. We would like to thank Professors Akio Matsumoto,
Toichiro Asada and Masahiro Yabuta of Chuo University for their suggestions and M. Hosoe ofKumamoto Gakuin University for their kind advice.
JEL classiﬁcation: L2, L4, L5
Keywords: vertical integration, diﬀerence in productivity, cost eﬀect, productivity eﬀect,
upstream market structure.
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

Vertical integration is a typical means to achieve market foreclosure. There is the fear, however,that vertical integration makes a market ineﬃcient by reducing competition through market
foreclosure. On the other hand, if it is competitively neutral or improves market eﬃciencies,
the integration is likely to lower market prices through eﬃciency gains of the merger. Thus, itis crucially important and controversial not only for the ﬁrm’s management strategies, but also
for courts and regulators to determine whether the merger is competition-neutral or not.
There have thus been two lines of thought about eﬀects of vertical integration; one is that
it is competitively neutral or promotes market eﬃciencies through its eﬃciency gains; see, forexample, Bork (1978) and Chen (2001). Chen (2001) in particular makes an important contribution on the subject; he develops an equilibrium theory of vertical integration and shows thatthe merger lowers downstream prices and that it can do so even if market foreclosure arises. The
other line of thouight is that vertical integration has an anticompetitive eﬀect on a market; see,for example, 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 an anticompetitive eﬀect, market price is increased because it
confers more monopolistic power on the integrated ﬁrm.
Following Hart and Tirole (1990), Chen and Riordan (2007) made major contributions on
the analysis of eﬀects of vertical integration in vertically related markets using the contract
theory. Their analysis focuses on the collusive behavior of ﬁrms in the vertically related markets
because a ﬁnal good is diﬀerentiated. Thus, downstream prices are increased by the collusionof downstream ﬁrms. By contrast, the present paper focuses on a market where a ﬁnal good
is homogeneous and then analyzes eﬀects of integration on the basis of competitive behaviorof ﬁrms without paying attention to incentives for integration. Firms compete à la Cournot
in the upstream and the downstream markets and we will show important features of this invertically related markets. The characteristics of the present model are that downstream ﬁrms
diﬀer in their productivity and that upstream market structure has an important inﬂuence on
the analysis. 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 equilibrium pricegoes down.1 Thus, our model shows that there are no trade-oﬀs between competitiveness and
1

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

of cements and ready-mixed industries in the US.

1market eﬃciency.
We model these markets as multi-stage games with perfect and complete information. Upstream ﬁrms supply inputs in the upstream market and downstream ﬁrms demand them. Downstream ﬁrm i, i = 1, 2, produces one unit of a consumer good with the use of the αi units of
inputs and sells a consumer good in the downstream market. Note that αi is a ﬁxed constantand α1 = α2 .2 Backward induction is used to analyze these games. Given the total demand fortheir outputs, the downstream ﬁrms compete to supply it. Once equilibrium outputs of downstream ﬁrms are determined, demand for inputs to be supplied by the upstream ﬁrms is derived
from equilibrium outputs of the downstream ﬁrms. With known total demand for inputs by thedownstream ﬁrms, the upstream ﬁrms compete to supply it in a market where demand for inputs is a function of input prices. Thus, equilibrium prices in the upstream and the downstream
market are determined in these markets. A detailed version of our model is shown in Figures 1
and 2 in Sections 2 and 3 respectively.
In what follows, we will distinguish two types of games; a Monopoly-Duopoly game (hereafter, MD game) and a Double-Duopoly game (hereafter, DD game). The MD game has an
upstream monopolist and two downstream ﬁrms, whereas the DD game has two independentﬁrms in both markets. We shall argue that in these games, the eﬀect of the integration depends
crucially upon the diﬀerence in productivities (α1 /α2 ) between the downstream ﬁrms. Introduction of the diﬀerence in productivity enables us to analyze some important features of verticallyrelated markets. Vertical integration has two distinct types of eﬀects on a market.3 One is acost-reducing eﬀect (hereafter, cost eﬀect) which reduces the downstream price. The other is the
productivity 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 ﬁrm. The impact of vertical
integration will be examined on the basis of these cost and productivity eﬀects. Downstream
equilibrium prices may go down and market foreclosure can arise, depending upon the relative
strength of these two eﬀects.
The foreclosure brought about by the integration depends crucially upon upstream market
structure. Although the vertical integration enables the upstream monopolist to exploit leverage
too eﬀectively for the ineﬃcient rival to stay in the downstream market, an eﬃcient competitor
can still be viable. However, the duopolist may fail to foreclose a rival ﬁrm even if the rival
2
3

Chen and Riordan (2007) assumes that αi = 1 for both downstream ﬁrms.
Chen (2001) also identiﬁes two eﬀects of the integration: the eﬃciency and the collusive eﬀects. The former

is our cost eﬀect and the latter, which is diﬀerent from the productivity eﬀect, comes solely from the fact that
downstream products are diﬀerentiated.

2

has lower-productivity facilities. Thus, the strength of the eﬀect of vertical integration depends
upon upstream market structure. Moreover, the model can investigate if the integrated upstream
duopolist can maximize proﬁts by not supplying inputs to an upstream market. The model may
also explain why “keiretsu” emerges in the DD game.4 These results obtained in this paper are
in stark contrast with those in Chen (2001) and Chen and Riordan (2007) in which the presenceof upstream rivals makes no diﬀerence to their analyses. Note that the present model does notconsider incentives for the merger.
Our paper is organized as follows. Section 2 sets forth the eﬀects of the integration in
the MD game. The question of possible foreclosure resulting from integration by a monopolist
is examined. The eﬀect of integration on the downstream prices is investigated to determine
if market eﬃciency is aﬀected. Section 3 explores the DD game to investigate when a rival
ﬁrm can continue to operate despite integration. Another facet of this investigation is whether
the upstream division of the integrated ﬁrm supplies the upstream market. These investigations
reveal the importance of upstream market structure. Our model also investigates the appearanceof “keiretu”. We end with conclusions and a discussion of limitations of the model in Section 4.

4

Salinger (1988) also reached the same conclusions under some assumptions.

32

Basic Ingredients and Upstream Monopoly

Consider a vertically related market in which two ﬁrms D1 and D2 supply a consumer good.
The consumer good is produced by using inputs which are provided by an upstream monopolist
U. The former market, supplied by D1 and D2, is called the downstream market and the latterthe upstream market which is monopolized by the monopolist U. This game is called the MDgame deﬁned above. 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 Di, i = 1, 2, and a
is a constant.
To simplify our analysis, assume that αi units of inputs are translated into a unit of output(or a consumer good) by Di, where the αi is a positive constant.5 It follows thatxi =

1yi , i = 1, 2,αi(2)where yi is the quantity of inputs for Di. These are the production functions of D1 and D2. To
proceed with our analysis, it will be assumed that productivity of D2 is higher than that of D1,
and that the ratio of their productivities is not larger than 4. Speciﬁcally, let α = α1 /α2 andAα = {α|1 < α ≤ 4}. Formally, our assumption can be expressed asAssumption 1. α ∈ Aα .
Together with this, it is also assumed thataci = αi p,

i = 1, 2,(3)where aci stands for the unit cost of the consumer good produced by Di and p for the price of
inputs. This means that Di using αi units of inputs produces one unit of the consumer good ata cost αi p.
To simplify our analysis, assume that the marginal cost c of U is a positive constant. To
proceed with our analysis, the following assumption is made:Assumption 2. a > 4α1 c.

5Words “inputs” and “outputs” are named from the viewpoint of downstream ﬁrms.

4

U bids to acquire D2J

D2 accepts U’s bidJ

J D2 rejects U’s bid
J
J
J
^
J

Pure monopoly

U bids to acquire D1J

D1 accepts U’s bid
Integrated ﬁrm I announces
prices pi for D2

J

J D1 rejects U’s bid
J
J
J
^
J

U announces prices pd
for D1 and D2

D2 will purchasefrom I

D1, D2 will purchasefrom Up?
ixI = xI (pi )x2 = x2 (pi)p?
dx1 = x1 (pd)x2 = x2 (pd)Figure 1: The Monopoly Duopoly game

Several interesting integrated market structures emerge depending upon the number of independent ﬁrms in the upstream and the downstream market. Our games structures are depicted
in Fig. 1. We consider our games with the following stages. First, U bids to acquire the eﬃcientﬁrm D2. When D2 accepts U’s bid, a vertical integration takes place and a downstream market
changes into a monopoly. In fact, it will be shown that integration of U and D2 results in apure monopoly in view of Assumptions 1 and 2. In stage 2 if D2 rejects U’s bid, U can bid toacquire the less eﬃcient ﬁrm D1. If D1 accepts U’s bid an integrated ﬁrm, called I, is formed.
This subgame is called the Partial-Monopoly game (i.e., PM game) because I is the upstream
monopolist and has a downstream competitor at the same time. The upstream division of I
announces monopoly quantities for D2. If D1 does not accept U’s bid, all ﬁrms are independent.
This subgame is the benchmark game against which the PM game is evaluated and is called the
MD game. U announces inputs prices pd .
In stage 3, the downstream division of I and D2 simultaneously choose outputs xi givena demand for consumer good, and inputs are purchased to produce their demanded outputs.

5In the MD game, D1 and D2 simultaneously purchase inputs from U and produce a consumer
good at Nash equilibrium quantities xi . In what follows, consider ﬁrst the MD game and thenthe PM game. In this section, we focus on analyzing the conditions under which the vertical
integration causes equilibrium prices in the downstream market to go down or not, together
with the question if the independent ﬁrm can be viable, and also if it exits from the market after
vertical integration by U.
After the MD game, we will consider a one-sided vertical integration in a Double-Duopoly(i.e.,
DD) game in which there are two independent ﬁrms in both, the upstream and the downstream,
markets. When one of the upstream ﬁrms merges with one of the downstream ﬁrms, the otherﬁrms remain independent. This new game is called the One-sided Integration game (hereafter,
OI game). The problem examined will be whether the independent ﬁrms can stay in the market
in the post-merger game. These games will be examined in the next section and structure forthese games will be shown in the next section.
To summarize, the following games will be considered and analyzed:(1) MD game where there are an upstream monopolist and downstream duopolists,(2) PM game where the monopolist merges with one of the downstream duopolists,(3) DD game where there are duopolists in both markets,(4) OI game where one of the upstream duopolists merges with one of the downstream duopolists.
The MD and the PM games will be examined in this section, while the DD and the OI games
will be taken up in the next section.
Consider ﬁrst the MD game in which an upstream monopolist supplies inputs to D1 andD2. Given a demand for a consumer good, the downstream ﬁrms play a Cournot game in themarket. Unit costs for them are given by aci, i = 1, 2. Their proﬁts are expressed as
πi = (P − aci)xi = (a − αi p − x1 − x2 )xi, i = 1, 2.
The objective of each ﬁrm is to maximize proﬁts, which requires
∂π1
∂x1
∂π2
∂x2

= a − α1 p − 2x1 − x2 = 0,
= a − α2 p − x1 − 2x2 = 0.

Solving these for xi yields
x̂1 =
x̂2 =

a − (2α1 − α2 )p,3a + (α1 − 2α2 )p,36where outputs of both ﬁrms are assumed positive and x̂2 > x̂1 because of Assumption 1. Thus,
total equilibrium output X̂ is given byX̂ = x̂1 + x̂∗2 =2a − (α1 + α2 )p.
3

On the other hand, total demand Ŷ for inputs by these ﬁrms is expressed asŶa − (2α1 − α2 )pa + (α1 − 2α2 )p+ α233a(α1 + α2 ) − 2(α21 − α1 α2 + α22 )p.
3= α1 x̂1 + α2 x̂2 = α1=

Now we turn to the formal statement of our analysis of the MD game:
Lemma 1. In the Monopoly-Duopoly game, the equilibrium prices in the upstream and thedownstream markets are given by the following expressions respectivelyp∗d =a(α1 + α2 ) + 2(α21 − α1 α2 + α22 )c> c,4(α21 − α1 α2 + α22 )

and
PD =

a(5α21 − 2α1 α2 + 5α22 ) + 2(α31 + α32 )c> α1 p∗d ,12(α21 − α1 α2 + α22 )where p∗d and PD stand for equilibrium prices in the upstream and the downstream markets.
Proof. The upstream monopolist U maximizes proﬁts, given a demand for inputs, which are
given byπm = (p − c)Ŷ = (p − c)a(α1 + α2 ) − 2(α21 − α1 α2 + α22 )p.
3

Then, diﬀerentiating πm with respect to p and solving for p,p∗d =a(α1 + α2 ) + 2(α21 − α1 α2 + α22 )c> c.
4(α21 − α1 α2 + α22 )Note that equilibrium outputs of D1 and D2 are given by x̂1 , x̂2 and that x̂2 > x̂1 because ofAssumption 1. Then, it is enough to show that x̂1 > 0 at p = p∗d . Substituting p∗d into outputx̂1 of D1,
x∗1 =
>
=

1 a(2α21 − 5α1 α2 + 5α22 ) − 2c(α1 − α2 )2 (2α1 − α2 )a − (2α1 − α2 )p∗b
= (
)
3
3
4(α21 − α1 α2 + α22 )1 4α1 c(2α21 − 5α1 α2 + 5α22 ) − 2c(α1 − α2 )2 (2α1 − α2 )()
34(α21 − α1 α2 + α22 )c 2α31 − 5α21 α2 + 6α1 α22 + α32(),
6α21 − α1 α2 + α227where the inequality above is due to Assumption 2 and the denominator is positive under
Assumption 1. The sign of x∗1 depends upon the numerator, which is expressed as
2α31 − 5α21 α2 + 6α1 α22 + α32 = (2α3 − 5α2 + 6α + 1)α32 > 0,
which is positive for α = α1 /α2 ∈ Aα . Then, outputs of the downstream ﬁrms are positive.
Next, we will show that p∗d is larger than c. In fact, the diﬀerence between them isp∗d − c =a(α1 + α2 ) + 2(α21 − α1 α2 + α22 )ca(α1 + α2 ) − 2(α21 − α1 α2 + α22 )c−c=.
4(α21 − α1 α2 + α22 )4(α21 − α1 α2 + α22 )This diﬀerence will be positive if the numerator is positive, because the denominator is positivefor α1 > α2 . It follows that the numerator is written asa(α1 + α2 ) − 2(α21 − α1 α2 + α22 )c > 4α1 c(α1 + α2 ) − 2(α21 − α1 α2 + α22 )c = 2c(α21 + 3α1 α2 − α22 )= 2α22 c(α2 + 3α − 1) > 0, f or α ∈ Aα ,where the inequality is due to Assumption 2. Thus, p∗d is larger than c. It follows from theseresults that U can make positive proﬁts by supplying inputs to this market. Noting that thedemand for the consumer good is given by (1), it follows from p∗d derived above thatPD =a(5α21 − 2α1 α2 + 5α22 ) + 2(α31 + α32 )c> α1 c.
12(α21 − α1 α2 + α22 )The proof will be given after we ﬁrst show that PD − α1 p∗d > 0.
Finally, consider if the downstream ﬁrms can be viable. Noting that α1 p∗d is the average costof D1, the diﬀerence between PD and α1 p∗d is given by

PD − α1 p∗d ===a(5α21 − 2α1 α2 + 5α22 ) + 2(α31 + α32 )ca(α1 + α2 ) + 2(α21 − α1 α2 + α22 )c−α112(α21 − α1 α2 + α22 )4(α21 − α1 α2 + α22 )1(a(2α21 − 5α1 α2 + 5α22 ) + 2c(−2α31 + 3α21 α2 − 3α1 α22 + α32 ))12(α21 − α1 α2 + α22 )1(a(2α2 − 5α + 5)α22 + 2c(−2α3 + 3α2 − 3α + 1)α32 ))212(α1 − α1 α2 + α22 )=1212(α1 − α1 α2 + α22 )=α2 c(2α2 − 5α + 5)6(α2 − α + 1)

=
>
=







(−2α3 + 3α2 − 3α + 1)a+(2α2 − 5α + 5)2α32 c2α2 c(2α2 − 5α + 5)(−2α3 + 3α2 − 3α + 1)
a
+
2α2 c
(2α2 − 5α + 5)

α2 c(2α2 − 5α + 5) a(
+ f (α))6(α2 − α + 1)2α2 c
α2 c(2α2 − 5α + 5) 4α1 c(+ f (α))6(α2 − α + 1)2α2 c
α2 c(2α2 − 5α + 5)
(2α + f (α)),
6(α2 − α + 1)
8



3

2

+3α −3α+1)
where f (α) = (−2α. Noting that 2α21 − 5α1 α2 + 5α22 = (2α2 − 5α + 5)α22 is positive(2α2−5α+5)for α ∈ Aα , the sign of (PD − α1 p∗d ) is determined by that of the second term (2α + f (α)).
Solving (2α + f (α) = 0) for α, we have one real solution; α = −0.1263. Thus, the second term
is positive for α ∈ Aα . Then, (PD − α1 p∗d ) > 0 for α ∈ Aα. Then, proﬁts π1∗ of D1 are positive
because π1∗ = (PD − α1 p∗d )x∗1 , in which x∗1 and (PD − α1 p∗d ) are positive for α ∈ Aα.
Finally, we show that (PD −α1 c) > 0. Noting that p∗d > c, it is easy to show that (PD −α1 c) >(PD − α1 p∗d ) > 0. Then, equilibrium price PD enables D1 and D2 to reap positive proﬁts and all
ﬁrms in vertically related markets can be viable for α ∈ Aα .
Next, we study the PM game in which the upstream monopolist U merges with D1, but
D2 remains independent. However, if the the upstream monopolist merges with D2, which has
superior production technology, downstream market structure changes into a pure monopoly.
This will be shown in the proof of Lemma 2. This is the reason why U merges with D1.
Note also that D1 and U form the integrated ﬁrm I and the marginal cost of inputs for the
upstream division of I is c. Thus, the proﬁts of the two ﬁrms, I and D2, are given byπI= πd + πu = (P − α1 c)xI + α2 (p − c)x2 = (a − α1 c − (xI + x2 ))xI + α2 (p − c)x2 ,

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

∂πd
= a − α1 c − 2xI − x2 = 0,∂xI=

= a − α2 p − xI − 2x2 = 0,

where xI stands for output of I and x2 for that of D2. The Nash equilibrium outputs of theseﬁrms are expressed as
x∗I =
x∗2 =

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

D2 has to purchase inputs from I and its demand is given by α2 x∗2 , which isα2 x∗2 = Y =α2 (a + α1 c − 2α2 p).
3Solving this equation for p, we have the inverse demand p for inputs:p=−3Y + α2 (a + α1 c)
.
2α22
9

(4)
(5)

Firm I faces demand for its products derived above. Its proﬁt is
πI= πd + πu = (P − α1 c)x∗I + (p − c)Y = (P − α1 c)x∗I + (=−5Y 2 + 4Y (α1 − α2 )α2 c + α22 (a − α1 c)2.
4α22−3Y + α2 (a + α1 c)
− c)Y
2α22

It follows that the condition for optimality is
5Y
α1
dπI
= − 2 + (−1 +)c = 0.
dY
2α2
α2
U supplies inputs to D2, which are given byY∗ =2(α1 − α2 )α2 c.
5Given the inverse demand p for input, price charged for this input by the supplier isp∗i =−3Y ∗ + α2 (a + α1 c)
5a − α1 c + 6α2 c
5a − (α1 − 6α2 )c
=
=
.
2
10α2
10α2
2α2

(6)

It would be of some interest to check whether the equilibrium price in the upstream market ishigher than the provider’s marginal cost c. Subtracting c from p∗i , we havep∗i − c =5a − (α1 − 6α2 )c5a − (α1 + 4α2 )c
−c =
> 0,
10α2
10α2

which will be positive under our Assumptions 1 and 2. Thus, I has cost advantage over its
competitor in the downstream market.
We can now summarize our analysis above as:
Lemma 2. Equilibrium prices in the upstream and the downstream market in the PartialMonopoly game are given by
p∗i =
PI

=

5a − (α1 − 6α2 )c,10α2
5a + c(3α1 + 2α2 ),10where PI is higher than α2 p∗i . Moreover, output of the independent ﬁrm D2 is given byx∗2 =2(α1 − α2 )c> 0.
5Proof. Substituting (6) into (4) and (5) yields,
x∗I

=
=

a − 2α1 c + α2
a − 2α1 c + α2 p∗i=335a − (7α1 − 2α2 )c,10105a−(α1−6α2 )c
10α 2

and

x∗2 =

a + α1 c − 2α2a + α1 c − 2α2 p∗i=335a−(α1−6α2 )c
10α 2

=

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

(7)

As noted before, I has cost advantage over its downstream competitor D2. However, it follows
from Assumption 1 that the competitor has more eﬃcient technology. Then, consider whether
this cost advantage is enough for ﬁrm I to drive D2 out of the market. Using PI and p∗i derived
above, we get:PI − α2 p∗i =2c(α1 − α2 )> 0.
5where the sign comes from Assumption 1 and the equality holds at α = 1. Thus, both I andD2 can make positive proﬁts and they can be viable.
Comparisons between PI and PD are much more complicated. It follows from Lemmas 1
and 2 thatPI − PD ===5a + (3α1 + 2α2 )c a(5α21 − 2α1 α2 + 5α22 ) + 2(α31 + α32 )c−1012(α21 − α1 α2 + α22 )5a(α21 − 4α1 α2 + α22 ) + 2(4α31 − 3α21 α2 + 3α1 α22 + α32 )c60(α21 − α1 α2 + α22 )5a(α2 − 4α + 1) + 2(4α3 − 3α2 + 3α + 1)α2 c.
60(α2 − α + 1)

For notational simplicity, we now use new deﬁnitions:f (α) = PI − PD ,4α3 − 3α2 + 3α + 1.
g(α) =
α2 − 4α + 1
Using these deﬁnitions, the diﬀerences in equilibrium prices are rewritten as
PI − PD = f (α)
=
=
=

5a
2α2 c
{
(α2 − 4α + 1) + (4α3 − 3α2 + 3α + 1)}
260(α − α + 1) 2α2 c
4α3 − 3α2 + 3α + 1
5a
α2 c
2
(α
+
}
−
4α
+
1){
30(α2 − α + 1)2α2 cα2 − 4α + 1
5aα2 c(α2 − 4α + 1){+ g(α)},230(α − α + 1)
2α2 c
11

where (α2 − α + 1) is positive for α ∈ Aα . It then follows that f (α) = 0 when the second and/or
the last terms in equation above is equal to zero. Then, consider ﬁrst the following equation:
5a+ g(α) = 0,
2α2 c

(8)

which in turn implies that f (α) = 0.
In view of Assumptions 1 and 2,
5 × 4α1 c
5 × 4α2 c
5a
>
>
= 10.
2α2 c
2α2 c
2α2 c
Next, consider the following equation:
10 + g(α) = 10 +4α3 − 3α2 + 3α + 1= 0.
α2 − 4α + 1

The approximate values of solutions to the equation are given by α = −4.142, 0.320, and 2.072.
Note also that the function g(α) is less than −2.5 for α > 1 because g(1) = −2.5, and that g(α)
is negative, monotonically decreasing and approaching −∞ as α increases in the open interval(1, 3.732). In the analysis to follow, the solution which is appropriate here is α = 2.072 because
α > 1 (Assumption 1). It follows from the properties of g(α) and 2α5a2 c > 10 that the solutionα∗ to equation (8) is larger than 2.072, but less than 3.732.
Finally, together with Lemmas 1 and 2, we can formally state:
Proposition 1. The Partial-Monopoly game yields lower equilibrium prices than the MonopolyDuopoly game if diﬀerences in productivity are relatively small. However, if productivity ofthe partial monopolist I becomes low enough, it provides a higher equilibrium price than the
Monopoly-Duopoly game does. Formally, we havePI ≤ PD ,f or 1 < α ≤ α∗ ,PI > PD ,f or

4 ≥ α > α∗ ,where 2.072 < α∗ < 3.732.
Proof. Consider the following:
PI − PD =α2 c5a(α2 − 4α + 1){+ g(α)},230(α − α + 1)
2α2 c

2c
is positive for α ∈ Aα and (α2 − 4α + 1) is negative for 0.2679 < α < 3.7320.
where 30(αα2−α+1)

However, the sign of the last term { 2α5a2c + g(α)} is indeterminate. As mentioned above,
5a
> 10.
2α2 c
12

Next, consider (8):10 + g(α) = 10 +4α3 − 3α2 + 3α + 1= 0,
α2 − 4α + 1

where the solution to the equation above which we consider here is 2.072. It follows from theproperties of g(α) that5a+ g(α) = 0
2α2 c
has a unique, positive solution α∗ in the open interval (2.072, 3.732).
Then, we encounter two cases depending upon whether the quadratic term (α2 − 4α + 1)
is negative or not. It is easy to show that the quadratic term is positive and g(α) > 0 for
α > 3.732. This means that 2α5a2 c + g(α) > 0 for α > 3.732. Then, it follows from (8) that
PI > PD for α > 3.732. It also follows from the property of g(α) that we have
5a
2α2 c
5a
g(α) +
2α2 cg(α) +α ≤ α∗ ,

≥ 0,

f or

< 0,

f or 3.732 > α > α∗ .

As (α2 − 4α + 1) < 0 for α in the open interval, (1, 3.732), it follows from (8) and thediscussion above thatα ≤ α∗ ,PI≤ PD ,f or

PI> PD ,

f or 3.732 > α > α∗ .

This completes the proof.
This proposition clearly shows one of the essential features of vertically related markets. The
vertical integration generally causes equilibrium prices to increase. However, our investigationshows that this is not necessarily true when the integration occurs in the MD game. The
integration has two distinct eﬀects on equilibrium prices in the downstream market: the costeﬀect and the productivity eﬀect. The partial monopolist I (or integrated ﬁrm ) can get accessto inputs at lower costs (i.e., the marginal costs), while the un-integrated, independent ﬁrm D2
is supplied inputs at market price. Decreases in production costs result in lower equilibrium
price and vice versa. Then, this eﬀect may be called the cost eﬀect.6 This is equivalent to thecancellation of the double marginalization. 7
6
7

Our cost eﬀect is called the eﬃciency eﬀect in Chen (2001).
See, for example, Spengler (1950).

13

Considering the technological aspects, productivity of the partial monopolist I exerts strong
inﬂuence on the cost eﬀect. For example, when productivity of the monopolist I is high, itstrengthens the cost eﬀect. By the same token, however, the cost eﬀect is weakened when
productivity is low. This new eﬀect may be called the productivity eﬀect, which comes from thediﬀerence in productivity between the partial monopolist I and the downstream ﬁrm D2.
When productivity of the monopolist I is high enough in the sense that α ≤ 1, the cost eﬀect
is strengthened by the productivity eﬀect so that equilibrium prices in the downstream market
become too low for the independent ﬁrm to make positive proﬁts. When productivity of themonopolist I becomes lower (or productivity of D2 becomes higher) so that α becomes larger
than 1, the productivity eﬀect works against the cost eﬀect caused by the low productivity, andequilibrium prices go up. However, the productivity eﬀect is dominated by the cost eﬀect so thatequilibrium prices are lower in the PM game than in the MD game. To be precise, equilibrium
prices become lower after integration as long as α ≤ α∗ . However, if the productivity of theintegrated ﬁrm is so low that α > α∗ , the productivity eﬀect dominates the cost eﬀect so that
downstream prices become so high that they are higher in the PM game than in the MD game.
These results are mainly due to our modeling which focuses on the diﬀerence in productivity ofdownstream ﬁrms.

14

3

One-sided Vertical Integration in Double-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 issue is whether the vertical integration is anticompetitive or not, and the other is whether it promotes market eﬃciencies. In this sectioneﬀects of one-sided vertical integration are examined in the DD game.
The DD game is a benchmark game against which the OI game ( one-sided integrationgame) is evaluated. This game structure is depicted in Fig. 2. In what follows, it will beconsidered whether the vertical integration by one of the upstream duopolist causes decreases
in downstream prices, and whether it results in excluding a competitor from the market. Fromthe viewpoint of antitrust, the integration by eﬃcient ﬁrms are taken seriously by authorities.
In fact, this type of merger is considered to pave the way for the monopolization of a market.
Thus, there are concerns that merger may lead to reduced competition. This is the reason why
the merger of eﬃcient ﬁrms is examined in what follows.
U1 bids to acquire D2J

D2 accepts U1’s bidJ

J D2 rejects U1’s bid
J
J
J
^
J

Integrated ﬁrm F announces
quantities yF for D1
D1 will purchasefrom FU1 bids to acquire D1

D1 accepts U1’s bid

=


?

yF
y2
xF = xF (yF , y2 )x1 = x1 (yF , y2 )







Z

Z
Z

Integrated ﬁrm f andU2 simultaneously announce
quantities yf = 0 and y2 for D2
D2 will purchasefrom U2

Z D1 rejects U1’s bid
ZZZ~

U1 and U2 simultaneously
announce quantities
yi for D1 and D2
D1 and D2 will purchasefrom U1 and U2

?
y

f
y2

xf = xf (yf , y2 )x2 = x2 (yf , y2 )Figure 2: The Double-Duopoly game

15

?

y1
y2
x1 = x1 (y1 , y2 )x2 = x2 (y1 , y2 )Game structure of the present model is as follows: In stage 1, upstream ﬁrm U1 can bid
to acquire the eﬃcient downstream ﬁrm D2. When D2 accepts U1’s bid, one-sided vertical
integration occurs, and the integrated ﬁrm is called F. If D2 rejects the bid, the verticallyrelated markets are a DD game. In stage 2, after integration, the upstream division of F andupstream ﬁrm U2 simultaneously announce quantities yi at Nash equilibrium. When D2 rejects
U1’s bid, U1 can bid to acquire the less eﬃcient ﬁrm D1.
In stage 3, in the OI game, the downstream division of F and D2 produce outputs at Nashquantities xi for a given demand for the consumer good. If D1 accepts U1’s bid, the U1D1 integration occurs, which is called f. The integrated ﬁrm f and the independent ﬁrm U2
simultaneously announce quantities yi at Nash equilibrium in the upstream market. If both D1and D2 reject U1’s bid, the vertically related markets are the DD game. In the U1-D1 merger
setting, the upstream division of f and U2 simultaneously announce Nash equilibrium quantitiesyi .
In stage 4, in the vertically integrated setting, the downstream division of f and the independent ﬁrm D2 purchase inputs to produce consumer goods and simultaneously choose Nashequilibrium quantities of outputs xi (yi ). In the DD game, D1 and D2 purchase inputs in the
upstream market and produce outputs at Nash quantities xi = xi (yi ) for a given demand forthe consumer good.
Consider the DD game, where upstream ﬁrms U1 and U2 supply their products to downstream ﬁrms. U1 and U2 produce inputs at constant marginal costs (c1 and c2 ), where c1 ≤ c2
without loss of generality. Demand for upstream products is derived from outputs by D1 and D2,
whose products are produced according to the demand function (1) and sold. Their production
functions are given by (2). To solve for the equilibrium in this game, we use backward induction.
It follows that ﬁrms in the downstream market play a simultaneous-move (or a static) game,
given the demand function (1), and maximize their respective proﬁts.
To proceed with our analysis, assume that all ﬁrms are constrained to compete à la Cournot
and that
Assumption 3. a > 4α1 c2 .
This new assumption replaces Assumption 2 in the MD game.
Using eqs. (1) and (3), proﬁts of ﬁrm i are given by
πi = (P − aci )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.
16

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,3where 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 − α22 ).
3Solving this for p yields inverse demand for inputs, which is given byp=

3Y − a(α1 + α2 ).
2(α1 α2 − α1 2 − α22 )(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 the upstream market. Then,Y = y1 + y2 .
It follows from this and (9) that proﬁt of U1 is
πU 1 = (p − c1 )y1 = (3Y − a(α1 + α2 )3(y1 + y2 ) − a(α1 + α2 )− c1 )y1 = (− c1 )y1 .
222(α1 α2 − α1 − α2 )2(α1 α2 − α21 − α22 )where c1 is the constant marginal cost of U1 and yi is supply of the inputs by Ui, i = 1, 2.
Diﬀerentiating proﬁts with respect to y1 yields
6y1 + 3y2 − a(α1 + α2 )∂πU 1
− c1 = 0.
=∂y12(α1 α2 − α21 − α22 )Similarly, we have
πU 2 = (p − c2 )y2 = (3Y − a(α1 + α2 )3(y1 + y2 ) − a(α1 + α2 )− c2 )y2 = (− c2 )y2 ,222(α1 α2 − α1 − α2 )2(α1 α2 − α21 − α22 )17where c2 is the constant marginal cost of U2. Diﬀerentiating πU 2 with respect to y2 and equating
the derivative to zero gives
3y1 + 6y2 − a(α1 + α2 )∂πU 2
− c2 = 0.
=∂y22(α1 α2 − α21 − α22 )Solving these equations for y1 and y2 , we get the equilibrium values of inputs supplied by
upstream ﬁrms:
y1∗ =
y2∗ =

a(α1 + α2 ) + 2(c2 − 2c1 )(α21 − α1 α2 + α22 )9a(α1 + α2 ) + 2(c1 − 2c2 )(α21 − α1 α2 + α22 ),9which are both positive. In fact, it follows from Assumption 1, 2 and c1 ≤ c2 that the numeratorof y1∗ is reduced to2(α21 − α1 α2 + α22 )(c2 − 2c1 ) + a(α1 + α2 )> 2(α21 − α1 α2 + α22 )(c2 − 2c1 ) + 4α1 c1 (α1 + α2 )> 2(α21 − α1 α2 + α22 )(−2c1 ) + 4α1 c1 (α1 + α2 )= c1 (−4α21 + 4α1 α2 − 4α22 + 4α21 + 4α1 α2 )= 4α22 c1 (2α − 1) > 0,
where the inequality is due to Assumption 1.
The numerator of y2∗ is reduced to2(α21 − α1 α2 + α22 )(c1 − c2 − c2 ) + a(α1 + α2 )> 2(α21 − α1 α2 + α22 )(−2c2 ) + 4α1 c2 (α1 + α2 )= 4c2 (−α21 + α1 α2 − α22 + α21 + α1 α2 )= 4c2 (2α1 α2 − α22 )= 4α22 c2 (2α − 1) > 0,
where again the inequality is due to Assumption 1.
Then, the total demand for inputs is
Y ∗ = y1∗ + y2∗ =2(α1 α2 − α21 − α22 )(c1 + c2 ) + 2a(α1 + α2 ).
9The equilibrium price p∗N of inputs is given by substituting Y ∗ into (9), and it isp∗N =a(α1 + α2 ) + 2(c1 + c2 )(α21 − α1 α2 + α22 ).
6(α21 − α1 α2 + α22 )18

It is easy to show that price p∗N is higher than c2 . In fact,p∗N − c2 =a(α1 + α2 ) + 2(c1 − 2c2 )(α21 − α1 α2 + α22 ),6(α21 − α1 α2 + α22 )where the denominator is positive under Assumption 1 and the numerator is positive because
it is the same as that of y2∗ in the duopoly game.
Next, consider if x∗1 is positive at p = p∗N . Substituting p∗N into x∗1 yieldsa(α1+α2 )+2(c1 +c2 )(α21 −α1 α2 +α22 )

x∗1

=
=

a − (2α1 − α2 )a − (2α1 − α2 )p∗N6(α21−α1 α2 +α22 )
=
3
3
2
2
6a(α1 − α1 α2 + α2 ) − (2α1 − α2 )(a(α1 + α2 ) + 2(c1 + c2 )(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(c1 + c2 )(α21 − α1 α2 + α22 ))= a(4α21 − 7α1 α2 + 7α22 ) − 2(2α1 − α2 )(α21 − α1 α2 + α22 )(c1 + c2 )> 4α1 c2 (4α21 − 7α1 α2 + 7α22 ) − 8α1 (α21 − α1 α2 + α22 )c2 = 4α1 c2 (2α21 − 5α1 α2 + 5α22 )= 4α1 c2 (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 show 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 )(c1 + c2 )a + (α1 − α2 )p∗N=,318(α21 − α1 α2 + α22 )(10)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 )(c1 + c2 )18(α21 − α1 α2 + α22 )a(4α2 − 7α + 7) − 2α2 (2α3 − 3α2 + 3α − 1)(c1 + c2 ).
18(α2 − α + 1)

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 and 2,
the numerator is given bya(4α2 − 7α + 7) − 2α2 (2α3 − 3α2 + 3α − 1)(c1 + c2 )> 4α1 c2 (4α2 − 7α + 7) − 4α2 c2 (2α3 − 3α2 + 3α − 1)
> 4α2 c2 (4α2 − 7α + 7) − 4α2 c2 (2α3 − 3α2 + 3α − 1)
= 4α2 c2 ((4α2 − 7α + 7) − (2α3 − 3α2 + 3α − 1))= 4α2 c2 (−2α3 + 7α2 − 10α + 8) > 0,19

where it should be noted that c1 ≤ c2 . 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- and thedown-stream markets.
We can now establish:
Lemma 3. In Double-Duopoly game the equilibrium prices in the upstream and downstream
markets are respectively given by
p∗N

=

PN

=

a(α1 + α2 ) + 2(c1 + c2 )(α21 − α1 α2 + α22 ),6(α21 − α1 α2 + α22 )a(7α21 − 4α1 α2 + 7α22 ) + 2(α31 + α32 )(c1 + c2 ).
18(α21 − α1 α2 + α22 )As we focus on eﬀects of merger on the downstream market, consider the DD game: Eﬃcientﬁrm D2 integrates with upstream ﬁrm U1, and the downstream division of the integrated ﬁrm F
is superior in technologies to the downstream rival D1. It follows that F can get access to inputs
at lower costs than D1. This game is called OI game. In the previous section, eﬀects of vertical
integration were examined in the game in which there is an upstream monopolist. Now, we will
focus on eﬀects of the vertical integration on a market in a DD game. To examine the featuresof the integration, consider a game in which eﬃcient D2 merges with the more productive U1.
Figure 3 depicts this OI game.
F may be able to make proﬁts by supplying outputs produced by using inputs not only fromthe upstream division, but also from an upstream market. However, assume that the integrated
ﬁrm does not purchase inputs from the upstream market because prices of inputs are higher
than the marginal costs of inputs. Thus, proﬁts of F and D1 are expressed as
πF= (P − α2 c1 )xF + (p − c1 )y1 = (a − α2 c1 − xF − x1 )xF + (p − c1 )y1 ,π1 = (P − α1 p)x1 = (a − α1 p − xF − x1 )x1 ,where xF stands for output of F produced by through-puts and p for input price.
The ﬁrst order conditions for maximum proﬁts are
∂πF
∂xF
∂π1
∂x1

= a − α2 c1 − 2xF − x1 = 0,
= a − α1 p − xF − 2x1 = 0.

Solving these equations for xF and x1 , we have
x∗F

=

x∗1 =

a − 2α2 c1 + α1 p,
3a + α2 c1 − 2α1 p
,
3
20

U1

U2

?

Upstream Market?

?
Cournot Competition

D2
Q

Q
Q

Q
Q


Q
+
s
Q







D1


Consumers

Figure 3: Market Structure of the OI Game

which are analogous to eqs. (4) and (5). As noted above, derived demand d for inputs by D1 isgiven byd = α1 x∗1 = α1

a + α2 c1 − 2α1 p.
3Solving this for p yields inverse demand for the inputs, and is given byp=(a + α2 c1 )α1 − 3d
.
2α21

(11)

Given this demand for the inputs, the upstream division of F and U2 supply the inputs tothe downstream ﬁrms, and their proﬁts areπF= (a − α2 c1 − x∗F − x∗1 )x∗F + (p − c1 )y1 = (a − α2 c1 − x∗F − x∗2 )x∗F + ((a + α2 c1 )α1 − 3d− c)y12α21(a + α2 c1 )α1 − 3(y1 + y2 )− c1 )y1 ,2α21(a + α2 c1 )α1 − 3d(a + α2 c1 )α1 − 3(y1 + y2 )− c2 )y2 = (− c2 )y2 ,πU 2 = (p − c2 )y2 = (
2
2α1
2α21
d = y1 + y2 .
= (a − α2 c1 − x∗F − x∗1 )x∗F + (Substituting x∗F , x∗2 , and p derived above into πF and πU 2 , the ﬁrst order condition for

21

maximum proﬁt yields
∂πF
∂y1

= −

5y1 + 2(y2 + α1 (α1 − α2 )c1 )≤ 0, f or y1 , y2 ≥ 0,
2α21

where the inequality is due to Assumption 1. It follows from the Kuhn-Tucker conditions that
optimal supply ŷ1 of inputs of F is equal to 0. Thus, the integrated ﬁrm F produces throughputs
and does not supply inputs to the upstream market. It follows from these arguments that U2
is the sole supplier of inputs, which are demanded by D1. U2 maximizes its proﬁts, given thederived demand for inputs. Then, we have
∂πU 2
∂y2

=

−6y2 + α1 (a + α2 c1 − 2α1 c2 )
= 0,
2α21

where (a + α2 c1 − 2α1 c2 ) is positive because of Assumption 3. Solving this equation for y2 , we
have
ŷ2 =

α1 (a + α2 c1 − 2α1 c2 )> 0,
6where the inequality is due to Assumption 3. Substituting this into (11) and noting thatd = y2 , equilibrium price p̂ of inputs is determined and then substituting this p̂ into x∗F , x∗1yields equilibrium outputs x̂F , x̂1 of the two ﬁrms.
Now we can summarize our arguments above as:
Lemma 4. If the more productive D2 in a downstream market merges with the more eﬃcient
upstream ﬁrm U1, the one-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 byŷ1 = 0,α1 (a + α2 c1 − 2α1 c2 )
,
ŷ2 =
6
a + α2 c1 + 2α1 c2
,
p̂ =
4α1
5a − 7α2 c1 + 2α1 c2
,
x̂F =
12
a + α2 c1 − 2α1 c2
,
x̂1 =
6
5a + 5α2 c1 + 2α1 c2
.
P̂ =
12
Proof. In the post-merger game, U2 maximizes proﬁts by supplying products for the given
demand (11) for inputs. Note also that d = y2 because the upstream division does not supply
inputs to the upstream market. Then, proﬁts of U2 are given by
πU 2 = (p − c2 )y2 = (α1 (a + α2 c1 ) − 3d
− c2 )y2 .
2α2122

Solving the ﬁrst order condition for y2 yieldsŷ2 =α1 (a + α2 c1 − 2α1 c2 ).
2α21Substituting this into (11), we havep̂ =a + α2 c1 + 2α1 c2.
4α1

It is easy to check that p̂ > c2 under Assumption 3. In fact,p̂ − c2 =a + α2 c + 2α1 c2
a + α2 c1 − 2α1 c2− c2 =
> 0.
4α1
4α1

As ŷ2 is positive, proﬁts of U2 are positive and U2 can supply inputs to the upstream market.
Substituting p̂ into eqs. (4) and (5) yields
x̂F

=

x̂1 =

5a − 7α2 c1 + 2α1 c2
a − 2α2 c1 + α1 p̂
=
> 0,
3
12
a + α2 c1 − 2α1 c2
a + α2 c1 − 2α1 p̂
=
> 0.
3
6

Finally, together with (1), equilibrium price in the downstream market isP̂ = a − x̂F − x̂2 =5a + 5α2 c1 + 2α1 c2.
12

It is easy to show that P̂ is larger than α1 p̂. In fact, per-units proﬁts (P̂ − α1 p̂) areP̂ − α1 p̂ =5a + 5α2 c1 + 2α1 c2
a + α2 c1 − 2α1 c2
a + α2 c1 + 2α1 c2
− α1
> 0,
=
12
4α1
6

where the sign comes from Assumption 3. This means that ﬁrm F can reap positive proﬁts andsell outputs in the downstream market because equilibrium output x̂1 of ﬁrm D1 is positive.
It is of some interest to note that the vertical integration does not exclude a rival ﬁrm
from the market in spite of the fact that the integrated ﬁrm is more competitive because thedownstream division (ﬁrm 2) is more productive in view of Assumption 1 and the upstream
division (U1) has advantages in costs. Thus, these results are in a striking contrast with theeﬀects of the vertical integration by the upstream monopolist, who can foreclose a rival fromthe market even if the downstream division does not have advantages in production technologies(see, for example, Lemma 2). In other words, even if the cost eﬀect and the eﬃciency eﬀect workfor the integrated ﬁrm, a downstream rival can be viable in the OI game. On the other hand,the cost eﬀect is so strong in the PM game that a rival has to exit from the market even if theeﬃciency eﬀect is inactive. However, as was shown above, even if both the cost and eﬃciency
23

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
because ŷ1 = 0 in Lemma 4. As the downstream division does not purchase inputs in the
upstream market, this result is similar to that from Salinger (1988) model, where the integrated
ﬁrms do not participate in the upstream market.8 This reminds us of “ keiretu ” in Japanesemanufacturing industries. For example, Toyota has purchased inputs (or parts) solely fromits subcontractors and so has Nissan. The subcontractors of Toyota have not supplied theirproducts to Nissan and vice versa, though the situation has recently been changing. Our results
above may be similar to this situation. The model explains the occurrence of “ keiretu ”, inwhich parties to the contract are not allowed to trade with ﬁrms which are not parties to thecontract. If ﬁrm D1 integrates with upstream ﬁrm U1, it will be shown that the integrated ﬁrm
may maximize proﬁts by not supplying inputs to the upstream market.9
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
these questions. We ﬁrst examine if a vertical merger is anti-competitive. Our Lemma 4
shows that the merger is not anti-competitive in the DD game because rival ﬁrms are notexcluded by the merger. However, it follows from Lemma 2 that the vertical merger enables
the upstream monopolist to drive a rival out of the downstream market when an integrated
upstream monopolist 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 its Proposition 2 that the answer is independent of upstream market structure.
It is also possible to examine whether the one-sided vertical merger improves market eﬃciency. 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 in aggravation of themarket eﬃciency. However, it is not easy to show which is correct. More precisely, we can state:
Proposition 2. One-sided vertical integration of D2 and U1 is neutral in competitiveness, but
8Chen and Riordan (2007) observed that vertical integration enables the integrated ﬁrm to preempt an up-

stream independent ﬁrm.
9 ∂πI
> 0 for positive y1 , y2 . If the best response functions of two ﬁrms intersect in the ﬁrst quadrant, then∂y1ŷ1 > 0 and the upstream division can maximize proﬁts by supplying inputs to the upstream market. However,calculations similar to previous ones will show that ŷ1 = 0.

24

it does promote market eﬃciency for α ≤ 2.618.
Proof. As shown in Lemma 4 and the argument above, U2 can reap positive proﬁts by supplying inputs in the upstream market. Moreover, it will be shown that D1 can be viable in thedownstream market. Proﬁts of D1 are given by x̂1 × (P̂ − α1 p̂), where x̂1 is positive in view ofLemma 4, which shows per-unit proﬁts (P̂ − α1 p̂) to be positive..
Thus, the independent ﬁrm D1 can be viable in the post-merger game. The one-sided merger
is neutral from the viewpoint of the competitiveness of the market.
Noting that 4α2 ≥ α1 > α2 and c2 ≥ c1 , it follows from Lemmas 3 and 4 that we cancompare equilibrium prices in pre- and post-merger game:
PN − P̂

=
=
=
=
>
>
=
≥
>
=
=
=

a(7α21 − 4α1 α2 + 7α22 ) + 2(α31 + α32 )(c1 + c2 ) 5a + 5α2 c1 + 2α1 c2−18(α21 − α1 α2 + α22 )
12
2
2
2
2
−(α1 − 7α1 α2 + α2 )a + (α1 − α1 α2 + α2 )(−6α1 c2 − 15α2 c + 4(1 + α)α2 (c1 + c2 ))36(1 − α + α2 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(−6α1 c2 − 15α2 c1 + 4α2 (c1 + c2 ) + 4α1 (c1 + c2 ))36(1 − α + α2 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(−11α2 c1 + 4α1 c1 + 4α2 c2 − 2α1 c2 )36(α21 − α1 α2 + α22 )222−(α1 − 7α1 α2 + α2 )a + (α1 − α1 α2 + α22 )(4α1 c1 − 4α1 c1 − 7α2 c1 + 4α2 c2 − 2α1 c2 ))36(α21 − α1 α2 + α22 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(4α1 c1 − 4α1 c1 − 7α1 c2 + 4α2 c2 − 2α1 c2 ))36(α21 − α1 α2 + α22 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(−9α1 c2 + 4α2 c2 )36(α21 − α1 α2 + α22 )−(α21 − 7α1 α2 + α22 )a + (α21 − α1 α2 + α22 )(−9α1 c2 + α1 c2 )36(α21 − α1 α2 + α22 )2−4α1 c2 (α1 − 7α1 α2 + α22 ) − 8α1 c2 (α21 − α1 α2 + α22 )36(α21 − α1 α2 + α22 )α1 c2 (28α1 α2 − 4α21 − 4α22 − 8α21 + 8α1 α2 − 8α22 )36(α21 − α1 α2 + α22 )α1 c2 (36α1 α2 − 12α21 − 12α22 )36(α21 − α1 α2 + α22 )α1 c2 (3α − α2 − 1)
≥ 0,
3(1 − α + α2 )where the ﬁrst two inequalities are due to α1 > α2 , the third inequality is due to 4α2 ≥ α1 , thefourth inequality is due to a > 4α1 c2 and (3α − α2 − 1) ≥ 0 for 1 < α ≤ 2.618. The calculations
above show that equilibrium price is higher in the pre-merger game than in the post-merger
game; PN > P̂ for 1 < α ≤ 2.618 to be precise. This means that the merger promotes market
eﬃciency.
25

AS was shown in P roposition 1, in the Integrated Monopoly game the vertical integration
causes equilibrium downstream prices to go down. Together with P roposition 2, whether the
upstream market is a monopoly or not, the answer to the second question is that the integration
can result in higher market eﬃciency. Note, however, that the integrated duopolist can not
exclude a rival ﬁrm out of the market. This is in striking contrast with merger by a monopolist,
who can drive a rival ﬁrm out of a market, but only if the monopolist has superior productivity.
Vertical integration has two distinct eﬀects: the cost eﬀect and the productivity eﬀect. The
productivity eﬀect is weaker in the DD game so that the rival can be viable even if the rival has
inferior productivity in the DD game. Thus. the exclusion of a downstream rival ﬁrm can arise
only if upstream market structure is a monopoly.
Even if a downstream ﬁrm can stay in the market, the integrated ﬁrm will have incentives tomonopolize a market. There are several means to exclude rivals from a market. One of the typicalmeans is a price squeeze by the integrated ﬁrm. It is shown in Yang and Kawashima (2011) thatthe integrated ﬁrm does not have incentives to monopolize the market by a price squeeze in thePartial-Monopoly game (or PM game), but that it does in the One-sided Integration game (orOI game).

26

4Conclusions

The proposed 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 productivity eﬀect which comes from thefact that the integrated ﬁrm owns a downstream division. Both the cost and productivity 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 therival from the market. The cost eﬀect is strong enough for the integrated monopolist to exclude
the rival from a market. On the other hand, when the merged ﬁrm has advantages in costs andproduction relative to an independent ﬁrm in the duopoly game, market foreclosure does nottake place because the cost eﬀect of the duopolist is not strong and hence the integration does
not exclude the independent from the market.
From the viewpoint of welfare, the vertical integration results in lower downstream prices
even if foreclosure occurs. Even if integration is neutral in competitiveness in a market because
the number of rivals is not 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, the upstream division of the merged ﬁrm provides
throughputs only to the downstream division. This is similar to “keiretu” of Japanese industries.
This is a game in which an eﬃcient upstream ﬁrm and an eﬃcient downstream ﬁrm merge.
Moreover, as mentioned in Footnote 5, our results are independent of the pair of ﬁrms whichmerge.
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 demand
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, hence this remains a subject for future investigation.

27

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