GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.116

How do market structures affect decisions on vertical
integration/separation?

Noriaki Matsushima and Tomomichi Mizuno

February 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

How do market structures aﬀect decisions on vertical
integration/separation?∗
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Tomomichi Mizuno
Competition Policy Research Center, Japan Fair Trade Commission
January 31, 2010

Abstract
We provide a simple model to investigate decisions on vertical integration/separation. The
key feature of this model is that more than one input is required for the ﬁnal products of the local
downstream monopolists. Depending on their cost structure, downstream ﬁrms’ decisions on verticalseparation can be both strategic complements and strategic substitutes. As a result, the equilibrium
number of vertically integrated ﬁrms depends on the cost structure. When the local downstream
monopolists merge, vertical separation tends to appear in equilibrium. When an upstream ﬁrm canprice discriminate, the downstream ﬁrms vertically separate. When the downstream ﬁrms competewith each other, vertical integration tends to appear if the degree of product diﬀerentiation is lower.

JEL Classiﬁcation Codes: D43, L13, L22, M11.
Key Words: vertical integration, vertical separation, local monopolists, inputs, technology.

∗ The ﬁrst author gratefully acknowledges ﬁnancial support from the Ministry of Education, Science, Sports and Culture,
Grant-in-Aid for Scientiﬁc Research (C) (21530243) and Young Scientists (B) (21730193). Of course, any errors are theresponsibility of the authors. The views expressed herein are those of the authors and should not be interpreted as thoseof the Japan Fair Trade Commission.
† Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, Mihogaoka
6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6879-8571. E-mail: nmatsush@iser.osaka-u.ac.jp1

1

Introduction

Determinants of vertical structure have long been discussed in the management and economics literature.1 Investigating this problem is diﬃcult because there can be many determinants of verticalstructure. The following reﬂects the diﬃculty of this problem. The extent of vertical specialization
varies considerably among countries and industries (Hummels et al. (1998)). For instance, in the apparel industry, some brands (The Gap, L.L. Bean, and Eddie Bauer) are distributed through verticallyintegrated specialized retailers. Others (Tommy Hilﬁger and Calvin Klein) are distributed primarily ona nonexclusive basis through department stores and other nonintegrated retailers (Gertner and Stillman(2001)). In the assembly industries, companies in Western European countries are less integrated thanthose in the US, but they are still far more integrated than those in Japan. There are also signiﬁcantdiﬀerences in vertical industry structure among individual European economies (Hemmert (1999)). We
therefore think that providing an analytic framework to investigate this problem is important from theviewpoints of management and economics.
We provide a simple model to investigate decisions about vertical separation (integration) and show
several results that have not been explained in previous research (we mention the diﬀerence between
this model and those in related previous studies below). The key feature of this model is that more than
one input is required for the ﬁnal products of the local downstream monopolists.2 The model can beapplied to many industries. For instance, in the aircraft industry, two major ﬁrms, Airbus and Boeing,
rely heavily on ﬁrm-speciﬁc inputs (e.g., engines, wings, horizontal stabilizers) produced by independent

1 Macher and Mowery (2004) clearly summarize the research on the relation between vertical specialization and industry

structure.
2

This setting is related to models with complementary suppliers (Economides and Salop (1992), Nalebuﬀ (2000),

Baldwin and Woodard (2007), Casadesus-Masanell et al. (2007), and Maruyama and Minamikawa (2009)). These papers
discuss how mergers among complementary suppliers appear and/or how these mergers change equilibrium outcomes.
Such complementary suppliers provide their products directly to consumers. This setting is quite diﬀerent from ours.
Note that the meaning of the term ‘vertical integration’ in these papers is diﬀerent from that in our paper. Although amerger among complementary suppliers is called ‘vertical integration’ in these papers, in our model the term indicates amerger between an upstream and a downstream ﬁrm.

2manufacturers, and then sell their aircraft to airline companies, which are the ﬁnal customers (Beelaerts
van Blokland et al. (2008)).
The model structure is as follows. There are two independent local downstream monopolists D1 andD2 that use two inputs: A, and either B1 or B2 .3 The inputs are produced by an independent upstream
supplier A and a production unit inside the downstream ﬁrm Di (i = 1, 2). The nondiscriminatory
wholesale price of input A is unilaterally determined by supplier A.4 To produce a ﬁnal product, Di
needs ci units of input A, and one unit of input B (i = 1, 2). D1 is more eﬃcient than D2 , that is,
c2 > c1 . In this model, c2 /c1 represents the eﬃciency diﬀerence between the downstream ﬁrms. Under
this condition, we consider four cases: (1) both downstream ﬁrms produce units of input B, (2) bothdownstream ﬁrms separate from their input production units, and (3,4) one of the downstream ﬁrms
separates from its input production unit. The separated unit (supplier Bi ) supplies to the downstreamﬁrm Di to maximize its own proﬁt (i = 1, 2).5
[Figure 1 here]
We show that depending on the production technologies of the downstream ﬁrms (c2 /c1 ), the following three situations concerning the vertical structures of the downstream ﬁrms appear: both downstreamﬁrms vertically integrate or neither do when c2 /c1 is small; only the eﬃcient downstream ﬁrm verticallyintegrates when c2 /c1 is large. In this model, depending on the eﬃciency diﬀerence, decisions on vertical
integration can be both strategic complements and strategic substitutes.
This result is related to Buehler and Schmutzler (2005), who examine the reasons that asymmetricvertical structures are established and that integrated ﬁrms tend to be large in many industries.6 Usinga reduced-form successive Cournot model (see Salinger (1988)), they show the two following results:
3 To discuss decisions of vertical integration, Laussel (2008) and Matsushima and Mizuno (2009) provide models withmultiple inputs. In these two papers, there is only one downstream ﬁrm that determines its vertical structure.
4 In Section 5, we brieﬂy discuss price discrimination by suppler A.
5 We assume that the downstream ﬁrm cannot merge with supplier A.

A more detailed discussion of the assumption

is provided in Section 2.
6 Buehler and Schmutzler (2005) provide several examples related to asymmetric vertical structures.

3

(1) there may be asymmetric equilibria where only one of the symmetric ﬁrms vertically integrates; and(2) eﬃcient ﬁrms are more likely to integrate when downstream ﬁrms diﬀer with respect to their initialeﬃciency levels.7 The latter result suggests that large ﬁrms are more likely to integrate vertically.8
Our paper contributes to the literature on vertical integration. First, we provide a market structurewith the following feature: depending on the cost structure of downstream ﬁrms, the downstream ﬁrms’
decisions on vertical separation can be both strategic complements and strategic substitutes, whereas
vertical integration decisions of downstream ﬁrms in Buehler and Schmutzler (2005) are strategic substitutes. As a result, one of our models can derive the following three situations concerning the verticalstructures of the downstream ﬁrms: both downstream ﬁrms vertically integrate and no downstreamﬁrm vertically integrates when the decisions on vertical integration are strategic complements; only one
downstream ﬁrm vertically integrates when the decisions on vertical integration are strategic substitutes.9 Second, our model shows that downstream competition in itself is not essential in decisionsof vertical integration/separation. In our model, the two downstream ﬁrms are local monopolists andprocure their inputs from a common supplier.10 This feature of our model is quite diﬀerent from thoseof the previous studies.
We extend the basic model in several directions. First, we consider a downstream merger. When thedownstream ﬁrms merge, full integration does not appear in equilibrium but rather vertical separation
tends to do so. This tendency reduces social welfare. This property can explain why merged ﬁrms
tend to spin oﬀ their previously integrated supplier divisions. For instance, in the late 1990s, Mercedes
Benz and Chrysler merged and then restructured their supplier relations around a minimum number of
7 Linnemer (2003) and Dufeu (2004) also discuss vertical integration under Cournot competition.
8 Several papers also derive asymmetric equilibria related to vertical integration.

See Ordover et al. (1990), Gaudet

and Long (1996), Abiru et al. (1998), Choi and Yi (2000), Church and Gandal (2000), Chen (2001), Elberfeld (2002),
Jensen (2003), and Matsushima (2004, 2009).
9 Using a Hotelling linear city model, Matsushima (2009) also shows these three vertical structures, which are aﬀectedby exogenous parameters. The mechanism used to derive those results is quite diﬀerent from ours. In Matsushima (2009),
vertical integration aﬀects the locations of downstream ﬁrms in the Hotelling linear city, and the locations aﬀect decisions
on vertical integration.
10 In Section 6, we brieﬂy discuss the strategic interaction between the downstream ﬁrms.

4full-service-development and logistics partners (Olin et al. (1999)). To our knowledge, in the literatureof industrial organization, the relation between a downstream merger and vertical structure has notbeen discussed (we brieﬂy mention this in Section 4).
Second, we allow supplier A to price discriminate. This is related to the literature of input pricediscrimination (De Graba (1990), Yoshida (2000), Inderst and Valletti (2009)). We show that bothdownstream ﬁrms vertically separate when supplier A price discriminates. As a result, allowing pricediscrimination reduces social welfare. Third, we investigate the case in which the downstream ﬁrms compete. We show that the downstream ﬁrms tend to integrate when the degree of product diﬀerentiation
is lower.
The remainder of this paper is organized as follows. Section 2 formulates the model. Section 3
presents the results. Sections 4, 5, and 6 extend the basic model. Section 4 discusses the case ofdownstream merger. Section 5 discusses the case of input price discrimination. Section 6 introduces
downstream market competition into the basic model. Section 7 concludes the paper.

2

The model

Consider two independent downstream markets. In each market, there is a downstream ﬁrm that uses
two inputs, A and B. The two inputs are produced by a common upstream supplier A and a production
unit within each downstream ﬁrm i (i = 1, 2). We call the internal production unit Bi (i = 1, 2). To
simplify the analysis, we assume that the common wholesale price of input A is unilaterally determinedby the supplier. In each market, the demand for the product is linear:pi = 1 − Qi ,(1)where pi is the market price, and Qi is the output supplied by the downstream ﬁrm Di (i = 1, 2).
An initial simplifying assumption is that the downstream ﬁrms are located in separate geographic
markets. Thus, the downstream ﬁrms compete for inputs from upstream supplier A, and upstream
suppliers indirectly compete to supply them but downstream ﬁrms do not compete directly with one

5another. This assumption clariﬁes the mechanism behind the main results. Thus, having built intuition
and analysis based on an independent ﬁnal good market assumption, we will (in Section 4) amend ouranalysis to include head-to-head competition between downstream ﬁrms.
We consider two cases concerning production technologies. (i) Downstream ﬁrm i produces a unit of
ﬁnal product using ci units of input A and one unit of input Bi. (ii) Downstream ﬁrm i produces a unitof ﬁnal product using one unit of input A and ci units of input Bi. Hence, each production technology
is denoted by:case (i): Qi = min{qA /ci , qBi },(2)case (ii): Qi = min{qA , qBi /ci },(3)where qi is the amount of input j (j = A, Bi ) and ci is a positive constant (i = 1, 2). Each downstream
ﬁrm uses Leontief production technology. No ﬁrm incurs an additional marginal cost for a unit ofproduct except for the input prices set by the suppliers. Per unit production cost of Di (wi , i = 1, 2) iswritten by (wBi = 0 if Di and Bi are not separated)case (i): wi = ci wA + wBi ,(4)case (ii): wi = wA + ci wBi .

(5)

To discuss the decisions on vertical separation, we must consider the following three cases: (1) bothdownstream ﬁrms separate from their respective input production units (full separation), (2) only one
downstream ﬁrm separates (partial separation), and (3) no downstream ﬁrm separates (no separation).
We assume that the downstream ﬁrm cannot merge with ﬁrm A. We now explain why the downstream ﬁrm cannot merge with ﬁrm A. In general, downstream ﬁrms compete, and usually procure their
inputs from common suppliers (Ahmadjian and Lincoln (2001)). In this situation, a vertical merger between ﬁrm A and a downstream ﬁrm tends to be prohibited by antitrust authorities because of theforeclosure problem (Rey and Tirole (2007)). Although the two markets are independent in our model,
our setting is also applicable to the case in which two markets are interdependent and represented by6

standard diﬀerentiated inverse demand functions pi = 1 − Qi − γQj (i = 1, 2, i ̸= j). Therefore, thisassumption is plausible.
To develop an equilibrium theory of vertical integration, I follow the method of Pepall et al. (2004,
p. 439) and analyze a stage game. The game runs as follows. First, each downstream ﬁrm determines
whether to separate from its input unit. Second, given the decisions of the downstream ﬁrms, thesupplier(s) determine the wholesale price(s). Third, given the wholesale price(s), each downstream ﬁrm
sets the quantity it supplies.

3

Results

We now calculate two cases concerning production technologies.

3.1

Case (i): Qi = min{qA /ci , qBi }Given the wholesale prices wA , wB1 , and wB2 , the maximization problem of Di is:max(1 − Qi − ci wA − wBi )Qi .
Qi

The ﬁrst-order conditions lead to:Qi =1 − ci wA − wBi(1 − ci wA − wBi )2, πDi =.
2
4

As mentioned earlier, the wholesale prices are unilaterally determined by the suppliers. The maximization problems of suppliers A and Bi are:max wA (c1 Q1 + c2 Q2 ) ⇒ max wA
wA

wA

2
∑
i=1

ci

1 − ci wA − wBi,
2

1 − ci wA − wBi
max wBi, (i = 1, 2).
wBi
2Here we have three cases of vertical structure: (1) full separation, (2) partial separation, and (3) noseparation. Taking into account that wBi = 0 if ﬁrm i does not separate from Bi, we derive the7ﬁrst-order condition(s) of the supplier(s). These condition(s) lead to:

(2a)

(2b)

(3)

2c2i − ci cj + 3c2j
c1 + c2
SS
,
w
=
,
Bi
3(c21 + c22 )6(c21 + c22 )(2c2i − ci cj + 3c2j )2(2c2i − ci cj + 3c2j )2
SS
SS
,
π
=
,
πDi
=
Bi
144(c21 + c22 )2
72(c21 + c22 )2c2 − c1 c2 + 2c22
c1 + 2c2
SI
SI
, wB1
= 1 2
,
wA
= 2
2
3c1 + 4c2
3c1 + 4c22
(c2 − c1 c2 + 2c22 )2(3c21 − c1 c2 + 2c22 )2(c2 − c1 c2 + 2c22 )2
SI
SI
SI
= 1
=
, πB1
, πD2
,
πD1
= 1
2
2
2
2
2
2
4(3c1 + 4c2 )2(3c1 + 4c2 )4(3c21 + 4c22 )2
2c1 + c22c2 − c1 c2 + c22
IS
IS
wA
= 2
, wB2
= 1 2
,
2
4c1 + 3c2
4c1 + 3c22
(2c21 − c1 c2 + 3c22 )2(2c21 − c1 c2 + c22 )2(2c21 − c1 c2 + c22 )2
IS
IS
IS
=
, πD2
=
, πB2
=
,
πD1
2
2
2
2
2
2
4(4c1 + 3c2 )4(4c1 + 3c2 )2(4c21 + 3c22 )2
c1 + c2(c2 − c1 c2 + 2c22 )2(2c21 − c1 c2 + c22 )2
II
II
II
wA
=
, πD1
= 1
, πD2
=
,
2
2
2
2
2
2(c1 + c2 )16(c1 + c2 )16(c21 + c22 )2
SS
wA
=

(1)

where the superscript SS denotes the full-separation setting; the superscript SI (resp. IS) denotes that
D1 (resp. D2 ) separates and D2 (resp. D1 ) integrates; the superscript II denotes the no-separation
setting.
From the above results, we can make the following payoﬀ matrix related to the ﬁrst-stage decisionsof the downstream ﬁrms.
D1 /D2IS(2c21 − c1 c2 + c22 )2(2c21 − c1 c2 + c22 )2+4(4c21 + 3c22 )22(4c21 + 3c22 )2(2c21 − c1 c2 + c22 )2
16(c21 + c22 )2I(c21 − c1 c2 + 2c22 )2
16(c21 + c22 )2(2c21 − c1 c2 + 3c22 )24(4c21 + 3c22 )2(3c21 − c1 c2 + 2c22 )24(3c21 + 4c22 )2(2c22 − c1 c2 + 3c21 )2(2c22 − c1 c2 + 3c21 )2+144(c21 + c22 )2
72(c21 + c22 )2S(c21 − c1 c2 + 2c22 )2(c2 − c1 c2 + 2c22 )2
+ 1
2
2
2
4(3c1 + 4c2 )2(3c21 + 4c22 )2(2c21 − c1 c2 + 3c22 )2(2c21 − c1 c2 + 3c22 )2+2
2
2
144(c1 + c2 )72(c21 + c22 )2

From the payoﬀ matrix, we have the following proposition.
8

Proposition 1 D1 (resp. D2 ) vertically separates but D2 (resp. D1 ) does not separate vertically inequilibrium if c2 ≤ (3/4)1/4 c1 ≃ 0.9306c1 (resp. c2 ≥ (4/3)1/4 c1 ≃ 1.0746c1 ). D1 and D2 both verticallyseparate or neither do in equilibrium if (3/4)1/4 c1 ≤ c2 ≤ (4/3)1/4 c1 .
The diﬀerence between the eﬃciencies of their production technologies is large (the absolute value of|c2 /c1 − 1| is large), and the relatively ineﬃcient ﬁrm separates from its production unit B.
Basically, vertical separation induces supplier A to lower its wholesale price wA . This is because supplier A faces the diﬃculty of fully exerting monopoly power because of the additional double marginalization problem caused by the separated unit (Cournot (1838) and Sonnenschein (1968)). This diminishes
the proﬁt of supplier A, and a portion of the monopoly proﬁt of supplier A is transferred to supplier B
through competition between the suppliers. Note that because vertical separation enhances the standard double marginalization problem, it diminishes the quantity supplied by the separated downstreamﬁrm.
The ratio of the marginal costs c2 /c1 aﬀects the extent to which vertical separation decreases thejk(j, k ∈ {I, S}) decreases with ci
wholesale price of supplier A (wA ). We can easily show that wA

given that ci ≥ cj (i = 1, 2, j ̸= i). This is because the derived demand of supplier A becomes moreelastic as the value of ci increases. As a result, a higher value of ci induces supplier A to lower its
wholesale price. Now suppose that ﬁrm 1 is more eﬃcient without loss of generality (c1 < c2 ). Verticalseparation by ﬁrm 1 does not similarly decrease the wholesale price of supplier A because the price levelSSIS
is already low. We can easily check this fact with the following calculus: ∂wA/∂c2 < 0, ∂wA/∂c2 < 0,ISSSand ∂(wA− wA)/∂c2 < 0.11 This calculus means that the relatively eﬃcient downstream ﬁrm does

not suﬀer such an increase in wA caused by its integration decision given that both downstream ﬁrms
separate from their input units. Therefore, SS does not appear if c2 /c1 is suﬃciently large.
To discuss the incentive of the ineﬃcient downstream ﬁrm, we now mention the relations among
11 For any c ≥ c ,2
1
IS − w SS )
∂(wA
∂
A
=
∂c2
∂c2

„

c1 (2c21 − c1 c2 + 3c22 )3(c21 + c22 )(4c21 + 3c22 )«=−c1 (4c41 (c1 + c2 ) + c21 c22 (24c2 − 7c1 ) + 9c42 (2c2 − c1 ))< 0.
3(c1 + c2 )2 (4c1 + 3c2 )29jk
wA , wBi , and ci . We have already mentioned that wA(j, k ∈ {I, S}) decreases with ci if ci ≥ cj

(i = 1, 2, j ̸= i). We have also mentioned that a lower wA allows separated suppliers Bi to set higher
prices wBi (i = 1, 2), and that the converse also holds. For ﬁrm i, as the eﬃciency of ﬁrm j (j ̸= i)
improves, the wholesale price set by Bi decreases because of the increase in wA . This pricing behavior
by Bi mitigates the negative eﬀect of the additional double marginalization problem caused by verticalseparation. This eﬀect induces the ineﬃcient downstream ﬁrm to separate from its input unit. From
the discussion, we can conclude that only the ineﬃcient downstream ﬁrm i separates from its input unit
if ci /cj is large.

3.2

Case (ii): Qi = min{qA , qBi /ci }Given the wholesale prices wA , wB1 , and wB2 , the maximization problem of Di is:max(1 − Qi − wA − ci wBi )Qi .
Qi

The ﬁrst-order conditions lead to:Qi =1 − wA − ci wBi(1 − wA − ci wBi )2, πDi =.
2
4

As mentioned earlier, the wholesale prices are unilaterally determined by the suppliers. The maximization problems of supplier A and Bi are:max wAwA

2∑
1 − wA − ci wBi
i=1

2, max wBi ci
wBi

1 − wA − ci wBi.
2

As in case (i), we have three cases of vertical structure: (1) full separation, (2) partial separation, and(3) no separation. Taking into account that wBi = 0 if ﬁrm i does not separate from Bi, we derive theﬁrst-order condition(s) of the supplier(s). These condition(s) lead to:
(1)
(2a)
(2b)
(3)

1
1
1
1
SS
SS
SS
, wBi
=
, πDi
=
, πBi
=
,
3
3ci
36
18
3
2
1
2
4
SI
SI
SI
SI
SI
wA
= , wB1
=
, πD1
=
, πB1
=
, πD2
=
,
7
7c1
49
49
49
3
2
4
1
2
IS
IS
IS
IS
IS
wA
= , wB2
=
, πD1
=
, πD2
=
, πB2
=
,
7
7c2
49
49
49
1
1
II
II
wA
= , πDi
=
,
2
16
SS
wA
=

10

where the superscript SS denotes the full-separation setting; the superscript SI (resp. IS) denotes that
D1 (resp. D2 ) separates and D2 (resp. D1 ) integrates; and the superscript II denotes the no-separation
setting.
From the above results, we can make the following payoﬀ matrix related to the ﬁrst stage decisionsof the downstream ﬁrms.
D1 /D2

I

S
2
1
+
49 49

1
16

I
1
16

4
49
4
49

S
2
1
+
49 49

1
1
+
36 18
1
1
+
36 18

From the payoﬀ matrix, we have the following proposition.
Proposition 2 For any c1 and c2 , both D1 and D2 separate or neither separates in equilibrium.
Note that in this setting, ci does not aﬀect the proﬁtability of ﬁrm i (i = 1, 2). The strategic interaction
between the downstream ﬁrms through supplier A exists. The logic behind the proposition is similar tothat in the previous model.

4

Downstream merger

We now consider the case in which the independent downstream ﬁrms merge. We now assume thatthe eﬃciencies of the two downstream ﬁrms do not change even though they merge (we brieﬂy discuss
the case in which the eﬃciencies are equalized by the merger). At ﬁrst glance, under the assumption,
this merger does not aﬀect the welfare property in the markets because the downstream ﬁrms do notcompete. As explained below, this intuition is not true, but the merger tends to decrease social welfare.
11The timing of the game is as follows. Given that the downstream ﬁrms merge, the merged ﬁrm
determines its vertical structure. After that, each supplier sets its wholesale price. Finally, the merged
ﬁrm determines its quantities supplied to the two markets.
We now check the decision of the merged downstream ﬁrm on its vertical structure. Anticipating
the wholesale price(s) set by the supplier(s) and the quantities it supplies, the merged ﬁrm determines
its vertical structure. We have already derived the proﬁts under each vertical structure in the previous
section. Without loss of generality, we assume that c2 ≥ c1 . Using the proﬁts derived in the previous
section, we have the following proposition concerning the vertical structure of the merged downstream
ﬁrm.
Proposition 3 The merged downstream ﬁrm separates both input units B1 and B2 if and only if
1 ≤ c2 /c1 ≤ 2; and the merged downstream ﬁrm separates only input unit B2 if and only if c2 /c1 ≥ 2.
The merged downstream ﬁrm tends to separate its input units vertically. The following ﬁgure shows
the comparison between the two cases. Because vertical separation enhances the problem of double
marginalization, the decision of the merged ﬁrm is harmful from the viewpoint of social surplus.

[Figure 2 here]
When the downstream ﬁrms are independent, each considers only its own proﬁt, even though a verticalseparation by a downstream ﬁrm harms another downstream ﬁrm because of the increase in the wholesale
price of supplier A. On the other hand, the merged ﬁrm internalizes the negative eﬀect of verticalseparation. Therefore, the merged ﬁrm tends to separate vertically. When the eﬃciency levels of themerged downstream units are equalized, the merged ﬁrm always separates both input units because
c2 /c1 = 1.
We can interpret the result as follows. A larger downstream ﬁrm tends to separate from its production
units. In other words, a large ﬁrm concentrates its resources on its ﬁnal product activities, and to doso, it separates from its production units. This result may shed light on the discussion concerning the
optimal vertical and horizontal scope of the ﬁrm as mentioned in the introduction.
12

We now brieﬂy mention the relation between previous research and our paper. Because verticallyrelated industries have recently attracted considerable attention from policymakers, antitrust authorities
and economists, several researchers investigate downstream mergers with vertical relations (Horn and
Wolinsky (1988), Ziss (1995), Symeonidis (in press) and references therein). Our paper discusses therelation between a downstream merger and the vertical structure in the market. This has not been
discussed in previous research.12

5

Price discrimination

We now consider the case in which supplier A is able to price discriminate. When it does so, the twomarkets become independent because the strategic interaction through the supplier A disappears. As
in the previous section, we calculate two cases concerning production technologies.

5.1

Case (i): Qi = min{qA /ci , qBi }We consider market i. Given the wholesale prices wAi and wBi , the maximization problem of Di is:max(1 − Qi − ci wAi − wBi )Qi .
Qi

The ﬁrst-order conditions lead to:Qi =1 − ci wAi − wBi(1 − ci wAi − wBi )2, πDi =.
2
4

As mentioned earlier, the wholesale prices are unilaterally determined by the suppliers. The maximization problems of suppliers A and Bi are:max wAi ci
wAi

1 − ci wAi − wBi
1 − ci wAi − wBi, max wBi
.
wBi
2
2

Here, we have two cases of vertical structure: (1) vertical separation, and (2) no separation. Taking
into account that wBi = 0 if ﬁrm i does not separate from Bi, we derive the ﬁrst-order condition(s) of
12 Ziss (2007) discusses the relation between downstream mergers and the vertical structure within ﬁrms. In his model,

however, the decision of vertical separation is not discussed.

13the supplier(s). These condition(s) lead to:
1
1
1
1
S
S
S
, wBi
= , πDi
=
, πBi
=
,
3ci
3
36
18
1
1
I
I
, πDi
=
,
(2) wAi=
2ci
16S(1) wAi=

where the superscript S denotes the vertical separation setting; and the superscript I denotes the
no-separation setting. From the above results, we have the following proposition.
Proposition 4 For any ci , Di vertically separates from Bi (i = 1, 2).
This case is simple. To reduce wAi , downstream Di separates from its input unit Bi (i = 1, 2). In
this case, the positive eﬀect caused by the decrease in wAi dominates the negative eﬀect caused by theadditional double marginalization problem.
We now compare the consumer and the social surplus in the two cases. The consumer surplus andthe social surplus in the case where supplier A price discriminates are:
CSD =

1
11
, SWD =.
36
36

When supplier A does not price discriminate, if each Di vertically separates from Bi (note that thiscase is the worst among the three vertical structures), the consumer surplus and the social surplus are:CSU =107c21 − 38c1 c2 + 107c22
13c21 − 10c1 c2 + 13c22, SWU =.
22288(c1 + c2 )288(c21 + c22 )The diﬀerences between them are:
CSU − CSD =5(c1 − c2 )219(c1 − c2 )2
≥
0,
SW
−
SW
=
≥ 0.
U
D
288(c21 + c22 )288(c21 + c22 )In this setting, allowing price discrimination is harmful from the viewpoints of consumer and social
welfare.

5.2

Case (ii): Qi = min{qA , qBi /ci }Given the wholesale prices wAi and wBi , the maximization problem of Di is:max(1 − Qi − wAi − ci wBi )Qi .
Qi

14

The ﬁrst-order conditions lead to:Qi =1 − wAi − ci wBi(1 − wAi − ci wBi )2, πDi =.
2
4

As mentioned earlier, the wholesale prices are unilaterally determined by the suppliers. The maximization problems of suppliers A and Bi are:max wAiwAi

1 − wAi − ci wBi
1 − wAi − ci wBi, max wBi ci
.
wBi
2
2

As in case (i), we have two cases of vertical structure: (1) vertical separation, and (2) no separation. Taking into account that wBi = 0 if ﬁrm i does not separate from Bi, we derive the ﬁrst-order condition(s)of the supplier(s). These condition(s) lead to:
1
1
1
1
S
S
S
, wBi
=
, πDi
=
, πBi
=
,
3
3ci
36
18
1
1
I
II
=
,
(2) wAi= , πDi2
16
S(1) wAi=

where the superscript S denotes the full separation setting; and the superscript I denotes the noseparation setting. From the above results, we have the following proposition.
Proposition 5 For any ci , Di vertically separates from Bi (i = 1, 2).
When supplier A does not price discriminate, vertical integration can appear in equilibrium. In thismodel, vertical separation always appears in equilibrium. This never improves social welfare.

6

Market competition

To analyze the eﬀect of competition in the downstream market, we assume the diﬀerentiated demandis pi = 1 − Qi − γQj , where pi is the retail price of downstream ﬁrm i, Qi (resp. Qj ) is the amount of
product sold by downstream ﬁrm i (resp. j), and γ ∈ [0, 1] denotes a degree of product diﬀerentiation.
In the same way as in the previous sections, we calculate two cases.

15

6.1

Case (i): Q = min{qA /ci , qBi }Given the wholesale prices wA , wB1 , and wB2 , the maximization problem of Di is:max(1 − Qi − γQj − ci wA − wBi )Qi .
Qi

The ﬁrst-order conditions lead to:[]22(1 − ci wA − wBi ) − γ(1 − cj wA − wBj )2(1 − ci wA − wBi ) − γ(1 − cj wA − wBj )Qi =, πDi =,
(2 − γ)(2 + γ)
(2 − γ)(2 + γ)
where i, j = 1, 2 and i ̸= j. As mentioned earlier, the wholesale prices are unilaterally determined by
the suppliers. The maximization problems of suppliers A and Bi are:max wA (c1 Q1 + c2 Q2 ) ⇒ max wA
wA

wA

2
∑
i=1

ci

2(1 − ci wA − wBi ) − γ(1 − cj wA − wBj ),(2 − γ)(2 + γ)

2(1 − ci wA − wBi ) − γ(1 − cj wA − wBj )max wBi, (i, j = 1, 2 and i ̸= j).
wBi(2 − γ)(2 + γ)
Here, we have three cases: (1) full separation, (2) partial separation, and (3) no separation. Taking
into account that wBi = 0 if ﬁrm i does not separate from Bi, we derive the ﬁrst-order condition(s) of
the supplier(s). Solving the ﬁrst-order conditions, we have the proﬁts of the ﬁrms. Using the result, we
can ascertain what vertical structure appears in equilibrium as in the previous sections. We have thefollowing proposition (∆i (i = 1, 2, 3, 4) is deﬁned in the Appendix).
Proposition 6 D1 (resp. D2 ) vertically separates but D2 (resp. D1 ) does not in equilibrium if ∆2 isplus (resp. ∆4 is plus). D1 and D2 vertically separate or neither of them do in equilibrium if ∆1 isplus, ∆2 is minus, ∆3 is plus, and ∆4 is minus. D1 and D2 do not vertically separate in equilibrium,otherwise.
This proposition is summarized in Figure 3.
[Figure 3 here]
We have already explained the intuition behind the result that γ = 0. In this setting, we add thecompetition between the downstream ﬁrms. In this competitive environment, a lower procurement16

cost is an advantage for a downstream ﬁrm. To lower its procurement cost, downstream ﬁrm i has
an incentive not to separate from its input unit Bi. This tendency becomes strong as the degree ofcompetition that is represented by γ increases. Therefore, as the value of γ increases, the range of (I, I)(none of the downstream ﬁrms separates from its input unit) is enlarged.

Case (ii): Qi = min{qA , qBi /ci }6.2

Given the wholesale price wA , wB1 , and wB2 , the maximization problem of Di is:max (1 − Qi − γQj − wA − ci wBi )Qi .
Qi

The ﬁrst-order condition leads to:
Qi =

[
]2
2(1 − wA − ci wBi ) − γ(1 − wA − cj wBj )2(1 − wA − ci wBi ) − γ(1 − wA − cj wBj ), πDi =.
(2 − γ)(2 + γ)
(2 − γ)(2 + γ)

As mentioned earlier, the wholesale prices are unilaterally determined by the suppliers. The maximization problems of suppliers A and Bi are:max wAwA

2∑
2(1 − wA − ci wBi ) − γ(1 − wA − cj wBj )(2 − γ)(2 + γ)

i=1

max wBi ci Qi ⇒ max wBi ci
wBi

wBi

,

2(1 − wA − ci wBi ) − γ(1 − wA − cj wBj ), (i, j = 1, 2 and i ̸= j).
(2 − γ)(2 + γ)

Here, we have three cases: (1) full separation, (2) partial separation, and (3) no separation. Consideringthat wBi = 0 if ﬁrm i does not separate from Bi, we derive the ﬁrst-order condition(s) of the supplier(s).
As in the previous subsection, we derive the equilibrium proﬁts in each vertical structure. From theresult, we have the following proposition.
Proposition 7 Both D1 and D2 separate or neither of them separates in equilibrium if γ ≤ 0.3708.
Both D1 and D2 integrate in equilibrium if γ > 0.3708.

7

Concluding remarks

We provide a simple model to investigate decisions of vertical separation. The key feature of this modelis that more than one input is required for the ﬁnal products of the local downstream monopolists. Depending on the cost structure of the downstream ﬁrms, their decisions on vertical separation can be both17

strategic complements and strategic substitutes. As a result, one of our models can derive the following
three situations concerning the vertical structures of the downstream ﬁrms: both downstream ﬁrms
vertically integrate; no downstream ﬁrm vertically integrates when the decisions on vertical integration
are strategic complements; and only one downstream ﬁrm vertically integrates when the decisions onvertical integration are strategic substitutes. Our model also shows that downstream competition in
itself is not essential in decisions of vertical integration/separation. In our model, the two downstreamﬁrms are local monopolists and procure their inputs from a common supplier. These features of ourmodel are quite diﬀerent from those of previous studies.
We have extended the basic model in several directions. First, we consider a downstream merger.
When the downstream ﬁrms merge, full integration does not appear in equilibrium but rather verticalseparation tends to appear. This tendency reduces social welfare. Second, we allow a common supplier
to price discriminate. We show that both downstream ﬁrms vertically separate when the common
supplier price discriminates. As a result, allowing price discrimination reduces social welfare. Third, weinvestigate the case in which the downstream ﬁrms compete. We show that the downstream ﬁrms tend
to integrate when the degree of product diﬀerentiation is lower.
In this paper, we have assumed that the production technology of downstream ﬁrms is of the Leontief
type. That is, two inputs are perfectly complementary. Although this assumption simpliﬁes the analyses
executed here, it may be restrictive. Incorporating other assumptions of the production technology into
our basic model is a consideration for future research. We have set the number of downstream ﬁrms at
two in this paper. When the number of downstream ﬁrms is more than two, the strategic interaction
among the downstream ﬁrms and the common supplier would be more complicated. This is also asigniﬁcant topic for future research.

18

Appendix
We now discuss the results in Section 6.

Case (i)Solving the optimization problem(s) of the supplier(s), we have the following result (we now

set c = c2 /c1 without loss of generality):(1)(c1 + c2 )(8 − 2γ − γ 2 ),22(c1 + c22 )(12 − γ 2 ) − c1 c2 γ(20 − γ)
(2 − γ)[c2i (4 + γ) + 2c2j (3 + γ) − ci cj (2 + 5γ + γ 2 )]
SS
wBi
=
,
2(c2i + c2j )(12 − γ 2 ) − ci cj γ(20 − γ 2 )

SS
wA
=

2(6 − γ 2 )(4 − 2c + 6c2 + γ − 5cγ + 2c2 γ − cγ 2 )2,
(2 + γ)2 (24 + 24c2 − 20cγ − 2γ 2 − 2c2 γ 2 + cγ 3 )22(6 − γ)(6 − 2c + 4c2 + 2γ − 5cγ + c2 γ − cγ 2 )2
SS
SS
,
πD2
+ πB2
=
(2 + γ)2 (24 + 24c2 − 20cγ − 2γ 2 − 2c2 γ 2 + cγ 3 )2
(2 − γ)[2c1 + c2 (4 + γ)](2 − γ)[c1 c2 (2 + 3γ) − 2c21 − 2c22 (4 + γ)]
SI
SI
wA
=
,
w
=
,
B1
12c21 + c22 (16 − γ 2 ) − 12c1 c2 γ
12c21 + c22 (16 − γ 2 ) − 12c1 c2 γ2(2 − 2c + 4c2 − 3cγ + c2 γ)2 (6 − γ 2 )
SI
SI
πD1
+ πB1
=
,
(2 + γ)2 (12 + 16c2 − 12cγ − c2 γ 2 )24(6 − 2c + 4c2 + 2γ − 5cγ + c2 γ − cγ 2 )2
SI
πD2
=
,
(2 + γ)2 (12 + 16c2 − 12cγ − c2 γ 2 )2
(2 − γ)[2c22 + c21 (4 + γ) − c1 c2 (2 + 3γ)]
(2 − γ)[2c2 + c1 (4 + γ)]
IS
IS
=
, wB2
,
=
wA
2
2
2
12c2 + c1 (16 − γ ) − 12c1 c2 γ
12c22 + c21 (16 − γ 2 ) − 12c1 c2 γ
4(4 − 2c + 6c2 + γ − 5cγ + 2c2 γ − cγ 2 )2
IS
πD1
=
,
(2 + γ)2 (16 + 12c2 − 12cγ − γ 2 )22(4 − 2c + 2c2 + γ − 3cγ)2 (6 − γ 2 )
IS
IS
πD2
+ πB2
=
,
(2 + γ)2 (16 + 12c2 − 12cγ − γ 2 )2
(2 − γ)(c1 + c2 )
II
,
wA
=
4(c21 + c22 − c1 c2 γ)(2 − 2c + 4c2 − 3cγ + c2 γ)2(4 − 2c + 2c2 + γ − 3cγ)2
II
II
πD1
=
, πD2
=
.
2
2
2
16(2 + γ) (1 + c − cγ)
16(2 + γ)2 (1 + c2 − cγ)2
SS
SS
πD1
+ πB1
=

(2a)

(2b)

(3)

From the above result, we can make the following payoﬀ matrix related to the ﬁrst-stage decisions of
the downstream ﬁrms.

19

D1 /D2

I

S
II
πD2

IS
IS
πD2
+ πB2

I
II
πD1

IS
πD1

SI
πD2

SS
SS
πD2
+ πB2

S
SI
SI
+ πB1
πD1

SS
SS
+ πB1
πD1

To derive Proposition 6, we have to check the following four symbols ∆i (i = 1, 2, 3, 4).
SS
SS
IS
Sign[(πD1+ πB1) − πD1] = Sign[384 − 256γ 2 + 30γ 4 − γ 6 + c(−384γ + 272γ 3 − 16γ 5 )+ c2 (−80γ 2 − 56γ 4 − 2γ 6 ) + c3 (192γ + 32γ 3 + 8γ 5 ) + c4 (−288 + 48γ 2 − 8γ 4 )] ≡ ∆1 ,II
SI
SI
] = Sign[48 − 32γ 2 + c(−96γ + 64γ 3 )) − πD1+ πB1
Sign[(πD1+ c2 (8γ 2 − 32γ 4 ) + 40c3 γ 3 + c4 (−64 − γ 4 )] ≡ ∆2 ,SS
SS
SISign[(πD2+ πB2) − πD2] = Sign[−288 + 48γ 2 − 8γ 4 + c(192γ + 32γ 3 + 8γ 5 )+ c2 (−80γ 2 − 56γ 4 − 2γ 6 ) + c3 (−384γ + 272γ 3 − 16γ 5 )+ c4 (384 − 256γ 2 + 30γ 4 − γ 6 )] ≡ ∆3 ,IS
ISIISign[(πD2+ πB2) − πD2] = Sign[−64 − γ 4 + 40cγ 3+ c2 (8γ 2 − 32γ 4 ) + c3 (−96γ + 64γ 3 ) + c4 (48 − 32γ 2 )] ≡ ∆4 .

Case (ii)

Solving the optimization problem(s) of the supplier(s), we have the following result:22−γ2(6 − γ)
SS
SS
SS
, wBi
=
, πDi
+ πBi
=
,
6−γ
ci (6 − γ)
(6 − γ)2 (2 + γ)2
4 − 2γ8(6 − γ)4(4 + γ)
6+γ
SI
SI
SI
SI
, wB1
=
, π SI + πB1
=
, πD2
=
,
(2a)wA=
14 + γc1 (14 + γ) D1(2 + γ)2 (14 + γ)2(2 + γ)2 (14 + γ)2
6+γ
4 − 2γ4(4 + γ)28(6 − γ 2 )
IS
IS
IS
IS
IS
(2b)wA
=
, wB2
=
, πD1
=
,
π
+
π
=
,
D2
B2
14 + γc2 (14 + γ)
(2 + γ)2 (14 + γ)2(2 + γ)2 (14 + γ)2
1
1
II
II
=
.
(3) wA= , πDi24(2 + γ)2SS(1) wA=

20

From the above results, we can make the following payoﬀ matrix related to the ﬁrst-stage decisions of
the downstream ﬁrms.
D1 /D2IS
14(2 + γ)28(6 − γ 2 )(2 + γ)2 (14 + γ)2I
4(4 + γ)2(2 + γ)2 (14 + γ)214(2 + γ)24(4 + γ)
(2 + γ)2 (14 + γ)2S

8(6 − γ)
(2 + γ)2 (14 + γ)22(6 − γ)(6 − γ)2 (2 + γ)22(6 − γ)(6 − γ)2 (2 + γ)2

From the payoﬀ matrix, we have Proposition 7.

21

References
Abiru, M., Nahata, B., Raychaudhuri, S., and Waterson, M., 1998. Equilibrium structures in verticaloligopoly. Journal of Economic Behavior and Organization 37, 463–480.
Ahmadjian, C. L. and Lincoln, J. R., 2001, Keiretsu, governance, and learning: case studies in change
from the Japanese automotive industry, Organization Science 12, 683–701.
Baldwin, C.Y., and Woodard, C.J., 2007, ‘Competition in Modular Clusters’, Working Paper, Harvard
Business School.
Beelaerts van Blokland, W.W.A., Verhagen, W.J.C., and Santema, S.C., 2008. The eﬀects of coinnovation on the value-time curve: a quantitative study on product level, Journal of Business
Market Management 2, 5–24.
Buehler, S. and Schmutzler, A., 2005. Asymmetric vertical integration. Advances in Theoretical
Economics 5(1), Art. 1.
Casadesus-Masanell, R., Nalebuﬀ, B., and Yoﬃe, D., 2007. Competing complements, Working Paper,
Harvard Business School.
Chen, Y., 2001. On vertical mergers and their competitive eﬀects, RAND Journal of Economics 32,
667–685.
Choi, J. P. and Yi, S.-S., 2000. Vertical foreclosure with the choice of input speciﬁcations. Rand
Journal of Economics 31, 717–743.
Church, J. and Gandal, N., 2000. Systems competition, vertical merger, and foreclosure. Journal of
Economics and Management Strategy 9, 25–51.
Cournot, A. 1838, Researches into the Mathematical Principles of the Theory of Wealth, (Macmillan,
New York); English translation, N. Bacon, 1897.
DeGraba, P., 1990. Input market price discrimination and the choice of technology, American Economic
Review 80, 1246–1253.
Dufeu, I., 2004. Market size and vertical equilibrium in the context of successive Cournot oligopolies.
Topics in Theoretical Economics 4(1), Art. 2.
Economides, N. and Salop, S.C., 1992. Competition and integration among complements, and network
market structure, Journal of Industrial Economics 40, 105–123.
Elberfeld, W., 2002. Market size and vertical integration: Stigler’s hypothesis reconsidered. Journalof Industrial Economics 50, 23–42.

22

Gaudet, G. and Long, N.V., 1996. Vertical integration, foreclosure, and proﬁts in the presence of
double marginalization. Journal of Economics and Management Strategy 5, 409–32.
Gertner, R. H. and Stillman, R. S., 2001. Vertical integration and internet strategies in the apparelindustry. Journal of Industrial Economics 49, 417–440.
Hemmert, M., 1999. ‘Intermediate organization’ revisited: a framework for the vertical division oflabor in manufacturing and the case of the Japanese assembly industries. Industrial and Corporate
Change 8, 487–517.
Hummels, D., Rapoport, D., and Yi, K.M., 1998. Vertical specialization and the changing nature ofworld trade. FRBNY Economic Policy Review 4, 79–99.
Horn, H. and Wolinsky, A., 1988. Bilateral monopolies and incentives for merger. RAND Journal of
Economics 19, 408–419.
Inderst, R. and Valletti, T., 2009. Price discrimination in input markets, RAND Journal of Economics
40, 1–19.
Jansen, J., 2003. Coexistence of strategic vertical separation and integration. International Journal ofIndustrial Organization 21, 699–716.
Laussel, D., 2008. Buying back subcontractors: the strategic limits of backward integration. Journalof Economics and Management Strategy 17, 895–911.
Linnemer, L., 2003. Backward integration by a dominant ﬁrm. Journal of Economics and Management
Strategy 12, 231–259.
Macher, J.T. and Mowery, D.C., 2004. Vertical specialization and industry structure in high technology
industries. Advances in Strategic Management 21, 317–356.
Maruyama, M. and Minamikawa, K., 2009. Vertical integration, bundled discounts and welfare. Information Economics and Policy 21, 62–71.
Matsushima, N., 2004. Technology of upstream ﬁrms and equilibrium product diﬀerentiation. International Journal of Industrial Organization 22, 1091–1114.
Matsushima, N., 2009. Vertical mergers and product diﬀerentiation, Journal of Industrial Economics
57, 812–834.
Matsushima, N. and Mizuno, T. 2009. Vertical separation as a defense against strong suppliers, ISER
Discussion Paper No. 755, Osaka University.
Nalebuﬀ, B.J., 2000. Competing against bundles, in Peter J. Hammond and Gareth D. Myles, eds.,
Incentives, Organization, and Public Economics: Papers in Honour of Sir James Mirrlees, 323–336,(Oxford University Press, Oxford, UK).
23

Olin, J.G, Greis, N.P., and Kasarda, J.D., 1999. Knowledge management across multi-tier enterprises:
The promise of intelligent software in the auto industry. European Management Journal 17,
335–347.
Ordover, J., Saloner, G., and Salop, S., 1990. Equilibrium vertical foreclosure. American Economic
Review 80, 127–142.
Pepall, L., Richards, D.J., and Norman, G., 2004. Industrial Organization: Contemporary Theory andPractice, Third Edition (South-Western, Ohio).
Rey, P. and Tirole, J., 2007. A primer on foreclosure, in Armstrong, M. and Porter, R.H. (eds.),
Handbook of Industrial Organization, Vol. 3 (Elsevier, Amsterdam, The Netherlands).
Salinger, M. A., 1988. Vertical mergers and market foreclosure, Quarterly Journal of Economics 103,
345–356.
Sonnenschein, H., 1968. The dual of duopoly is complementary monopoly: or, two of Cournot’s theories
are one, Journal of Political Economy 76, 316–318.
Symeonidis, G., in press. Downstream merger and welfare in a bilateral oligopoly, International Journalof Industrial Organization.
Yoshida, Y., 2000. Third-degree price discrimination in input markets: output and welfare, American
Economic Review 90, 240–246.
Ziss, S., 1995. Vertical separation and horizontal mergers, Journal of Industrial Economics 43, 63–75.
Ziss, S., 2007. Hierarchies, intra-ﬁrm competition and mergers, International Journal of Industrial
Organization 25, 237–260.

24

A

B1

A

B2

w

wB1

w

wB2

D1 B1

B2 D2

Consumers

Consumers

D1

Consumers

Both ﬁrms integrate

D1 B1

Consumers

No ﬁrm integrates

A

B2

B1

A

w

wB2

wB1

w

D2

Consumers

D2

B2 D2

D1

Consumers

Consumers

One ﬁrm integrates (1)Consumers

One ﬁrm integrates (2)Figure 1: Four possible vertical structures

25

No merger(I, I)(S, S)

Merger 1 (S, S)1.07(I, S)(I, S)(S, S)

2 (I, S)c2 /c1Figure 2: Comparison of the vertical structures in the two cases

26

c
2

∆4 = 0(I,S)

1.5∆3 = 0(I,I)(S,S) and (I,I)

(I,I)

1

∆1 = 0(I,I)∆2 = 0

(S,I)

0.5
0.2

0.4

0.6

0.8

1 ǫ

Figure 3: Vertical structure with downstream competition.

27