2007・34
Why do large firms tend to integrate vertically?
-asymmetric vertical integration reconsideredNoriaki Matsushima
Tomomichi Mizuno

Why do large firms tend to integrate vertically?
— asymmetric vertical integration reconsidered —Noriaki Matsushima∗
Graduate School of Business Administration, Kobe University
andTomomichi Mizuno
Graduate School of Economics, Kobe University
July 23, 2007

Abstract
We provide a theoretical framework to discuss the relation between firm size andvertical structures. The framework is based on a Hotelling model with three downstream
and three upstream firms. Each downstream firm procures its input from each upstream
firm and the procurement problems affect the locations of the firms. We show that thedownstream firm that has the largest market share is more likely to integrate vertically.
In other words, integrated firms tend to have a large market share. We also show thatvertical integration enhances the degree of product differentiation. As a result, vertical
integration mitigates the competition among the downstream firms. We briefly discuss
whether inefficient downstream firms tend to integrate vertically. We conclude that thisis true because those downstream firms tend to be far away from those rival firms andvertical integration enables downstream firms to escape tough competition.

JEL classification: L22, L13, R32
Key words: vertical integration, asymmetry, product differentiation, location∗

Correspondence author: Noriaki Matsushima, Graduate School of Business Administration, Kobe University, Rokkodai 2-1, Nada, Kobe, Hyogo 657-8501, Japan. Phone: +81-78-803-6981, Fax: +81-78-803-6977,
E-mail: nmatsush@kobe-u.ac.jp1

1

Introduction

In many industries, vertically integrated and separated firms coexist. For instance, in theapparel industry, some brands (The Gap, L.L. Bean, Eddie Bauer, etc.) are distributed
through vertically integrated specialized retailers. Other apparel brands (Tommy Hilfiger,
Calvin Klein, etc.) are distributed primarily on a 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 than those in theUSA, but still far more integrated than those in Japan. There are also significant differencesin vertical industry structure between individual European economies (Hemmert (1999)).
Along this line of enquiry, Buehler and Schmutzler (2005) examine how asymmetricvertical structures come about and why integrated firms tend to be large in many industries.1
Using a reduced-form successive Cournot model (see Salinger (1988)), they show the following
two main results: (i) there may be asymmetric equilibria where only one of the symmetricfirms vertically integrates; (ii) efficient firms are more likely to integrate when downstream
firms differ with respect to their initial efficiency levels.2 The latter result suggests that large
firms are more likely to integrate vertically.3
Although the results in Buehler and Schmutzler (2005) are interesting and important in
the literature of industrial organization, they mention several limitations. One limitation isthat in reality, high-cost integrated firms and low-cost separated firms coexist in the foodretail distribution sectors of several European countries. In their paper, since a firm isreferred to as a large firm because of its cost advantage, the model cannot explain such a
coexistence pattern in the retail sector. If we derive results that are similar to the two main
results in Buehler and Schmutzler (2005) without any cost asymmetry, the model could be1

Buehler and Schmutzler (2005) provide several examples related to asymmetric vertical structures (theU.K. beer industry, the U.S. cable television industry, etc.).
2

Linnemer (2003) and Dufeu (2004) also discuss vertical integration under Cournot competition.

3

Several papers also derive those asymmetric equilibria related to vertical integration. See Ordover etal. (1990), Gaudet and Long (1996), Abiru et al. (1998), Church and Gandal (2000), Choi and Yi (2000),
Elberfeld (2002), Jensen (2003), Matsushima (2004, 2006).

2

a complement for the model of Buehler and Schmutzler (2005). Furthermore, if we derivea result that high-cost integrated firms and low-cost separated firms coexist, we could saythat the model provides an additional contribution to the literature on vertical integration.
To derive complementary results to those of Buehler and Schmutzler (2005), we also
provide a theoretical framework to discuss the relation between firm size and vertical structures. To discuss this topic, we extend the models of Matsushima (2004, 2006), which provide
frameworks for considering the relation between vertical structures and product differentiation. In these models, each downstream firm procures its input from each upstream firm
and the procurement problems affect the locations of the firms.4
We now explain the basic structure of this paper. There exist three downstream firms
and three upstream firms. One of the downstream firms and one of the upstream firms locate
at the center of a linear city à la Hotelling (we call these national firms). On each side, one
downstream and one upstream firm exists and the locations of the firms are determined
endogenously (we call these local firms). All downstream firms have the same marginalcost.5 The differences among the downstream firms stem from product positioning: one isat the center and the others are around the peripheral points. According to Matsushima(2004, 2006), local upstream firms are differentiated along a linear city corresponding to thedownstream product space. That is, if a downstream firm’s product is differentiated toward
the left, there is an efficiency benefit to be had by specializing its input accordingly. This isformalized by the location of the input in its product space. We can interpret transportation
costs as the loss of conversion from an upstream firm’s product into a suitable input for adownstream firm. After purchasing input from an upstream firm, each downstream firm sets
its retail price.6
The setting is suitable for the following market structures. Suppose that there are several
4

In our model, the marginal costs of downstream firms depend on their locations. Using different frame-works, Karlson (1985) and Aiura and Sato (2007) also discuss those situations.
5

We also discuss the case of asymmetric costs in Section 5.

6

The location structure is similar to DeGraba (1987). In his model, however, there are no upstream firms.

3

firms. Large (mainstream) firms compete with many firms including small ones because theymainly produce basic and standard products, which are often recognized as the original onesin the market. On the other hand, small (marginal) firms mainly compete only with large(mainstream) firms that produce standard products, but do not compete with other small
firms, because they produce quite different products and the substitutability of their products
is quite low. We can apply the model to competition between firms in rural and urban areas.
In this case, we can call national firms (resp. local firms) urban area firms (resp. rural area
firms).
The paper shows that a national firm vertically integrates to mitigate price competition
under any condition. When the per-unit transport cost of upstream firms is large, one of
the local firms integrates vertically to commit not to locate near the national downstream
firm. When the per-unit transport cost of upstream firms is large enough, the other local
firms integrate vertically to commit not to locate near the national downstream firm. The
first result of Buehler and Schmutzler (2005) also appears in our model, and the second one,
where large firms are more likely to integrate vertically, is derived from a different mechanism.
That is, without cost asymmetry among the downstream firms, we derive the second resultin Buehler and Schmutzler (2005). We think that our result is a complementary for those in
Buehler and Schmutzler (2005).
We now mention the intuition behind the result. When the national downstream andupstream firms are separate ones, each local downstream firm tends to locate near the center
to decrease its procurement cost. The shorter distances between the national and local firms
enhance the elasticity of demand for the national downstream firm (the shorter distancemeans that products are less differentiated). Because of the high elasticity of demand,the increase in the input price for the national downstream firm significantly decreases thequantity supplied by the national downstream firm. Vertical integration is the best way
to avoid a significant decrease in the quantity supplied by the national firm, because theintegrated firm is able to produce without variable costs. Given that the national firms
vertically integrate, as the per-unit procurement costs increases, each local firm moves toward4the center. Vertical integration by a pair of local downstream and upstream firms mitigates
price competition because the integrated local firm does not have to take into account thepotential supplier. This induces vertical integration. The nonintegrated firms, however, do
not react with a counter-merger of their own unless the per-unit transport cost of upstream
firms is large enough.
The remainder of this paper is organized as follows. Section 2 presents the basic model.
Section 3 shows the results under several integration patterns. Section 4 shows the main
result regarding the decisions on vertical integration. Section 5 presents a brief discussion ofasymmetric product costs. Section 6 concludes.

2

The Model

There are three downstream firms, DA , DB , and D0 , which produce the same physicalproduct. There is a linear city of length 2 ([−1, 1]), which lies on the abscissa of a line, andconsumers are uniformly distributed with density 1 along the interval. We call [−1, 0] market
A and [0, 1] market B. Suppose that in market A (resp. market B), DA (resp. DB ) is locatedat point lA (resp. lB ), which is a positive distance from zero. Suppose that D0 is located atzero. In market A (resp. market B), a consumer living at y, which is a positive distance fromzero, incurs a transportation cost of (lA − y)2 (resp. (lB − y)2 ) when purchasing a productfrom DA (resp. DB ). The consumers have unit demands, i.e., each consumes one or zero
units of the product. Each consumer derives a surplus from consumption (gross of price andtransportation costs) equal to s. We assume that s is so large that every consumer consumes
one unit of the product. This downstream market structure is also discussed in DeGraba(1987); however, he does not consider the vertical relation, which is the main concern of ourpaper.
Three upstream firms, UA , UB , and U0 , supply inputs to three downstream firms. Suppose that UA (resp. UB ) is located at point hA (resp. hB ), which is a positive distance fromzero. Suppose that U0 is located at zero. Upstream firms engage in price competition for thebusiness of downstream firms. Each input of a downstream firm produced by an upstream
5firm is a perfect substitute. To supply an x distant downstream firm, an upstream firm
incurs a transport cost τ x. We assume that τ < 1/2.
We analyze a four-stage game. In the first stage, upstream firms (UA and UB ) anddownstream firms (DA and DB ) simultaneously choose their locations. In the second stage,
each upstream firm, Ui (i = 0, A, B), simultaneously chooses its wholesale prices, wij ∈ [0, ∞(j = 0, A, B), where j is the index of the downstream firm, Dj (j = 0, A, B). For instance,
wAB is UA ’s wholesale price for DB . Each upstream firm engages in price competition forthe business of downstream firms. In the third stage, observing the wholesale prices, eachdownstream firm chooses its supplier among UA , UB , and U0 and then sets its retail pricepi ∈ [0, ∞) (i = 0, A, B) simultaneously. In the fourth stage, observing the retail prices,
consumers select between the sellers DA , DB , and D0 .

3

The results

In this section, we show the main result of the paper. To derive the results, we consider six
cases: (1) no integration; (2) integration of the national firms; (3) integration of one pair of
local firms; (4) integrations of the national firms and one pair of local firms; (5) integrationsof two pairs of local firms; (6) full integration.

3.1

No integration

This case is the basic setting in our model. The cases presented below are variants of thebasic setting. After we explain the case thoroughly, we simply mention the difference between
the basic setting and the remaining cases.
The third and fourth stages

For a consumer living at
l2 − p0 + pi
xi = i
,
2li

6

(1)

the full price (transport cost plus price) is the same at either of the two firms in market i.7
The profit of each downstream firm is given byµ¶l 2 − p0 + pA
πdA ≡ (pA − wA ) 1 − A,
2lAµ
2 −p +p ¶lB0BπdB ≡ (pB − wB ) 1 −
,
2lB
µ 2
2 −p +p ¶lA − p0 + pA lB0B
πd0 ≡ (p0 − w0 )
+
.
2lA
2lB

(2)
(3)
(4)

In eqs. (2), (3), and (4), wA = min{wAA , wBA , w0A }, wB = min{wAB , wBB , w0B }, and w0 =min{wA0 , wB0 , w00 }. These mean that DA (DB or D0 ) procures its input from the upstream
firm that bids the lowest wholesale price. From the first-order conditions of downstream
firms, we obtain equilibrium price p∗i (lA , lB , wA , wB , w0 ) (i = 0, A, B) in the third stage.
The second stage

For each downstream firm, each upstream firm sets its price at the more

efficient rival firm’s transport cost if its cost is lower than the rival’s (Bertrand competition),otherwise it sets at its own transport cost. The prices set by Ui (i = 0, A, B) are as follows:7

UA : wAA = max{τ |hA − lA |, min{τ lA , τ (hB + lA )}}(5)

wAB = max{τ (hA + lB ), min{τ lB , τ |hB − lB |}}(6)

wA0 = max{τ hA , min{0, τ hB }}(7)

UB : wBA = max{τ (hB + lA ), min{τ |hA − lA |, τ lA }}(8)wBB = max{τ |hB − lB |, min{τ (hA + lB ), τ lB }}(9)

wB0 = max{τ hB , min{0, τ hA }}(10)

U0 : w0A = max{τ lA , min{τ |hA − lA |, τ (hB + lA )}}(11)

w0B = max{τ lB , min{τ (hA + lB ), τ |hB − lB |}}(12)

w00 = max{0, min{τ hA , τ hB }}(13)

Eq. (1) is derived from the following equation: −pi − (li − xi )2 = −p0 − x2i .

7The first stage

We discuss the location choices of UA and UB . In this case |hA − lA | < lA

must be satisfied in equilibrium and then the wholesale price is wA = τ lA .8 A similarargument can be applied to the location of UB and we obtain wB = τ lB . The upstream
firm’s profits areµ
2 − p∗ + p∗ ¶lA
0AπuA = (τ lA − τ |hA − lA |) 1 −,2lAµ
2 − p∗ + p∗ ¶lB
0BπuB = (τ lB − τ |hB − lB |) 1 −
.
2lB

(14)
(15)

U0 supplies its product to D0 at the price w0 = min{τ hA , τ hB }. The retail prices p∗i (i =0, A, B) depend on hA and hB through the wholesale price w0 . We can derive the following
lemma:9
Lemma 1 hi = li (i = A, B) in equilibrium.
We can show that the intuition that Ui chooses hi = li (i = A, B) in equilibrium. We now
suppose that hi = li and lA ≥ lB (i = A, B). In this case, the wholesale prices of D0 , DA ,and DB are τ hB , τ lA , and τ lB , respectively. Moreover, |hi − li | < li (i = A, B) must be
satisfied because each upstream firm earns a positive profit.
First, we consider the location choice of UA . If UA chooses hA < lA , the wholesale priceof D0 is τ hB (if hB ≤ hA ) or τ hA (if hB > hA ) and the per-unit transport cost of UA isτ (lA − hA )(> 0). Moreover, the wholesale price of DA does not change. The first and second
changes decrease the profitability of UA and the third one is neutral. If UA chooses hA > lA ,the wholesale prices of D0 and DA do not change and the per-unit transport cost of UA is
hA −lA (> 0). The second change decreases the profitability of UA and the first one is neutral.
Since the profit of UA does not increase in either case, hA = lA is the optimal location of UA .
Second, we consider the location choice of UB . When UB chooses hB < lB or hA < hB ,the properties in the case of UA also hold. When UB chooses lB < hB ≤ hA , the wholesale
8

When UA chooses hA = 0, it cannot gain any profit by selling to downstream firms, because UA ’s

wholesale price is equal to its own transport cost (wAA = τ lA , wAB = τ lB , and wA0 = 0). From eqs. (5), (6),and (7), if |hA − lA | ≥ lA in equilibrium, then UA cannot supply their product to any downstream firms.
9

We can apply the lemma to the remaining cases.

8

prices of D0 increase and the per-unit transport cost of UB is hB − lB (> 0). The first effectis the indirect positive effect through the increase in the quantity supplied by DB and UB .
The second is, however, the direct negative effect. In the setting, the latter effect dominates
the former one. hB = lB is the optimal location of UB .
From eqs. (2), (3), and (4), and solutions in stages 2 and 3, the profits of the downstream
firms are:µ2 − p∗ + p∗ ¶
lA
0
A
1−
,
2lA
µ
2 − p∗ + p∗ ¶
lB
0
∗
B
.
πdB ≡ (pB − τ lB ) 1 −2lB
πdA ≡ (p∗A − τ lA )

(16)
(17)

The first-order conditions lead to the following results:
6 − 5τ
,8
τ (−6 + 5τ )(26 + 5τ ) ∗τ (−22 + 5τ )(−6 + 4τ )
∗
∗
πuA
= πuB
=−
, πu0 =,
384
192(−6 + 5τ )(26 + 5τ )2(−22 + 5τ )2 (−6 + 5τ )
∗
∗
∗
πdA
= πdB
=−
, πd0
=−
.
9216
4608
∗
∗
h∗A = h∗B = lA
= lB=

Those results is summarized in Table 1.10 As the value of τ (per unit transport cost of input)
τ
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50

∗
h∗A = lA
0.718750.68750
0.656250.62500
0.59375
0.56250
0.531250.50000
0.46875
0.43750∗h∗B = lB
0.718750.68750
0.656250.62500
0.59375
0.56250
0.531250.50000
0.46875
0.43750∗πuA
0.019650.037960.05486
0.070310.08427
0.09668
0.10750
0.11667
0.124150.12988∗
πuB
0.019650.037960.05486
0.070310.08427
0.09668
0.10750
0.11667
0.124150.12988∗πu00.03257
0.06159
0.087160.10938
0.12834
0.144140.15688
0.16667
0.17358
0.17773∗
πdA
0.42992
0.419090.40763
0.39551
0.38272
0.36926
0.355120.340280.32473
0.30847∗πdB
0.42992
0.419090.40763
0.39551
0.38272
0.36926
0.355120.340280.32473
0.30847∗πd00.59030
0.55173
0.514480.47852
0.443830.410400.378200.347220.317430.28882

Table 1: Locations and profits (no integration)
10

When only one pair of local firms vertically integrate (there exist two cases), we cannot explicitly solve

the equilibrium outcomes. In those cases, we use a numerical simulation to derive the results. To compare
the results in each subsection, we show these tables in the subsections.

9increases, the values of li = hi (i = A, B) decrease. To understand the benchmark case, we
now show the intuition behind the result.11 In this model, given the locations of the otherfirms, when a downstream firm moves further away from its rival, three effects occur. First,
price competition between downstream firms is softened (the “price effect”). The price effectenhances the profit of the downstream firm (DA ). Second, the demand for the downstream
firm falls (the “demand effect”). The demand effect diminishes the profit of the downstream
firm (DA ). Third, the wholesale price of the downstream firm rises (the “input price effect”)
because the distance between the downstream firm and the other supplier increases. (e.g.,
DA and U0 ).
The input price effect is affected by the value of τ . The larger τ is, the larger are
the wholesale prices. On the other hand, the price and demand effects are unaffected by
the value of τ because the cost of each downstream firm is equal to that of its rival inequilibrium. Therefore, when τ is large, the third effect (the input price effect) is importantfor downstream firms. To lower the input price, each downstream firm shortens the distanceto the rival’s supplier.
The profits of each downstream firm decrease as the value of τ increases. On the other
hand, given that the value of τ is small, the profits of each upstream firm increase as thevalue of τ increases. The increase in τ enables the upstream firms to set their prices at higher
levels because the rivals of each upstream firm have to incur higher transport costs, which are
positively related to the value of τ . The increase in τ , however, induces the distances between
the national and local downstream firms to be shortened. This enhances competition among
the upstream firms. When τ is large (small), the latter negative (the former positive) effectdominates another one.
We now briefly mention the reason that in this case the location strategy of each downstream firm accelerates the tendency to locate near the center. That is, there exists complementarity concerning the location strategies of downstream firms. Although DeGraba (1987)
does not point out the property of complementarity, we believe that it also exists in his study.
11

The intuition mentioned in the paper is also mentioned in Matsushima (2004, 2006).

10

We also believe that this point is important in the literature of industrial organization andregional science.
Given that both downstream and upstream firms choose the same value of l and h, when
one of the downstream firms (now denoted as DA ) moves toward the center, the competition
between DA and the national firm intensifies. Because of the intense competition between
them, the national firm does not take into account the price of the other downstream firm(now denoted as DB ) to any significant extent. That is, from the viewpoint of DB , whichhave not yet moved, the demand for DB becomes less elastic. The change in the national
firm’s pricing strategy softens the “price effect” and then induces DB to be near the center.
Therefore, the movement toward the center by one of the downstream firms induces anotherdownstream firm to move near the center. We will discuss this characteristic in Section 3.3.

3.2

Integration of the national firms

The remaining cases are variants of the basic setting mentioned above. We briefly mention
what is the difference between the rest of the cases and the basic setting.
The third and fourth stages

Suppose that U0 and D0 merge. Let I0 be an integrated

firm 0 and p0 be I0 ’s retail price. The integrated upstream firm sells its product to theintegrated downstream firm at a wholesale price that is equal to own transport cost. Hence,
w00 = 0, since D0 and U0 are located at zero. Then, w0 = 0. The profit of I0 isµ 2
2 −p +p ¶lA − p0 + pA lB0B
πI0 ≡ p0
+
.
2lA
2lB

(18)

From the first-order conditions of the firms, we obtain equilibrium price p∗i (lA , lB , wA , wB , w0 )(i = 0, A, B) in the third stage.
The second stage

The pricing strategies of UA and UB are similar to those in the former

subsection. The only difference is that U0 sets w00 = 0.

11

The first stage

As mentioned in Section 3.1, we obtain wA = τ lA and wB = τ lB . By

the integration, w0 = 0 and then p∗i (lA , lB , wA , wB , w0 ) (i = A, B, 0) depends only on thedownstream firm’s location (li , i = A, B). Ui (i = A, B) is able to locate at hi = li without
the changes in the quantities supplied (D0 , DA , and DB ). Therefore, hi = li (i = A, B).
The first-order conditions lead to the following result:3(2 − τ ),8
τ (−2 + τ )(−26 + 5τ )
∗
∗
πuA
= πuB
=
,
128
(2 − τ )(−26 + 5τ )2(2 − τ )(22 + 5τ )2πdA = πdB =, πI0 =
.
3072
1536
∗
∗
h∗A = h∗B = lA
= lB=

Those results are summarized in Table 2. The integrated national firm is a tough (efficient)
τ
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50

∗
h∗A = lA
0.731250.71250
0.69375
0.67500
0.65625
0.63750
0.618750.60000
0.58125
0.56250∗h∗B = lB
0.731250.71250
0.69375
0.67500
0.65625
0.63750
0.618750.60000
0.58125
0.56250∗πuA
0.01961
0.037850.05474
0.070310.08459
0.097620.109410.12000
0.129420.13770∗πuB
0.01961
0.037850.05474
0.070310.08459
0.097620.109410.12000
0.129420.13770∗πdA
0.420890.402170.38395
0.366210.34895
0.332170.315850.30000
0.28460
0.26965∗πdB
0.420890.402170.38395
0.366210.34895
0.332170.315850.30000
0.28460
0.26965∗
πI00.62850
0.626220.62337
0.61992
0.61588
0.611210.605930.60000
0.593420.58618

Table 2: Locations and profits (integration of the national firm)competitor. To avoid competition with the national firm, each local downstream firm is
far away from it. Because the distance between the national firm and each local firm islonger than in the no integration case, each nonintegrated upstream firm is able to set its
price at a higher level. Moreover, the greater distance is a commitment device to discourage
competition among the downstream firms. When the value of τ is sufficiently large, thecommitment device is functioning well.

12

3.3

Integration of one pair of local firms

Suppose that UA and DA merge (note that we can apply the same logic to a merger of UB
and DB ). Let IA be a integrated firm A and pA be IA ’s retail price. The discussion of thethird and fourth stages is similar to that in the former subsections.
The second stage

The way to determine the input price of the integrated firm is the only

difference between this case and the basic setting. The input price of the integrated local
firm does not depend on the location of the downstream firm relative to the two upstream
units (its own and the central units). If its own upstream unit is closer, then this is clearly
the cost incurred by the integrated firm. Even if the central upstream firm is closer so thatthe integrated firm procures input from that firm, this remains the relevant expression of
the cost, since the upstream firm will charge the next best price facing the integrated firm,
which is the cost incurred. Therefore, wA = τ |lA − hA |.12
The first stage

Since the argument is the same as that of section 3.1, wB = τ lB . Because

wA = τ |lA − hA |, the optimal location of UI is hA = lA . Therefore wA = 0.
In this case, the local firms are asymmetric with respect to the procurement conditions.
The asymmetry affects the locations of the firms and then lA > lB in equilibrium. We
now mention that the intuition lA > lB holds. Suppose that lA = lB . When IA increases lA(which moves to the edge), the marginal cost does not increase. On the other hand, when DB
increases lB (which moves to the edge), the marginal cost increases because wB = τ lB . Hence,
IA has more incentive to move farther from zero than DB , that is lA > lB . By numericalcalculation, we can show that lA > lB in equilibrium. Applying the same reasoning in theformer subsection, we have hB = lB . Since it is difficult to show the result analytically, we
use numerical expression to show the result summarized in Table 3. The integration mainly
has two effects on the interaction between the integrated and the national firms. First, the12Note that we implicitly exclude the case in which the integrated upstream firm locates at the furthest

point from the integrated downstream firm because the location is obviously inefficient from the viewpoint of
the integrated firm.

13

τ
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50

∗
h∗A = lA
0.734170.71862
0.703360.688420.673830.659610.645810.632450.61957
0.60720∗h∗B = lB
0.722310.694850.667650.64072
0.614100.58783
0.56193
0.53644
0.511410.48687∗
πuB
0.01974
0.03832
0.05574
0.071990.08706
0.10096
0.11368
0.125240.13565
0.14493∗
πu00.03255
0.06157
0.08727
0.10984
0.129480.146370.16072
0.17272
0.18256
0.19041∗πdB
0.43155
0.42275
0.41371
0.404450.39498
0.385310.375470.365480.35535
0.34512∗πd00.591430.554790.520220.487650.45703
0.428270.40133
0.376120.35258
0.33064∗πIA
0.447770.454470.460160.46482
0.468430.47097
0.472440.472850.472210.47054

Table 3: Locations and profits (integration of one pair of local firms)

integrated firm does not take into account its procurement cost and does not need to locate
at a closer point to the national firm. The greater distance between the integrated andnational firms mitigates the competition between them. Second, the efficiency improvementof the integrated firm enhances competition. In this setting, the former effect dominates thelatter one (see the downstream prices of those firms). The integrated and the national firms
set those prices at higher levels.
As mention in Section 3.1, for the nonintegrated downstream firm, the increment in the
distance between the national and the integrated firms are similar to the enhancement in theelasticity of demand, which is related to the price effect. The nonintegrated downstream firm
is far away from the center. This is reflected in the value of lB under the first and the thirdcases. The location strategy of the nonintegrated downstream firm affects the nonintegrated
upstream firm. To meet the location decision by the nonintegrated downstream firm, thenonintegrated upstream firm is also far away from the center. The location of the upstream
firm increases the procurement cost of the national firm, which mitigates the competition
between the national and the integrated firms.

3.4

Integrations of the national firms and one pair of local firms

Suppose that UA and DA merge and U0 and D0 do likewise. Let IA (resp. I0 ) be an
integrated firm A (resp. firm 0) and pA (resp. p0 ) be IA ’s (resp. I0 ’s ) retail price. Each

14

integrated upstream firm sells its product to its own integrated downstream firm at its
wholesale price, which is equal to its transport cost. Hence, wAA = τ |hA − lA | and w00 = 0.
Then, w0 = 0. The pricing strategies of upstream firms are similar to those in the former
subsections. We obtain wA = τ |hA − lA |, wB = τ lB , and w0 = 0. As mentioned earlier, we
can show that hA = lA and hB = lB in equilibrium. Since it is difficult to show the result
analytically, we use numerical expression to show the result summarized in Table 4. This
τ
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50

∗
h∗A = lA
0.746530.743030.739500.73595
0.73237
0.728770.725150.721500.717850.71417∗h∗B = lB
0.73475
0.71960
0.70455
0.689610.674770.66005
0.645440.630960.61661
0.60238∗πuB
0.019700.03818
0.05549
0.07167
0.08674
0.10074
0.11371
0.12569
0.136700.14678∗πdB
0.42236
0.405170.38852
0.37239
0.35678
0.34168
0.327070.31295
0.299310.28613∗πIA
0.438620.43704
0.435390.43365
0.43183
0.429930.42795
0.42588
0.42374
0.42151∗πI00.629400.628420.62726
0.625930.624410.62272
0.620850.61880
0.61658
0.61418

Table 4: Locations and profits (integrations of the national and one local firms)

case is the combination of the cases 2 and 3. The integration of a pair of local downstream
and upstream firms increases the distances between the local and national firms, respectively(lA = hA and lB = hB ). It mitigates the competition among them. The integration by thenational firms also enlarges the distances. Because it decreases the marginal cost of thenational firm, it harms the regional firms.

3.5

Integrations of two pairs of local firms

Suppose that UA and DA , and UB and DB merge, respectively. Let IA (resp. IB ) be an
integrated firm A (resp. firm B) and pA (resp. pB ) be IA ’s (resp. IB ’s) retail price. The
pricing strategies of upstream firms are similar to those in the former subsections. We obtainwA = τ |lA − hA |, wB = τ |lB − hB |, w0 = min{τ hA , τ hB }. As mentioned earlier, we can showthat hA = lA and hB = lB in equilibrium and we have wA = wB = 0. The first-order15

conditions lead to the following result:11 − 5τ
(3 − τ )(13 + 5τ )2
∗
∗
, πIA
= πIB
=
,
24
1152
(3 − τ )τ (11 − 5τ ) ∗(3 − τ )(11 − 5τ )2∗πu0=, πd0 =
.
48
576
∗
∗
h∗A = h∗B = lA
= lB=

Those results are summarized in Table 5. The integrated local firms do not have to locate
τ
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50

∗
h∗A = lA
0.73750
0.72500
0.712500.70000
0.68750
0.67500
0.662500.65000
0.63750
0.62500∗h∗B = lB
0.73750
0.72500
0.712500.70000
0.68750
0.67500
0.662500.65000
0.63750
0.62500∗πu00.03303
0.06344
0.091290.11667
0.139650.160310.17874
0.19500
0.209180.22135∗πd00.59186
0.555080.51984
0.486110.45386
0.42305
0.393650.36563
0.33895
0.31359∗πIA
0.44957
0.45879
0.46773
0.476390.48474
0.492770.500470.507810.51479
0.52138∗πIB
0.44957
0.45879
0.46773
0.476390.48474
0.492770.500470.507810.51479
0.52138

Table 5: Locations and profits (integrations of the local firms)

near the center. The distances between the national and each local firm increases. The longer
distances mitigate the competition among them, but increase the procurement costs of thenational firm. When the value of τ is small enough, the former positive effect dominates thelatter one. The additional integration enhances the profit of the national firm.

3.6

Full integration

Suppose that UA and DA , UB and DB , and U0 and D0 merge, respectively. Let IA (resp. IB
and I0 ) be an integrated firm A (resp. B and 0) and pA (resp. pB and p0 ) be IA ’s (resp. iB ’s
and I0 ’s) retail price. The pricing strategies of upstream firms are similar to those in theformer subsections. We obtain wA = τ |lA − hA |, wB = τ |lB − hB |, and w0 = 0. As mentioned
earlier, we can show that hA = lA and hB = lB in equilibrium and we have wA = wB = 0.
The first-order conditions lead to the following result:
169
121
3 ∗
∗
∗
∗
∗
= πIB
=
≈ 0.440104, πI0=≈ 0.630208.
h∗A = h∗B = lA
= lB
= , πIA
4
384
192
16

(19)

The above equilibrium outcomes do not depend on τ . Since all firms merge and h∗A = h∗B =∗ = l∗ , all wholesale prices are zero. Then, transport cost τ |h − l | (i = A, B) is always
lA
i
i
B

zero. Therefore, τ has no effect on the equilibrium outcome.

4

Integration decisions

We shall discuss decisions regarding vertical integration. The integration decisions are represented by the following 2 × 2 × 2 matrix. Π∗j (HA H0 HB ) is the joint profit of upstream andFirm 0 : No integration

Firm ANo integration
Integration

Firm BNo integration
IntegrationΠ∗A (N N N ), Π∗B (N N N ), Π∗0 (N N N ) Π∗A (N N I), Π∗B (N N I), Π∗0 (N N I)
Π∗A (IN N ), Π∗B (IN N ), Π∗0 (IN N )Π∗A (IN I), Π∗B (IN I), Π∗0 (IN I)
Firm 0 : Integration

Firm ANo integration
Integration

Firm BNo integration
IntegrationΠ∗A (N IN ), Π∗B (N IN ), Π∗0 (N IN )Π∗A (N II), Π∗B (N II), Π∗0 (N II)∗∗∗ΠA (IIN ), ΠB (IIN ), Π0 (IIN )Π∗A (III), Π∗B (III), Π∗0 (III)
Table 6: Payoff matrix

downstream firms j (j = A, 0, B) and Hj ∈ {N, I} is the index of firm j’s vertical structure(N : nonintegrated, I: integrated).13 After tedious calculations, we have the following result:
ResultWhen vertical integration is determined endogenously. For any τ , integration of

the national firm occurs. Integration of one local firm occurs if and only if 0.30 ≤ τ ≤ 0.36.
13

Given the values of τ = 0.01, 0.02, . . . , 0.50, Π∗j (· · · ) (j = A, B, 0) are derived by the following calculations:∗∗∗∗1. Π∗A (N N N ) = Π∗B (N N N ) is πuA+ πdA
and Π∗0 (N N N ) is πu0+ πd0
in Section 3.1.
∗∗∗
2. Π∗A (N IN ) = Π∗B (N IN ) is πuA+ πdA
and Π∗0 (N IN ) is πI0
in Section 3.2.
∗∗∗
3. Π∗A (IN N ) = Π∗B (N N I) is πIA; Π∗B (IN N ) = Π∗A (N N I) is πuB+ πdB; and Π∗0 (IN N ) = Π∗0 (N N I) is
∗
∗
πu0
+ πd0
in Section 3.3.
∗∗∗∗
4. Π∗A (IIN ) = Π∗B (N II) is πIA; Π∗B (IIN ) = Π∗A (N II) is πuB+ πdB; and Π∗0 (IIN ) = Π∗0 (N II) is πI0in

Section 3.4.
∗∗∗
5. Π∗A (IN I) = Π∗B (IN I) is πIAand Π∗0 (IN I) is πu0+ πd0
in Section 3.5.
∗∗
6. Π∗A (III) = Π∗B (III) is πIAand Π∗0 (III) is πI0
in Section 3.6.

17

Full integration occurs if and only if 0.37 ≤ τ .14
For any τ , the national downstream firm vertically integrates with the upstream firm
U0 . As mentioned in Section 3.1, when they are separate firms, the price effect is weak, inwhich case each local downstream firm tends to locate near the center. The shorter distances
between the national and the local firms enhance the elasticity of demand for the national
downstream firm. Because of the high elasticity of demand, the increase in the input pricefor the global downstream firm significantly decreases the quantity supplied by the global
firm. To avoid the significant decrease in the quantity supplied by the global firm, vertical
integration is the best way because the integrated firm is able to produce its product without
variable costs.
Given that national firms vertically integrate, as the value of τ increases, each local
firm moves toward the center. Vertical integration mitigates price competition because theintegrated local firm does not have to access the potential supplier. This induces vertical
integration.
The nonintegrated firms, however, do not react with a counter-merger of their own unlessthe value of τ is large enough.15 Because of the complementarity of location, the locationof the nonintegrated firms is far away from the center because the vertical integration is acommitment device not to locate near to the center. The equilibrium locations mitigate thecompetition between the national and the other local firms. Both profits of the nonintegrated
firms (πuB and πdB ) increase. Therefore, the regional asymmetry concerning the verticalstructure appears in equilibrium.
The profit of the nonintegrated upstream firm increases with the value of τ because thenonintegrated upstream firm can set its wholesale price at τ li . On the other hand, the profitof the nonintegrated downstream firm decreases with the value of τ because it moves toward14

To check the result, we present several payoff matrixes under some values of τ in Appendix.

15

Matsushima (2006) also shows this type of asymmetric vertical structure. The reason why the asymmetricvertical structure appears is different from that in this paper. In Matsushima (2006), the nonintegrated
downstream firm locates near to the center and the central position is superior to the edge. In this paper,
the complementarity of location is the driving force.

18

the center. Because of the complementarity of location, the nonintegrated downstream firm’s
movement to the center induces the integrated local firm to move toward the center eventhough the integrated firm need not take into account the procurement cost (see Sections
3.3 and 3.4). The movements of the local firms enhance competition among the local andnational firms. To avoid this competition, vertical integration by another pair appears whenthe value of τ is large.

5

Cost asymmetry among the downstream firms

To consider the relation between product inefficiency and vertical mergers, we add asymmetries of marginal costs to the basic setting. We briefly discuss the following two cases: (i)the national downstream firm is less efficient than the local downstream firms; (ii) one of thelocal downstream firms is less efficient than the other two downstream firms.
In the first case, if the inefficiency is not significant, we can apply the result in theformer section, that is, the inefficient national downstream firm integrates unless the otherdownstream firms do not integrate. Because of its product positioning, the national firm
essentially tends to integrate and the asymmetry does not change the property when theinefficiency is not significant.
In the second case, the inefficient local firm tends to integrate vertically, that is, for
an intermediate range of τ , the following integration pattern appears in equilibrium: thenational and the inefficient downstream firms vertically integrate, respectively. We now
explain the intuition behind the result. The inefficient firm tends to be far away from thenational firm because of its inefficiency. In other words, the inefficient firm’s incentive tolocate near the center is weaker than that of the efficient local downstream firm. As the valueof τ increases, each local downstream firm moves head for the center. The complementarityof their locations strengthens the effect of the increment in τ . Because of its inefficiency,
the inefficient firm’s incentive to avoid moving to the center is stronger than that for theefficient firm. Therefore, for an intermediate range of τ , the integration pattern mentioned
above appears in equilibrium.
19

As pointed out by Buehler and Schmutzler (2005), in the food retail distribution sector ofseveral European countries, high-cost integrated firms and low-cost separated firms coexist.
The integration pattern in the discussion is consistent with this fact. We believe that ourmodel could complement that of Buehler and Schmutzler (2005).

6

Concluding remarks

We provide a theoretical framework to discuss the relation between firm size and verticalstructures. The framework is based on a Hotelling model with three downstream and three
upstream firms. We show that the downstream firm that has the largest market share is more
likely to integrate vertically. In other words, integrated firms tend to have a large market
share. We briefly discuss whether inefficient downstream firms tend to integrate vertically.
We conclude that this would be true because those downstream firms tend to be far awayfrom those rival firms and vertical integration enables downstream firms to escape fromtough competition. Those results could explain the fact that high-cost integrated firms andlow-cost separated firms coexist in the food retail distribution sector of several European
countries. Therefore, we think that our model could be a complement for the model of
Buehler and Schmutzler (2005).
In our paper, we assume that a national firm sets a uniform price. As discussed in
DeGraba (1987), it could employ discriminatory pricing under some market conditions. The
pricing would change the location strategies of local firms and the vertical structures offirms. Whether discriminatory pricing affects the locations and the vertical structure is aconsideration for future research.
[2007.7.23 825]

20

References
Abiru, M., Nahata, B., Raychaudhuri, S., and Waterson, M., 1998. Equilibrium structuresin vertical oligopoly. Journal of Economic Behavior and Organization 37, 463–480.
Aiura, H. and Sato, Y., 2007. Welfare properties of spatial competition with locationdependent costs. Regional Science and Urban Economics, forthcoming.
Buehler, S. and Schmutzler, A., 2005. Asymmetric vertical integration. Advances in Theoretical Economics 5(1), Art. 1.
Available at: http://www.bepress.com/bejte/advances/vol5/iss1/art1
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.
Dufeu, I., 2004. Market size and vertical equilibrium in the context of successive Cournot
oligopolies. Topics in Theoretical Economics 4(1), Art. 2.
Available at: http://www.bepress.com/bejte/topics/vol4/iss1/art2
DeGraba, P. J., 1987. The effects of price restrictions on competition between national andlocal firms. Rand Journal of Economics 18, 333–347.
Elberfeld, W., 2002. Market size and vertical integration: Stigler’s hypothesis reconsidered.
Journal of Industrial Economics 50, 23–42.
Gaudet, G. and Long, N.V., 1996. Vertical integration, foreclosure, and profits in thepresence 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 theapparel industry. Journal of Industrial Economics 49, 417–440.
Hemmert, M., 1999. ‘Intermediate organization’ revisited: a framework for the verticaldivision of labor in manufacturing and the case of the Japanese assembly industries.
Industrial and Corporate Change 8, 487–517.
Jansen, J., 2003. Coexistence of strategic vertical separation and integration. International
Journal of Industrial Organization 21, 699–716.
Karlson, S. H., 1985. Spatial competition with location-dependent costs. Journal of Regional Science 25, 201–214.

21

Linnemer, L., 2003. Backward integration by a dominant firm. Journal of Economics andManagement Strategy 12, 231–259.
Matsushima, N., 2004. Technology of upstream firms and equilibrium product differentiation. International Journal of Industrial Organization 22, 1091–1114.
Matsushima, N., 2006. Vertical mergers and product differentiation, GSBA Kobe University
Discussion Paper Series: 2006-9.
http://www.b.kobe-u.ac.jp/publications/dp/2006/2006 9.pdf
Ordover, J., Saloner, G., and Salop, S., 1990. Equilibrium vertical foreclosure. American
Economic Review 80, 127–142.
Salinger, M. A., 1988. Vertical mergers and market foreclosure, Quarterly Journal of Economics 103, 345–356.

22

Appendix
τ = 0.10
Firm 0:

Nointegration

Firm B

Firm ANo integratin
Integratin

0.45705,
0.45447,

Firm 0:Integration

Firm B

Firm ANo integratin
Integratin

0.44002,0.43704,

Firm 0:

Nointegration

Firm B

Firm ANo integratin
Integratin

0.46582,
0.46482,

Firm 0:Integration

Firm B

Firm ANo integratin
Integratin

0.43652,
0.43365,

Firm 0:

Nointegration

Firm B

Firm ANo integratin
Integratin

0.46633,
0.47054,

Firm 0:Integration

Firm B

Firm ANo integratin
Integratin

0.4306,0.43032,

No integratin
0.45705,
0.46107,0.61332
0.616360.46107,0.45879,

Integratin
0.45447,0.45879,

0.616360.61852

No integratin
0.44002,0.44335,

0.626220.628420.44335,0.4401,

Integratin
0.43704,0.4401,0.628420.63021

No integratin
0.46582,
0.47644,0.58789
0.59749

0.47644,0.47639,

Integratin
0.46482,
0.47639,0.59749
0.60278

No integratin
0.43652,0.44406,

0.61992
0.625930.44406,
0.4401,

Integratin
0.43365,0.4401,0.625930.63021

No integratin
0.46633,
0.48553,

0.558220.577080.48553,
0.49119,

Integratin
0.47054,0.49119,0.57708
0.58546

No integratin
0.4306,0.44268,

0.61220.623070.44268,0.4401,

Integratin
0.43032,0.4401,0.623070.63021

τ = 0.20

τ = 0.2923

τ = 0.30
Firm 0:

Nointegration

Firm B

Firm ANo integratin
Integratin

0.46594,0.47097,

Firm 0:Integration

Firm B

Firm ANo integratin
Integratin

0.42979,
0.42993,

Firm 0:

Nointegration

Firm B

Firm ANo integratin
Integratin

0.46167,
0.47261,

Firm 0:Integration

Firm B

Firm ANo integratin
Integratin

0.42427,0.42754,

Firm 0:

Nointegration

Firm B

Firm ANo integratin
Integratin

0.46063,0.47273,

Firm 0:Integration

Firm B

Firm ANo integratin
Integratin

0.42325,
0.42713,

No integratin
0.46594,
0.48627,0.554540.57465

0.48627,
0.49277,

Integratin
0.47097,
0.49277,

0.57465
0.58336

No integratin
0.42979,
0.44242,0.611210.62272

0.44242,0.4401,

Integratin
0.42993,0.4401,0.62272
0.63021

No integratin
0.46167,
0.48957,

0.530990.559450.48957,
0.50197,

Integratin
0.47261,0.50197,0.559450.57009

No integratin
0.42427,0.44039,0.60479
0.620450.44039,0.4401,

Integratin
0.42754,0.4401,0.620450.63021

No integratin
0.46063,0.48993,

0.526810.55683

0.48993,
0.50345,

Integratin
0.47273,
0.50345,0.55683
0.56777

No integratin
0.42325,
0.43998,

0.60363
0.62005

0.43998,0.4401,

Integratin
0.42713,0.4401,0.62005
0.63021

τ = 0.36

τ = 0.37

24

τ = 0.40
Firm 0:

Nointegration

Firm B

Firm ANo integratin
Integratin

0.45694,
0.47285,

Firm 0:Integration

Firm B

Firm ANo integratin
Integratin

0.42,
0.42588,

Firm 0:

Nointegration

Firm B

Firm ANo integratin
Integratin

0.43835,0.47054,

Firm 0:Integration

Firm B

Firm ANo integratin
Integratin

0.40735,
0.42151,

No integratin
0.45694,
0.49072,

0.51389
0.548840.49072,
0.50781,

Integratin
0.47285,0.50781,0.54884
0.56063

No integratin
0.42,
0.43864,0.6
0.61880.43864,0.4401,

Integratin
0.42588,0.4401,0.61880.63021

No integratin
0.43835,
0.49005,

0.46655
0.52106

0.49005,
0.52138,

Integratin
0.47054,0.52138,

0.52106
0.53494

No integratin
0.40735,
0.43291,

0.586180.614180.43291,0.4401,

Integratin
0.42151,0.4401,0.614180.63021

τ = 0.50

25