Discussion Paper No. 754

HORIZONTAL MERGERS,FIRM HETEROGENEITY,
AND R&D INVESTMENTS
Noriaki Matsushima
Yasuhiro Sato
Kazuhiro Yamamoto

September 2009

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Horizontal mergers, ﬁrm heterogeneity, and R&D investments∗
Noriaki Matsushima†

Yasuhiro Sato‡

Kazuhiro Yamamoto§

September 8, 2009

Abstract
We investigate the incentive and the welfare implications of a merger when heterogeneous oligopolists compete both in process R&D and on the product market. We examine
how a merger aﬀects the output, investment, and proﬁts of ﬁrms, whether ﬁrms havemerger incentives, and, if so, whether such mergers are desirable from the viewpointof social welfare. We also derive equilibrium conﬁgurations and explore their welfare
properties.

JEL classiﬁcation: L41, L13, O32
Key words: mergers, oligopoly, R&D, heterogeneity∗
We would like to thank seminar participants at Nagoya University for their helpful comments. Our
research has been ﬁnancially supported by the Grand-in-Aid for Scientiﬁc Research.
†
Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,
Ibaraki, Osaka 567-0047, Japan. E-mail: nmatsush@iser.osaka-u.ac.jp‡
Yasuhiro Sato, Graduate School of Economics, Osaka University, 1-31 Machikaneyama, Toyonaka, Osaka
560-0043, Japan. E-mail: ysato@econ.osaka-u.ac.jp§
Kazuhiro Yamamoto, Graduate School of Economics, Osaka University, 1-31 Machikaneyama, Toyonaka,
Osaka 560-0043, Japan. E-mail:yamamoto@econ.osaka-u.ac.jp

1

Introduction

When ﬁrms face decisions on mergers, they expect to acquire greater market shares and/or toachieve greater eﬃciencies. These eﬃciency gains include operating synergies, diversiﬁcation,ﬁnancial synergies, and asset divestiture (Burgelman [4], Jensen and Ruback [13], Larsson
and Finkelstein [15], and Tremblay and Tremblay [23]). The prospect of economies of scaleis also an important motive for a horizontal acquisition (Capron [5]).1 Economies of scale
arise as a result of a horizontal merger if the merged ﬁrm achieves unit cost savings when it
increases the scale of a given activity. They can also be achieved in functional areas including
R&D, distribution, sales, and administrative activities through the spread of ﬁxed costs.
Among the elements related to economies of scale, technological innovation has often
been mentioned as one of the main reasons for M&As (de Man and Duysters [7] and Zhao[24]). One reason for this is that M&As may raise the overall R&D budgets of the companies
involved (de Man and Duysters [7]). This allows them to reap economies of scale and enables
them to tackle larger R&D projects than each individual ﬁrm could have done independently.
For instance, since the 1990s, Alcoa has sequentially taken over Alumax and Reynolds, whichwere US aluminum companies, while Alcan, Pechiney, and Alusuisse carried out a three-way
merger in 2003. Those mergers might enable the companies to enlarge their capacities
and/or innovation activities. On the other hand, M&As may be executed in order to lessen
innovation expenditure if the innovative activities of merged ﬁrms overlap strongly witheach other. For instance, when oil companies merge, they expect to yield savings in costs,
including transportation and plant construction costs, because they can use their installed oil
plants cooperatively and can eliminate the overlaps of their oil plants functions (Matsushima[17]).
Eﬃciency gains from M&As are also achieved by merging with a ﬁrm with higher production eﬃciency if a technology spillover occurs within the merged ﬁrm. This can be astrong merger incentive when ﬁrms are heterogeneous in terms of their production eﬃciency.
1

Capron [5] pointed out the signiﬁcant risk of damaging acquisition performance when the divested assets

and redeployed resources are those of the target.

2

This implies that, for a low-productivity ﬁrm, merging with a high-productivity ﬁrm canbe considered as an alternative to R&D. As a result, the eﬃciency levels of ﬁrms are determined endogenously, and the achieved levels depend crucially on combinations of merged
ﬁrms. Therefore, it would be worth analyzing M&As in the presence of both R&D activities
related to ﬁrm eﬃciencies and heterogeneity of ﬁrms.
Several existing studies have examined the eﬀects of R&D activities on M&As (Davidson
and Ferrett [6] and Stenbacka [22]). In addition, some previous research concerning M&As
took into account the heterogeneity of ﬁrms (Perry and Porter [19], Farrell and Shapiro [8],Barros [3], and Fumagalli and Vasconcelos [9]). None of these studies, however, incorporates
both eﬀects (R&D and heterogeneity) into their analytical models.
We investigate horizontal mergers in a Cournot model with ﬁrm heterogeneity and R&D
investment. We ﬁrst explore in detail the possible impacts of a merger on the output,
investment, and proﬁts of ﬁrms.2 We then examine whether ﬁrms have merger incentives
and, if so, whether such mergers are desirable from the viewpoint of social welfare. Our
results imply that although the importance of R&D and ﬁrm heterogeneity leads to merger
incentives, such mergers are desirable only when the share of eﬃcient ﬁrms is high. In our
setting, assuming suﬃciently large cost diﬀerences between eﬃcient and ineﬃcient ﬁrms,
if R&D is not costly, a merger between homogeneous ﬁrms tends to occur in equilibrium;otherwise, a merger between heterogeneous ﬁrms tends to occur. Moreover, if R&D is notcostly, it is likely that we will observe a discrepancy between the equilibrium and optimal
conﬁgurations.
We brieﬂy review three closely related articles. Davidson and Ferrett [6] developed a
Cournot model involving a cost-reducing R&D investment. In their model, ﬁrms with thesame ex ante marginal cost produce diﬀerentiated goods, and a merged ﬁrm produces two
types of goods. Investment is good speciﬁc, but investment has spillover eﬀects. They
showed that investment increases the proﬁtability of mergers and also explored how thedegree of the spillover aﬀects this proﬁtability. Stenbacka [22] considered the case in which2Note that we cannot discuss the eﬀect of the spillover of the investment, as in Davidson and Ferrett [6],because we assume that ﬁrms produce a homogeneous good.

3the results of a cost-reducing investment are uncertain and the realization of the results is
private information belonging to the innovating ﬁrm. Stenbacka then examined whether
merger anticipations lead to ex ante incentives for innovating ﬁrms to reveal their costreducing information.3 Barros [3] investigated a simple model highlighting the basic economic
intuition about the relationship between initial market concentration and the size asymmetryof merger participants. He showed that, in more concentrated markets, mergers involve lessasymmetric ﬁrms, and merger participants rank high in terms of the size distribution ofﬁrms. However, he did not take into account R&D activities.
The remainder of the paper is organized as follows. Section 2 presents the basic modeland some results. Section 3 provides the results in the heterogeneous ﬁrm case. Section 4
concludes the paper.

2

Basic setup

We ﬁrst provide a basic model that involves investments and merger decisions, and explain
the basic characteristics of the eﬀects of investment on mergers. Consider an industry consisting of N (≥ 3) ﬁrms initially. The model has three stages: ﬁrst, it is determined whether
or not predetermined pairs of ﬁrms merge; second, after the merger decisions, each (merged)ﬁrm chooses its investment level, which subsequently determines its marginal cost; and in
the third stage, each ﬁrm engages in Cournot competition, given the decisions on the merger
and the realized marginal costs.
More precisely, in the second stage, each (merged) ﬁrm determines how much to invest
in cost-reducing R&D activities. Let xi denote the investment level chosen by ﬁrm i. A unit
increase in investment decreases the ﬁrm’s marginal cost by the same margin. Thus, thetotal production cost incurred by the ﬁrm is given by (zi − xi )qi , where qi denotes the output
level chosen by ﬁrm i. In this speciﬁcation, zi signiﬁes the ex ante marginal cost (before theinvestment), whereas zi − xi denotes the ex post marginal cost (after the investment). The
cost of the investment is assumed to be γx2i . γ represents the degree of cost required for
3

Atallah [2] also analyzed merger proﬁtability in the case of cost-reducing R&D. In his numerical example,

however, R&D investment does not provide incentives for mergers in most of cases.

4

R&D investment.
We ﬁrst consider an equilibrium without mergers. In the third stage, upon observing xi ,the ﬁrms engage in standard Cournot (quantity) competition. The inverse demand functionis speciﬁed as follows:p = 1 − Q,(1)where Q is the total output. Each ﬁrm simultaneously chooses qi so as to maximize its own
proﬁt. Here, we assume that zi is the same across ﬁrms (zi = z, 0 < z < 1). In the next
section, we allow zi to diﬀer across ﬁrms in a particular way. The proﬁt of ﬁrm i is given by:
πi = [P − (z − xi )] qi − γx2i .
The Cournot competition leads to the following output levels:∑
1 − z + (N + 1)xi − N
j=1 xjqi =,n+1∑
N (1 − z) + N
j=1 xj
Q=
.
N +1

(2)

Substituting this and (1) into the proﬁt function, we can derive the optimal investment level
chosen by ﬁrm i in the second stage:xi =N (1 − z).
γ (N + 1)2 − N(3)This yields dxi /dN < 0 and readily proves that a merger (i.e., a reduction in the number N
of ﬁrms) increases the investment level. A merger reduces the competition among ﬁrms andincreases the market share of each ﬁrm, which leads to stronger incentives for cost-reducing
investment. Equations (3) and (2) indicate the proﬁt of ﬁrm i without the merger:[]
γ (1 − z)2 γ (N + 1)2 − N 2
.
πi =
[
]2
γ (N + 1)2 − N
From this, we can see that a ﬁrm supplies goods if and only if:4
γ>
4

N2
.
(N + 1)2This is also the second-order condition for the investment decision.

5(4)

When this condition does not hold, competition in investment is so intense that no ﬁrm canmake positive proﬁts.
We now examine how the existence of the investment decision aﬀects the proﬁtability
and desirability of a merger. Here, in order to make the analysis as simple as possible, we
consider a pairwise merger; i.e., a merger between two ﬁrms only. A pairwise merger only
reduces the number of ﬁrms from n to n − 1, implying that the proﬁt πM of a merged ﬁrm
becomes:πM =][γ (1 − z)2 γN 2 − (N − 1)2(γN 2 − N + 1)2.

In this section, we use simple gains from the merger as a criterion of a merger incentive.5
Therefore, when we consider a pairwise merger, we compare πM with the joint proﬁt of ﬁrms
involved in the merger described in (4). If the former is larger than the latter, we consider
that this merger is proﬁtable and that these two ﬁrms have an incentive to merge. Moreformally, a pairwise merger is proﬁtable if:
πM − 2πi > 0.

(5)

We can show that whether this inequality holds depends on the (premerger) number n ofﬁrms and the investment cost parameter γ:
 [

]
2


2
 2 N − γ (N + 1)
(1 − N )2 − γN 2 2.
πM − 2πi = γ (1 − z)[]2 −
(1 − N + γN 2 )2 
 N − γ (N + 1)2
(6)

Because 0 < z < 1, whether (5) holds true depends on the sign of the term in curly
brackets, which depends only on γ and N . The following ﬁgure describes the region wherea pairwise merger is proﬁtable in the γ − N plane.
[Figure 1 here]

5

This criterion is consistent with the equilibrium concept of the core (see Horn and Persson [11]). In fact,

when ﬁrms are homogeneous, an allocation involving a merger that satisﬁes this criterion is in the core under
the restriction of a pairwise merger. In the next section, where we introduce ﬁrm heterogeneity, we use thecore as an equilibrium concept and provide an analysis of the equilibrium.

6

The shaded areas of Figure 1-(i) represent the combinations of the investment cost parameterγ and the number of ﬁrms N under which a pairwise merger is proﬁtable. Note ﬁrst that in
our model, a merger changes (a) the degree of competition and (b) the level of investment.
As shown by Salant et al. [21], in standard Cournot competition, only the eﬀect on (a) isrelevant, and a merger is not proﬁtable unless a merged ﬁrm occupies more than 80% of
the market share. In contrast, once we introduce the cost-reducing investment, a pairwise
merger is more likely to be proﬁtable because of the eﬀect on (b). Equation (3) implies
that a pairwise merger increases the investment of each ﬁrm in cost-reducing R&D. A larger
investment leads to a lower marginal cost and raises the proﬁt of each ﬁrm for a given
market share. Because of this eﬀect, a pairwise merger that is not proﬁtable under standard
Cournot competition may become proﬁtable. For example, when γ = 1, a pairwise merger is
proﬁtable for N ≤ 4, and the merged ﬁrm occupies one-third (+33%) at most of the market
share.
Second, for a given γ, a pairwise merger is more likely to be proﬁtable for a smaller N .
This is because the fewer the number of ﬁrms, the more a pairwise merger increases themarket share, which is likely to increase the additional proﬁts of a merger.6
Finally, when investment is less costly, a pairwise merger is proﬁtable for a larger N .
In fact, a pairwise merger becomes proﬁtable for any number N (≥ 3) of ﬁrms if the cost
parameter γ becomes suﬃciently small (The proof is given in Appendix A).
Summarizing the above arguments, we have the following proposition.
Proposition 1 The cost-reducing R&D investment makes it more likely that a pairwise
merger will be proﬁtable. A pairwise merger is proﬁtable for any number of ﬁrms if theinvestment cost is suﬃciently small.
We next examine the desirability of a pairwise merger. We use the social surplus W as
6

A change in the market share due to a pairwise merger is given by:
1
1
1
− =
,
n−1
n
n(n − 1)

which is larger for a smaller number of ﬁrms.

7

the criterion of welfare:
Q2 ∑
+
πi .
2
N

W = consumer surplus (CS) + sum of ﬁrms’ proﬁts =

(7)

i=1

Denoting the premerger and postmerger surpluses as W and WM , respectively, a merger
is desirable if and only if WM − W > 0. The shaded areas of Figure 1-(ii) describe the
combinations of γ and N under which a merger is desirable from the viewpoint of social
welfare. For a given γ, a pairwise merger is less likely to be desirable for a smaller N , as
the merger eﬀect on the degree of competition is larger. Moreover, when investment is lesscostly, a pairwise merger is desirable for any number N (≥ 3) of ﬁrms if the cost parameterγ is suﬃciently small (The proof is given in Appendix B):
Proposition 2 The cost-reducing R&D investment makes it more likely that a pairwise
merger will be desirable. A pairwise merger is desirable for any number of ﬁrms if theinvestment cost is suﬃciently small.
Figure 1-(iii) shows the areas in which proﬁtability and desirability do not go together.
A pairwise merger is proﬁtable but undesirable when investment is costly and the number ofﬁrms is small (γ > 1.5 and N is close to 3). In this case, the eﬀect of reducing competition
among ﬁrms is large enough for a merger to be proﬁtable, but the merger eﬀect of enhancing
investment is small, leading to decreases in the consumer surplus. The latter eﬀect dominates
the former, and the social surplus decreases. A pairwise merger is unproﬁtable but desirable
when the number of ﬁrms is large and the cost of investment is not small (the upper-rightarea of Figure 1-(iii)). With a large number of ﬁrms, the eﬀect of a merger on the degreeof competition is small, and it scarcely reduces the consumer surplus. Hence, because ofincreases in the investment, the consumer surplus increases. In contrast, a small impact on
the degree of competition leads to an unproﬁtable merger. The former eﬀect dominates thelatter, and a merger increases the social surplus.

8

3

Heterogeneous ﬁrms

Now, we introduce the heterogeneity of ﬁrms: we allow z to diﬀer across ﬁrms in the most
simple way. Firms are either eﬃcient and equipped with a low ex ante marginal cost zl orineﬃcient and equipped with a high ex ante marginal cost zh , where zl < zh .
We assume that if a pair of ﬁrms merge, the value of zi becomes the minimum value
between them. In this setting, a merger between heterogeneous ﬁrms is similar to the elimination of the less eﬃcient of the merging ﬁrms. Therefore, the incentive for a merger canpossibly come from two sources: one is the eﬃciency gain that arises in the case of merger
between heterogeneous ﬁrms, and the other is the gain from investment that is explained in
the previous section.
Suppose that zl = c and zh = c + δ, where c and δ are positive constants. δ represents thediﬀerence between eﬃcient and ineﬃcient ﬁrms. We can apply the same calculation method
to derive the equilibrium outcomes in the two cases. The ex ante marginal cost of a ﬁrm canbe represented by c + ci , where ci (i = 1, 2, . . . , , N ) takes value of either 0 or δ.
When there are N ﬁrms, the proﬁt functions of the ﬁrms are denoted by:π(N, c; ci ) = [1 − Q − (c + ci − xi )] qi − γx2i ,where c ≡ (c1 , c2 , . . . , cN ). Given the values of x, the ﬁrst-order conditions lead to:∑
1 − c − N (ci − xi ) + j̸=i (cj − xj ).
q(N, c; ci ) =N +1
Substituting q into the proﬁt functions, we derive the optimal investment levels x:{}∑
N [γ(N + 1) − N ] (1 − c) − N [γ(N + 1) − 1] ci + γ(N + 1) j̸=i cj
x(N, c; ci ) =.
[γ(N + 1)2 − N ] [γ(N + 1) − N ]

(8)

(9)

¿From the optimal investment levels, we have the equilibrium quantities supplied by theﬁrms:q(N, c; ci ) ={}∑γ(N + 1) [γ(N + 1) − N ] (1 − c) − N [γ(N + 1) − 1] ci + γ(N + 1) j̸=i cj[γ(N + 1)2 − N ] [γ(N + 1) − N ]9.

The proﬁts and the consumer surplus are given by:[]
γ γ(N + 1)2 − N 2π(N, c; ci ) =[γ(N + 1)2 − N ]2 [γ(N + 1) − N ]2× [γ(N + 1) − N ] (1 − c) − N [γ(N + 1) − 1] ci + γ(N + 1)CS =γ 2 (N + 1)2 [N (1 − c) −

∑

∑

2
cj  ,

j̸=i
2

i ci ].

2 [γ(N + 1)2 − N ]2

n eﬃcient and m ineﬃcient ﬁrms exist

We consider the case in which n eﬃcient andm ineﬃcient ﬁrms exist (the number of ﬁrms is N = n + m). Using (8) and (9), we knowthat an eﬃcient ﬁrm invests and produces more than an ineﬃcient ﬁrm:7
δN> 0,γ(N + 1) − Nγδ(N + 1)
q(n, m, c; 0) − q(n, m, c; δ) => 0.
γ(N + 1) − N

x(n, m, c; 0) − x(n, m, c; δ) =(10)

Moreover, the proﬁt functions and the consumer surplus are given by:[]
γ γ(N + 1)2 − N 2 {[γ(N + 1) − N ] (1 − c) + γδ(N + 1)(N − n)}2π(n, m, c; 0) =,[γ(N + 1)2 − N ]2 [γ(N + 1) − N ]2[]γ γ(N + 1)2 − N 2 {[γ(N + 1) − N ] (1 − c) − δ [γ(N + m − 1)(N + 1) − N ]}2π(n, m, c; δ) =,[γ(N + 1)2 − N ]2 [γ(N + 1) − N ]2CS =γ 2 (N + 1)2 [N (1 − c) − δ(N − n)]22 [γ(N + 1)2 − N ]2.

In this case, δ must satisfy the following inequality because each ﬁrm’s quantity suppliedmust be positive.
0<δ<(1 − c)((N + 1)γ − N )≡ δ̄.
(n + 1)(N + 1)γ − NThis determines the relevant interval of γ:
γ>

7

N
.
N +1

The following inequalities hold true because we assume that γ > 1.

10

(11)

3.1

Merger eﬀects

In this subsection, we analyze the eﬀects of a merger on each ﬁrm’s levels of investment,
output, and proﬁt. In our current framework, a pairwise merger takes one of the following
three forms.
Type (I) A merger of eﬃcient ﬁrms.
Type (II) A merger of heterogeneous ﬁrms.
Type (III) A merger of ineﬃcient ﬁrms.
Because we assumed that the marginal cost of a merged ﬁrm is equal to the lesser marginal
cost of the ﬁrms that compose the merger, a Type (I) merger is represented by a reductionin the number n of eﬃcient ﬁrms, and a merger of Types (II) and (III) is described by adecline in the number m = N − n of ineﬃcient ﬁrms.
We ﬁrst see how mergers of diﬀerent types of ﬁrms aﬀect ﬁrms. From Section 2, we knowthat a merger increases the investment level if ﬁrms are symmetric. In the following threecases, the relation between the investment level and a merger is satisﬁed:∂x(n, m, c; δ)∂x(n, m, c; δ)∂x(n, m, c; 0)
< 0,
< 0,
< 0.
∂n
∂n
∂m
However, this relation does not always hold:∂x(n, m, c; 0)(1 − c)(N 2 − 1)((N + 1)γ − N )2
> 0 iﬀ δ̄ > δ >,3∂m(N + 1) ((n + 2)N − n) − N 2 (N + 1)2 (n + 2)γ + N 2 (N 2 + n)(12)∂x(n, m, c; 0)< 0 otherwise.
∂m
When the cost diﬀerence between eﬃcient and ineﬃcient ﬁrms is large (δ is large), a merger
including an ineﬃcient ﬁrm decreases the investment levels of the eﬃcient ﬁrms.
We now mention the mechanism behind the inequalities (Ishida et al. [12]). When aﬁrm enlarges its eﬀort to reduce its marginal cost, two eﬀects exist: (1) direct gain and (2)strategic gain. The direct gain stems from the quantity supplied by the ﬁrm. In other words,
the signiﬁcance of the cost saving is correlated to the quantity supplied. The strategic gain11

stems from decreases in the quantities supplied by the ﬁrm’s rivals (because of an increase in
their market price). This is because the improvement of a ﬁrm’s eﬃciency reduces the rivals’
quantities supplied because of strategic substitutability. The two gains depend on the numberof competitors. In general, the direct gain decreases with the number of competitors (seeSection 2). However, the negative eﬀect of the increase in competitors is small when thosecompetitors are ineﬃcient. The strategic gain may increase with the number of competitors
because a unit decrease in a ﬁrm’s marginal cost can aﬀect more ﬁrms when the number ofcompetitors is large. Because the quantity supplied by each eﬃcient ﬁrm is large, the strategic
gain (the price increase) has a signiﬁcant impact on the incentive to invest. Therefore, if
an additional competitor is less eﬃcient, an increase in ineﬃcient competitors enhances the
R&D incentives of the eﬃcient ﬁrms.
We now check how diﬀerent the impact of a merger on the strategic variables is fordiﬀerent types of mergers:8()∂x(n, m, c; 0) ∂x(n, m, c; 0)
−
−
> 0,
∂n
∂m
(
)
∂x(n, m, c; δ) ∂x(n, m, c; δ)
−
−
> 0.
∂n
∂m

(13)

A Type (I) merger has a larger impact on ﬁrms’ investments than a merger of Type (II) or(III). An eﬃcient ﬁrm occupies a larger market share than an ineﬃcient ﬁrm, which implies
that a Type (I) merger, which reduces the number of eﬃcient ﬁrms, weakens the competition
among ﬁrms and increases the incentive for investment more than do mergers of other types.
A similar analysis is possible regarding the output of each ﬁrm:)
∂q(n, m, c; 0) ∂q(n, m, c; 0)
−
> 0,
−
∂n
∂m
(
)
∂q(n, m, c; δ) ∂q(n, m, c; δ)
−
−
> 0.
∂n
∂m
(

(14)

A Type (I) merger increases the output of each ﬁrm by more than does a merger of othertypes. This result arises from two eﬀects: ﬁrst, a Type (I) merger weakens the competition
among ﬁrms more do other types of merger; second, it increases the cost-reducing investment
8

All of the explicit derivations are given in Appendix C.

12

by more than do other types of mergers. As shown in Appendix C, a larger value of γ makesthese tendencies weaker, implying that the lower cost of investment strengthens the impacts
of a Type (I) merger relative to those of a merger of Type (II) or (III).
Next, we examine how a particular type of merger diﬀerently aﬀects diﬀerent ﬁrms:∂(x(n, m, c; 0) − x(n, m, c; δ)) < 0,∂n∂−(x(n, m, c; 0) − x(n, m, c; δ)) < 0,∂m∂−(q(n, m, c; 0) − q(n, m, c; δ)) < 0,∂n∂−(q(n, m, c; 0) − q(n, m, c; δ)) < 0.
∂m−

Any type of merger aﬀects ineﬃcient ﬁrms more than it aﬀects eﬃcient ﬁrms. This result
comes from the assumption of the quadratic cost function. As seen in (10), the level ofinvestment is larger for an eﬃcient ﬁrm than for an ineﬃcient ﬁrm. When the cost function isquadratic, even a small change in the investment level leads to large changes in the investment
cost for an eﬃcient ﬁrm, which makes an eﬃcient ﬁrm less sensitive to a particular change.
Appendix C shows that such diﬀerences between eﬃcient and ineﬃcient ﬁrms in terms of
their reactions to a particular merger become larger as the investment becomes less costly(as the value of γ becomes smaller).
Summarizing the above arguments, we obtain the following proposition.
Proposition 3 (a) A merger between eﬃcient ﬁrms (a Type (I) merger) has larger impacts
on the levels of investment and output than does a merger involving an ineﬃcient ﬁrm(a merger of Type (II) or (III)). (b) A reduction in the number of ﬁrms of a particular
type has larger impacts on an ineﬃcient ﬁrm than on an eﬃcient ﬁrm. (c) Thesetendencies become more prominent as the cost of investment declines.
Although we have uncovered the eﬀects of a merger on investment and output, it isvery diﬃcult to know whether a merger is proﬁtable and desirable for arbitrary values of δ
(ﬁrm diﬀerence) and γ (investment cost) with n eﬃcient and m ineﬃcient ﬁrms. In orderto obtain clear-cut results, we need to specify either a pair of δ and γ or a pair of n and m.
13

The following propositions shows the proﬁtability and desirability of a merger at the limitsof δ and γ, which are comparable to the results shown in Propositions 1 and 2 (the proof isgiven in Appendix D).
Proposition 4 Suppose that ﬁrm diﬀerences are suﬃciently large and that the investment
cost is suﬃciently small. A Type (I) or Type (II) merger is almost always proﬁtable.
A Type (III) merger is always proﬁtable. A merger of any type is desirable if the ratio
of eﬃcient ﬁrms is suﬃciently high.
As shown in Appendix D, a Type (I) or Type (II) merger is proﬁtable when the ratio ofeﬃcient ﬁrms is higher than some threshold values (TpI and TpII ), which are given as follows.
A Type (I) merger:n> TpI ≡N√

A Type (II) merger:n> TpII ≡N

N [N 2 (N + 2) − 1]/2.
N 2 (N − 1) + 1√

N [N 2 (N + 2) − 1].
N 2 (N − 1) + 1

We can readily check that TpI and TpII decline as the number N of total ﬁrms increases.
When N = 3, TpI + 0.14 and TpII + 0.2. Hence, in this case, any type of merger is
proﬁtable if at least one ﬁrm is eﬃcient. These threshold values decline to TpI + 0.04 and
TpII + 0.05 (TpI + 0.0085 and TpII + 0.012) when N = 5 (N = 10). Therefore, we can safely
say that, even if ﬁrms are heterogeneous, a pairwise merger is very likely to be proﬁtable
if the diﬀerence between ﬁrms is large and the investment cost is small even if ﬁrms areheterogeneous.
In contrast, we obtain very diﬀerent results in terms of the desirability of a merger whencomparing the case with heterogeneous ﬁrms and the case with homogeneous ﬁrms. From
Appendix D, the threshold value for desirability for any merger (Td ) is given as:
2[N 2 (N − 1) + 1]2n> Td ≡.
N
N (2N 5 − 4N 3 + N − 1)14

We can see that Td is increasing in N . When N = 3, Td + 0.63, implying that a merger isdesirable if there is more than one eﬃcient ﬁrm although it is undesirable if there is only
one eﬃcient ﬁrm. Td increases to 0.70 and 0.82 when N becomes 5 and 10, respectively. As
shown in Proposition 3-(b), a particular type of merger has larger impacts on the output of
an ineﬃcient ﬁrm than on that of an eﬃcient ﬁrm. This implies that the lower the shareof eﬃcient ﬁrms is, the more a merger reduces the total output. This eﬀect dominates theinvestment-enhancing eﬀect of a merger when the share of eﬃcient ﬁrms is lower than acertain threshold level, leading to welfare loss.
In the next subsection, we specify the number of ﬁrms as three and examine fully which
type of merger is likely to occur under particular values of δ and γ. In the case of threeﬁrms, we can go beyond the arguments on proﬁtability to arguments on the equilibrium.

3.2

Equilibrium and its eﬃciency—the case of three ﬁrms

Here, we demonstrate how the eﬀects uncovered in the previous section interact to yield
equilibrium conﬁgurations by using three ﬁrm cases. We follow Horn and Persson [11] anduse the core as an equilibrium concept. In so doing, we consider only a pairwise merger andassume that each merger consists of two ﬁrms. This assumption has been used frequently inprevious studies and is appropriate in analyzing merger decisions, especially in the three-ﬁrm
model, because monopoly is prohibited in most countries.9 The ownership structure of theindustry is assumed to be formed through a cooperative game of coalition formation. With
three ﬁrms, it is interesting to consider the following two cases: (i) one eﬃcient and two
ineﬃcient ﬁrms exist, and (ii) two eﬃcient and one ineﬃcient ﬁrms exist. In these cases, wehave four possible market structures.
Type (0) No merger.
Type (I) A merger of eﬃcient ﬁrms.
Type (II) A merger of heterogeneous ﬁrms.
9

See Davidson and Ferrett [6], Lommerud et al. [16], and Qiu and Zhou [20] for recent examples.

15

Type (III) A merger of ineﬃcient ﬁrms.
In case (i), there are one eﬃcient ﬁrm (labeled A) and two ineﬃcient ﬁrms (labeled E and
F ). Therefore, we have the following three possible market structures and four ownership
structures.
(i)Type (0) No merger: M0 = {A, E, F }.
(i)

(i)′

Type (II) A merger of heterogeneous ﬁrms: M2 = {AE, F }, M2= {AF, E}.

(i)Type (III) A merger of ineﬃcient ﬁrms: M3 = {A, EF }.
In case (ii), there are two eﬃcient ﬁrms (labeled A and B) and one ineﬃcient ﬁrms
(labeled E), and the following three possible market structures and four ownership structures
are relevant.
(ii)

Type (0) No merger: M0= {A, B, E}.
(ii)

Type (I) A merger of eﬃcient ﬁrms: M1= {AB, E}.
(ii)

Type (II) A merger of heterogeneous ﬁrms: M2(ii)′= {AE, B}, M2= {A, BE}.

The solution procedure is based on Horn and Persson [11]. They treat the merger process
as a cooperative game of coalition formation, where the players are free to communicate andto write binding contracts. When owners of ﬁrms agree on a merger, they can decide on
any division of the ﬁrm’s proﬁts. However, payments between coalitions are not allowed.
The approach then involves a comparison of any two possible ownership structures Mi and
Mj . Mi is said to “dominate” Mj if the combined proﬁts of the “decisive” group of owners
are larger in Mi than in Mj . The decisive group of owners consists of the owners that are
expected to be able to inﬂuence whether Mi will be formed instead of Mj and vice versa.
In our model, although owners belonging to identical coalitions in Mi and Mj cannot aﬀect
whether Mj will be formed instead of Mi , all remaining owners can inﬂuence this choice and(i)thus they are decisive.10 For instance, when we compare Type (0) (no merger, M0 ) and10

See Horn and Persson [11] for a detailed discussion on these points.

16

(i)

Type (III) (a merger of ineﬃcient ﬁrms, M3 ) in case (i), ﬁrm A does not merge in both
(i)

(i)

ownership structures, and ﬁrms E and F merge in M3 and do not merge in M0 . In this
(i)

(i)

case, E and F are the decisive owners, whereas A is not, implying that M3 dominates M0(i)

if the proﬁt of the merged ﬁrm in M3 is larger than the sum of proﬁts of ﬁrms E and F in(i)

M0 . The ownership structure is in the core if the structure is undominated by any otherstructures. We use the core as the equilibrium concept in the following analysis.

One eﬃcient and two ineﬃcient ﬁrms exist

There are two types of pairwise mergers:

Type (II), a merger between heterogeneous ﬁrms; and Type (III), a merger between ineﬃcientﬁrms. Figure 2 illustrates the combinations of the cost parameter (γ) and the degree of ﬁrm
diﬀerence (δ/(1 − c)) that lead to equilibrium.
[Figure 2 here]
Note ﬁrst that, because both types of mergers imply a decrease in the number of ineﬃcientﬁrms, neither of the ownership structures in the two types of mergers dominates the other.
Hence, a particular ownership is in the core if it is undominated by the ownership structurein the no-merger case. A Type (II) merger takes place in equilibrium for a broad area,
whereas we observe a Type (III) merger when the investment cost is not large, which comesfrom Proposition 3 (c). When ﬁrms are less diﬀerent and R&D investments are very costly,
the no-merger case is in the core because of the “merger paradox” (Salant et al. [21]). Whenﬁrms are very diﬀerent and R&D investments are not costly (see the upper left area in Figure2), a merger between ineﬃcient and eﬃcient ﬁrms diminishes the merged ﬁrm’s incentive toinvest (see (12)), and then this merger is not proﬁtable. On the other hand, it is likely that
two ineﬃcient ﬁrms will merge because the merger mitigates the eﬃcient ﬁrm’s incentive toinvest (see (12)).
We now discuss how those pairwise mergers change social welfare. The two types of
pairwise mergers lead to the same duopoly market with one eﬃcient and one ineﬃcient ﬁrm.
From the results in the previous subsections, we have the conditions under which thosepairwise mergers enhance/harm social welfare.
17[Figure 3 here]
We now mention the mechanism behind the eﬀects of mergers depicted in Figure 3. Whenthe cost diﬀerence is moderate, a merger is desirable because the elimination of an ineﬃcient
ﬁrm improves the production eﬃciency in this market (Lahiri and Ono [14]). When the costdiﬀerence is large enough and R&D investments are not costly, a merger is not desirable.
The existence of ineﬃcient ﬁrms induces the eﬃcient ﬁrm to engage in more R&D (see (12)),which then accelerates the scale economies of R&D investments engaged in by the eﬃcientﬁrm.
Two eﬃcient ﬁrms and one ineﬃcient ﬁrm exist

There are two types of pairwise

mergers: Type (I), a merger between eﬃcient Firms; and Type (II), a merger between
heterogeneous ﬁrms. The following two ﬁgures describe the equilibrium and its welfare
properties.
[Figure 4 here][Figure 5 here]
In this case, again, a Type (II) merger takes place in equilibrium over a broad area of
the Figure. In contrast, a Type (I) merger occurs only when the degree of ﬁrm diﬀerence is
suﬃciently large and the investment cost is suﬃciently small. A Type (I) merger leads to a
duopoly of eﬃcient and ineﬃcient ﬁrms, whereas a Type (II) merger results in a duopoly of
two eﬃcient ﬁrms. Proposition 3 (a) implies that the former has larger impacts on the pricethan the latter. This eﬀect is ampliﬁed as γ gets smaller (Proposition 3 (c)). Moreover, alarge ﬁrm diﬀerence indicates that the merged ﬁrm in a Type (I) merger is more likely tohave a higher market share, implying that this type of merger is more likely to be proﬁtable.
Meanwhile, from Proposition 3 (b), an outside ﬁrm (an ineﬃcient ﬁrm) in a Type (I) merger
gains more than the merged ﬁrm and this ownership structure is in equilibrium.
We now mention the mechanism behind the eﬀects of mergers depicted in Figure 5. The
mechanism behind a Type (II) merger is similar to the previous case in which only one
eﬃcient ﬁrm exists. When the cost diﬀerence is small and R&D investments are not costly,18

a Type (I) merger is desirable even though it does not take place. The desirability of themerger stems from the scale economies of R&D investments, but it does not take placebecause of the “merger paradox” (Davidson and Ferrett [6] and Salant et al. [21]). Whenthe cost diﬀerence is large and R&D investments are not costly, a Type (I) merger is notdesirable even though it takes place. The desirability of the merger stems from the increasein the degree of market concentration, but again the merger does not take place for thereasons discussed in the previous paragraph. Because the ineﬃcient ﬁrm is highly ineﬃcient,a Type (I) merger gives a strong market power to the merged ﬁrm.

3.3

Discussions on ﬁrm heterogeneity

In this subsection, we consider the case in which three ﬁrms are all diﬀerent: ﬁrms A andC, respectively, are eﬃcient and ineﬃcient, and their ex ante marginal costs are c and c + δ,
respectively. Firm B is the second most eﬃcient ﬁrm, and its ex ante marginal cost is c + αδ,
where 0 ≤ α ≤ 1.
[Figure 6 here]
Figure 6 shows that the desirability of a merger is nonmonotonic with respect to α when
R&D investments are not costly (γ = 1). That is, given the eﬃciencies of the most andthe least eﬃcient ﬁrms, the second most eﬃcient ﬁrm with an intermediate value of α does
not enhance social welfare. This is because it crowds out the incentive of the most eﬃcientﬁrm to invest. If it is less eﬃcient (the value of α is higher), it enhances the incentive of
the most eﬃcient ﬁrm to invest. Therefore, if R&D investments are not costly (γ is small),ineﬃcient ﬁrms can be more beneﬁcial than ones with intermediate eﬃciency levels. When
R&D investments are costly (γ = 4), the standard intuition mentioned by Lahiri and Ono[14] holds. As the eﬃciency of a ﬁrm becomes worse, the existence of the ﬁrm tends to beharmful from the viewpoint of social welfare.

19

4Conclusion

We investigated the incentive and the welfare implications of a merger when heterogeneous
oligopolists compete both in process R&D and on the product market. We showed that
although ﬁrms have merger incentives in the presence of R&D, such mergers improve social
welfare only when the share of eﬃcient ﬁrms is suﬃciently high. Under a suﬃcient degree ofﬁrm heterogeneity, a low R&D cost leads to a merger between homogeneous ﬁrms in equilibrium, whereas a high R&D cost implies a merger between heterogeneous ﬁrms. Moreover, if
R&D is not costly, we observe a discrepancy between the incentives for, and the desirabilityof, mergers.

20Appendix A:

Proﬁtability of a pairwise merger for a suﬃciently small γ.

¿From (6), πM − 2πi is positive if and only if:
[
](
][
]2
)2 [
∆ = 2 N 2 − (N + 1)2 γ 1 − N + N 2 γ + N 2 γ − (1 − N )2 N − (N + 1)2 γ > 0.
For a given number N (≥ 3) of ﬁrms, ﬁrms supply goods as long as γ > N 2 / (N + 1)2 .
Therefore, the lower bound of the relevant interval of γ is N 2 / (N + 1)2 . Evaluating ∆ at
this lower bound, we have:limγ→N 2 /(N +1)2 +0

Appendix B:∆=(N − 1)2 N 2 (2N 2 − 1)>0
(N + 1)2

for N ≥ 3.

Desirability of a pairwise merger for a suﬃciently small γ.

From (7), we have:
WM − W =[](N − 1) 4N − 2 + N 2 (γ − 2) + N 3 γ(1 − N + N 2 γ)2+

2N 3 − N (N + 2)(N + 1)2 γ[N − (N + 1)2 γ]2.

This is positive if and only if:[][]2Γ = (N − 1) 4N − 2 + N 2 (γ − 2) + N 3 γ N − (N + 1)2 γ()2+ [2N 3 − N (N + 2)(N + 1)2 γ] 1 − N + N 2 γ> 0.
Evaluating Γ at this lower bound, we have:
{
[
(
)]
}
N 3 N (2N − 3) 3 + 2N 2 N 2 − 2 − 2
2N 2
lim
Γ=
+
>0
(N + 1)4(N + 1)4
γ→N 2 /(N +1)2 +0

Appendix C:

Derivations of the eﬀects of mergers.

From (8) and (9), we obtain:

21

for N ≥ 3.

()∂x(n, m, c; δ) ∂x(n, m, c; δ)
=−
−
−
∂n
∂(N − n)γδN (N + 1)=
> 0,[γ(N + 1) − N ] [γ(N + 1)2 − N ]()()∂q(n, m, c; 0) ∂q(n, m, c; 0)∂q(n, m, c; δ) ∂q(n, m, c; δ)
−
−
=−
−
∂n
∂(N − n)∂n∂(N − n)22γ δ(N + 1)=
> 0.
[γ(N + 1) − N ] [γ(N + 1)2 − N ](∂x(n, m, c; 0) ∂x(n, m, c; 0)−∂n∂(N − n))A larger value of γ makes these tendencies weak because:∂∂γ(γδN (N + 1)[γ(N + 1) − N ] [γ(N + 1)2 − N ][]δN (N + 1) N 2 − γ 2 (N + 1)3

)
=

< 0,
[γ(N + 1) − N ]2 [γ(N + 1)2 − N ]2()γ 2 δ(N + 1)2γδN (N + 1)2 [2N − γ(N + 1)(N + 2)]∂=< 0.
∂γ [γ(N + 1) − N ] [γ(N + 1)2 − N ][γ(N + 1) − N ]2 [γ(N + 1)2 − N ]2
Moreover, diﬀerences between eﬃcient and ineﬃcient ﬁrms in their reactions to a particular
type of merger are given by:−∂∂(x(n, m, c; 0) − x(n, m, c; δ)) = −(x(n, m, c; 0) − x(n, m, c; δ))∂n∂(N − n)∂=−(q(n, m, c; 0) − q(n, m, c; δ))∂n∂(q(n, m, c; 0) − q(n, m, c; δ))=−∂(N − n)γδ=−< 0.
[γ(N + 1) − N ]2

These diﬀerences are smaller for a larger value of γ because:∂∂γ

Appendix D:(γδ
−[γ(N + 1) − N ]2)=δ [γ(1 + N ) + N ]> 0.
[γ(N + 1) − N ]3

Proﬁtability and desirability in the limits of δ and γ.

A pairwise merger is proﬁtable if (5) holds true. For each type of merger, this condition
becomes as follows.
A Type (I) pairwise merger (i.e., a merger of eﬃcient ﬁrms) is proﬁtable if π(n − 1, m, c; 0) −2π(n, m, c; 0) > 0.
22

A Type (II) pairwise merger (i.e., a merger of heterogeneous ﬁrms) is proﬁtable if π(n, m −1, c; 0) − π(n, m, c; 0) − π(n, m, c; δ) > 0.
A Type (III) pairwise merger (i.e., a merger of ineﬃcient ﬁrms) is proﬁtable if π(n, m −1, c; 0) − 2π(n, m, c; δ) > 0.
The upper bound of the relevant interval of δ is δ, given in (11), and the lower bound of
the relevant interval of γ is N/ (N + 1). Evaluating gains from a merger at these bounds,
we obtain:
lim π(n − 1, m, c; 0) − 2π(n, m, c; 0)lim

γ→N/(N +1)+0 δ→δ+0=(1 − z)2 {n2 N [N 2 (N + 2) − 1] − 2[N 2 (N − 1) + 1]2 },n2 (N + 1)[N 2 (N − 1) + 1]2
lim π(n, m − 1, c; 0) − π(n, m, c; 0) − π(n, m, c; δ)limγ→N/(N +1)+0 δ→δ+0=(1 − z)2 {n2 N [N 2 (N + 2) − 1] − [N 2 (N − 1) + 1]2 },n2 (N + 1)[N 2 (N − 1) + 1]2
lim π(n, m − 1, c; 0) − 2π(n, m, c; δ)limγ→N/(N +1)+0 δ→δ+0=(1 − z)2 N (N 2 + N − 1)
> 0.
[N 2 (N − 1) + 1]2

We thus see that when δ is suﬃciently large and γ is suﬃciently small, a pairwise merger ofType (I) is proﬁtable if:
n
>
N

√

N [N 2 (N + 2) − 1]/2.
N 2 (N − 1) + 1

Similarly, a pairwise merger of Type (II) is proﬁtable if:√
N [N 2 (N + 2) − 1]n>.
N
N 2 (N − 1) + 1
The desirability of a merger comes from the following results:
]
]
[
[
∂W
∂W
=
lim
lim −
lim
lim −
∂n
∂m
γ→N/(N +1)+0 δ→δ+0
γ→N/(N +1)+0 δ→δ+0=(1 − z)2 {n(2N 5 − 4N 3 + N − 1) − 2[N 2 (N − 1) + 1]2 }> 0.
2n(N + 1)[N 2 (N − 1) + 1]2

This implies that a merger is desirable if:n2[N 2 (N − 1) + 1]2>.
N
N (2N 5 − 4N 3 + N − 1)23

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26

6.0

6.0

5.5

5.5
no firms

no firms
supply

5.0

5.0

4.5

4.5

n

n

supply4.0

unprofitable

3.5

desirable

4.0

profitable

3.5

undesirable
3.0

3.0
0.5

1.0

1.5

2.0

0.5

1.0

Γ

1.5

2.0

Γ

(i) proﬁtability(ii) desirability

6.0

5.5

a merger is desirable but does not occur
no firms
supply

n

5.0

4.5

a merger is undesirable and does not occur

4.0

an undesirable merger occurs

3.5
a desirable merger occurs
3.0
0.5

1.0

1.5

2.0

Γ

(iii) discrepancy
Figure 1: Proﬁtability and desirability of a pairwise merger with investment.

27

0.4

no inefficientfirm supplies

HIIIL∆H1-cL

0.3

0.2

HIIL and HIIIL

0.1

HIIL
H0L
0.0
1.0

1.5

2.0

2.5

3.0

Γ

Figure 2: Equilibrium: one eﬃcient and two ineﬃcient ﬁrms.
Note:

Type (0)=No merger, Type (II)=A merger of heterogeneous ﬁrms,
Type (III)=A merger of ineﬃcient ﬁrms.

0.4

no inefficientfirm supplies∆H1-cL

0.3

A desirable merger occurs

0.2

An undesirable merger does not occur
An undesirable merger occurs

0.1

0.0
1.0

1.5

2.0

2.5

3.0

Γ

Figure 3: Social welfare and the desirability of mergers: one eﬃceint and two
ineﬃcient ﬁrms.

28

0.4

no inefficientfirm supplies
0.3∆H1-cL

No core
HIL

0.2

0.1

HIIL
H0L
0.0
1.0

1.5

2.0

2.5

3.0

Γ

Figure 4: Equilibrium: two eﬃcient and one ineﬃcient ﬁrms.
Note:

Type (0)=No merger, Type (I)=A merger of eﬃcient ﬁrms,
Type (II)=A merger of heterogeneous ﬁrms.

29

0.4

no inefficientfirm supplies∆H1-cL

0.3

0.2

An undesirable merger does not occur
An undesirable merger occurs

0.1

A desirable merger does not occur
0.0
1.0

1.5

2.0

2.5

3.0

Γ

Type (I) merger
0.4

no inefficientfirm supplies∆H1-cL

0.3

0.2

A desirable merger occurs
An undesirable merger does not occur
An undesirable merger occurs

0.1

A desirable merger does not occur
0.0
1.0

1.5

2.0

2.5

3.0

Γ

Type (II) merger
Figure 5: Social welfare and the desirability of mergers: two eﬃceint and one
ineﬃcient ﬁrms.

30

∆H1-cL∆H1-cL

0.4
0.4
0.3
0.3
0.2
0.2

The AB merger is beneficial

0.1

0.2

0.4

0.6

0.1

Α

1

0.8

The AC merger is beneficial

0.2

Welfare (AB)

0.4

0.6

Α

1

0.8

Welfare (AC, BC)(γ = 4)∆H1-cL∆H1-cL

0.2

0.2

0.15

0.15

0.1

0.1

The AB merger is beneficial.

The AC merger is beneficial

0.05

0.05

0.2

0.4

0.6

0.8

1

Α

0.2

Welfare (AB)

0.4

0.6

0.8

Welfare (AC, BC)(γ = 1)Figure 6: The desirability of mergers: three diﬀerent ﬁrms.

31

1

Α