A. Mantovani

D. Dragone

L. Lambertini

English

Research Papers in Economics

Horizontal Mergers with Scale Economies
Davide Dragone, Luca Lambertini and Andrea Mantovani
Department of Economics, University of Bologna
Strada Maggiore 45, 40125 Bologna, Italy
phone: +39-051-2092600; fax: +39-051-2092664
dragone@spbo.unibo.it; lamberti@spbo.unibo.it; mantovan@spbo.unibo.it
June 5, 2006
Abstract
We examine the proﬁtability and social eﬃciency of horizontal
mergers in a Cournot oligopoly with decreasing average costs. As-
suming the merger allows for a reduction in the total amount of ﬁxed
costs, we identify the conditions under which the merger is, respec-
tively, proﬁtable and socially desirable. There exists an admissible
parameter range wherein the merger is socially convenient but not
proﬁtable. In such a case, the policy maker may induce ﬁrms to merge
through subsidies ﬁnanced via a lump sum tax.
JEL Classiﬁcation: D43, G34, L13
Keywords: horizontal merger, oligopoly, eﬃciency
11I n t r o d u c t i o n
The bearings of production technology on the proﬁtability of horizontal merg-
ers in a Cournot oligopoly has attracted wide attention ever since Salant et
al. (1983). Under the assumption of constant average cost, they have shown
that, unless a vast majority of the ﬁrms in the industry merge, the merger
itself is not proﬁtable, and therefore the scope for antitrust action is limited.
This seminal contribution has generated a stream of literature investigating
other cases, such as that of a quadratic cost function (Perry and Porter,
1985; McAfee and Williams, 1992).1 To the best of our knowledge, the only
contribution where the case of increasing returns (i.e., economies of scale) is
illustrated is Farrell and Shapiro (1990), where, however, it is assumed that
the amount of ﬁxed costs retained by the ﬁrm resulting from the merger is
exactly equal to the sum of the pre-merger ﬁxed costs of the participating
ﬁrms. The usual argument put forward to justify a merger is the increase
in productive eﬃciency generated by the merger itself (see, e.g., Farrell and
Shapiro, 2000). Therefore, an interesting perspective is that where such ef-
ﬁciency gain is the outcome of an adjustment in ﬁxed costs via the merger.
This is precisely the view that we adopt in the present paper. We model a
situation where a synergy operates, in such a way that the overall ﬁxed cost
associated with the merger is a fraction of the sum of the ﬁxed costs borne ex
ante by the ﬁrms taking part into the merger. This setup allows us to draw
several conclusions. The ﬁrst is that, if the ﬁxed cost savings generated by
1Other routes taken by the literature include the analysis of horizontal mergers un-
der Bertrand competition (Davidson and Deneckere, 1985) or non-linear market demand
functions (Hennessy, 2000), inter alia.
2the merger are large enough, then the merger becomes proﬁtable irrespective
of the number of ﬁrms involved. Likewise, there exists a critical threshold of
such eﬃciency gain above which the merger is socially desirable. The ranking
of the two thresholds depend on the number of ﬁrms in the industry before
the merger takes place. If the size of the merger (that is, the number of ﬁrms
involved in it) is small enough, then we observe the counterintuitive and, thus
far, neglected situation where the merger is socially convenient while being
not proﬁtable. If this is the case, then the policy maker may induce ﬁrms
to merge by subsidising them (for instance, by using funds raised through
taxes) provided that the representative shareholder is suﬃciently atomistic
to ensure that future dividends compensate current taxation.
2T h e m o d e l
We examine a Cournot oligopoly where, ex ante, n ﬁrms sell a homogeneous
g o o dw h o s ed e m a n df u n c t i o ni sp = a−
Pn
i=1 qi. Each ﬁrm uses a technology
characterised by increasing returns to scale, summarised by the cost function
Ci = cqi + F, where c>0i st h em a r g i n a lc o s ta n dF>0i saﬁxed cost.
Firms compete simultaneously. Therefore, individual outputs and proﬁts
at the symmetric Nash equilibrium are:
q
∗ (n)=
a − c
n +1
; π
∗ (n)=
(a − c)
2
(n +1 )
2 − F (1)
and the corresponding consumer surplus is:
CS
∗ (n)=
[nq∗ (n)]
2
2
=
n2 (a − c)
2
2(n +1 )
2 . (2)
Hence, social welfare is SW∗ (n)=nπ∗ (n)+CS∗ (n).
3Now consider the situation where m ∈ [2,n− 1] ﬁrms evaluate the per-
spective of merging horizontally. We suppose that the merger gives rise to a
restructuration of production plants within the resulting ﬁrm; in particular,
we assume that the amount of ﬁxed cost be equal to b F ≡ (1 + b)F, with
b ∈ [0,m− 1]. Therefore, for all b<m− 1, the usual argument whereby a
merger is justiﬁed on eﬃciency grounds applies in the form of a reduction of
the total amount of ﬁxed cost borne by society.
As a consequence of the merger, outputs, proﬁts and consumer surplus
modify as follows:
q
∗ (n − m +1 ) =
a − c
n − m +2
;
b π
∗ (n − m +1 ) =
(a − c)
2
(n − m +2 )
2 − (1 + b)F
π
∗ (n − m +1 ) =
(a − c)
2
(n − m +2 )
2 − F (3)
CS
∗ (n − m +1 ) =
(n − m +1 )
2 (a − c)
2
2(n − m +2 )
2
where b π
∗ (n − m + 1) is the proﬁto ft h eﬁrm resulting from the merger while
π∗ (n − m + 1) is the proﬁt accruing to each of the n − m ﬁrms that have
remained independent. The associated social welfare is
SW
∗ (n − m +1 )=b π
∗ (n − m +1 ) + ( n − m)π
∗ (n − m +1 ) + CS
∗ (n − m +1 ).
(4)
Comparing ex ante and ex post proﬁts and welfare, we may prove the
following:
Proposition 1 For all admissible levels of {b,n,m}, there exist the thresh-
old values of ﬁxed costs Fπ and FSW above which the merger involving m
4ﬁrms is, respectively, privately and socially desirable. If n =3and m =2 ,
then Fπ <F SW. If instead n ≥ 4, then (i) Fπ >F SW if m is suﬃciently
small, and (ii) Fπ <F SW if m is suﬃciently large.
Proof. The private (proﬁt) incentive to carry out the merger requires:
b π
∗ (n − m +1 )
m
>π
∗ (n) ⇔ (5)
F>F π ≡
m(a − c)
2
m − 1 − b
∙
1
(n +1 )
2 −
1
m(n − m +2 )
2
¸
while the social incentive requires:
SW
∗ (n − m +1 )>S W
∗ (n) ⇔ (6)
F>F SW ≡
(a − c)
2
2(m − 1 − b)
∙
1
(n − m +2 )
2 −
1
(n +1 )
2
¸
.
Comparing Fπ and FSW, we obtain Fπ = FSW at
m1 =
5
4
+ n −
1
4
√
24n +3 3 ;m2 =
5
4
+ n +
1
4
√
24n +3 3 . (7)
Clearly, m2 >nand can be disregarded; m1 is always smaller than n, but
m1 ≥ 2 if and only if n ≥ 4. If n =3 , we have a special case where m1 =1 .688.
Therefore, we have two cases:
(i) for n =3a n dm =2
Fπ =
(a − c)
2
72(1 − b)
<F SW =
7(a − c)
2
288(1 − b)
;( 8 )
(ii) for all n ≥ 4,
Fπ >F SW for all m ∈ [2,m 1);
Fπ <F SW for all m ∈ (m1,n− 1].
(9)
5This concludes the proof.
N o wt a k et h es p e c i a lc a s ew h e r en =3a n dm =2 . Here, given that the a
priori degree of concentration of the industry is very high, the critical thresh-
old for F above which the antitrust agency permits the merger is necessarily
higher than that above which ﬁrms ﬁnd it proﬁtable.
The perspective changes signiﬁcantly for all n ≥ 4. Provided that, as n
increases, the market becomes progressively more competitive, then m1 be-
comes relevant and determines the presence of an interval where relatively
small mergers are socially desirable. Note that m1 is monotonically increas-
ing in n. To this regard, it is worth stressing that, for all F ∈ (FSW,F π),
any merger involving m ∈ [2,m 1) is socially convenient while it is not pri-
vately so. That is, we have the seemingly counterintuitive result whereby
the antitrust agency would like the m ﬁrms to merge, whereas they prefer
to remain independent. To the best of our knowledge, this possibility has
been overlooked so far in the existing literature. The reason of this result is
that m is small. This entails two related consequences: the ﬁrst is that, for
any given b, the reduction in ﬁx e dc o s t se n j o y e db yt h eﬁr m si n v o l v e di nt h e
merger is limited; the second is that the reduction in the overall number of
ﬁrms in the industry after the merger has taken place is also limited. The
ﬁrst fact makes the merger unattractive to ﬁrms, while the second makes it
appealing for the regulator. When this is the case, the policy maker may
design an income transfer from consumers to the merging ﬁrms by means of
a lump sum tax that becomes a subsidy to ﬁrms, so as to make the merger
attractive to them. The total subsidy amounts to
µ
π
∗ (n) −
b π
∗ (n − m +1 )
m
+ ε
¶
m (10)
6where ε is positive and arbitrarily small. This of course has no redistributive
eﬀects provided that all agents holds symmetric shares of the stock of those
ﬁrms that are to carry out the merger.
7References
[1] Deneckere, R. and C. Davidson (1985), “Incentives to Form Coalitions
with Bertrand Competition”, RAND Journal of Economics, 16,4 7 3 - 8 6 .
[2] Farrell, J. and C. Shapiro (1990), “Horizontal Mergers: An Equilibrium
Analysis”, American Economic Review, 80, 107-26.
[3] Farrell, J. and C. Shapiro (2000), “Scale Economies and Synergies in Hor-
izontal Merger Analysis”, University of California at Berkeley, Economics
Working Paper E00-291.
[4] Hennessy, D. (2000), “Cournot Oligopoly Conditions under Which Any
Horizontal Merger Is Proﬁtable”, Review of Industrial Organization, 17,
277-84
[5] McAfee, R.P. and M.A. Williams (1992), “Horizontal Mergers and An-
titrust Policy”, Journal of Industrial Economics, 40, 181-87.
[6] Perry, M.K. and R.H. Porter (1985), “Oligopoly and the Incentive for
Horizontal Merger”, American Economic Review, 75,2 1 9 - 2 7 .
[7] Salant, S.W., S. Switzer and R.J. Reynolds (1983), “Losses from Horizon-
tal Merger: The Eﬀects of an Exogenous Change in Industry Structure
on Cournot-Nash Equilibrium”, Quarterly Journal of Economics, 98,1 8 5 -
213.
8

Horizontal Mergers with Scale Economies

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