International Journal of Industrial Organization
19 (2001) 1245–1261
www.elsevier.com / locate / econbase
Endogenous mergers and market structure
Vasco Rodrigues*
ˆ
´
´
Portuguesa,
Faculdade de Ciencias
Economicas
e Empresariais, Universidade Catolica
Centro Regional do Porto, Rua Diogo Botelho 1327, 4169 -005 Porto, Portugal
Received 12 February 1999; received in revised form 1 November 1999; accepted 4 February 2000
Abstract
In this paper we use a two-stage game to model endogenous mergers. In the first stage of
the game firms decide whether or not to merge, and in the second stage they compete on the
product market. Merger occurrence is determined by the interplay of the initial number of
firms in the industry, the expected competitive intensity, and the possibility of economizing
on fixed costs through merger. It is shown that the equilibrium market concentration is
decreasing in the first of these factors and increasing in the other two. Welfare implications
are discussed.  2001 Elsevier Science B.V. All rights reserved.
JEL classification: L12; L13; L41
Keywords: Endogenous mergers; Market structure; Antitrust policy
1. Introduction
Formal modeling of merger processes is a relatively recent endeavor in
economics. The seminal work is Salant et al. (1983), which analyzes the impact of
a merger among an exogenously determined subset of the n firms in an industry,
assuming Cournot behavior and constant average and marginal costs. A surprising
conclusion is that no merger involving less than 80% of the firms in the industry
* Tel.: 1351-2261-962-45; fax: 1351-2261-962-91.
E-mail address: [email protected] (V. Rodrigues).
0167-7187 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved.
PII: S0167-7187( 00 )00059-X
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V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
would be privately profitable. The supervening literature (e.g. Perry and Porter,
1985; Deneckere and Davidson, 1985; Kwoka, 1989; Farrel and Shapiro, 1990)
has challenged the robustness of this conclusion and has examined the welfare
effects of exogenous mergers.
An even more recent strand of literature, with which this paper concurs,
investigates the pattern of mergers that can be expected to occur, by endogenizing
the merger decision. Kamien and Zang (1990) analyze the conditions that limit the
possibility of monopolization of an industry by intra-industry acquisition of firms.
Their main conclusion is that monopolization can only occur in industries
composed ex ante of a small number of firms, a conclusion that our model
reiterates. Gaudet and Salant (1992) extend the previous model and, among other
conclusions, show that not every privately profitable merger will occur when
mergers are endogenously determined. Gowrisankaran (1999) presents a particularly complex model in which firms make merger, exit, investment and entry
decisions before competing in the product market. Using dynamic programming
methods the author is able to characterize one, not necessarily unique, MarkovPerfect Nash Equilibrium for this game. Among his conclusions, the fact that an
increase in entry costs results in increased concentration through merger seems
particularly relevant.
Our model builds upon the existing literature, particularly Kwoka (1989), to
investigate how an industry’s endogenous level of concentration relates to
mergers’ capability to generate market power and efficiency gains, their conflicting
effects that drive much of the economists’ interest in the subject. In the two-stage
game presented, the mergers that occur endogenously are explained by the
interplay of the number of firms in the initial market structure, the expected
competitive intensity and the economies permitted by the merger, with the
equilibrium market concentration decreasing in the first of these factors and
increasing in the other two. Having characterized the equilibrium market structure,
we then show that in industries that are not, ex ante, very concentrated,
endogenously generated mergers will only be socially detrimental if the expected
competitive intensity is extremely strong or the opportunities for fixed cost savings
are very limited.
The paper is organized as follows: our model of endogenous mergers is
presented first; we then analyze the implications of mergers on the product market;
the next section investigates the equilibrium market structure and related welfare
implications; we then conclude.
2. A model of endogenous mergers
Consider a two-stage game in which, in the first stage the firms must decide
whether to merge or not, and in the second stage the remaining firms compete in
the product market. In what circumstances, if any, would firms decide to merge?
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1247
2.1. The first stage: the merger formation process 1
We consider a simplified process of endogenous merger formation. In the first
stage of the game, firms announce sequentially whether they are available to
participate in a merger. All firms having made their announcements, those that
declared themselves available to merge, merge in single firm, with the other firms
remaining as independent competitors.2
A condensed (since this is an n-firm game) extensive form representation of the
game is shown in Fig. 1.
We choose to model the first stage of the game in a sequential fashion on two
grounds. First, a simultaneous endogenous merger formation game, with the
decision rule we use, would always have n 1 1 trivial Nash equilibria, in which no
firm would merge whatever the conditions of the product market. These equilibria
correspond to the case where every firm announces that it is not prepared to merge
and to n situations where all but one of the firms say they are not prepared to
merge and the remaining firm says otherwise. Such situations are not equilibria in
the sequential game when the conditions of the product market are conducive to
merger, which we regard as an advantage. Second, the sequential process seems to
adjust better to the characteristics of the real merger processes in which, typically,
firms do observe their competitors’ moves.
2.2. The second stage: competition in the product market
We follow Kwoka (1989) in modeling competition on the product market using
a conjectural variation model.3
1
The literature on endogenous merger formation that we briefly reviewed does indeed deal with
acquisition games. Typically, each firm (each firm’s shareholders) defines the price it is prepared to pay
for each of the other firms and the price it will be asking for itself. Each firm is acquired by its highest
bidder, if the corresponding bid is above the price the firm is demanding for itself. The merger
formation process that we present is similar to Prokop’s (1999) sequential cartel formation process and
has more to do with the extensive literature on coalition formation that draws both on cooperative and
non-cooperative game theory. Recent examples of other endogenous coalition formation processes with
possible application to mergers can be found, among others, in Bloch (1996) and Yi (1998).
2
Although mergers do not, in general, have this open coalition structure, this sometimes happens. We
have particularly in mind situations in which an external entity (such as the State or an investment
bank) does initiate a merger process by inviting the firms in the industry to analyze the possibility of
merging. That this would happen in countries with strong antitrust laws, and with little industrial policy
tradition, such as the US, may be unlikely. But examples can be found in other parts of the world: in
his analysis of industrial conditioning in Portugal, Confraria (1990) describes some such situations.
And, even in the US, some of the mergers which occurred during the so-called ‘first merger wave’
seem not to differ much from this situation.
3
See Tirole (1988), or Shapiro (1989), for a critique of the static conjectural variation model and
Dockner (1992), Cabral (1995) and Pfaffermayr (1997) for reinterpretations as a reduced form for the
equilibrium of dynamic models of competition. It is in this spirit that we use it here. Besides, for our
purposes, the model has the advantage of providing a continuous measure of competitive intensity. Our
general conclusions would still emerge, however, if we proceeded by comparing the results of the
merger game for a menu of alternative formulations (Bertrand, Cournot, cartel, etc.) of the second stage
of the game.
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1248
Fig. 1. The (condensed) extensive form of the game. Legend: No, the firm commits not to participate
in the merger; Yes, the firm commits to participate in the merger; P (n9), the single firm profit in an n9
firm industry; n, the initial number of firms.
Assume an n-firm homogenous product oligopoly. Market demand and cost
functions are given by:
P5a2Q
(1)
TCi 5 cqi 1 F
(2)
where P is the market price, Q is the demanded quantity, and qi is the quantity
supplied by firm i. Of course, Q 5 qi 1 q2i , with q2i representing the quantity
supplied by the ith firm (n21) competitors. F and c are, respectively, fixed and
marginal costs parameters, which we assume are identical for all firms. The
consideration of fixed costs, contrary to much of the previous literature (e.g. Salant
et al., 1983), allows us to investigate the role of fixed cost economies in
motivating mergers.
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
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Let li 5 dq2i / dqi represent firm i’s conjectural variation, i.e. its expectation
regarding the change in its competitors production resulting from a change in its
own production level, and assume that this conjecture is identical for all firms
( li 5 l). We will refer to l as the competitive intensity of the industry 4 , with
lower values of l corresponding to more intense competition, since we can have
the industry equilibrium varying from a competitive equilibrium to a perfectly
collusive one as we move from l 5 2 1 to l 5 n 2 1.
Assuming quantity as the strategic variable, profit maximization requires that
≠Pi / ≠qi 5 0. It is then straightforward to show that, in equilibrium:
a2c
qi 5 ]]]
n1l11
(3)
l11
Pi 5 ]]]]2 sa 2 cd 2 2 F
sn 1 l 1 1d
(4)
and
where Pi stands for firm i’s profit.
We will restrict the parameter values to a 2 c . 0, n > 2, F > 0, and 2 1 < l <
n 2 1. For these values of the parameters, each firm’s profit is a decreasing
function of the number of firms in the industry, its competitive intensity and the
amount of fixed costs.
3. The impact of a merger on the product market 5
In this setting, two motives could lead firms to merge: market power and
efficiency.
Suppose that, resulting from the first stage of the game, m 1 1 of the n firms
comprising an industry merge. The new number of firms in the industry will be
n9 5 n 2 m. Assume, further, that the merger has no impact on firms’ conjectural
variations and marginal costs. The post-merger (pm) equilibrium will still be
characterized by identical production for each of the n 2 m firms, with:
a2c
q pm
5 ]]]]
i
n2m1l11
(5)
Comparing the post- and pre-merger equilibrium situations, it can be shown that
4
This is not accurate, however, since we will be allowing for changes in the number of firms,
through merger. For different numbers of firms in the industry, the same value of the conjectural
variation parameter will imply different competitive intensities: e.g. conjectural variations equal to
unity amounts to perfectly collusive behavior in a duopoly, but corresponds to less than perfectly
collusive behavior in less concentrated market structures.
5
Except for the consideration of fixed costs, this section follows Kwoka (1989) closely.
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V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
the firm resulting from the merger will produce less than the sum of the
pre-merger production of the merged firms, but more than any of them produced
separately, that each of the non-merging firms will increase its production, and that
the overall production level will decrease. This will lead to a price increase that
substantiates the market power motive for merger.
The efficiency motive is incorporated in the model through fixed cost
economies. Let u be a coefficient measuring the degree in which the merger allows
such economies, with 0 < u < m.6
Post-merger profit will be:
l11
m11
P pm
5 ]]]]]2 sa 2 cd 2 2 ]] F
i
u 11
sn 2 m 1 l 1 1d
(6)
A merger will, then, be privately profitable if g 5 P pm
2sm 1 1dPi is positive.
i
Replacing (4) and (6), g is shown to be:
F
G
1
m11
usm 1 1d
g 5s l 1 1dsa 2 cd 2 ]]]]]2 2 ]]]]2 1 ]]] F
u 11
sn 2 m 1 l 1 1d
sn 1 l 1 1d
(7)
It is now possible to check whether the merger of any subset of the n firms in
the industry is privately profitable. Start with l 5 2 1, equivalent to a competitive
industry. In this case, the first term in (7) is zero. Then, as could be expected, a
merger among a subset of the firms in the industry will only be profitable if it
allows for fixed cost economies. The only exception to this conclusion is, of
course, the case in which all the n firms in the industry merge, destroying its
competitive nature, which is always profitable.7 Similarly, if the firms expect a
perfectly collusive type of behavior, there is, excluding fixed cost economies, no
incentive to merge.8 In fact, even without any merger, the industry profit is already
maximized, there being room for no improvement. The merger of all the n firms,
creating a multi-plant monopoly, would only maintain the pre-merger industry
profit, while the merger of any smaller subset of the firms would imply a loss for
the merged firms because, transforming them in a single firm, it would reduce their
share in the same industry profit pie.
If 2 1 , l , n 2 1, firms’ behavior is somewhere between the extremes of
competitive and perfectly collusive behavior. In this case, it is a sufficient
6
With u 5 0, fixed costs have a sunk nature, and are not adjustable post-merger: the firm resulting
from the merger has the same fixed costs as the whole of the merged firms. At the other extreme, with
u 5 m, the firm resulting from the merger has the same fixed costs as one of the merged firms. This can
happen if those fixed costs result from an indivisibility in the production factors or if the merger is
purely anti-competitive, resulting in the closure of all but one of the merged firms. Intermediate cases
are, of course, conceivable.
7
The monopoly profit would be P pm
5 (a 2 c 2 ) / 4 2 (nF ) /(u 1 1). (6) no longer applies.
i
8
Note that assuming perfectly collusive expectations is equivalent to assuming l 5 n 2 1 before the
merger but l 5 n 2 m 2 1 after the merger. In this case (7) cannot be directly applied.
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1251
condition for the merger to be privately profitable that:
1
m11
]]]]]2 2 ]]]]2 . 0
sn 2 m 1 l 1 1d
sn 1 l 1 1d
(8)
In fact, if F 5 0 or u 5 0, (8) is also a necessary condition for a merger’s private
profitability. Given the restrictions on the parameter values, this is equivalent to:
m
n 1 l , ]]]]
1m21
(9)
sm 1 1d 0.5 2 1
as shown by Kwoka (1989). This condition implies that, for any given (m 1 1)
number of participants in the merger, there is always an industry size (n)
sufficiently large for the merger to be unprofitable, that this industry size is
increasing in the industry’s competitive intensity, and that a merger to monopoly is
always profitable (and the most profitable of all possible mergers).9
The effect of the consideration of fixed costs, and of the possibility that mergers
generate fixed cost economies, is to enlarge the range of privately profitable
mergers. In fact, inspection of (7) reveals that if those economies are sufficiently
important any merger will be profitable.10
We would further note that the preceding analysis assumes that conjectural
variations are unchanged by the merger. But both the theoretical and the empirical
literature on collusion show that this becomes easier as the number of firms in the
industry gets smaller. It would, then, seem reasonable to assume that l 5 fsnd with
dl / dn , 0. Again, this possibility would enlarge the range of privately profitable
mergers. In the analysis of the endogenous formation of mergers, presented in the
next section, we maintain, however, the invariant conjectural variations hypothesis.
4. The endogenous market structure
We use the sub-game perfect Nash equilibrium concept to analyze the game
described. In what follows, we assume u 5 m, i.e. the most favorable situation with
regard to the possibility of obtaining fixed cost economies through merger. This is
not restrictive since we analyze what happens for different values of fixed costs
and, for our purposes, lower values of u would be tantamount to lower fixed costs.
In particular, u 5 0, i.e. the case in which the merger does not allow any fixed cost
economies, is equivalent to the case where firms have no fixed costs. We will also
restrict the conjectural variation coefficient to the range 2 1 < l < 1.11
9
For a thorough analysis of this condition’s implications, see Kwoka (1989).
Failure to consider fixed cost economies can partially explain the paradoxical results of Salant et
al. (1983), as the authors acknowledge.
11
This excludes the possibility of l being larger than the value that would imply perfectly collusive
behavior post-merger.
10
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1252
Monopoly, it would seem, should be firms’ preferred market structure, since it
excludes all competition. However, mergers to monopoly are seldom observed,
even in economies with lax antitrust laws. Thus, it is natural that we start our
analysis by studying what deters such mergers.
4.1. Merger to monopoly
The analysis of the extensive form of the game shows that:
Ps1d
Ps2d , ]]
n
(10)
is a necessary and sufficient condition for monopolization.12 This condition is
equivalent 13 to:
S
l11
F
1 1
F
]]]2 2 ]]]2 , ] ] 2 ]]]2
n 4 sa 2 cd
sl 1 3d
sa 2 cd
D
(11)
We now ask, when will industries be monopolized through merger?
4.1.1. Absence of fixed costs
If fixed costs are zero, solving (11) for n results in:
1 sl 1 3d 2
n , ] ]]]
4 l11
(11a)
The second member in this inequality is strictly decreasing in l, when this
parameter lies in the above-defined range. So, excluding fixed costs, the weaker
the competitive intensity, the larger the pre-merger market concentration will have
to be for a monopoly to emerge through merger. In fact, except for very
competitive industries, i.e. industries in which l is very close to 21, monopolization will only occur in very concentrated industries, in line with the conclusions of
Kamien and Zang (1990).
As an example, for l 5 0, equivalent to Cournot behavior, monopolization will
only occur for n , 2.25, implying that no industry in which at least three firms
coexist will be monopolized. And for sufficiently cooperative industries, as
12
A standard backward induction argument proves this: note that, (10) being true, if every other firm
has said ‘yes’, the last firm will do the same; knowing this, at the preceding node, the next-to-last firm
will also say ‘yes’; etc. A supplement including the complete version of this proof, and other material,
is available from the author on request.
13
In the following inequality, duopoly and monopoly profits have been divided by (a 2 c)2 . Given
the linear specification of the demand curve in (1), (a 2 c) is the quantity that would be demanded at a
competitive price, allowing us to interpret (a 2 c)2 as a measure of the market size. For the sake of
simplicity, we will, henceforth, refer to the ratio between fixed costs and market size (F /(a 2 c)2 ) as
‘fixed costs’.
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1253
implied by l 5 1, merger to monopoly will never occur. Condition (11a) becomes
less restrictive, however, as the industry becomes more competitive: with l 5 2
0.7 monopolization would occur if there were four firms, or fewer; with l 5 2
0.8, if there were six firms, or fewer; and, with l 5 2 0.9 if there were 11 firms,
or fewer. In the limit case of l 5 2 1, (11a) is always verified, implying that
merger would occur whatever the number of firms in the industry: small as a
firm’s share of monopoly profit can be, it will always be larger than the zero profit
it would earn if the merger to monopoly did not happen.
4.1.2. Low fixed costs
In the general case, in which fixed costs are positive, the conclusions depend on
their value. We will analyze the case of low fixed costs first, this being closest to
the case of zero fixed costs just examined. In particular, suppose that:
F
l11
]]]2 , ]]]2
sa 2 cd
sl 1 3d
(12)
This is equivalent to saying that fixed costs are such that both duopoly and
monopoly profits are positive. This being so, both members of (11) are also
positive, and solving that condition for n we get:
1
F
] 2 ]]]2
4 sa 2 cd
n , ]]]]]]
l11
F
]]]2 2 ]]]2
sl 1 3d
sa 2 cd
(11b)
In this case, the conclusion obtained in the preceding subsection, that the weaker
the competitive intensity the larger the pre-merger concentration will have to be
for a monopoly to emerge through merger, continues to hold. But, as fixed costs
approach the upper limit allowed by (12), condition (11b) becomes less and less
restrictive, as its second member tends to infinity. As an example, for l 5 0,
equivalent to Cournot behavior, monopoly would emerge endogenously for
industries with no more than two firms, if fixed costs are 0.04, in industries with
no more than seven firms, if fixed costs are 0.09, and in industries with no more
than 125 firms, if fixed costs are 0.11.
4.1.3. High fixed costs
Suppose now that fixed costs are sufficiently high for both duopoly and
monopoly profits to be negative.14 That is, suppose:
F
1
]]]2 . ]
4
sa 2 cd
14
Declining industries being a possible example of this type of situation.
(13)
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V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
In this case, solving (11) for n we get:
1
F
] 2 ]]]2
4 sa 2 cd
n . ]]]]]]
l11
F
]]]2 2 ]]]2
sl 1 3d
sa 2 cd
(11c)
Note that, since ( l 1 1) /( l 1 3)2 , 1 / 4, the critical value for n is always in the
interval 0–1. This implies, fixed costs being sufficiently high, that any industry
with at least two firms, whatever its competitive intensity, will be monopolized
through merger.
4.1.4. Intermediate fixed costs
When neither (12) nor (13) hold, fixed costs are sufficiently high for the
duopolist’s profits to be negative but the monopolist’s to be positive. Simple
inspection of (11) shows that its first member is then negative, while the second is
positive, implying that, as in the preceding case, whatever the number of firms in
the industry and its competitive intensity, a monopoly would always emerge
through merger.
Given these results, the fact that mergers to monopoly are unusual is not
surprising: if fixed costs are not too high, they are only to be expected in very
concentrated or very competitive industries, even in the absence of an antitrust
policy.
4.2. The general case
We now claim that, in the above-presented setting, there is a uniquely defined
equilibrium market structure. Particularly, let n* 5 n, if there is no positive integer
(n9) smaller than n such that:
Psn9d
Psn9 1 1d , ]]]
n 2 n9 1 1
(14)
and let n* be the smallest of any such values, otherwise. Then, the first n* 2 1
firms will refuse to merge while the remaining n 2 n* 1 1 merge in a single firm,
leading to an n*-firm market structure.
The proof is lengthy and we merely sketch it here.15 We proceed in two steps:
first, we prove that the equilibrium market structure will necessarily be composed
of n* firms; having shown this, we then prove that, since for any equilibrium
market structure it pays more to stay out of the merger than to participate,
first-mover advantages allow the first n* 2 1 firms to say ‘no’, leading the
15
The complete proof is included in the supplement to this paper.
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1255
remaining firms to merge. The first step of the proof is itself composed of three
sub-steps. We start by proving that, if the number of firms that have yet to speak
equals the number of those that would have to say ‘yes’ for an n*-firm market
structure to emerge, then every such firm will say ‘yes’. This is just a
generalization of the result already established for the monopoly case and the
proof follows the same lines as the one presented in Footnote 12. We then show
that if sufficient firms have already said ‘yes’ for an n*-firm market structure to
emerge, all the remaining firms will say ‘no’. To prove this, we note that, since
(14) does not hold for any integer smaller than n*, if sufficient firms have already
said ‘yes’, independently of what any other remaining firms do, the last firm will
say ‘no’. Standard backward induction reasoning then extends this result to the
previous firms. Finally we complete the proof showing that, taken together, the
two results just presented imply that the equilibrium market structure will be
comprised of n* firms.
Condition (14) is equivalent to (11) in the case where n9 5 1 and, using (4), it is
otherwise equivalent to: 16
F
l11
F
1
l11
F
]]]]2 2 ]]] , ]]] ]]]]2 2 ]]]
2
n 2 n9 1 1 sn9 1 l 1 1d
sn9 1 l 1 2d
sa 2 cd
sa 2 cd 2
G
(15)
We now examine the equilibrium market structure for different ranges of fixed
costs.
4.2.1. Absence of fixed costs
When there are no fixed costs, (15) implies that it is a necessary condition for
the existence of n9 firms to constitute the equilibrium that:
2sn9 1 l 1 1d 1 1
n 2 n9 , ]]]]]
sn9 1 l 1 1d 2
(15a)
The second member in this condition is strictly decreasing in n9 and l, and, for
n9 . 2 and l . 2 1, is always a value in the range 0–1. Hence, any market
structure comprising more than two firms can only be the equilibrium market
structure if it is also the initial market structure. But it can be shown that the same
is true regarding duopoly.17 So, we can now state that, excluding fixed costs, the
16
Note that, since n* is defined as the smallest of the values of n9 that conform to (14), it must be
the case that any smaller value of n9, like n*21, falsifies that condition. The two necessary conditions
for n* to be an equilibrium market structure defined by (14) and its denial for n*21 are very much in
the spirit of the conditions for cartel stability presented by d’Aspremont et al. (1983), that Prokop
(1999) shows coincide with the conditions that define the equilibrium in a cartel formation game
similar to our merger formation game.
17
This proof is also included in the supplement to this paper.
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V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
only mergers that will occur endogenously will be mergers to monopoly.18 And,
even these will only occur if the number of firms in the industry is sufficiently
small, or the competitive intensity sufficiently strong for (11a) to hold. Otherwise,
every single firm will prefer not to merge, since merger implies a market share loss
that is not sufficiently compensated for by the increased price.
4.2.2. Low fixed costs
However, this conclusion is only valid in the absence of fixed costs. If these
exist, and depending on their amount, the equilibrium market structure may
include any number of firms, and may arise through merger. If fixed costs are low,
such that:
F
l11
0 , ]]]2 , ]]]]2
sa 2 cd
sn9 1 l 1 2d
(16)
it is a necessary condition for n9 > 2 to be the equilibrium, given the initial number
of firms, the expected competitive intensity and the fixed costs, that:
l11
F
]]]2 2 ]]]
sn9 1 ld
sa 2 cd 2 2 2 , n 2 n9
]]]]]]]
l11
F
]]]]2 2 ]]]
n9
1
l
1
1
a
2
cd 2
s
d
s
l11
F
]]]]2 2 ]]]
sn9 1 l 1 1d
sa 2 cd 2 2 1
, ]]]]]]]
l11
F
]]]]2 2 ]]]
sn9 1 l 1 2d
sa 2 cd 2
(15b)
in which the first inequality corresponds to (15) and the second to its denial for
n9 2 1.
4.2.3. High fixed costs 19
If fixed costs are such that:
l11
F
l11
]]]]2 , ]]] , ]]]]2
2
sn9 1 l 1 2d
sa 2 cd
sn9 1 l 1 1d
(17)
18
In a sense, this reinforces the paradoxical result of Salant et al. (1983) presented above. A
corollary of our result, that subject to the restrictions implied by the model’s assumptions may be
relevant to antitrust policy, is that the occurrence of a merger that does not join all the firms in an
industry signals the existence of cost economies.
19
Unlike the case of monopoly, in the case of less concentrated market structures, there are only two
relevant ranges for fixed costs, which we call ‘high’ and ‘low’. If fixed costs exceeded the upper limit
defined by condition (17), corresponding to the third range of fixed costs we analyzed for monopoly,
(14) would hold for a number of firms smaller than the one under consideration and a more
concentrated market structure would result.
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1257
the profit of a firm is positive in an n9-firm market structure but negative in an
(n9 1 1)-firm market structure. This implies that (15) always holds and the
existence of n9 firms will be the equilibrium market structure regardless of the
initial number of firms in the industry. This supposing that the conditions for an
even more concentrated market structure to be an equilibrium are not fulfilled.
Two numerical examples will help to clarify the implications of the above
conditions for the equilibrium market structure and the occurrence of mergers.
4.2.4. The endogenous market structure in a 10 -firm industry
Table 1 presents the equilibrium number of firms, for various values of
competitive intensity and fixed costs, assuming that the industry is, ex ante,
composed of 10 firms. The market structure tends to become more concentrated
the stronger the competitive intensity and the higher the fixed costs.
At very low levels of fixed costs (e.g. F /(a 2 c)2 5 0.001), if the competitive
intensity is not extremely strong ( l > 2 0.8), the equilibrium market structure
coincides with the initial market structure. But, if it is extremely strong, the firms
will merge to monopoly: no ‘intermediate’ market structure is sustainable in
equilibrium. Firms will also merge to monopoly if fixed costs are very high (e.g.
F /(a 2 c)2 5 0.12), whatever the competitive intensity. For intermediate values of
the fixed costs and the competitive intensity, the equilibrium market structure
tends to become more concentrated as the competitive intensity increases.
Consequently, increases in competitive intensity are, for some values of the
parameters, accompanied by decreases in consumers’ surplus: as an example, with
F /(a 2 c)2 5 0.05, an increase in competitive intensity from l 5 0 to l 5 2 0.2
will change the market structure from a triopoly to a duopoly, leading to a 9.25%
reduction in consumers’ welfare.
Table 1
Number of firms in the equilibrium market structure, for an industry which initially had 10 firms,
assuming various levels of competitive intensity ( l) and fixed costs (F /(a 2 c)2 )
l
21.0
20.8
20.6
20.4
20.2
0.0
0.2
0.4
0.6
0.8
1.0
Fixed costs
0.001
0.010
0.050
0.100
0.120
1
10
10
10
10
10
10
10
10
10
10
1
4
5
6
7
8
9
9
9
10
10
1
1
1
2
2
3
3
3
3
3
4
1
1
1
1
1
1
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1258
Table 2
Deadweight loss a resulting from the endogenous increase in market concentration, described in Table 1,
assuming various levels of competitive intensity ( l) and fixed costs (F /(a 2 c)2 )
l
Fixed costs
21.0
20.8
20.6
20.4
20.2
0.0
0.2
0.4
0.6
0.8
1.0
a
0.001
0.010
0.050
0.100
0.120
20.125
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
20.125
20.001
20.002
20.003
20.003
20.002
20.001
20.002
20.002
0.000
0.000
20.125
20.125
20.124
20.025
20.038
20.027
20.035
20.043
20.051
20.059
20.042
20.125
20.125
20.124
20.123
20.122
20.121
20.065
20.077
20.089
20.101
20.111
20.125
20.125
20.124
20.123
20.122
20.121
20.119
20.117
20.115
20.113
20.111
The deadweight loss is (a 2 c)2 times the figures presented.
Table 2 quantifies the deadweight losses associated with the endogenous
changes in concentration presented in Table 1, as compared to a situation in which
mergers were not allowed. Clearly, these losses are greater where the competitive
intensity is stronger. There is no direct relation between the two, however: when
the increases in competitive intensity are insufficient to induce additional concentration, they result in smaller deadweight losses, by constraining market power.
It has been pointed out in the literature (e.g. Salant et al., 1983; Farrel and
Shapiro, 1990) that merger generated fixed cost economies might offset these
deadweight losses. Table 3 shows that this is indeed the case, for endogenously
Table 3
Social gain (fixed cost savings plus deadweight loss) resulting from the endogenous increase in market
concentration, described in Table 1, assuming (a 2 c)2 5 1, for various levels of competitive intensity
( l) and fixed costs (F /(a 2 c)2 )
l
21.0
20.8
20.6
20.4
20.2
0.0
0.2
0.4
0.6
0.8
1.0
Fixed costs
0.001
0.010
0.050
0.100
0.120
20.116
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
20.035
0.059
0.048
0.037
0.027
0.018
0.009
0.008
0.008
0.000
0.000
0.325
0.325
0.326
0.375
0.362
0.323
0.315
0.307
0.299
0.291
0.258
0.775
0.775
0.776
0.777
0.778
0.779
0.735
0.723
0.711
0.699
0.689
0.955
0.955
0.956
0.957
0.958
0.959
0.961
0.963
0.965
0.967
0.969
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1259
determined mergers in a 10-firm industry, except where fixed costs are small and,
simultaneously, competitive intensity is particularly strong.20
In fact, numerical calculation shows that, for any industry size, there is a
competitive intensity sufficiently weak ( l sufficiently high), and a fixed costs
value sufficiently large, to make endogenously generated mergers socially beneficial, and that these limit values decrease in the industry size: in a three-firm
industry, either l > 2 0.1 or F /(a 2 c)2 > 0.05 will ensure that any endogenously
generated merger is socially desirable; in a 10-firm industry, as seen above, this
happens for l > 2 0.8 or F /(a 2 c)2 > 0.05; and, in a 20-firm industry, for
l > 2 0.9 or F /(a 2 c)2 > 0.01.
4.2.5. A two-firm merger in an n-firm industry
Given certain values for the number of firms in the industry (n) and the fixed
costs (F /(a 2 c)2 ), conditions (11), (15b) and (17) were used to compute the
ranges of competitive intensity ( l) that would lead to such a merger. Although the
interaction of conditions (11), (15b) and (17) does not allow any simple
formulation of the conditions in which two firms will merge, intermediate degrees
of competitiveness seem to be more conducive to this: at one extreme, conditions
of extreme competitiveness, with l close to 21, will lead to monopoly, or some
other very concentrated market structure; at the other extreme, when the values of
l approach 1, the pre-merger industry profit is already close to the one that would
be generated by perfect collusion and a two-firm merger will be of no interest if
the fixed costs are not significant or the industry is not very small (Table 4).
It comes as no surprise that two-firm mergers are seldom observed when fixed
costs are very low (e.g. F /(a 2 c)2 5 0.001): we are, in these situations, very close
to the case of no fixed costs in which, we have already seen, only mergers to
monopoly would happen, and even these only if the industry was strongly
Table 4
Ranges of competitive intensity ( l) for which two-firm mergers would be observed, given various
industry sizes (n) and fixed costs (F /(a 2 c)2 ) levels, as implied by conditions (11), (15b) and (17)
Fixed costs
n
3
4
5
10
20
44
20
0.001
0.010
0.050
0.100
hj
hj
hj
hj
[20.602; 2 0.532]
hj
[20.384; 20.152]
[20.595; 20.423]
[20.691; 20.394]
[0.168; 0.703]
hj
hj
[0.179; 1.000]
[0.506; 1.000]
hj
hj
hj
hj
hj
hj
hj
hj
hj
hj
Table 3 assumes (a 2 c)2 5 1. This assumption is not restrictive, however: (a 2 c)2 is a scalar
relative to the entries in the table.
1260
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
competitive. In fact, conditions (15b) and (17) imply that two-firm mergers are
profitable for some ranges of intense industry competitiveness. But it turns out
that, in every one of these, condition (11) indicates that monopoly will prevail. For
different reasons, monopoly, or at least some very concentrated market structure,
will also prevail at high levels of fixed costs (e.g. F /(a 2 c)2 5 0.1). In this case,
any unconcentrated market structure will lead to negative profits and, so, two-firm
mergers will be unattractive.
Two-firm mergers are more common at intermediate levels of fixed costs. This
will only happen up to a certain industry size, though: we have already seen that
for any number of participants in the merger, there is always an industry size
sufficiently large to make it unprofitable; and, in our model (if not in reality) firms
do not carry out unprofitable mergers.
5. Concluding remarks
In the model presented in this paper, three factors interact to determine whether
or not firms will merge: the initial number of firms in the industry, the expected
competitive intensity and the possibility of economizing on fixed costs through
merger. The model shows that the equilibrium market concentration, and firms’
propensity to merge, is decreasing in the first of these factors and increasing in the
other two, a conclusion capable of empirical refutation.
This direct relation between competitive intensity and concentration implies
that, when firms have the ability to alter the market structure, (expected)
competitive intensity is inversely related to the consumer surplus. This result, that
seems to run contrary to the direct relation between competition and consumer
surplus often admitted in industrial organization, echoes other results in the
two-stage game literature. We have shown, however, that, when fixed cost
economies are taken into consideration, endogenously generated mergers will be
socially beneficial, if the competitive intensity is not extremely strong or the
industry is not, ex ante, very concentrated. This does provide some guidance for
where merger policy should focus.
It remains to be seen whether our results are robust to changes in the model’s
assumptions: different, or more general, functional forms for cost and demand,
firm heterogeneity, and different endogenous merger formation processes, all seem
to deserve future research. The fact that, while providing new insights, our model
reiterates some of the results in the literature is nevertheless encouraging.
Acknowledgements
´
Funding from the PRAXIS XXI program, from the Portuguese Ministerio
da
ˆ
Ciencia
e Tecnologia, under Grant No. PRAXIS / PCSH / C / CEG / 30 / 96, is
V. Rodrigues / Int. J. Ind. Organ. 19 (2001) 1245 – 1261
1261
acknowledged. I have benefited from the comments of Jose´ Manuel Amado da
˜ Mario
´
Silva, Margarida Catalao,
Alexandre Silva, T.S. Ho, Victor Mendes,
participants at the 7th Encontro Nacional de Economia Industrial, at Lamego,
´
Portugal and an internal seminar at Universidade Catolica
Portuguesa, Porto, and
of an anonymous referee. Errors and omissions rest with the author.
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