This article examines the incentive to merge in a Bertrand competition model with generalized substitutability and price competition. The model suggests that acquisition of firms by their rivals can result in maximal concentration of the industry.endogenous mergers, concentration, price competition

Emilie Dargaud

English

Research Papers in Economics

Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
1. Introduction
This article investigates the incentive for ﬁrms to merge and the market structures which
could involve maximal concentration through mergers. Literature on the incentives to merge
is extensive but there are two major ways to model this : exogenous mergers (see for example
Salant Switzer and Reynolds (1983), Deneckere and Davidson (1985), Perry and Porter
(1985) or Farrell and Shapiro (1990)) and endogenous mergers. In this article, we present a
model of endogenous mergers as in Kamien and Zang (1990), considering a three-stage game
with a simultaneous bidding stage. Other contributions have been made about simultaneous
bidding as in Gaudet and Salant (1992), Gonzalez-Maestre and Lopez-Cunat (2001) or Ziss
(2001). In their paper, Kamien and Zang (1990) study the incentives to merge by considering
internal competition between ﬁrms owned by a same owner, but this is at odds with reality
because if contracts were renegotiable ex-post, ﬁrms may act cooperatively to maximize the
proﬁt of the owner. Our study, by contrast, consider that ﬁrms belonging to the same owner
act cooperatively between themselves.
In this perspective, we consider a three-stage game. The ﬁrst stage is a simultaneously
bidding stage among some sellers and buyers. In the second stage, each buyer chooses the
number of its active ﬁrms considering that ﬁrms belonging to the same owner act coopera-
tively between themselves. Finally price competition follows between active ﬁrms. We can
characterize subgame perfect Nash equilibrium and we show that maximal concentration of
the industry is an equilibrium.
The article is organized as follows. Section 2 presents the model. Characterizations of
equilibrium are provided in section 3. Concluding remarks follow. Proofs of results appear
in the appendix.
2. The model
We consider the following utility function derived from H ackner (2000):
U(q;I) =
n X
i=1
qi  
1
2
"
n X
i=1
q
2
i + 2
X
i6=j
(qiqj)
#
+ I (1)
The parameter  2 (0,1) is a measure of the substitutability between products. Utility
is quadratic in the consumption of the n horizontally diﬀerentiated products and linear in
the consumption of other goods: I, which price is normalized to one.
The demand function is given by:
qi(pi;pj;n) =
1
1 + (n   1)
"
1  
1 + (n   2)
1   
pi +

1   
X
j6=i
pj
#
(2)
We assume that entry into the industry is diﬃcult and that each producer operates at a
constant and identical marginal and average cost c which is normalized to 0. All the relevant
variables and strategies available to the ﬁrms are common knowledge.
We posit an initial industry consisting of n independent ﬁrms. Two ﬁrms (1 and 2) are
buying ﬁrms and the others play the role of seller. Since antitrust authorities make eﬀorts to
138Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
inhibit monopolization through the issuance of merger guidelines, the maximal concentration
in this paper is duopoly.
Let us now turn to the formal description of our three-stage game.
Stage 1 : bidding stage.
The buying ﬁrms make oﬀers simultaneously to others and each of them sets a ceasing price
for its own ﬁrm. This stage is a one-shot interaction situation. Let Kj be the number of
ﬁrms owned by a merged entity Mj (j = 1;2) and Z the number of outsiders (hereafter
"out" in mathematical computations), ﬁrms which have not been bought. We suppose that
a ﬁrm is sold to its willingness to sell.
A market structure is a Nash equilibrium in this subgame if no ﬁrm is able to purchase one
or several ﬁrms and the others accept and if the “net” proﬁt of the buyer is maximal.
The equilibrium conditions are then deﬁned by: 
WTPMi(K1;K2;Z)  WTS
Mi
out(K1;K2;Z);8i = 1;2
Ki 2 ArgmaxMi(Ki;Kj;Z)
where WTPMiand WTS
Mi
out design respectively the willingness to pay of the merged entity
Mi and the willingness to sell by an outsider to the merged entity Mi.
Stage 2: merger stage.
Each merged entity decides the optimal number (k
j) of its active ﬁrms to maximize its proﬁt
(0 < k
j  Kj).
A SPNE (Subgame Perfect Nash Equilibrium) in this acquisition game is said to be “merged”
if the number of active ﬁrms in the last stage is fewer than the initial number of ﬁrms.
Stage 3: competition stage.
Firms belonging to a same merged entity act cooperatively amongst one another but face
price competition with each other. Therefore, the maximization programs of the merged
entity M1 (symetric one for M2) and an outsider are respectively :
(
maxp
M1
1 ;p
M1
2 ;:::;p
M1
k1
M1(k1;k2;Z;;
Pk1
i=1 p
M1
i ;
Pk2
i=1 p
M2
i ;
P
j2out pj)
maxpout out(k1;k2;Z;;
Pk1
i=1 p
M1
i ;
Pk2
i=1 p
M2
i ;
P
j2out pj)
3. Analysis of equilibria
3.1. Equilibrium prices
Resolving each maximization program, equilibrium prices are determined by :
lemma 1. 8
> <
> :
p
1=
(1 )(2+2(Z 1)+2k1+k2)(2+(2Z 3)+2(k1+k2))
A
p
2=
(1 )(2+(2Z 3)+2(k1+k2))(2+2(Z 1)+(k1+2k2))
A
p=
(1 )(2+2(Z 1)+2k1+k2)(2+2(Z 1)+(k1+2k2))
A
139Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
With A = 22k2
1(4   4 + 3Z + 3k2) + 2(1 + (Z   1) + k2)(2(2 + (Z   3))(1 + (Z   1)) + (4   4 + 3Z)k2) +
k1(2(8 + 5(Z   2))(1 + (Z   1)) + k2(22   25 + 17Z + 6k2))
p
i (8i = 1;2) designs the equilibrium price of the merged entity Mi and p the one of an
outsider.
Note that :
 p
i > p iﬀ ki > 1, 8i = 1;2
 p
i > p
j iﬀ ki > kj, 8i = 1;2 ; i 6= j
Equilibrium proﬁt of the merged entity Mi is then given by:

Mi =
1
(1 + (ki + kj + Z   1))A2(1   )ki(1 + (Z   1) + kj)
(2 + 2(Z   1) + 2ki + kj)
2(2 + (2Z   3) + 2(ki + kj))
2;8(i;j) = (1;2);i 6= j
3.2. Merger phase
To determine if an owner of several ﬁrms will choose to close some of them, we have
to compute the value of @M1
@k1 jk1=K1. If @M1
@k1 jk1=K1  0 then k
1 < K1 (according that
@2M1
@k2
1 jk1=K1  0): Numerical simulations ﬁxing the value of n allows us to obtain :
proposition 1. Merged equilibria can occur in this game if  is high enough and Z relatively
low. Especially, for Z  2, merged equilibria can not occur in this game.
Proof. See Appendix A.
The presence of outside ﬁrms increases competitive pressure, so that when the number
of outsiders is high enough (Z  2), a merged entity could choose to let active all its ﬁrms
to maintain its market power.
In the remainder of this paper, we assume  = 0:9 in order to consider all the diﬀerent
cases (merged or unmerged equilibria).
The objective by now is to analyse if the number of ﬁrms owned by the merged entities
inﬂuences the number of their active ﬁrms.
lemma 2. The reaction function of the merged entity Mi is given by (8(i;j) = (1;2);i 6= j):
k

i(kj;Z)
8
<
:
= Ki if Z  2
= f(kj)  Ki if Z = 1
= g(kj)  Ki if Z = 0
Proof. See Appendix B.
The number of active ﬁrms plays a major role: since products are horizontally diﬀerenti-
ated, demand increases with this number so a merger can gain market shares, but equilibrium
price is lower. Active ﬁrms create internal competition but reinforce competition with the
others.
140Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
3.4. Bidding stage
Each of the two shareholders simultaneously sets a vector of bids facing the number of ﬁrms
owned by the other. At the same time, the selling ﬁrms decide simultaneously whether to
accept or not.
The equilibrium conditions are deﬁned by:

WTPMi(K1;K2;Z)  WTS
Mi
out(K1;K2;Z);8i = 1;2
Ki 2 Argmax Mi(Ki;Kj;Z)
with :

WTPMi(Ki;Kj;Z) = Mi(Ki;Kj;Z)   Mi(1;Kj;Z + Ki   1)
WTS
Mi
out(Ki;Kj;Z) = out(Ki   1;Kj;Z + 1)  (Ki   1)
When the outsider sets its selling price, it forestalls its proﬁt in the last stage if it de-
clines the oﬀer considering all the other ﬁrms the owner wants to buy have accepted and
considering that the owner which buy others can close some of them after.
lemma 3.
 Mergers M1 and M2 buy all the outside ﬁrms so as to get : K1+K2=n (Z=0).
 Since K1+K2=n only market structures wherein one owner lets all his ﬁrms active and
the other closes some of his ﬁrms can occur.
Proof. See Appendix C.
4. Concluding Remarks
A three-stage game is considered in which the ﬁrms initially bid for merger forming
coalitions, then the merged entities decide how many of the original varieties will be oﬀered
and price competition follows.
Our main conclusion is that maximal concentration of the industry occurs at equilibrium
even if we consider a high number of ﬁrms. This result depends on the degree of diﬀerentia-
tion. This variable is widely used as a measure of the intensity of competition in industrial
organization model. In this context, we show that when the competition is really ﬁerce in
the market, then merged equilibria can occur.
Our model could be extended to the case of coalitions structures in which a coalition,
maximizing its joint payoﬀ, decides the number of active ﬁrms which compete. In this case,
the non-competing ﬁrms are not closed, they do not compete but still exist by receiving, for
example, an allowance from the active ﬁrms.
141Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
Appendix
Appendix A.
Analytical expression of @M1
@k1 jk1=K1 is too much complex to be used so we have to make nu-
merical simulations1.
First, we can easily prove that for Z  2 then merged equilibria can not occur in this game.
To do this, we vary the values of K1 2 [1;n   1] and Z with Z  2. We have to ﬁx n and
we make a 3D graphic of @M1
@k1 jk1=K1 depending of the values of k2 (k2 2 (0;n   K1   Z]) and 
( 2 (0;1)). We then select only the point for which @M1
@k1 jk1=K1  0.
Second, we search for the negative values of @M1
@k1 jk1=K1 setting the values of  and Z = 0 or
Z = 1.
We plot a 3D graphic which depends on the values of 0 < k1  n   1 and 0 < k2  n   1. We
then observe that if  = 0:1 then @M1
@k1 jk1=K1 is always strictly greater than 1. Moreover, the space
of parameters under which @M1
@k1 jk1=K1  0 is always greater with the case  = 0:9 than with the
case  = 0:5, so merged equilibria can occur if the products are not too much diﬀerentiated.
For each case, we have veriﬁed that: @2M1
@k2
1
jk1=K1  0
Appendix B
For Z  2 then no merged equilibrium can occur so k
i = Ki;8i = 1;2 (see Appendix A).
For Z = 1 and Z = 0, we compute the value of @M1
@k1 jk1=K1 for each value of K1. Example : the
following table is the values for Z = 1 and n = 16 (K1 2 (1;14))
K1
  
@M1
@k1
  
k1=K1
1 > 0; 8 k2
2 > 0; 8 k2 < 13:564
3 > 0; 8 k2 < 9:04082
4 > 0; 8 k2 < 7:85105
5 > 0; 8 k2 < 7:38937
6 > 0; 8 k2 < 7:19915
7 > 0; 8 k2 < 7:13668
8 > 0; 8 k2 < 7:14278
9 > 0; 8 k2 < 7:18883
10 > 0; 8 k2 < 7:25945
11 > 0; 8 k2 < 7:34571
12 > 0; 8 k2 < 7:44215
13 > 0; 8 k2 < 7:54526
14 > 0; 8 k2 < 7:65272
Table I: For Z = 1, sign of @M1
@k1 jk1=K1 for each possible value of K1
Remarks:
For Z = 1, then merged equilibria can occur. For example, for K1 = 9 (K2 = 16 K1 Z = 6),
k2 must be less than 7.14 in order to have k
1 = K1. By deﬁnition, k
2  K2 , k2  6 so k
1 = K1.
Facing k
1 = K1 = 9 we maximize M2 to ﬁnd that at equilibrium k
2 < K2 and consequently we
face a merged equilibrium.
1We present the results obtained for n = 16 but we have proved all the results for each n  16 (As regards
the maximal value, we have to set this arbitrarily because we use numerical simulations)
142Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
The following table gives the best response function of the merged entity depending of its number
of ﬁrms.
K1 K2 k1 k2
1 14 1 14
2 13 2 13
3 12 2.19 12
4 11 2.36 11
5 10 2.61 10
6 9 3.02 9
7 8 3.8 8
8 7 8 3.8
9 6 9 3.02
10 5 10 2.61
11 4 11 2.36
12 3 12 2.19
13 2 13 2
14 1 14 1
Table II: Reaction functions of the two merged entities for Z = 1 and n = 16
Applying the same reasoning for Z = 0, we obtain the following table:
K1 K2 k1 k2 k1 k2
1 15   0.156969 15
2 14   0.157632 14
3 13 3 0.203337 0.158403 13
4 12 4 0.186461 0.159311 12
5 11 5 0.177512 0.160394 11
6 10 6 0.171958 0.161711 10
7 9 7 0.168174 0.163346 9
8 8 8 0.165428 0.165428 8
9 7 9 0.163346 0.168174 7
10 6 10 0.161711 0.171958 6
11 5 11 0.160394 0.177512 5
12 4 12 0.159311 0.186461 4
13 3 13 0.158403 0.203337 3
14 2 14 0.157632  
15 1 15 0.156969  
Table III: Reaction functions of the two mergers for Z = 0 and n = 16
Note that for Z = 0 and Ki > 2,8i = 1;2, two cases are possible for each structure (K1,K2).
Appendix C. Example for n = 16.
143Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
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144Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
For each value of K2 2 [1;n   2], we compute the willingness to pay of M1 to buy 1, 2, ... ,
n   K2 ﬁrms and the total willingness to sell of outside ﬁrms.
WTPMi(Ki;Kj;Z) = Mi(Ki;Kj;Z)   Mi(1;Kj;Z + Ki   1) (3)
WTS
Mi
out(Ki;Kj;Z) = out(Ki   1;Kj;Z + 1)  (Ki   1) (4)
Example for K = 1 :
number of ﬁrms bought K1 WTP(104) WTS(104)
1 2 5.06 4.99
2 3 10.27 10.08
3 4 15.76 15.39
4 5 21.69 21.16
5 6 28.26 27.60
6 7 35.76 35.04
7 8 44.57 44.10
8 9 55.35 55.60
9 10 69.07 70.92
10 11 87.54 92.80
11 12 114.21 126.39
12 13 156.9 183.84
13 14 237.68 298.09
14 15 1559.51 591.64
Table VI: K2 = 1
The willingness to pay of the buyer M1 is strictly greater than the total willingness to sell of
outside ﬁrms if the number of ﬁrms belonging to M1 is :K1 with K1 2 [2;3;4;5;6;7;8;15]:
Moreover, the “net” proﬁt of M1 is maximal for K1 = 15 (, Z = 0).
We do this exercise for each possible value of K2.
145Economics Bulletin, 2012, Vol. 32 No. 1 pp. 137-146
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146

A note on price and quantity competition in diﬀerentiated oligopolies

Endogenous mergers and maximal concentration: a note

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