We discuss horizontal mergers in a linear, homogeneous, symmetric Cournot market where the new entity repeatedly competes with outside firms over an indefinite horizon and efficiency gains are ruled out. If the degree of collusion among the outside firms is large enough, then, despite the large payoff of each outsider, we obtain output configurations solving both the profitability and the free riding issues. Such a result requires that mergers involve a sufficiently small number of firms, which is in sharp contrast with the findings in the literature and rationalize the empirical fact that relatively small mergers, even in absence of synergies, do actually occur and that, although outside firms may benefit from the merger of their rivals, insiders end up being better off. Finally, we show that merging can often be a more advantageous alternative than a fully collusive agreement, in which, moreover, the free riding component is not solved.
We are grateful to Carmen Bevia, Berardino Cesi, Lapo Filistrucchi, Alberto Iozzi, Antonio Nicoló, Luca Panaccione, Francesco Ruscitti, Francois Salanié, Helder Vasconcelos and the seminar audience at the University of Rome Tor Vergata.
A Stick and Carrot Strategies
Grim trigger strategies are unforgiving and an everlasting punishment may seem unrealistic. Stick and carrot strategies are forgiving and overcome the issue. We here show that all output configurations in Proposition 2, and not just the restriction to those of Proposition 3, can be produced in equilibrium under stick and carrot strategies (Abreu 1986, 1988). In these strategies, if a firm deviates from the cooperative path, firms expand output for one period and then go back to cooperation, provided that no further deviation from the punishment path occurred. Although here firms do not produce the same output in the cooperative phase, we show that an equilibrium in which all firms produce the same punishment quantity exists. Let:
denote the present discounted values of future profits in the punishment period and along the following cooperative paths for the merged entity and each outsider respectively. The incentive compatibility constraints write:
Constraints ensuring that firms do not deviate from the punishment path are also needed. These write:
The following proposition describes the equilibrium of the repeated game:
Provided that, then if, an output configurations whereandis produced in equilibrium in the infinitely repeated Cournot game under stick and carrot strategies.
Proof of Proposition 1.
Consider the system:
The first two conditions ensure that both the merged entity and each outsider obtain at least the pre-merger CN profit and the third condition ensures that the single insider obtains at least the same profit of an outsider. System (23) can be rewritten as:
System (24) is solved by the output configuration of Proposition 1. □
Proof of Proposition 2.
The first step is to solve [P]. The Lagrangian of the problem is:
where (λ, γ, ω) are the Lagrange multipliers. Since we focus on an interior solution, then . The Karush–Kuhn–Tucker first order conditions write:
First notice that condition (31) is just a simpler way of rewriting . From (25):
Plugging (33) in (27), yields:
Plugging (33) in (29), yields:
Plugging (33) in (31), yields:
System (34)–(36) has solutions if and . It remains to show the existence of a discount factor δ ∈ (0, 1) such that (7) and (8) hold. First remember that (7) is always satisfied. Turning to (8), pick any and also remember that, according to [P], . We now compute each outsider’s deviation profit. Let:
from which . Substituting in and plugging back into the deviation profit, one obtains that . Hence, (8) becomes:
Solving (38) w.r.to δ, yields:
Proof of Proposition 3.
From the output configurations provided in (12) and (13), the profit of an outside firm can be written as , with A ≡ m – n. It is easy to check that increases with δ if , decreases with δ if and, therefore, reaches its maximum at . Then, from Proposition 2, we can evaluate at and respectively to obtain the range of the discount factor such that the merger is profitable and the free riding issue is solved. We obtain that and . Finally, we just have to check that is smaller than , as, otherwise, we know from (12) that outside firms would not produce within the required range. It is immediate to check that iffn ≥ 3m. □
Proof of Proposition 4.
From the output configurations provided in (12) and (13), the market share of the insiders . First we check that sI always increases with δ, as whenever n ≥ 3m. Finally, we just have to check that sI = m/n if and sI = 1/3 if . □
Proof of Proposition 5.
From the output configurations provided in (12) and (13), . From (14), . Then, since both and Πfc(n,δ) increase monotonically with δ if , we just have to check that the equation has a unique root in δ, which is always larger than if n ≥ 8. This can be easily proved with a standard mathematical software. We used the program Wolfram Mathematica 7.0. Details are available from the authors upon request. □
Proof of Proposition 6.
When the representative outsider and the merged entity deviate from the punishment phase, they choose their deviation quantities and in order to maximize:
Thus, when the merged entity deviates from the punishment phase its production is and its profit is . This profit is equal to zero if .
When the representative outsider deviates from the punishment phase its production is and its payoff is . Substituting in , yields . This profit is equal to zero if . Moreover, straightforward calculation shows that if , then the punishment profits of the representative outsider and the merged entity are .
As , (17) becomes , which is always satisfied. Constraints (18)–(20) write:
By construction , thereby (42) is redundant. Thus, provided that , when collusion is feasible. □
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