Horizontal Mergers in a Dynamic Cournot Market: Solving the Free Riding Issue Without Efficiency Gains

Marc Escrihuela-Villar; Walter Ferrarese

doi:10.1515/bejeap-2018-0321

Published by De Gruyter August 6, 2019

Horizontal Mergers in a Dynamic Cournot Market: Solving the Free Riding Issue Without Efficiency Gains

Marc Escrihuela-Villar and Walter Ferrarese

From the journal The B.E. Journal of Economic Analysis & Policy

https://doi.org/10.1515/bejeap-2018-0321

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Abstract

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.

Keywords: horizontal mergers; repeated interaction; free riding issue

JEL Classification: L11; L13; C73

Acknowledgements

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.

Appendix

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:

(15)ΠIp=πIp+δ1−δΠIc;

(16)ΠIp=πIp+δ1−δΠIc,o = 1,...,n−m,

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:

(17)11−δΠIc(QI,qo)≥ΠId(QId,qo)+δ(ΠIp(QIp,qop)+δ1−δΠIc(QI,qo));

(18)11−δΠoc(QI,qo)≥Πod(QIc,q−o,qod)+δ(Πop(QIp,qop)+δ1−δΠoc(QI,qo)),o=1,…,n−m.

Constraints ensuring that firms do not deviate from the punishment path are also needed. These write:

(19)ΠIp≥ΠIdp+δΠIp;

(20)Πop≥Πodp+δΠop,o=1,...,n−m.

The following proposition describes the equilibrium of the repeated game:

Proposition 6.

Let:

(21)δoIC(n,m,qo)=(1−(n−m−2)qo)2−8qo(1−(n−m)qo16((1−(n−m)qo2)qo+1(n−m)2);

(22)δoP(n,m,qo)=1(n−m)2(1−(n−m)qo2)qo+1(n−m)2.

Provided thatmax{δoIC,δoP}<1, then ifδ∈(max{δoIC,δoP},1), an output configurations whereqo∈(qo_,qo¯)andQI(qo)=1−(n−m)qo2is produced in equilibrium in the infinitely repeated Cournot game under stick and carrot strategies.

B

Proof of Proposition 1.

Consider the system:

(23){(1−QI−(n−m)qo)QI≥1(n+1)2;(1−QI−(n−m)qo)qo≥1(n+1)2,o=1,...,n−m;(1−QI−(n−m)qo)QIm≥(1−QI−(n−m)qo)qo.

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:

(24){(1−QI−(n−m)qo)QI≥1(n+1)2;(1−QI−(n−m)qo)qo≥1(n+1)2,o=1,...,n−m;QI≥mqo.

System (24) is solved by the output configuration (qo,QI(qo)) of Proposition 1. □

Proof of Proposition 2.

The first step is to solve [P]. The Lagrangian of the problem is:

L=ΠIc+λ(Πoc−ΠiCN(n−m+1))+γ(ΠIc−ΠiCN(n−m+1))+ω(ΠIc−mΠoc),

where (λ, γ, ω) are the Lagrange multipliers. Since we focus on an interior solution, then λ=γ=ω=0. The Karush–Kuhn–Tucker first order conditions write:

(25)∂L∂QI=1−(n−m)qo−2QI=0;

(26)QI∂L∂QI=0;

(27)∂L∂λ=(1−(n−m)qo−QI)qo−(1n−m+2)2>0;

(28)λ∂L∂λ=0;

(29)∂L∂γ=(1−(n−m)qo−QI)QI−(1n−m+2)2>0;

(30)γ∂L∂γ=0;

(31)∂L∂ω=QI−mqo>0;

(32)ω∂L∂ω=0.

First notice that condition (31) is just a simpler way of rewriting ΠIc−mΠoc>0. From (25):

(33)QI=1−(n−m)qo2.

Plugging (33) in (27), yields:

(34)(1−(n−m)qo2)qo−(1n−m+2)2>0.

Plugging (33) in (29), yields:

(35)(1−(n−m)qo2)2−(1n−m+2)2>0.

Plugging (33) in (31), yields:

(36)qo<1n+m.

System (34)–(36) has solutions if m≥2,n>n¯(m) and qo∈(qo_,qo¯). 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 qo∈(qo_,qo¯) and also remember that, according to [P], Πoc=(1−(n−m)qo2)qo. We now compute each outsider’s deviation profit. Let:

(37)qod=argmaxqod¯(1−QI−(n−m−1)qo−qod¯)qod¯,

from which qod=1−QI−(n−m−1)qo2. Substituting QI=1−(n−m)qo2 in qod and plugging back into the deviation profit, one obtains that Πod=(1−(n−m−2)qo4)2. Hence, (8) becomes:

(38)11−δ(1−(n−m)qo2)qo≥(1−(n−m−2)qo4)2+δ1−δ(1n−m+2)2.

Solving (38) w.r.to δ, yields:

(39)δ≥(n−m+2)2(1−qo(n−m+2))(m−n+2)(m−n−6+qo((m−n)2−4))=δv(n,m;qo).

□

Proof of Proposition 3.

From the output configurations provided in (12) and (13), the profit of an outside firm can be written as Πoc(n,m,δ)=(A−2)4+2(A2−4)2δ−(A−6)(2+A)2(3A−2)δ2((A−2)3−(A−2)(2+A)2δ)2, with A ≡ m – n. It is easy to check that Πoc(n,m,δ) increases with δ if δ<δ^(n,m), decreases with δ if δ>δ^(n,m) and, therefore, reaches its maximum at δ=δ^(n,m). Then, from Proposition 2, we can evaluate δv(n,m;qo) at qo_ and qo¯ 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 δv(n,m;qo_)=1 and δv(n,m;qo¯)=δ~. Finally, we just have to check that δ~ is smaller than δ^(n,m), as, otherwise, we know from (12) that outside firms would not produce within the required range. It is immediate to check that δ~<δ^(n,m)iffn ≥ 3m. □

Proof of Proposition 4.

From the output configurations provided in (12) and (13), the market share of the insiders sI=δ(3m−3n−2)(2+m−n)+(2−m+n)2δ(2+m−n)(m2−2+n(3+n)−m(3+2n))−(m−n−1)(2−m+n)2. First we check that s_I always increases with δ, as ∂sI∂δ=4(m−n−2)3(n−m)(2+m−n)((m−n−1)(2−m+n)2−δ(2+m−n)(m2−2+n(3+n)−m(3+2n)))2>0 whenever n ≥ 3m. Finally, we just have to check that s_I = m/n if δ=δ~ and s_I = 1/3 if δ=δ^. □

Proof of Proposition 5.

From the output configurations provided in (12) and (13), Π^Ic(n,2,δ)=(n2+δ(n−4)(3n−4))2n2(n2−δ(n−4)2)2. From (14), Πfc(n,δ)=((1+n)2−δ(n−1)(3+n))((1+n)2+δ(n−1)(1+3n))(1+n)2((1+n)2−δ(n−1)2)2. Then, since both Π^Ic(n,2,δ) and Π^fc(n,δ) increase monotonically with δ if δ<δ^(n,m), we just have to check that the equation Π^Ic(n,2,δ)2=Πfc(n,δ) has a unique root in δ, which is always larger than δ~ 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 qodp and QIdp in order to maximize:

(40)Πodp=(1−(n−m−1)qop−QIp−qodp)qodp;

(41)ΠIdp=(1−(n−m)qop−QIdp)QIdp.

Thus, when the merged entity deviates from the punishment phase its production is QIdp=1−(n−m)qop2 and its profit is ΠIdp=(1−(n−m)qop2)2. This profit is equal to zero if qop=1n−m.

When the representative outsider deviates from the punishment phase its production is qodp=1−(n−m−1)qop−QIp2 and its payoff is Πodp=(1−QIp−(n−m−1)qop2)2. Substituting qop=1n−m in Πodp, yields Πodp=((n−m)(1−QIp)−(n−m−1)2(n−m))2. This profit is equal to zero if QIp=1n−m. Moreover, straightforward calculation shows that if QIp=qop=1n−m, then the punishment profits of the representative outsider and the merged entity are Πop=ΠIp=−1(n−m)2.

As ΠIc=ΠId, (17) becomes ΠIc≥ΠIp, which is always satisfied. Constraints (18)–(20) write:

(42)δ≥1(n−m)2(1−(n−m)qo2)2+1(n−m)2=δIP;

(43)δ≥(1−(n−m−2)qo)2−8qo(1−(n−m)qo)16((1−(n−m)qo2)qo+1(n−m)2)=δoIC;

(44)δ≥1(n−m)2(1−(n−m)qo2)qo+1(n−m)2=δoP.

By construction ΠIc>Πoc, thereby (42) is redundant. Thus, provided that max{δoIC,δoP}<1, when δ∈(max{δoIC,δoP},1) collusion is feasible. □

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Published Online: 2019-08-06

Horizontal Mergers in a Dynamic Cournot Market: Solving the Free Riding Issue Without Efficiency Gains

Abstract

Acknowledgements

Appendix

A Stick and Carrot Strategies

Proposition 6.

B

Proof of Proposition 1.

Proof of Proposition 2.

Proof of Proposition 3.

Proof of Proposition 4.

Proof of Proposition 5.

Proof of Proposition 6.

References

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