## Abstract

I study a merger model among symmetric Cournot firms where—before a merger occurs—firms choose output simultaneously and in which a merged entity acquires the market leadership. I find conditions under which a single or multiple mergers are profitable and solve the free-riding problem. The model connects to Liu and Wang (Econ Lett 129:1–3, 2015), who show that a single leading entity can profitably merge with an arbitrary number of firms. The current paper extends their results in two directions: first, I find the conditions under which the free-riding issue is solved; second, I study the implications of multiple mergers, in which the merged entities are allowed to be heterogeneous in the number of merging firms. A welfare analysis shows that mergers may be welfare-enhancing—even without efficiency gains. Moreover, the set of welfare-enhancing mergers is the same irrespective of the measure that is used: consumer surplus only, or the sum of consumer surplus and industry profits. This suggests caution for the antitrust authorities in evaluating the overall effect of these mergers.

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## Notes

As a merger that involves only leaders/followers is profitable when the ratio between the number of merging leaders/followers and the total number of leaders/followers is above 0.8, a profitable bilateral merger can occur only if it involves all the leaders/followers in the market. Already with three leaders/followers, such a ratio drops to 0.66. This explains the results in Huck et al. (2001) and Feltovich (2001).

Restricting attention to the linear case is an obvious limit. However, especially for multiple heterogeneous mergers, it allows analytical tractability and highlights the main features of the model. A natural generalization would be a model with an implicit market inverse demand.

Although the results in Sect. 4 can be applied to those of this section by setting \(L=1\), a deep analysis of the scenario with a unique leader is required in order to:

*i)*properly discuss the extension of LW (2015) in terms of the resolution of the free-riding issue; and*ii)*provide a comparison between the current setting and a vast strand of literature of mergers in Stackelberg markets that assumes a single merger that, moreover, involves only two firms.The notation \(\llcorner \lrcorner \) means approximating down to the largest integer and \(\ulcorner \urcorner \) means approximating up to the lowest integer. This notation is required to tackle cases where

*n*is even and \(\frac{n+1}{2}\) is not an integer.This can formally be captured by evaluating \(\frac{\partial (n-m)q_{f}}{\partial m}=-\frac{1}{2(n-m+1)^{2}}<0\).

This can formally be captured by evaluating \(\frac{\partial p_1^{post}}{\partial m}=\frac{1}{2(n-m+1)^{2}}>0\).

This can formally be captured by evaluating \(\frac{\partial \pi _f}{\partial m}=\frac{1}{2(n-m+1)^{3}}>0\).

The resolution of the free-riding problem has, so far, been obtained by assuming either a change in the firms’ cost structure (Heywood and McGinty 2008; Gelves 2008; Brito and Catalão-Lopes 2011) or strong enough cost synergies (Cunha and Vasconcelos 2015). The current paper instead shows that the free-riding component can be solved leaving all of these hypotheses aside.

More generally, in a game with one leader and \(n-1\) followers, the leader obtains \(\pi _{l}=\frac{1}{4n}>\frac{1}{(n+1)^{2}} ,\; \forall n\ge 3\), while each follower obtains \(\pi _{f}=\frac{1}{4n^{2}}<\frac{1}{(n+1)^{2}} ,\; \forall n\ge 3\).

As when seven firms merge in two leaders, a trilateral merger is profitable, while a four-firm merger is not, one may wonder whether the latter could take place. Actually, both the theoretical literature (Brito 2003; Fridolfsson and Stennek 2005; Budzinski and Kretschmer 2016) and the empirical literature (Andrade et al. 2001; Tichy 2001; Gugler et al. 2003; Röller et al. 2006) have shown that unprofitable mergers often occur and that they can be the result of rational actions. Although rationalizing unprofitable mergers is beyond the scope of the paper, this is why I decided not to limit the analysis only to multiple profitable mergers.

For example, consider the case of a unique leader. In this case, when

*m*firms merge, the maximal size is exactly \(\overline{m}_{l}=m\). Thus, according to Proposition 3, the free-riding issue is (weakly or strictly) solved only if \(n-m+1\ge m\), that is \(m\le \llcorner \frac{n+1}{2}\lrcorner \). This coincides with point*ii)*in Proposition 1.The welfare implications are clearly valid provided that quantity competition is an appropriate description of the industry. Although antitrust authorities tend to prefer price competition as the standard model, the welfare results that are presented here are still relevant, in the light of the large literature on mergers in Cournot markets and of the good fit of the Cournot hypothesis to some specific industries. A good example—as is also discussed in Escrihuela-Villar and Faulí-Oller (2007) and Cunha and Vasconcelos (2015)—is the DRAM industry, where firms have differential strategic power and multiple mergers are observed.

One of the main results in Farrell and Shapiro (1990) is that mergers that generate no synergies unambiguously rise market price. This is no longer true in this setting.

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## Acknowledgements

This paper is part of the first chapter of my Ph.D. thesis. I am extremely grateful to the Editor Lawrence J. White, two anonymous referees, my advisor Alberto Iozzi and the members of the Ph.D. defence committee Carmen Bevia, Luca Panaccione and Helder Vasconcelos. I am also grateful to Berardino Cesi, Lapo Filistrucchi, Antonio Nicoló, Francesco Ruscitti, Francois Salanié, the audience at seminars at the University of Rome Tor Vergata, the Max Planck Institute for Tax Law and Public Finance in Munich and the 2017 EARIE Conference in Maastricht.

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## Appendix

### Appendix

### Proof of Proposition 1

For point (i), see Proposition 1 in LW (2015).

For point (ii), let:

be the gain of an insider with respect to an outsider with a unique leading entity. Since the denominator of (13) is positive, \({\textit{sign}}(\triangle )={\textit{sign}}(n-2m+1)\), which is positive if \(n\ge 4\) and \(m<\frac{n+1}{2}\) and the expression is equal to 0 if \(m=\frac{n+1}{2}\). \(\square \)

### Proof of Proposition 2

Equation (5) is non negative if \(m_l\le \frac{(n+1)^2}{(L+1)^2(n-m+1)}\). Thus, \({\widehat{m_{l}}}\equiv \frac{(n+1)^2}{(L+1)^2(n-m+1)}\). Multiple mergers are profitable only if:

If \(\pi _{l}\) is the same for all leaders \(l=1,\ldots,L\), \(\pi ^{CN}(n)\) fixed and \(g_{l}\) decreasing in \(m_{l}\), then if \(g_{l}\ge 0\) for the leader that is formed by the largest number of insiders, for some redistribution of the *m* merging firms into the \(L\ge 2\) entities, then \(g_{l}\ge 0, \forall l=1,\ldots,L\). Apart from which of the *n* identical firms act as insiders, the redistribution in which it is harder for all mergers to be profitable when *m* out of *n* firms merge into \(L\ge 2\) leaders is the one in which a leader is formed by \(\overline{m}_{l}\) insiders and the other mergers involve two firms. Thus, if \({\widehat{m_l}}\ge \overline{m}_{l}\) all mergers are simultaneously profitable in the originally picked distribution and in all other distributions as well. When instead \({\widehat{m_l}}<\overline{m}_{l}\), a merger is profitable only if a leader is formed by at most \(\llcorner {\widehat{m_l}}\lrcorner \) insiders. \(\square \)

### Proof of Proposition 3

This proof follows the same reasoning of Proposition 2 and it is therefore omitted. \(\square \)

### Proof of Proposition 4

The explicit form of the output variation is:

Simple algebra shows that (15) is positive if \(m<\mathbf {M}(n,L)\). \(\square \)

### Proof of Proposition 5

(15) is: positive if \(m<\mathbf {M}(n,L)\); equal to zero if \(m=\mathbf {M}(n,L)\); and negative if \(m>\mathbf {M}(n,L)\). The explicit form of the social welfare variation is:

After some algebra, one gets that (16) is: positive if \(m<\mathbf {M}(n,L)\); equal to zero if \(m=\mathbf {M}(n,L)\); and negative if \(m>\mathbf {M}(n,L)\). This proves the claim. \(\square \)

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### Cite this article

Ferrarese, W. When Multiple Merged Entities Lead in Stackelberg Oligopolies.
*Rev Ind Organ* **56, **131–142 (2020). https://doi.org/10.1007/s11151-019-09678-6

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DOI: https://doi.org/10.1007/s11151-019-09678-6

### Keywords

- Horizontal mergers
- Stackelberg markets
- Welfare

### JEL Classification

- L11
- L13
- L22
- L41