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Mergers in Stackelberg Markets with Efficiency Gains

Abstract

This paper analyzes the profitability of mergers and their induced welfare effects in a setting where: (i) firms compete `a la Stackelberg; and (ii) mergers may give rise to efficiency gains. The results contrast with the ones obtained by previous literature where merger’s induced efficiency gains are assumed away. In particular, we find that under certain conditions regarding the cost benefits resulting from mergers, the so called “free-riding problem” is eliminated and mergers are not only profitable but also welfare enhancing, even with linear costs.

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Notes

  1. Other strands in the literature also tried to solve the (Salant et al. 1983)’s merger’s paradox and studied the induced welfare effects and the free-riding problem of horizontal mergers by adopting Bertrand competition with product differentiation (Deneckere and Davidson 1985), Cournot competition with convex costs (Perry and Porter 1985), spatial competition (Rothschild et al. 2000) or by changing the properties of the demand function (Faulí-Oller 2002).

  2. Heywood and McGinty (2008) showed that the combination of convex costs and leadership can largely eliminate the merger paradox. Daughety (1990) showed that a merger between two followers is potentially profitable and that this merger may be welfare-improving. Huck et al. (2001) and Feltovich (2001) concluded that if a leader merges with a follower, the merger is unambiguously profitable. Further, Escrihuela-Villar and Faulí-Oller (2007) assumed that followers are less efficient than leaders and showed that leaders rarely have incentives to merge and that mergers among followers become profitable when the followers are inefficient enough.

  3. For details, see Escrihuela-Villar and Faulí-Oller (2007) and the references cited therein.

  4. This is a simplified version of the cost structure proposed by Motta and Vasconcelos (2005) and captures the specific case studied in Horn and Persson (2001a, b).

  5. Differently from this paper, Catalão Lopes (2007, 2008) studied mergers’ effects when firms have symmetric strategic roles but are pre-merger cost asymmetric. Similar to this paper, however, Catalão Lopes (2007, 2008) assumes that, after the merger, the merged firm becomes more efficient than outsiders.

  6. This feature of a merger was proposed by Perry and Porter (1985). In their framework firms’ marginal cost is linear in output and mergers reduce the variable costs of production. For a discussion on the literature that models mergers as the pooling of capacities see, for instance, Catalão-Lopes (2007).

  7. Usually, in constant marginal cost models it is assumed that all firms in the market have identical costs and that there are no fixed costs. Differently, in a Cournot setting, Catalão-Lopes (2008) studies the formation and stability of mergers where firms are asymmetric and face fixed production costs. After a two firms merger, only one fixed cost subsists. Catalão-Lopes (2008) shows that the existence of fixed costs increases the desirability of merging and may also increase its stability. Perry and Porter (1985), on the other hand, showed that assuming a positive fixed cost does not change the incentives to merge, since the insider firm would keep the fixed costs of each of its pre-merger constituent firms.

  8. For α=0, these results are the same as those obtained by Huck et al. (2001) and Feltovich (2001).

  9. See Appendix A for further information on post-merger equilibrium results.

  10. Note that in order for the insider’s quantity to be positive, the level of efficiency gains should be higher than a negative threshold. Since we have assumed that α>0, we exclude this threshold. Also, it is straightforward to show that the quantity of insider firms is always positive for any level of efficiency gains. However, in order for both outsider leader or follower firms to produce, α must be lower than α 1. See Remark 1, Appendix A.

  11. By solving g 2L(n, m, α)=0, g 2F(n, m, α)=0 and g LF(n, m, α)=0, we obtain the roots of α as a function of n and m, that ensure merger profitability. However, some of these roots are negative or higher than \(\overline {\alpha }\) and, therefore, we exclude them.

  12. This result is similar to the one obtained by Perry and Porter (1985), where firms compete à la Cournot and costs are convex. This is also similar to Heywood and McGinty (2007)’s paper, where they conclude, under convex costs, that mergers involving two leaders or two followers are profitable if the costs savings are sufficiently high.

  13. See Appendix C for further information.

  14. Note that for all merger cases the \(\frac {\partial g(n,m,\alpha )}{\partial \alpha }>0\) and that \(\frac {\partial ^{2} g(n,m,\alpha )}{\partial \alpha ^{2} }>0\). See Appendix B.

  15. More details on calculations for a Cournot market can be provided upon request to the authors.

  16. For instance, the model of Brito and Catalão-Lopes (2011) is significantly different from ours. In particular, they study the effects of the merger between two followers that become a leader. In addition, they consider a convex cost function and that after the merger firms have asymmetric costs. They show that under convex costs there is a free riding problem. The merger between two followers that become leader decreases the number followers. Now, this decrease of the number of followers has a negative effect on the profit of the merging followers because one of them is eliminated, however it has a positive effect for outsider followers. Also, by turning a follower into a leader induces a negative effect for outsider followers since increasing the number of leaders by one always reduces each follower’s profit. Hence, Brito and Catalão-Lopes (2011) show that if the cost effect is such that the negative effect on outsiders’ profits due to the appearance of more leaders is higher than the positive effect due to the elimination of one follower, the free-riding problem emerges.

  17. Note that, initially, \(\frac {\partial {\Delta } CS}{\partial \alpha }>0\) when α<α A2. Also,\(\frac {\partial ^{2} {\Delta } CS}{\partial \alpha ^{2}}\) is always negative. See Appendix E.

  18. This is a similar result to the one obtained by Salant et al. (1983), under Cournot: when firms have the same strategic power, a merger that generates cost savings may also increase social welfare.

  19. In a Cournot setting with asymmetric firms and fixed costs, Catalão-Lopes (2007) shows that mergers may be socially desirable as long as the level of asymmetry is high enough. Also, if firms are not sufficiently asymmetric, there is a decrease in social welfare, unless the fixed cost savings are high enough.

  20. The lower threshold is obtained from setting insider firm’s quantity greater than zero \(\left (q^{2L}_{L_{I}}>0\right )\). This threshold is always negative, then we can exclude it since α>0. Hence, for any level of efficiency gains, insider leader firms always produce a positive quantity. The upper threshold is obtained by setting the quantity of both outsider leader and follower firms greater than zero. This α upper threshold is the same for both types of outsider firms.

  21. Again, the lower threshold is always negative, and therefore we can exclude it because α>0.

  22. Since the lower threshold is always negative, we can exclude it because α>0.

  23. Note that this root is always negative or greater than \(\overline {\alpha }\) for m>2. Also, for m=2 this root is higher than α 1 imposed in A2. Thus, we exclude it.

  24. Since formal expressions are very long, we indicate only the signs of the derivatives. A mathematical appendix with all the details is, however, available upon request to the authors.

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Acknowledgments

The author Mariana Cunha acknowledges Fundação para a Ciência e Tecnologia for financial support (FCTDFRH – SFRH / BD / 70000 / 2010).

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Correspondence to Mariana Cunha.

Appendices

Appendix A: Post-Merger Equilibrium Results

a) Merger of Two Leaders

After the merger of two leaders, the equilibrium quantities, profits, the consumer surplus, the producer surplus and the social welfare are given by:

$$ q^{2L}_{L_{I}}=\frac{1}{2}\frac{n\alpha \left( n(m-1)-m(m-2)-3\right) +2}{m}, \text{for the insider leader firm;} $$
$$ q^{2L}_{L_{i}}=\frac{1}{2}\frac{n\alpha \left( m-n-3\right) +2}{m},\text{for the \(i=1,...,m-1\) outsider leader firms;} $$
$$ q^{2L}_{F_{j}}=\frac{1}{2}\frac{n\alpha \left( m-n-3\right) +2}{\left( n-m+1\right) m},\text{ for the \(j=1,...,n-m\) outsider follower firms;} $$
$$ \pi^{2L}_{L_{I}}\left( n, m,\alpha \right) =\frac{1}{4}\frac{\left[2-n\alpha(n-m(n-m+2)+3)\right]^{2}}{m^{2}\left( n-m+1\right) }, \text{for the insider leader firm;} $$
$$ \pi_{L_{i}}^{2L}\left( n, m,\alpha \right) =\frac{1}{4}\frac{\left[2-n\alpha\left( n-m+3\right)\right]^{2}}{m^{2}\left( n-m+1\right) }, \text{ for the \(i=1,...,m-2\) outsider leader firms;} $$
$$ \pi_{F_{j}}^{2L}\left( n, m,\alpha \right) {}={}\frac{1}{4}\frac{\left[ 2-n\alpha \left(n{}-{}m{}+{}3\right) \right]^{2}}{m^{2}\left( n{}-{}m{}+{}1\right)^{2}},\text{ for the \(j{}={}1,...,n-m\) outsider follower firms;} $$
$$ CS^{2L}=\frac{1}{8}\frac{\left[2(m(n-m+1)-1)+ n\alpha(n-m(2(n-m)+3)+3)\right]^{2}}{m^{2}\left( n-m+1\right)^{2}} $$

\(PS^{2L}=\frac {1}{4}\frac {\left [2-n\alpha (n-m(n-m+2)+3)\right ]^{2}}{m^{2}\left (n-m+1\right )}+\frac {1}{4}\frac {\left [-n+m(n-m+2)-2\right ]\left [2-n\alpha (n-m+3)\right ]^{2}}{m^{2}(n-m+1)^{2}}\)

S W 2L=C S 2L+P S 2L

Remark 1

Again, we assume that:Footnote 20

$$ \frac{2}{n\left[m\left( m-2\right) -n\left( m-1\right) +3\right] }<\alpha <\frac{2}{n\left( n-m+3\right) }\equiv \alpha_{1}. $$

These conditions are imposed to exclude the case where both outsider follower and leader firms do not produce.

b) Merger of Two Followers

After the merger of two followers, where the resultant firm is still follower, the equilibrium quantities, profits, the consumer surplus, the producer surplus and the social welfare are given by:

$$ q^{2F}_{F_{I}}=\frac{1}{2}\frac{n\alpha \left( \left( m+1\right) \left( n-m\right) -3\right) +2}{\left( n-m\right) \left( m+1\right)}, \text{ for the insider follower firm;} $$
$$ q^{2F}_{F_{j}}=\frac{1}{2}\frac{2-3n\alpha }{\left( n-m\right) \left( m+1\right) },\text{ for \(j=1,...,n-m-2\) outsider follower firms;} $$
$$ q^{2F}_{L_{i}}=\frac{1}{2}\frac{2-3n\alpha }{m+1}, \text{ for \(i=1,...,m\) outsider leader firms;} $$
$$ \pi_{F_{I}}^{2F}\left( n, m,\alpha \right) =\frac{1}{4}\frac{\left[2- n\alpha \left( 3-\left( m+1\right) \left( n-m\right)\right)\right]^{2}}{\left( m+1\right)^{2}\left( n-m\right)^{2}}, \text{ for the insider follower firm;} $$
$$ \pi_{L_{i}}^{2F}\left( n, m,\alpha \right) =\frac{1}{4}\frac{\left( 2-3n\alpha \right)^{2}}{\left( m+1\right)^{2}\left( n-m\right) }, \text{ for \(i=1,...,m\) outsider leader firms;} $$
$$ \pi_{F_{j}}^{2F}\left( n, m,\alpha \right) =\frac{1}{4}\frac{\left( 2-3n\alpha \right)^{2}}{\left( m+1\right)^{2}\left( n-m\right)^{2}},\text{ for \(j=1,...,n-m-2\) outsider follower firms;} $$
$$ CS^{2F}=\frac{1}{8}\frac{\left[ 2(n+m(n-m-1)-1)+n\alpha(3-2(m+1)(n-m))\right]^{2}}{\left( m+1\right)^{2}\left( n-m\right)^{2}} $$
$$ PS^{2F}=\frac{1}{4}\frac{\left( 2-3n\alpha \right)^{2}\left[n+m(n-m-1)-2\right] +\left[ n\alpha \left(3-(m+1)(n-m)\right)-2\right]^{2}}{\left( m+1\right)^{2}\left( n-m\right)^{2}} $$
$$ SW^{2F}=CS^{2L}+PS^{2L} $$

Remark 2

In order to exclude the case where firms do not produce, we impose the following conditions:Footnote 21

$$ \frac{2}{n\left[3-\left(m+1\right) \left( n-m\right) \right]}<\alpha <\frac{2}{3n}\equiv \alpha_{2}. $$

c) Merger of One Leader and One Follower

After the merger of one leader and one follower, where the resultant firm is now a leader, the equilibrium quantities, profits, the consumer surplus, the producer surplus and the social welfare are given by:

$$ q^{LF}_{I}=\frac{1}{2}\frac{2+n\alpha \left( m\left( n-m\right) -2\right) }{m+1}, \text{ for the insider firm;} $$
$$ q^{LF}_{L_{i}}=\frac{1}{2}\frac{2-n\alpha \left( n-m+2\right) }{m+1}, \text{ for \(i=1,..,m-1\) outsider leader firms;} $$
$$ q^{LFF_{j}}=\frac{1}{2}\frac{2-n\alpha \left( n-m+2\right) }{\left( n-m\right) \left( m+1\right) },\text{ for \(j=1,..,n-m-1\) outsider follower firms;} $$
$$ \pi_{I}^{LF}\left( n, m,\alpha \right) =\frac{1}{4}\frac{\left[2-n\alpha \left(2- m\left( n-m\right)\right)\right]^{2}}{\left( m+1\right)^{2}\left( n-m\right) }, \text{for the insider firm;} $$
$$ \pi_{L_{i}}^{LF}\left( n, m,\alpha \right) =\frac{1}{4}\frac{\left[ 2-n\alpha \left( n-m+2\right)\right]^{2}}{\left( m+1\right)^{2}\left( n-m\right) }, \text{ for \(i=1,..,m-1\) outsider leader firms;} $$
$$ \pi_{F_{j}}^{LF}\left( n, m,\alpha \right) {}={}\frac{1}{4}\frac{\left[2{}-{}n\alpha \left( n{}-{}m{}+{}2\right)\right]^{2}}{\left( m{}+{}1\right)^{2}\left( n{}-{}m\right)^{2}},\text{ for \(j{}={}1,..,n-m-1\) outsider follower firms;} $$
$$ CS^{LF}= \frac{1}{8}\frac{\left[(2(n+m(n-m-1)-1)- n\alpha(n+m(2(n-m)-1)-2))\right]^{2}}{\left( m+1\right)^{2}\left( m-n\right)^{2}} $$
$$\begin{array}{@{}rcl@{}} PS^{LF}&&=\frac{1}{4}\frac{\left[2-n\alpha \left(2- m\left( n-m\right)\right)\right]^{2}}{\left( m+1\right)^{2}\left( n-m\right) }+\frac{1}{4}\frac{(m-1)\left( 2-n\alpha \left( n-m+2\right)\right)^{2}}{\left( m+1\right)^{2}\left( n-m\right) }\\ &&+\frac{1}{4}\frac{\left(n-m-1)(2- n\alpha \left( n-m+2\right)\right)^{2}}{\left( m+1\right)^{2}\left( n-m\right)^{2}} \end{array} $$
$$ SW^{LF}=CS^{LF}+PS^{LF} $$

Remark 3

In order to exclude the situation where firms do not produce, we impose the following conditions: Footnote 22

$$ \frac{2}{n\left[ m\left( m-n\right) +2\right]}<\alpha <\frac{2}{n\left( n-m+2\right) }\equiv \alpha_{3}. $$

Appendix B: Proofs

Proof Proposition 1

The merger between two leaders is profitable if \(g^{2L}\left (n, m,\alpha \right )\equiv \pi ^{2L}_{L_{I}}\left (n, m,\alpha \right ) -2\pi _{L_{i}}^{BM}\left (n, m,\alpha \right ) >0\), that is, iff:

\(g^{2L}\left (n, m,\alpha \right )\equiv \frac {\left (m+1\right )^{2}\left [2-n\alpha \left (n-m(n-m+2)+3\right ) -2\right ]^{2}-8m^{2}\left (1-n\alpha \right )^{2}}{4m^{2}\left (n-m+1\right ) \left (m+1\right )^{2}}>0\)

Since the denominator of g 2L is always positive, 4m 2(nm+1)(m+1)2>0, we need to solve the numerator with respect to α in order to get the roots that satisfy the merger profitability condition.

Let \(\alpha _{4}\left (n,m\right ) \equiv \frac {2(m+1)^{2}n(n-m(n-m+2)+3)-8m^{2}n+2\sqrt {2}mn(m-1)(m+1)(n-m+1)}{(m+1)^{2}n^{2}(n-m(n-m+2)+3)^{2}-8m^{2}n^{2}}\) be the only positive root obtained from the solving the inequality above (g 2L>0) that is lower than α 1 and than \(\overline {\alpha }\). Hence, two leaders have incentives to merge for all m≥2, iff:

  • i)  α>0, for m=2; and

  • ii)  α>α 4, for all m>2.

The first and the second derivatives of g 2L with respect to α are, respectively, given by: \(\frac {\partial g^{2L}}{\partial \alpha }=\frac {n\alpha \left (n\left (m+1\right )^{2}\left (n-m(n-m+2)+3\right )^{2}-8m^{2}n\right ) -2n\left (m+1\right )^{2}\left (n-m(n-m+2)+3\right ) +8m^{2}n}{2m^{2}\left (n-m+1\right ) \left (m+1\right )^{2}}>0\) and \(\frac {\partial ^{2}g^{2L}}{\partial \alpha ^{2}}=\frac {n\left (n\left (m+1\right )^{2}\left (n-m(n-m+2)+3\right )^{2}-8m^{2}n\right ) }{ 2m^{2}\left (n-m+1\right ) \left (m+1\right )^{2}}>0\).

Proof Proposition 2

The merger between two followers is profitable if \(g^{2F}\left (n, m,\alpha \right )\equiv \pi ^{2F}_{F_{I}}\left (n, m,\alpha \right ) -2\pi _{F_{j}}^{BM}\left (n, m,\alpha \right ) >0\), that is, iff:

\(g^{2F}\left (n, m,\alpha \right )\equiv \frac {\left (2-n\alpha \left (\left (m+1\right ) \left (m-n\right ) +3\right )\right )^{2}\left (n-m+1\right )^{2}-8\left (1-n\alpha \right )^{2}\left (n-m\right )^{2}}{4\left (m+1\right )^{2}\left (n-m\right )^{2}\left (n-m+1\right )^{2}}>0\)

Let \(\alpha _{5}\left (n,m\right ) \equiv \frac {2n(3-(m+1)(n-m))(n-m+1)^{2}-8n(n-m)^{2}+2\sqrt {2}n(n-m)(n+m(n-m-1)-1)(n-m+1) }{n^{2}(3-(m+1)(n-m))^{2}(n-m+1)^{2}-8(n-m)^{2}n^{2}}\) be the only positive roots obtained from the solving the inequality above (g 2F>0) that is also lower than α 2 and than \(\overline {\alpha }\). Hence, two followers have incentives to merge for all nm≥2, iff:

  • i)  α>0, for nm=2; and

  • ii)  α 5<α<, for nm>2.

The first and the second derivatives of g 2F with respect to α are, respectively, given by: \(\frac {\partial g^{2F}}{\partial \alpha }= \frac {8n\left (1-n\alpha \right ) \left (n-m\right )^{2}-n\left (2-n\alpha \left (3-\left (m+1\right ) \left (n-m\right ) \right ) \right ) \left (n-m+1\right )^{2}\left (3-\left (m+1\right ) \left (n-m\right ) \right ) }{2\left (m+1\right )^{2}\left (n-m\right )^{2}\left (n-m+1\right )^{2}}>0\) and \(\frac {\partial ^{2}g^{2F}}{\partial \alpha ^{2}}=\frac {n^{2}\left (n-m+1\right )^{2}\left (3-\left (m+1\right ) \left (n-m\right ) \right ) \left (3-\left (m+1\right ) \left (n-m\right ) \right ) -8n^{2}\left (n-m\right )^{2}}{2\left (m+1\right )^{2}\left (n-m\right )^{2}\left (n-m+1\right )^{2}}>0\).

Proof Proposition 3

A merger between a leader and a follower is profitable if \(g^{LF}\left (n, m,\alpha \right )\equiv \pi ^{LF}_{I}\left (n, m,\alpha \right ) -\left [ \pi _{F_{j}}^{BM}\left (n, m,\alpha \right ) +\pi _{L_{i}}^{BM}\left (n, m,\alpha \right ) \right ] >0\), that is, iff:

\(g^{LF}\left (n, m,\alpha \right )\equiv \frac {\left (2-n\alpha \left (m\left (n-m\right )\right ) +2\right )^{2}\left (n-m+1\right )^{2}-4\left (1-n\alpha \right )^{2}\left (n-m+2\right ) \left (n-m\right ) }{4\left (m+1\right )^{2}\left (n-m\right ) \left (n-m+1\right )^{2}}>0\)

In this case, since both α obtained for solving g LF>0 are always negative we exclude them. Hence, the merger between one follower and one leader is always profitable for all α>0, all nm>1 and m>1. □

The first and the second derivatives of g LF with respect to α are, respectively, given by: \(\frac {\partial g^{LF}}{\partial \alpha }=\frac {n\left (n\alpha \left (m\left (n-m\right ) -2\right ) +2\right ) \left (n-m+1\right )^{2}\left (m\left (n-m\right ) -2\right ) +4n\left (1-n\alpha \right ) \left (n-m+2\right ) \left (n-m\right ) }{2\left (m+1\right )^{2}\left (n-m\right ) \left (n-m+1\right )^{2}} >0\) and \(\frac {\partial ^{2}g^{LF}}{\partial \alpha ^{2}}=\frac {n^{2}\left (n-m+1\right )^{2}\left (m\left (n-m\right ) -2\right ) \left (m\left (n-m\right ) -2\right ) -4n^{2}\left (n-m+2\right ) \left (n-m\right ) }{2\left (m+1\right )^{2}\left (n-m\right ) \left (n-m+1\right )^{2}}>0\).

Appendix C: No-synergies Benchmark Results

Insiders’ Profitability

By assuming that α=0, the merger profitability conditions that we obtain and summarize in Proposition 1 are the same as those presented in Huck et al. (2001)’s and Feltovich (2001)’s papers. That is, two leaders have only incentives to merge if there are m=2 leaders and, similarly, two followers have only incentives to merge if there are nm=2 followers (Huck et al. (2001) ’s Proposition 1 and Feltovich (2001)’s Result 1). Also, we obtain Huck et al. (2001) ’s Proposition 2, that is, a merger between a leader and a follower is always profitable.

Lemma 1

For all n>m, when mergers do not create any synergies (α=0):

  • i)  The merger between two leaders is profitable, for all m≥2, iff:

    \(g^{L}\left (n, m,\alpha =0\right ) >0\Leftrightarrow \frac {-\left (m^{2}-2m-1\right ) }{m^{2}\left (n-m+1\right ) \left (m+1\right )^{2}}>0.\) This is true only for m=2.

  • ii)  The merger between two followers is profitable, for all nm≥2, iff:

    \(g^{F}\left (n, m,\alpha =0\right ) >0\Leftrightarrow \frac { 2n-n^{2}+2mn-m^{2}-2m+1}{\left (n-m\right )^{2}\left (m+1\right )^{2}\left (n-m+1\right )^{2}}>0\). This is true only for nm=2.

  • iii)  The merger of a follower and a leader is profitable, for all nm≥1 and m≥1, iff:

    \(g^{I}\left (n, m,\alpha =0\right ) >0\Leftrightarrow \frac {1}{\left (n-m\right ) \left (m+1\right )^{2}\left (n-m+1\right )^{2}}>0\). This is always true.

Consumer Surplus

Lemma 2 sums up, for each merger, the results obtained for the consumer surplus impact without efficiency gains.

Lemma 2

For all n>m, when mergers do not create any synergies, α=0:

  • i)  The merger between two leaders always decreases consumer surplus iff:

    \(\frac {1}{2}\frac {2m\left (-n-mn+m^{2}\right ) +1}{m^{2}\left (m+1\right )^{2}\left (m-n-1\right )^{2}}<0\). This condition is true for all m≥2 and nm.

  • ii)  The merger between two followers decreases consumer surplus iff:

    \(\frac {1}{2}\frac {2\left (m-n\right ) \left (n+mn-m^{2}\right ) -1}{\left (m-n\right )^{2}\left (m+1\right )^{2}\left (m-n-1\right )^{2}}<0\). This condition is true for all m and nm≥2.

  • iii)  The merger of one follower and one leader decreases consumer surplus iff:

    \(\frac {2\left (n-m\right ) \left (-n-mn+m^{2}\right ) +1}{2\left (m-n\right )^{2}\left (m+1\right )^{2}\left (m-n-1\right )^{2}}<0.\) This condition is true for all m>1 and nm>1.

Social Welfare

Lemma 3 sums up, for each merger, the results obtained for the social welfare without efficiency gains.

Lemma 3

For all n>m, when mergers do not create any synergies, α=0:

  • i)  The merger between two leaders decreases social welfare iff:

    \(-\frac {1}{2}\frac {2m+1}{m^{2}\left (m+1\right )^{2}\left (m-n-1\right )^{2}} <0\). This condition is true for all m≥2 and nm.

  • ii)  The merger between two followers decreases social welfare iff:

    \(\frac {2m-2n-1}{2\left (m-n\right )^{2}\left (m+1\right )^{2}\left (m-n-1\right )^{2}}<0\). This condition is true for all m and nm≥2.

  • iii)  The merger of one follower and one leader decreases social welfare iff:

    \(\frac {2m-2n-1}{2\left (m-n\right )^{2}\left (m+1\right )^{2}\left (m-n-1\right )^{2}}<0\). This condition is always true for all m>1 and nm>1.

Free-riding Problem

Lemma 4

For all n>m, when mergers do not create any synergies, for α=0:

  • i)  The merger between two leaders has a free-riding problem if outsider leader firms earn more than the insiders, that is, iff:

    \({\pi _{I}^{L}}(n, m)<2{\pi _{O}^{L}}(n, m)\Leftrightarrow {f_{L}^{L}}(n, m)=\frac {1}{ m^{2}(n-m+1)}>0\). This is always true for all m≥2.

  • ii)  The merger between two followers has a free-riding problem if outsider follower firms earn more than the insiders, that is, iff:

    \({\pi _{I}^{F}}(n, m)<2{\pi _{O}^{F}}(n, m)\Leftrightarrow {f_{F}^{F}}(n, m)=\frac {1}{ (n-m)^{2}(m+1)^{2}}>0\). This is always true for all m>1 and nm>1.

  • iii)  The merger of one follower and one leader has a free-riding problem if outsider firms earn more than the insiders, that is, iff:

    \(\pi _{I}(n, m)<{\pi _{O}^{L}}(n, m)+{\pi _{O}^{F}}(n, m)\Leftrightarrow f_{FL}^{I}(n, m)=\frac {1}{(n-m)^{2}(m+1)^{2}}>0\). This is always true.

Appendix D: Free-riding Effects

a) Merger Between Two Leaders

There is free-riding problem if \(\pi ^{2L}_{L_{I}}(n, m,\alpha ) <2\pi _{L_{i}}^{2L}(n, m,\alpha ) \), that is, iff:

$$ f^{2L}\left( n, m,\alpha \right)\equiv \frac{2\left( 2-n\alpha \left( n-m+3\right)\right)^{2}-\left(2- n\alpha\left(n-m(n-m+2)+3\right)\right)^{2}}{4m^{2}\left( n-m+1\right)}>0 $$
(11)

Let \(\alpha _{6} \equiv \frac {4n(n-m+3)-2n(n-m(n-m+2)+3)-2\sqrt {2}mn(n-m+1)}{2n^{2}(n-m+3)^{2}-n^{2}(n-m(n-m+2)+3)^{2}}\) and

\(\alpha \equiv \frac {4n(n-m+3)-2n(n-m(n-m+2)+3)+2\sqrt {2}mn(n-m+1)}{2n^{2}(n-m+3)^{2}-n^{2}(n-m(n-m+2)+3)^{2}}\) Footnote 23 be the two roots obtained from solving the inequality (11). For all α>0, leader firms have incentives to free-ride on the merger of two leaders iff:

α<α 6(n, m), for m≥2.

The first and the second derivatives of f 2L with respect to α are, respectively, given by: \(\frac {\partial f^{2L}\left (n,m,\alpha \right ) }{\partial \alpha }=\frac {-4n\left (-n\left (m-3\right ) +m\left (m-4\right ) +9\right ) +2\alpha n^{2}\left (\left (n-m(n-m+2)+3\right )^{2}+2\left (n-m+3\right )^{2}\right ) }{4m^{2}\left (n-m+1\right ) }>0\) if \(\alpha >\frac {2\left (-n\left (m-3\right ) +m\left (m-4\right ) +9\right ) }{n\left (\left (n-m(n-m+2)+3\right )^{2}+2\left (n-m+3\right )^{2}\right ) }\) and

\(\frac {\partial ^{2}f^{2L}\left (n,m,\alpha \right ) }{\partial \alpha ^{2}}=\frac {2n^{2}\left (\left (n-m(n-m+2)+3\right )^{2}+2\left (n-m+3\right )^{2}\right ) }{4m^{2}\left (n-m+1\right ) }>0\).

b) Merger Between Two Followers

There is free-riding problem if \(\pi ^{2F}_{F_{I}}(n, m,\alpha ) <2 \pi _{F_{j}}^{2F}(n, m,\alpha ) \), that is, iff:

$$ f^{2F}\left( n, m,\alpha \right)\equiv\frac{2\left( 2-3n\alpha\right)^{2}-\left[ 2-n\alpha \left(3- \left( m+1\right) \left(n-m\right)\right)\right]^{2}}{4\left( m+1\right)^{2}\left( n-m\right)^{2}}>0 $$
(12)

Let \(\alpha _{7} \equiv \frac {2n((m+1)(n-m)-3)+12n-2\sqrt {2}n(m+1)(n-m)}{18n^{2}-n^{2}((m+1)(n-m)-3)^{2}}\) be the only root that is positive and that satisfies Assumption 1 obtained from solving the inequality (12). For all α>0, follower firms have incentives to free-ride on the merger of two followers iff:

α<α 7(n, m), for all nm>2 and m>1.

The first and the second derivatives of f 2F with respect to α are, respectively, given by: \(\frac {\partial f^{2F}\left (n,m,\alpha \right ) }{\partial \alpha }=\frac {2n(-(m+1)(n-m)-3)-n^{2}\alpha \left (\left (3-\left (m+1\right ) \left (n-m\right ) \right )^{2}-18\right ) }{2\left (m+1\right )^{2}\left (n-m\right )^{2}}>0\) if \(\alpha >\frac {2(-(m+1)(n-m)-3)}{n\left (\left (3-\left (m+1\right ) \left (n-m\right ) \right )^{2}-18\right ) }\) and \(\frac {\partial ^{2}f^{2F}\left (n,m,\alpha \right ) }{\partial \alpha ^{2}}=\frac {-n^{2}\left (\left (3-\left (m+1\right ) \left (n-m\right ) \right )^{2}-18\right ) }{2\left (m+1\right )^{2}\left (n-m\right )^{2}}>0\), except when m=1 and nm=2,3 and m=nm=2.

c) Merger Between a Leader and a Follower

There is free-riding problem if \(\pi _{I}^{LF}(n, m,\alpha ) <\pi ^{LF}_{F_{j}}(n, m,\alpha ) +\pi ^{LF}_{L_{i}}(n, m,\alpha ) \), that is, iff:

$$ f^{LF}\left( n, m,\alpha \right)\equiv\frac{\left(2{}-{} n\alpha \left( n{}-{}m{}+{}2\right)\right)^{2}\left( n{}-{}m{}+{}1\right) {}-{}[2{}-{}n\alpha(2{}-{}m(n{}-{}m))]^{2}(n{}-{}m) }{4\left( m{}+{}1\right)^{2}\left( n{}-{}m\right)^{2}}>0 $$
(13)

Let \(\alpha _{8} \equiv \frac {2\left (n-m+2\right ) \left (n-m+1\right ) +2\left (m\left (n-m\right ) -2\right ) \left (n-m\right ) +2(m+1)\sqrt [2]{\left (n-m+1\right ) \left (n-m\right )^{3}}}{n^{2}\left (n-m+2\right )^{2}\left (n-m+1\right ) -n^{2}\left (m\left (n-m\right ) -2\right )^{2}\left (n-m\right )}\) be the only root that is positive and that satisfies Assumption 1 obtained from solving the inequality (13). For all α>0, outsider firms have incentives to free-ride on the merger of a leader and a follower iff:

α<α 8(n, m), for all nm>1 and m>1.

The first and the second derivatives of f LF with respect to α are, respectively, given by: \(\frac {\partial f^{LF}\left (n,m,\alpha \right ) }{\partial \alpha }=\frac { -2n\left (\left (n-m\right ) \left (\left (m+1\right ) \left (n-m\right ) +1\right ) +2\right ) +\alpha n^{2}\left (\left (n-m+2\right )^{2}\left (n-m+1\right ) -(2-m(n-m))^{2}(n-m)\right ) }{2\left (m+1\right )^{2}\left (n-m\right )^{2}}>0\) if \(\alpha >\frac {2\left (\left (n-m\right ) \left (\left (m+1\right ) \left (n-m\right ) +1\right ) +2\right ) }{n\left (\left (n-m+2\right )^{2}\left (n-m+1\right ) -(2-m(n-m))^{2}(n-m)\right ) }\). Since this \(\alpha >\overline {\alpha }\) in some cases and is negative for the others, then \(\frac {\partial f^{LF}\left (n,m,\alpha \right ) }{\partial \alpha }<0\); \(\frac {\partial ^{2}f^{LF}\left (n,m,\alpha \right ) }{\partial \alpha ^{2}}=\frac {n^{2}\left (\left (n-m+2\right )^{2}\left (n-m+1\right ) -(2-m(n-m))^{2}(n-m)\right ) }{2\left (m+1\right )^{2}\left (n-m\right )^{2}} >0\), which is true for m=2 and nm=2,3, and m=3 and nm=2.

Appendix E: Consumer Surplus Results

a) Merger Between Two Leaders

If the merger between two leaders creates synergies, the consumer surplus increases iff ΔC S 2LC S 2LC S BM>0, that is, iff:

$$ \frac{\left[n\alpha(n{\kern-2.5pt}+{\kern-2.5pt}m(n{\kern-2.5pt}-{\kern-2.5pt}m){\kern-2.5pt}+{\kern-2.5pt}3){\kern-2.5pt}-{\kern-2.5pt}2\right]\left[n\alpha((1{\kern-2.5pt}-{\kern-2.5pt}4m)(n{\kern-2.5pt}+{\kern-2.5pt}m(n{\kern-2.5pt}-{\kern-2.5pt}m)){\kern-2.5pt}+{\kern-2.5pt}3)+4m(n+m(n-m))-2\right]}{8m^{2}(m+1)^{2}(n-m+1)^{2}}>0 $$
(14)

Solving the inequality (14) we obtain one threshold for α that is positive and satisfies Assumption 1, which is given by: \(\alpha _{9} \equiv \frac {2}{n\left (3+n+m(n-m)\right ) }\).

Hence, the merger between two leaders increases consumer surplus iff:

α>α 9(n, m), for all m≥2 and all nm>1.

The first and the second derivatives of ΔC S 2L with respect to α are, respectively, given by:

\(\frac {\partial {\Delta } CS^{2L}}{\partial \alpha }=\frac {\left (n(n+m(n-m)+3)\right ) \left (n\alpha ((1-4m)(n+m(n-m))+3)+4m(n+m(n-m))-2\right ) +\left (n\alpha (n+m(n-m)+3)-2\right ) n((1-4m)(n+m(n-m))+3)}{8m^{2}(m+1)^{2}(n-m+1)^{2}}>0\) if \(\alpha <\frac {2n\left (\left (n-m\left (m-n\right ) \right ) \left (4m-1\right ) -3\right ) +n\left (4m\left (n-m\left (m-n\right ) \right ) -2\right ) \left (n-m\left (m-n\right ) +3\right ) }{2n^{2}\left (\left (n-m\left (m-n\right ) \right ) \left (4m-1\right ) -3\right ) \left (n-m\left (m-n\right ) +3\right ) }\) and

\(\frac {\partial ^{2}CS^{2L}}{\partial \alpha ^{2}}=\frac { n^{2}(n+m(n-m)+3)((1-4m)(n+m(n-m))+3)+n^{2}(n+m(n-m)+3)((1-4m)(n+m(n-m))+3)}{ 8m^{2}(m+1)^{2}(n-m+1)^{2}}<0\).

b) Merger Between Two Followers

If the merger between two follower creates synergies the consumer surplus increases if ΔC S 2FC S 2FC S BM>0, that is, iff:

$$ \frac{(2-n\alpha((n-m)(1-4(n+m(n-m)))+3)-4(n-m)(n+m(n-m)))(2-n\alpha(n-m+3))}{8(m+1)^{2}(n-m+1)^{2}(n-m)^{2}}>0 $$
(15)

From solving the inequality (15), we obtain one threshold for α that is positive and satisfies Assumption 1, given by \(\alpha _{10}\equiv \frac {2}{n\left (n-m+3\right ) }\).

Hence, the merger between two followers increases consumer surplus iff:

α>α 10(n, n), for all nm≥2 and all m.

The first and the second derivatives of ΔC S 2F with respect to α are, respectively, given by:

\(\frac {\partial {\Delta } CS^{2F}}{\partial \alpha }=\frac { -n((n-m)(1-4(n+m(n-m)))+3)(2-n\alpha (n-m+3))-n(n-m+3)(2-n\alpha ((n-m)(1-4(n+m(n-m)))+3)-4(n-m)(n+m(n-m)))}{8(m+1)^{2}(n-m+1)^{2}(n-m)^{2}}>0\) if \(\alpha <-\frac {-2n\left (\left (m-n\right ) \left (-4n+4m\left (m-n\right ) +1\right ) -3\right ) +n\left (4\left (n-m\left (m-n\right ) \right ) \left (m-n\right ) +2\right ) \left (-m+n+3\right ) }{2n^{2}\left (\left (m-n\right ) \left (-4n+4m\left (m-n\right ) +1\right ) -3\right ) \left (-m+n+3\right ) }\) and

\(\frac {\partial ^{2}{\Delta } CS^{2F}}{\partial \alpha ^{2}}=\frac {n^{2}((n-m)(1-4(n+m(n-m)))+3)((n-m+3))+n^{2}(n-m+3)((n-m)(1-4(n+m(n-m)))+3)}{8(m+1)^{2}(n-m+1)^{2}(n-m)^{2}}<0\).

c) Merger Between a Leader and a Follower

If the merger between a leader and a follower creates synergies the consumer surplus is improved if ΔC S LFC S LFC S BM>0, that is, iff:

$$ \frac{\left[2(1-(m+1)(n-m))- n\alpha(2-(2m+1)(n-m))\right]^{2}(n-m+1)^{2}-4(n+m(n-m))^{2}(1-n\alpha)^{2}(n-m)^{2}}{8(n-m)^{2}(m+1)^{2}(n-m+1)^{2}}>0 $$
(16)

Solving the inequality (16) we obtain one threshold for α that is positive and satisfies Assumption 1, which is given by: \(\alpha _{11}\left (n,m\right ) \equiv \frac {2}{n\left (\left (n-m+1\right ) \left (n-m\right ) +2\right ) }\).

Hence, the merger between a leader and a follower increases consumer surplus if:

α>α 11(n, m), for m>1 and nm>1.

The first and the second derivatives of ΔC S LF with respect to α , respectively, given by:

\(\frac {\partial {\Delta } CS^{LF}}{\partial \alpha }=\frac { -n(2-(2m+1)(n-m))\left (2(1-(m+1)(n-m))-n\alpha (2-(2m+1)(n-m))\right ) (n-m+1)^{2}+4n(n+m(n-m))^{2}(1-n\alpha )(n-m)^{2}}{ 4(n-m)^{2}(m+1)^{2}(n-m+1)^{2}}>0\) if \(\alpha <\frac {-4n\left (n-m\left (m-n\right ) \right )^{2}\left (m-n\right )^{2}+n\left (\left (2m+1\right ) \left (m-n\right ) +2\right ) \left (2\left (m-n\right ) \left (m+1\right ) +2\right ) \left (-m+n+1\right )^{2}}{n^{2}\left (\left (2m+1\right ) \left (m-n\right ) +2\right )^{2}\left (-m+n+1\right )^{2}-4n^{2}\left (n-m\left (m-n\right ) \right )^{2}\left (m-n\right )^{2}}\) and

\(\frac {\partial ^{2}{\Delta } CS^{LF}}{\partial \alpha ^{2}}=\frac { n^{2}(2-(2m+1)(n-m))(2-(2m+1)(n-m))(n-m+1)^{2}-4n^{2}(n+m(n-m))^{2}(n-m)^{2} }{4(n-m)^{2}(m+1)^{2}(n-m+1)^{2}}<0.\)

Appendix F: Social Welfare Results

a) Merger Between Two Leaders

If the merger between two leaders creates synergies, the social welfare increases iff ΔS WS W 2LS W BM>0. Solving this inequality, the only threshold for α that is positive and lower than \(\overline {\alpha }\) is the following:

\(\alpha _{12} \equiv \frac {-A+\sqrt {A^{2}+8B\left (2m+1\right ) }}{2B}\),where A=2n(6m+n+m n 2+3m 2 n−2m 3 n−2m 4 n+2m 2 n 2+m 3 n 2+4m nm 2−3m 3+m 5+3) and \(B=-\frac {1}{2}n^{2}(18m+6n+12mn^{2}+4m^{2}n+2mn^{3}-22m^{3}n+4m^{4}n+6m^{5}n-6m^{6}n+9m^{2}n^{2}+2m^{2}n^{3} -8m^{3}n^{2}-2m^{3}n^{3}-2m^{4}n^{3}+6m^{5}n^{2}+24mn-10m^{2}-14m^{3}+9m^{4}+n^{2}+2m^{5}-4m^{6}+2m^{7}+9)\)

Hence, the merger between two leaders increases social welfare iff:

α 12(n, m), for the remaining all m>1 and nm>1.

The signs of the first and the second derivatives of ΔS W 2L are, respectively, given by: \(\frac {\partial {\Delta } SW^{2\text {L}}}{\partial \alpha }>0\) and \(\frac {\partial {\Delta }^{2}SW^{2L}}{\partial \alpha ^{2}}>0\), except when m=2 and nm=1.Footnote 24

b) Merger Between Two Followers

If the merger between two followers creates synergies the social welfare increase iff ΔS WS W 2FS W BM>0. Solving this inequality, the only threshold for α that is positive and lower than \(\overline {\alpha }\) is the following:

\(\alpha _{13}\left (n,m\right ) \equiv \frac {C+\sqrt {C^{2}-16D\left (2m-2n-1\right ) }}{2D}\),where C=4n(7m−7n+m n 2+m 2 nm n 3−3m 3 n+3m 2 n 2+5m n−2m 2m 3+m 4−3n 2n 3−3) and D=18n 2(−m+n+m nm 2−2)(nm+1)2−8(n+m nm 2)(nm)2 n 2+(mn+4m n 2−8m 2 n−4m n+4m 3+4n 2−3)(mn−3)n 2+

2((m+1)(mn)+3)2(nm+1)2 n 2

Hence, the merger between two followers increases social welfare iff:

α>α 13(n, m), for all nm≥2 and m>1.

The signs of the first and the second derivatives of ΔS W 2F are, respectively, given by: \(\frac {\partial {\Delta } SW^{2\text {F}}}{\partial \alpha }>0 \) and \(\frac {\partial {\Delta }^{2}SW^{2\text {F}}}{\partial \alpha }>0 \), for all α.

c) Merger Between a Leader and a Follower

If the merger between a leader and a follower creates synergies the social welfare is improved iff ΔS WS W LFS W BM>0. Solving the inequality, the only threshold for α that is positive and lower than \(\overline {\alpha }\) is the following:

\(\alpha _{14}\left (n,m\right ) \equiv \frac {-E+\sqrt {E^{2}-16G\left (2m-2n-1\right ) }}{2G}\),where E=4n(−5m+5n−8m n 2+7m 2 n−2m n 3+2m 3 n+m n 4−4m 4 n−4m 2 n 3+6m 3 n 2−6m n+3m 2−2m 3m 4+3n 2+m 5+3n 3+n 4+2) and G=−4n 2(−nm n+m 2)2(mn)2−8(n+m nm 2)(nm)2 n 2+2(nm)(nm+1)2(m(nm)−2)2 n 2+

(mn−2m n+2m 2+2)2(mn−1)2 n 2+2(nm)(m−1)(nm+1)2(nm+2)2 n 2+2(nm+2)2(nm−1)(nm+1)2 n 2

Hence, the merger between a leader and a follower increases social welfare iff:

α>α 14(n, m) , for all m>1 and nm>1.

The signs of the first and the second derivatives of ΔS W LF are, respectively, given by: \(\frac {\partial {\Delta } SW^{L\text {F}}}{\partial \alpha }<0\) and \(\frac {\partial {\Delta }^{2}SW^{L\text {F}}}{\partial \alpha ^{2}}<0\), for all α.

Appendix G: Figures

Figure 3 graphs the critical parameters of α that are presented in propositions 1, 4, 5 for the two leader merger case with n=12 and identifies five different regions. Also, in these regions, the outsider firms do not exit the market since \(\alpha ^{\ast }_{1}\) is always greater than \(\alpha ^{\ast }_{9}\) and therefore, Assumption 2 is always satisfied in the regions obtained.

Figure 4 graphs the critical parameters of α that are presented in propositions 2, 4, 5 and 6 for the two follower merger case with n=12 and identifies six different regions. Also, in these regions, the outsider firms do not exit the market since \(\alpha ^{\ast }_{2}\) is always greater than \(\alpha ^{\ast }_{10}\) and therefore, Assumption 2 is always satisfied in the regions obtained.

Figure 5 graphs the critical parameters of α that are presented in propositions 3 to 6 for the leader-follower merger case with n=12 and identifies four different regions. Also, in these regions, the outsider firms do not exit the market since \(\alpha ^{\ast }_{3}\) is always greater than \(\alpha ^{\ast }_{11}\) and therefore, Assumption 2 is always satisfied in the regions obtained.

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Cunha, M., Vasconcelos, H. Mergers in Stackelberg Markets with Efficiency Gains. J Ind Compet Trade 15, 105–134 (2015). https://doi.org/10.1007/s10842-014-0182-4

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Keywords

  • Mergers
  • Efficiency gains
  • Stackelberg

JEL Codes

  • L13
  • D43
  • L40
  • L41