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.

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: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: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: 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(n−m+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 n−m≥2, iff:

• i)  α>0, for n−m=2; and

• ii)  α ₅<α<, for n−m>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 n−m>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 n−m=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 n−m≥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 n−m=2.

• iii)  The merger of a follower and a leader is profitable, for all n−m≥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 n−m.

• 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 n−m≥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 n−m>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 n−m.

• 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 n−m≥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 n−m>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 n−m>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

$\alpha \equiv \frac {4n(n-m+3)-2n(n-m(n-m+2)+3)+2\sqrt {2}mn(n-m+1)}{2n^{2}(n-m+3)^{2}-n^{2}(n-m(n-m+2)+3)^{2}}$ 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 n−m>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 n−m=2,3 and m=n−m=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 n−m>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 n−m=2,3, and m=3 and n−m=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 ^2L≡C S ^2L−C 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 n−m>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 ^2F≡C S ^2F−C 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 n−m≥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 ^LF≡C S ^LF−C 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 n−m>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 W≡S W ^2L−S 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 n−m 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 n−m>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 n−m=1.²⁴

b) Merger Between Two Followers

If the merger between two followers creates synergies the social welfare increase iff ΔS W≡S W ^2F−S 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 n−m n ³−3m ³ n+3m ² n ²+5m n−2m ²−m ³+m ⁴−3n ²−n ³−3) and D=18n ²(−m+n+m n−m ²−2)(n−m+1)²−8(n+m n−m ²)(n−m)² n ²+(m−n+4m n ²−8m ² n−4m n+4m ³+4n ²−3)(m−n−3)n ²+

2((m+1)(m−n)+3)²(n−m+1)² n ²

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

α>α ₁₃(n, m), for all n−m≥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 W≡S W ^LF−S 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 ³−m ⁴+3n ²+m ⁵+3n ³+n ⁴+2) and G=−4n ²(−n−m n+m ²)²(m−n)²−8(n+m n−m ²)(n−m)² n ²+2(n−m)(n−m+1)²(m(n−m)−2)² n ²+

(m−n−2m n+2m ²+2)²(m−n−1)² n ²+2(n−m)(m−1)(n−m+1)²(n−m+2)² n ²+2(n−m+2)²(n−m−1)(n−m+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 n−m>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.