Privatization and merger in a mixed oligopoly with spatial price discrimination

Abstract

This paper is the first to investigate the impact of the presence of a public firm on the profitability of two-firm mergers in a spatial price discrimination model. The presence of the public firm may increase the set of mergers for two private firms to profitably merge. Merger increases social welfare in the presence of a public follower alone. However, privatizing the public firm never increases social welfare.

Appendix

Proof of Proposition 1

When there is no merger between private firms, the equilibrium profits and welfare become $\Pi _{1}^{N}=0.105t$, $\Pi _{2}^{N}=0.038t$, $W^{N}=r-0.095t$ (Heywood and Ye 2009).

(1) For $\alpha \in [0,0.672]$, solving $\Pi _1^N -\Pi _1^M =0$ generates a unique root $\alpha =0.432$. It is easy to verify that $\Pi _1^N \le \Pi _1^M$for $\alpha \in [0.432,1]$. And,

$\frac{\partial (\Pi _2^M -\Pi _2^N )}{\partial \alpha }=-\frac{4\left( {2-\alpha } \right) \left( {3-2\alpha } \right) \left( {2\alpha ^{5}+\alpha ^{4}-55\alpha ^{3}+264\alpha ^{2}-648\alpha +576} \right) }{\left( {48-48\alpha +17\alpha ^{2}-\alpha ^{3}} \right) ^{3}}<0;$ thus, at the edge of the range $\Pi _2^M -\Pi _2^N |_{\alpha =0.672} =0.046t>0$, proving that $\Pi _2^M >\Pi _2^N$ for $\alpha \in [0.432,0.672]$. For $\alpha \in [0.672,1]$,$\frac{\partial (\Pi _2^M -\Pi _2^N )}{\partial \alpha }<0$, and the profit difference reaches its minimum at $\Pi _2^M -\Pi _2^N |_{\alpha =1} =0$, implying that $\Pi _2^M \ge \Pi _2^N$. Exact expressions for the constrained case are available upon request.

(2) For $\alpha \in [0.432,0.672]$, solving

$\frac{\partial \left( {W^{M}-W^N} \right) }{\partial \alpha }=\frac{2\left( {5040\alpha -8280\alpha ^{2}+7229\alpha ^{3} -3662\alpha ^{4}+1116\alpha ^{5}-199\alpha ^{6}+16\alpha ^{7}-1152} \right) }{\left( {-48+48\alpha -17\alpha ^{2}+\alpha ^{3}} \right) ^{3}}=0$ generates $\alpha =0.493$. It can be confirmed that $\frac{\partial \left( {W^{M}-W^N } \right) }{\partial \alpha }>0$ when $\alpha \in [0.432,0.493]$, and $\frac{\partial \left( {W^{M}-W^N } \right) }{\partial \alpha }<0$ when $\alpha \in [0.493,0.672]$. Thus, the welfare difference reaches its maximum at $W^{M}-W^{N}|_{\alpha =0.493} =-0.003t$, implying that $W^{M}<W^{N}$. Similarly, for $\alpha \in [0.672,1]$, $\frac{\partial \left( {W^{M}-W^N } \right) }{\partial \alpha }>0$, and the welfare difference reaches its maximum at $W^{M}-W^{N}|_{\alpha =1} =0$, implying that $W^{M}\le W^{N}$ for $\alpha \in [0.672,1]$.

Proof of Proposition 2

When there is no merger between private firms, the public firm locates at the middle and the equilibrium profits and welfare become $\Pi _1^N =t/12$, $\Pi _3^N =t/12$, $W^{N}=r-t/12$ (Heywood and Ye 2009).

(1) From above, we know that $\Pi _1^N \le \Pi _1^M$ for $\alpha \in [0.325,1]$. For $\alpha \in [0.325,1]$, solving $\Pi _3^N =\Pi _3^M$ generates a unique root $\alpha =0.640$. It can be confirmed that $\Pi _1^N \le \Pi _1^M$ and $\Pi _3^N \le \Pi _3^M$ when $\alpha \in [0.325, 640]$. Thus, private firms would always merge when $\alpha \in [0.325, 640]$.

(2) For $\alpha \in [0.325,0.640]$,

$\frac{\partial \left( {W^{M}-W^N } \right) }{\partial \alpha }=\frac{3791\alpha -7446\alpha ^{2}+7396\alpha ^{3}-4036\alpha ^{4} +1335\alpha ^{5}-270\alpha ^{6}+26\alpha ^{7}-816}{\left( {-34+33\alpha -14\alpha ^{2}+\alpha ^{3}} \right) ^{3}}>0;$ thus, the welfare difference reaches its maximum at $W^{M}-W^{N}|_{\alpha =0.640} =-0.012t$, implying that $W^{M}<W^{N}$ for $\alpha \in [0.325,0.640]$.

Proof of Proposition 3

When there is no merger between two private firms, the public firm locates at the middle and the equilibrium profits and welfare become $\Pi _1^N =0.075t$, $\Pi _2^N =0.075t$, $W^{N}=r-0.101t$ (Heywood and Ye 2009).

(1) For $\alpha \in [0, 1]$, solving $\Pi _1^N =\Pi _1^M$ generates a unique root $\alpha =0.119$; solving $\Pi _2^N =\Pi _2^\mathrm{M}$ generates a unique root $\alpha =0.783$. Then, $\Pi _1^N \le \Pi _1^M$ for $\alpha \in [0.119, 1]$, $\Pi _2^N \le \Pi _2^\mathrm{M}$, for $\alpha \in [0, 0.783]$. Thus, private firms would merge when $\alpha \in [0.119, 0.783]$.

(2) For $\alpha \in [0.119,0.783]$, solving

$\frac{\partial \left( {W^{M}-W^N } \right) }{\partial \alpha }=\frac{2\left( {-423\alpha +558\alpha ^{2}-404\alpha ^{3} +72\alpha ^{4}+90} \right) }{\left( {30-15\alpha +2\alpha ^{2}} \right) ^{3}}=0$ generates $\alpha =0.316$. It can be confirmed that $\frac{\partial \left( {W^{M}-W^N } \right) }{\partial \alpha }>0$ when $\alpha \in [0.119,0.316]$, and $\frac{\partial \left( {W^{M}-W^N} \right) }{\partial \alpha }<0$ when $\alpha \in [0.316,0.783]$. Thus, the welfare difference reaches its minimum at $\alpha =0.119$ or $\alpha =0.783$, and we find that $W^{M}-W^{N}|_{\alpha =0.119} =0.015t$, $W^{M}-W^{N}|_{\alpha =0.783} =0.012t$, implying that $W^{M}>W^{N}$ for $\alpha \in [0.119,0.783]$.

Proof of Proposition 4

(1) From Heywood and Ye (2013), the range of profitable mergers without a public firm is $\alpha \in [0.440,1]$ . Therefore, the presence of a public firm increases the range of profitable mergers from $\alpha \in [0.440,1]$ to $\alpha \in [0.432,1]$.

(2) By comparing post-merger welfare with a public firm to that of pure triopoly in Heywood and Ye (2013), we can find that privatization always decreases social welfare. For $\alpha \in [0,0.432]$, there is no merger between private firms in both pure private and mixed oligopoly cases. We have $W^{PUB}-W^{PRI} =0.0036t>0$, where superscript PUB denotes the case with a public firm and superscript PRI denotes the case after privatization.

For $\alpha \in [0.432,0.672]$, the private leader merges with the private follower and NJC from the public firm is not binding.

• (1) For $\alpha \in [0.432,0.440]$, firms in the pure private case still have no incentives to merge and it can be confirmed that in range $\alpha \in [0.432,0.672]$,

  $\frac{\partial (W^{PUB} -W^{PRI} )}{\partial \alpha }=-\frac{2\left( {-1152+5040\alpha +7229\alpha ^{3}-199\alpha ^{6}+ 1116\alpha ^{5}-3662\alpha ^{4}-8280\alpha ^{2}+16\alpha ^{7}} \right) }{\left( {48-48\alpha +17\alpha ^{2}-\alpha ^{3}} \right) ^{3}}>0;$ thus, at the edge of the range $W^{PUB}-W^{PRI} |_{\alpha =0.432} =0.0036t>0$, proving that $W^{PUB}-W^{PRI} >0$ for $\alpha \in [0.432,0.440]$.

• (2) For $\alpha \in [0.440,0.672]$, merger takes place in both with or without a public firm and no NJC is binding. Thus, $W^{PUB}-W^{PRI} =0$.

For $\alpha \in [0.672,1]$, the private leader merges with the private follower and NJC from the public firm is binding.

• (1) For $\alpha \in [0.672,0.767]$, merger takes place in both cases and NJCs in the pure private case are not binding. It can be confirmed that $\frac{\partial \left( {W^{PUB}-W^{PRI} } \right) }{\partial \alpha }>0$ when $\alpha \in [0.672,0.767]$; thus, the welfare difference reaches its minimum at $W^{PUB}-W^{PRI}|_{\alpha =0.672} =0$, implying that $W^{PUB}-W^{PRI} >0$ for $\alpha \in (0.672,0.767]$.

• (2) For $\alpha \in [0.767,1]$, firms in both cases would merge and NJCs are binding in both cases. It can be confirmed that $\frac{\partial \left( {W^{PUB}-W^{PRI} } \right) }{\partial \alpha }=0$ when $\alpha =0.886$, $\frac{\partial \left( {W^{PUB}-W^{PRI} } \right) }{\partial \alpha }<0$ when $\alpha \in [0.767,0.886]$, and $\frac{\partial \left( {W^{PUB}-W^{PRI} } \right) }{\partial \alpha }>0$ when $\alpha \in [0.886,1]$. Thus, the welfare difference reaches its minimum at $W^{PUB}-W^{PRI}|_{\alpha =0.886} =0.0027t$, implying that $W^{PUB}-W^{PRI} >0$.

Exact expressions are available upon request.

Proof of Proposition 5

(1) The range of profitable mergers without a public firm is $\alpha \in [0.515, 1]$ (Heywood and Ye 2013). Therefore, the presence of a public firm decreases the range of profitable mergers from $\alpha \in [0.515, 1]$ to $\alpha \in [0.325, 640]$.

(2) By comparing welfare with a public firm to that of pure triopoly (Heywood and Ye 2013), we find that privatization always decreases social welfare.

For $\alpha \in [0,0.325]$, there is no merger between private firms. $W^{PUB}-W^{PRI} =0.018t>0$.

For $\alpha \in [0.325,0.640]$, the private leader merges with the private follower and NJC from the public firm is not binding.

• (1) For $\alpha \in [0.325,0.515]$, firms in the pure private case still have no incentives to merge. And it can be confirmed that in range $\alpha \in [0.325,0.515]$,

  $\frac{\partial (W^{PUB} -W^{PRI} )}{\partial \alpha }=\frac{-816+26\alpha ^{7}-4036\alpha ^{4}+7396\alpha ^{3} -7446\alpha ^{2}+3791\alpha +1335\alpha ^{5}-270\alpha ^{6}}{\left( {\alpha ^{3}-14\alpha ^{2}+33\alpha -34} \right) ^{3}}>0;$ thus, at the edge of the range $W^{PUB}-W^{PRI} |_{\alpha =0.325} =0.003t>0$, proving that $W^{PUB}-W^{PRI} >0$ for $\alpha \in [0.325,0.515]$.

• (2) For $\alpha \in [0.515,0.640]$, firms in both cases would merge and NJCs are all not binding. It can be confirmed that $\frac{\partial \left( {W^{PUB}-W^{PRI} } \right) }{\partial \alpha }<0$ when $\alpha \in [0.515,0.640]$; thus, the welfare difference reaches its minimum at $W^{PUB}-W^{PRI}|_{\alpha =0.640} =0.022t$, implying that $W^{PUB}-W^{PRI} >0$ for $\alpha \in [0.515,0.640]$.

For $\alpha \in [0.640,1]$, merger does not occur in the mixed oligopoly case.

• (1) For $\alpha \in [0.640,0.749]$, firms in the pure private case still have incentives to merge. It can be confirmed that $\frac{\partial \left( {W^{PUB}-W^{PRI} } \right) }{\partial \alpha }<0$ when $\alpha \in [0.640,0.749]$; thus, the welfare difference reaches its minimum at $W^{PUB}-W^{PRI}|_{\alpha =0.749} =0.030t$, implying that $W^{PUB}-W^{PRI} >0$ for $\alpha \in [0.640,0.749]$.

• (2) For $\alpha \in [0.749,1]$, firms in the pure private case still have incentives to merge and NJC from firm 3 to firm 2’s right corner is binding. Similarly, $\frac{\partial \left( {W^{PUB}-W^{PRI} } \right) }{\partial \alpha }<0$ when $\alpha \in [0.749,1]$; thus, the welfare difference reaches its minimum at $W^{PUB}-W^{PRI}|_{\alpha =1} =0.015t$, implying that $W^{PUB}-W^{PRI} >0$ for $\alpha \in [0.749,1]$.

Proof of Proposition 6

(1) The range of profitable mergers without a public firm is $\alpha \in [0.119, 0.783]$ (Heywood and Ye 2013). Therefore, the presence of a public firm has no impact on the range of profitable mergers $\alpha$.

(2) The welfare with a public firm is the same as that of pure triopoly (Heywood and Ye 2013) because the locations of firms are not affected by the presence of a public firm.