Privatization in a Stackelberg Mixed Oligopoly

Abstract

This chapter examines whether privatization improves social welfare in a Stackelberg mixed oligopoly. Extending the pioneering study of De Fraja and Delbono (Oxf Econ Pap 41(1):302–311, 1989) to Stackelberg competitions between a public firm and private firms, we investigate whether privatization increases social welfare in a sequential-move game. We consider the different competitive environments in which even after privatization, a public firm’s Stackelberg position is maintained. We demonstrate the following results: First, when a public firm acts as a Stackelberg leader before and after privatization, privatization necessarily decreases social welfare irrespective of the number of private firms. Second, even when a public firm acts as a Stackelberg follower before and after privatization, privatization decreases social welfare if the number of private firms is relatively small.

Appendix

1.1 A.3.1 Proof of Proposition 3.1

By simple calculation, the following equation is satisfied:

$$ {W}^{LB}\ge {W}^{LA}\iff \frac{X+nk\left(Y+1\right)}{X+k{Y}^2}\ge \frac{n{X}^2\left(Y+1\right)+2\left(k+1\right)X{Y}^2+\left(1-k\right){\left(1+k\right)}^2{Y}^2}{{\left(X+k+1\right)}^2{Y}^2} $$

$$ \begin{array}{l}\iff {\left(X+k+1\right)}^2{Y}^2\left[X+nk\left(Y+1\right)\right]\left(X+k{Y}^2\right)\hfill \\ {}\times \left[n{X}^2\left(Y+1\right)+2X{Y}^2\left(k+1\right)\left(1-k\right){\left(k+1\right)}^2{Y}^2\right]\ge 0\hfill \end{array} $$

$\iff {\left[{\left(k+1\right)}^3-{n}^2k\right]}^2\ge 0$. The equality holds only when $n={\left(\frac{{\left(k+1\right)}^3}{k}\right)}^{\frac{1}{2}}$.

1.2 A.3.2 Proof of Proposition 3.2

By direct calculation, the following relationship is satisfied: ${W}^{FB}\gtrless\ {W}^{FA}\iff$

$$ \frac{\left(k+1\right){n}^2+\left(k+1\right)\left(k+4\right)n+{\left(k+2\right)}^2}{\left(k+1\right){\left(Y+1\right)}^2}\gtrless \frac{{\left(k+1\right)}^2{\left(k+2\right)}^2{n}^2+\left(k+1\right){\left(k+2\right)}^2\left({k}^2+5k+2\right)n+\left(k+3\right){\left({k}^2+3k+1\right)}^2}{{\left(k+2\right)}^2{\left(X+n+k\right)}^2} $$

$\iff D\left(n,k\right)\equiv \left(k+1\right)\left({k}^3+2{k}^2 - k-1\right){n}^2-\left(k+1\right)\left(k+2\right)\left(3{k}^2+8k+2\right)n-{\left(k+2\right)}^2{\left({k}^2+3k+1\right)}^2\lessgtr 0$. ${k}^{+}$ denotes the positive real-valued solution of the cubic equation, $f(k)\equiv {k}^3+2{k}^2-k-1=0$, which is approximately 0.8019. When $k={k}^{+}$, because $f\left({k}^{+}\right)=0$ holds, $D\left(n,{k}^{+}\right)$ is a linear function of $n$ and $D\left(n,{k}^{+}\right)=-52.23n-128.71<0\forall n$. Thus, when $k={k}^{+}$, it is necessarily satisfied that ${W}^{FB}>{W}^{FA}$. When $k\ne {k}^{+}$, i.e., $f(k)\ne 0$, $D\left(n,k\right)$ is a quadratic equation of $n$. Denote two solutions of $D\left(n,k\right)=0$ with respect to $n$ as ${n}^{+}(k)$ and ${n}^{-}(k)$, where ${n}^{+}(k)>{n}^{-}(k)$.⁵ Note that ${n}^{-}(k)<0$ for all $k>0$. As $f(k)<0$ holds when $k\in \left(0,\;{k}^{+}\right)$, $D\left(n,k\right)$ is a quadratic function of $n$ in which the coefficient of the square of $n$ is negative and ${n}^{+}(k)<0$ also holds. Thus, when $k\in \left(0,\;{k}^{+}\right)$, $D\left(n,k\right)$ is always negative for all $n>0$ and ${W}^{FB}>{W}^{FA}$ holds. However, as $f(k)>0$ holds when $k>{k}^{+}$, $D\left(n,k\right)$ is a quadratic function of $n$ in which the coefficient of the square of $n$ is positive and ${n}^{+}(k)>0$ holds. In this case, if $n\in \left(0,{n}^{+}(k)\right)$, $D\left(n,k\right)$ is negative and as a result, ${W}^{FB}>{W}^{FA}$ is satisfied; if $n>{n}^{+}(k)$, $D\left(n,k\right)>0$ and ${W}^{FB}<{W}^{FA}$ is possible to occur. Note that ${n}^{+}(k)$ has the minimum value when $k>{k}^{+}$. When $k={k}^{\min}\approx 3.489$, the minimum value is ${n}^{+ \min}\equiv {n}^{+}\left({k}^{\min}\right)\approx 11.216$. Therefore, for all values of $k$, if $n=\left\{1,2,\cdots,\;11\right\}$, it is necessarily satisfied that $D\left(n,k\right)$ is negative, that is, ${W}^{FB}>{W}^{FA}$. When $n\ge 12$, ${W}^{FB}>{W}^{FA}$ if $n<{n}^{+}(k)$ and otherwise vice versa.

1.3 A.3.3 Proof of Proposition 3.3

By direct calculation, the following equations are satisfied: ${W}^{FB} > {W}^{LB}\iff \frac{\left(k+1\right){n}^2+\left(k+1\right)\left(k+4\right)n+{\left(k+2\right)}^2}{\left(k+1\right){\left(Y+1\right)}^2}>\frac{X+nk\left(Y+1\right)}{X+k{Y}^2}\iff nk\left(n+2k+3\right)>0$ and ${W}^{LB} > {W}^{CB}$ $\iff \frac{X+nk\left(Y+1\right)}{X+k{Y}^2}>\frac{{\left(k+1\right)}^3+nk\left(nk+{k}^2+4k+2\right)}{X^2}\iff {n}^2{k}^2>0$.