Privatization of state holding corporations

Abstract

Many countries have privatized part of their state holding corporations in recent years. However, the literature on this issue has analyzed mainly the privatization of uniproduct public firms. We consider a state holding corporation with two plants that may produce complement or substitute goods. Assuming that private firms are more efficient than the state holding corporation we find the following: If the marginal cost of the state corporation is low, it is not privatized either if goods are substitutes or if they are complements. However, if the marginal cost of the state corporation is high the two plants of the state holding corporation are sold to a single private investor if goods are complements, and to different investors if goods are substitutes. If goods are close substitutes and the marginal cost of the state corporation takes an intermediate value only one plant is privatized. We extend the model to consider that firms are equally efficient, that they face quadratic cost functions and that there are n uniplant private firms producing each good. We find two differences with the previous result: The government never privatizes just one plant of the state corporation, and when goods are complements the two plants of the state corporation are sold to different investors if n is high.

Appendices

Appendix 1: Proof of Proposition 1.

It can be shown that $CS^S-CS^D=-b(12+15b+4b^2)/((1+b)(3+b)^2(3+2b)^2)>0$ if and only if $b<0; PS^D-PS^S=-2b(3+6b+2b^2)/((1+b)(3+b)^2(3+2b)^2)>0$ if and only if $b_1<b<0$, where $b_1 =(-3+\sqrt{3} )/2 \approx -0.63$.

Next, we compare the welfare obtained in the different cases. It can be shown that:

$$\begin{aligned} W^{S}-W^{N}= & {} (-1+2( {3+b})^2c-( {3+b})^2( {3+2b})c^2)/((1+b)(3+b)^2),\,\end{aligned}$$

(4)

$$\begin{aligned} W^S-W^D= & {} -3b(2+b)/((1+b)(3+b)^2(3+2b)^2),\end{aligned}$$

(5)

$$\begin{aligned} W^S-W^O= & {} -(9+3b-19b^2+b^3+8b^4-2( {3+b})^2( {9+b-12b^2+4b^4})c\nonumber \\&+\, (1+b)( {3+b})^2(27-28b^2+8b^4)c^2)/(2(1+b)( {3+b})^2( {3-2b^2})^2),\end{aligned}$$

(6)

$$\begin{aligned} W^N-W^D= & {} (1-2( {3+2b})^2c+( {3+2b})^3c^2)/((1+b)(3+2b)^2),\end{aligned}$$

(7)

$$\begin{aligned} W^N-W^O= & {} (1-b+2( {1-b})( {4b^2+4b^3-9-8b})c+(27+9b-44b^2-20b^3\nonumber \\&+\, 16b^4+8b^5)c^2)/(2(1+b)(3-2b^2)^2,\end{aligned}$$

(8)

$$\begin{aligned} W^O-W^D= & {} (9-12b-4b^2+8b^3-2( {3+2b})^2( {9-8b-4b^2+4b^3})c\nonumber \\&+\, ( {3+2b})^2(27-28b^2+8b^4)c^2)/(2( {3+2b})^2( {3-2b^2})^2). \end{aligned}$$

(9)

We consider first that goods are complements $( {b<0})$. From (A2) we obtain that $W^S>W^D$ if $b<0$, so the government does not choose D for $b<0$. It can be shown from (A1) that $W^S-W^N$ is strictly concave in c for $b<0$. This expression equals to zero for two values of c. One of them is $c=c_1 =(9+6b+b^2-( {3+b})\sqrt{6+4b+b^2} )/(( {3+b})^2(3+2b))$, where $c_1 <\overline{\overline{c}}$ for $b<0$. The other one is greater that $\overline{\overline{c}}$ so we exclude it. We obtain that $W^S>W^N$ if $c>c_1$ and $W^S<W^N$ if $c<c_1$.

It can be shown from (A3) that $W^S-W^O$ is strictly concave in c for $b<0$. This expression equals to zero for two values of c. One of them is $c=c_{SO} <\overline{\overline{c}}$. The other one is greater that $\overline{\overline{c}}$ so we exclude it. We obtain that $W^S>W^O$ if $c>c_{SO}$ and $W^S<W^O$ if $c<c_{SO}$, where:

$$\begin{aligned}&c_{SO}\\&=\frac{81+63b-93b^2-71b^3+24b^4+24b^5+4b^6-(9+3b-6b^2-2b^3)\sqrt{2(27+18b-23b^2-13b^3+3b^4+4b^5+2b^6)} }{(3+b)^2(27+27b-28b^2-28b^3+8b^4+8b^5)}. \end{aligned}$$

Moreover, it can be checked that $c_1 >c_{SO}$, so for $c>c_1$ we obtain that $W^S>W^O$ and $W^S>W^N$.

It can be shown from (A5) that $W^N-W^O$ is strictly convex in c for $b<0$. This expression equals to zero for two values of c. One of them is $c=c_3 <\overline{\overline{c}}$. The other one is greater that $\overline{\overline{c}}$ so we exclude it. We obtain that $W^N>W^O$ if $c<c_3$ and $W^N<W^O$ if $c>c_3$, where $c_3 =(9-b-12b^2+4b^4-\sqrt{2( {1-b^2})( {3-2b^2})^3} )/(27+9b-44b^2-20b^3+16b^4+8b^5)$. Moreover, it can be checked that $c_1 <c_3$ for $b<0$, so when $c<c_{1}$ we obtain that $W^N>W^O$ and $W^N>W^S.$

Next we consider that goods are substitutes $( {b>0})$. Let $c_2 =((3+2b)-\sqrt{2(1+b)(3+2b)} )/( {3+2b})^2$ be the value of c such that $W^D=W^N$, where $W^D>W^N$ if $c>c_2$. Let $c_3$ be the value of c such that $W^O=W^N$, where $W^O>W^N$ if $c>c_3$. Let $c_4 =(81+36b-96b^2-44b^3+32b^4+16b^5-(9+6b-6b^2-4b^3)\sqrt{2(27-26b^2+8b^4)} )/((3+2b)^2(27-28b^2+8b^4))$ be the value of c such that $W^D=W^O$, where $W^D>W^O$ if $c>c_4$.

It can be shown that $c_2 =c_3 =c_4$ for $b=0.9971,c_3>c_2 >c_4$ for $b<0.9971,c_4>c_2 >c_3$ for $b>0.9971$ and that $c_i <\bar{c}( {i= 2, 3, 4})$ for $b>0$. The comparisons between $c_2 ,c_3$ and $c_4$ are illustrated in Figure 3. Thus, since $W^D>W^S$ when $b>0$ from (A2), the definitions of $c_2 ,c_3$ and $c_4$ imply that $W^D>max\{W^N,W^S,W^O\}$ if $c>\max \{c_2 ,c_4 \}$, $W^N>max\{W^S,W^D,W^O\}$ if $c<min\{c_2 ,c_3 \}$ and $W^O>max\{W^S,W^D,W^N\}$ if $c_{3}<c<c_{4}$.

Fig. 3

Comparison of $c_2 ,c_3$, and $c_4$

Appendix 2: Quadratic cost functions

When the government does not privatize there is a state corporation and two private uniplant firms. In the second stage of the game, private firm iBk sets the output level $q_{iBk}$ that maximizes its profit. The state corporation chooses the output levels $q_{1A}$ and $q_{2A}$ that maximize social welfare. Solving these problems, we obtain the following result. Denote $H_1 =4+n+b(2+n)$.

Lemma 5

When the government does not privatize the state corporation, in equilibrium:

$$\begin{aligned}&q_{iA}^N = 2/H_1 , \quad q_{iBk}^N =1/H_1 ,\quad \pi _{iA}^N =2/(H_1 )^2,\quad \pi _{iBk}^N =3/(2(H_1 )^2),\quad CS^N=(1+b)(2+n)^2/(H_1 )^2, \\&PS^N= (4+3n)/(H_1 )^2,\quad W^N=(8+b( {2+n})^2+n( {7+n}))/(H_1 )^2,\quad i =1, 2;\quad k= 1, \ldots ,n. \end{aligned}$$

When the government sells the state corporation to a single private investor there are one multiplant private firm and n private uniplant firms. In the second stage of the game, private multiplant firm A chooses the output levels $q_{1A}$ and $q_{2A}$ that maximize the joint profit of its two plants. Private firm iBk sets the output level $q_{iBk}$ that maximizes its profit. Solving these problems, we obtain the following result. Denote $H_2 =6+4b+2n+3bn+b^2n$.

Lemma 6

When the government sells the state corporation to a single private investor, in equilibrium:

$$\begin{aligned}&q_{iA}^S =2/H_2 ,\quad q_{iBk}^S =(2+b)/H_2, \pi _{iA}^S =(6+4b)/(H_2 )^2,\quad \pi _{iBk}^S =3( {2+b})^2/(2(H_2 )^2), \\&CS^S=( {1+b})( {2+( {2+b})n})^2/(H_2 )^2,\quad PS^{S} =(12+8b+3( {2+b})^2n)/(H_2 )^2, \\&W^S=(4( {4+3b})+( {2+b})( {10+7b})n+( {1+b})( {2+b})^2n^2)/(H_2 )^2,\quad i =1, 2;\quad k= 1, \ldots ,n. \end{aligned}$$

When each plant of the state corporation is sold to a different private investor, there are n+1 firms producing each good. In the second stage of the game, private firms iA and iBk set the output levels $q_{iA}$ and $q_{iBk}$ that maximize their profits, respectively. Solving these problems, we obtain the following result. Denote $H_{3} =3+b+n+bn$.

Lemma 7

When the government sells each plant of the state corporation to a different private investor, in equilibrium:

$$\begin{aligned}&q_{iA}^D =q_{iBk}^D =1/H_3, \pi _{iA}^D =\pi _{iBk}^D =3/(2(H_3 )^2),\quad CS^D=( {1+b})( {1+n})^2/(H_3 )^2, \\&PS^D=3(1+n)/(H_3 )^2,\quad W^D=(1+n)(4+b+n+bn)/(H_3 )^2,\quad i =1, 2;\quad k= 1, \ldots ,n. \end{aligned}$$

When the government privatizes only one plant of the state corporation we assume, without loss of generality, that the government sells plant 1A. In the second stage of the game, each private firm sets the output level that maximizes its profit. The public firm chooses the output level $q_{2A}$ that maximizes social welfare. Solving these problems, we obtain the following result. Denote $H_4 =12+7n+n^2-b^2(2+3n+n^2)$.

Lemma 8

When the government privatizes only one plant of the state corporation, in equilibrium:

$$\begin{aligned} q_{1A}^O= & {} q_{1Bk}^O =(4+n-b( {2+n}))/H_4 ,\,q_{2A}^O =2(3-b+n-bn)/H_4 ,\\&\,q_{2Bk}^O \,(3-b+n-bn)/H_4 ,\, \\ \pi _{1A}^O= & {} \pi _{1Bk}^O =3( {4+n-b( {2+n})})^2/(2(H_4 )^2),\,\pi _{2A}^O =2(3-b+n-bn)^2/(H_4 )^2,\,\pi _{2Bk}^O \\= & {} 3( {3-b+n-bn})^2/(2(H_4 )^2),\,CS^{O} =(26+50n+35n^2+10n^3+n^4\\&+\,b^3( {2+3n+n^2})^2-b( {-4+9n^2+6n^3+n^4})-b^2( {16+38n+31n^2+10n^3+n^4}))/(H_4 )^2,\, \\ PS^O= & {} (84+123n+49n^2+6n^3+b^2( {16+35n+25n^2+6n^3})\\&-\,2b( {36+67n+37n^2+6n^3}))/(2(H_4 )^2), \\ W^O= & {} (136+223n+119n^2+26n^3+2n^4+2b^3(2+3n+n^2)^2\\&-\,2b(32+67n+46n^2+12n^3+n^4) -b^2(16+41n+37n^2+14n^3\\&\quad +2n^4))/(2(H_4 )^2),\quad k= 1, \ldots ,n. \end{aligned}$$

Proof of Proposition 2

It can be shown that $CS^S-CS^D=-4b(1+b)(b^2n(1+n)+2(3+4n+n^2)+b(3+7n+3n^2))/(H_3 )^2(H_2 )^2>0$ if and only if $b<0; PS^D-PS^S=-b(3b^3n(1+n)+8n(3+n)+2b^2(4+11n+7n^2)+b(12+43n+19n^2))/(H_3 )^2(H_2 )^2>0$ if and only if $n>n_4$ when $b<0$, where $n_4 =-(24+43b+22b^2+3b^3-( {1+b})\sqrt{3( {192+176b+35b^2+6b^3+3b^4})} )/(2(1+b)^2(8+3b))$; otherwise $PS^D<PS^S$.

Let $n_1 =(-1+2b+b^2+\sqrt{33+44b+18b^2+4b^3+b^4})/(2(1+b))$ be the value of n such that $W^S=W^N$, where $W^S>W^N$ if $n>n_1$. Let $n_2 =-(1+b-\sqrt{33+18b+b^2} )/(2+2b)$ be the value of n such that $W^D=W^N$ , where $W^D>W^N$, if $n>n_2$. Finally, $W^S-W^D=b(b^3n(1+n)+8(3+n)+2b^2(2+5n+n^2)+b(24+17n+n^2))/(H_3 )^2(H_2 )^2$. It can be shown that $W^S>W^D$ for $b>0$. Let $n_3 =-(8+9b+b^2+\sqrt{64+48b+b^2+2b^3+b^4} )/(2b(1+b))$ be the value of n such that $W^{S}=W^{D}$ when $b\le 0$, where $W^S>W^D$ if $n>n_3$.

Finally, it can be shown that max$\{W^D,W^S,W^N\} >W^O$ for all b. $\square$