International cross-ownership of firms and strategic privatization policy

Abstract

We consider how the international cross-ownership of firms affects the privatization of a public firm competing with foreign firms. We show that when firms compete á la Cournot in a third market under a linear demand function, the domestic ownership of foreign firms can impede privatization, whereas the foreign ownership of the domestic firm can promote privatization. Moreover, the domestic ownership of foreign firms can render neither complete privatization nor complete nationalization optimal under moderate conditions. Conversely, when firms compete á la Bertrand, we demonstrate that it is always optimal to pursue complete nationalization.

Appendices

Appendix A: A Cournot competition in a third market

1.1 Appendix A.1: Stage 2

We first consider stage 2. Given $\theta$, the first-order condition of firm 1 is:

$$\begin{aligned} ( {1-\theta +\theta \delta })\underbrace{( {{p}'q_1 +p-{c}'_1 })}_{\pi _1^1 }=-\theta \beta \underbrace{q_2 {p}'}_{\pi _1^2 }, \end{aligned}$$

(33)

and the second-order condition is $( {1-\theta +\theta \delta })( {2{p}'+q_1 {p}''-{c}''_1 })+\theta \beta q_2 {p}''<0$. For firm 2, the first-order condition is:

$$\begin{aligned} \underbrace{{p}'q_2 +p-{c}'_2 }_{\pi _2^2 }=0, \end{aligned}$$

(34)

and the second-order condition is $2{p}'+q_2 {p}''-{c}''_2 <0$. To consider the effects of an increase in $\theta$, we totally differentiate the systems of (A1) and (A2):

$$\begin{aligned} \left[ {{\begin{array}{ll} {v_{11}^1 } &{} {v_{12}^1 } \\ {\pi _{21}^2 } &{} {\pi _{22}^2 } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {{dq_1 }/{d\theta }} \\ {{dq_2 }/{d\theta }} \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {\frac{-\beta \pi _1^2 }{1-\theta +\theta \delta }} \\ 0 \\ \end{array} }} \right] . \end{aligned}$$

(35)

The determinant of the matrix is $D\equiv v_{11}^1 \pi _{22}^2 -v_{12}^1 \pi _{21}^2$, for which we assume $D>0$. Making use of Assumption 1, we derive:

$$\begin{aligned} \left\{ {{\begin{array}{l} {\frac{dq_1 }{d\theta }=-\frac{\beta \pi _1^2 }{( {1-\theta +\theta \delta })D}\pi _{22}^2 <0,} \\ {\frac{dq_2 }{d\theta }=\frac{\beta \pi _1^2 }{( {1-\theta +\theta \delta })D}\pi _{21}^2 >0,} \\ {\frac{{dq_2 }/{d\theta }}{{dq_1 }/{d\theta }}=-\frac{\pi _{21}^2 }{\pi _{22}^2 }<0.} \\ \end{array} }} \right. \end{aligned}$$

(36)

1.2 Appendix A.2: Stage 1

In stage 1, the home government chooses the optimal privatization policy that maximizes its own welfare. The welfare of the home country is given by:

$$\begin{aligned} w^1( {q_1 ( \theta ),q_2 ( \theta )})&= \delta \left[ {p( {q_1 ( \theta ),q_2 ( \theta )})q_1 ( \theta )-c_1 ( {q_1 ( \theta )})} \right] \nonumber \\&+\beta \left[ {p( {q_1 ( \theta ),q_2 ( \theta )})q_2 ( \theta )-c_2 ( {q_2 ( \theta )})} \right] . \end{aligned}$$

(37)

Together with (A1) and (A2), the first-order condition is then:

$$\begin{aligned} \frac{\partial w^1}{\partial \theta }={p}'\left[ {\frac{( {1-\theta })\beta q_2 }{( {1-\theta +\theta \delta })}\frac{dq_1 }{d\theta }+\delta q_1 \frac{dq_2 }{d\theta }} \right] =0. \end{aligned}$$

(38)

Let $\eta _\theta \equiv -\frac{{\frac{dq_2 }{d\theta }} /{q_2 }}{{\frac{dq_1 }{d\theta }} /{q_1 }}=\frac{\pi _2^1 \pi _{21}^2 }{\pi _1^2 \pi _{22}^2 }>0,$ where $\eta _\theta$ is the elasticity of substitution between $q_1$ and $q_2$. We see that the optimal level of state ownership is given by:

$$\begin{aligned} \theta ^*=1-\frac{\pi _2^1 \pi _{21}^2 }{\beta \pi _1^2 \pi _{22}^2 }=1-\frac{\eta _\theta }{\beta }. \end{aligned}$$

(39)

Clearly, the optimal level of $\theta ^*$ cannot attain a value of one because $\left. {\frac{\partial w^1}{\partial \theta }} \right| _{\theta =1} =\delta {p}'q_1 \frac{dq_2 }{d\theta }<0$. Moreover, $\theta ^*>0$ as long as $\beta >\eta _\theta$.

From (A7), we see that $\frac{d\theta ^*}{d\beta }=( {\frac{\eta _\theta }{\beta ^2}})\left[ {1-{( {\frac{d\eta _\theta }{d\beta }})}/{( {\frac{\eta _\theta }{\beta }})}} \right] \frac{>}{<}0$. However, note that when the inverse demand function under consideration is a linear one, we have $\frac{d\eta _\theta }{d\beta }=0,$and $\frac{d\theta ^*}{d\beta }=\frac{\eta _\theta }{\beta ^2}>0.$ Similarly, although in general we have $\frac{d\theta ^*}{d\delta }\frac{<}{>}0,$ we have $\frac{d\theta ^*}{d\delta }<0$ under a linear inverse demand function.

Using firm 1’s first-order condition, rewriting (A7), we have

$$\begin{aligned} \delta \pi _1^1 +\beta \pi _1^2 =\delta \pi _2^1 \left( {\frac{\pi _{21}^2 }{\pi _{22}^2 }}\right) . \end{aligned}$$

(40)

However, this is precisely the condition that determines the pair $( {q{ }_1^L ,q_2^F })$, which would be obtained if firm 1 was the quantity-setting leader (whose objective function would be $\delta \pi ^1( {q_1 ,q_2 })+\beta \pi ^2( {q_1 ,q_2 }))$ and firm 2 was the quantity-setting follower. Hence, we see that:

Remark 1

The optimal level of state ownership always ensures that the stage 2 equilibrium output is identical to that obtainable if firm 1 was the quantity-setting leader and firm 2 was the quantity-setting follower.²¹ Hence, we have ${d( {\frac{\pi _2^1 \pi _{21}^2 }{\pi _1^2 \pi _{22}^2 }})} /{d\theta }=0$.

Appendix B: Bertrand competition in a third market

1.1 Appendix B.1: Stage 2

Given $\Theta$, the first-order condition of firm 1 is:

$$\begin{aligned} ( {1-\Theta +\Theta \Delta })\underbrace{( {X^1+( {P_1 -{C}'_1 })X_1^1 })}_{\Pi _1^1 }=-\Theta \mathrm{B}\underbrace{( {P_2 -{C}'_2 })X_1^2 }_{\Pi _1^2 }, \end{aligned}$$

(41)

and the second-order condition is $( {1-\Theta +\Theta \Delta })\Pi _{11}^1 +\Theta \mathrm{B}X_{11}^2 ( {P_2 -{C}'_2 })<0$.

For firm 2, the first-order condition is:

$$\begin{aligned} \underbrace{X^2+( {P_2 -{C}'_2 })X_2^2 }_{\Pi _2^2 }=0, \end{aligned}$$

(42)

and the second-order condition is $\Pi _{22}^2 <0$. Next, we consider the effects of an increase in $\Theta$ on the equilibrium outputs. We totally differentiate the systems of (A9) and (A10):

$$\begin{aligned} \left[ {{\begin{array}{ll} {V_{11}^1 } &{} {V_{12}^1 } \\ {\Pi _{21}^2 } &{} {\Pi _{22}^2 } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {{dP_1 }/{d\Theta }} \\ {{dP_2 }/{d\Theta }} \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {\frac{-\mathrm{B}\Pi _1^2 }{1-\Theta +\Theta \Delta }} \\ 0 \\ \end{array} }} \right] . \end{aligned}$$

(43)

The determinant of the matrix is $J\equiv V_{11}^1 \Pi _{22}^2 -V_{12}^1 \Pi _{21}^2$, for which we assume $J>0$. Making use of Assumptions 2 and 3, we derive:

$$\begin{aligned} \left\{ {{\begin{array}{l} {\frac{dP_1 }{d\Theta }=-\frac{\mathrm{B}\Pi _1^2 }{( {1-\Theta +\Theta \Delta })J}\Pi _{22}^2 >0,} \\ {\frac{dP_2 }{d\Theta }=\frac{\mathrm{B}\Pi _1^2 }{( {1-\Theta +\Theta \Delta })J}\Pi _{21}^2 >0,} \\ {\frac{{dP_2 }/{d\Theta }}{{dP_1 } / {d\Theta }}=-\frac{\Pi _{21}^2 }{\Pi _{22}^2 }>0.} \\ \end{array} }} \right. \end{aligned}$$

(44)

1.2 Appendix B.2: Stage 1

In stage 1, the home government chooses the optimal privatization policy to maximize its own welfare. The welfare of the home country is given by:

$$\begin{aligned} W^1( \Theta )&= \Delta \left[ {P_1 ( \Theta )X^1( {P_1 ( \Theta ),P_2 ( \Theta )})-C_1 \left[ {X^1( {P_1 ,P_2 })} \right] } \right] \nonumber \\&+\mathrm{B}\left[ {P_2 ( \Theta )X^2( {P_1 ( \Theta ),P_2 ( \Theta )})-C_2 \left[ {X^2( {P_1 ,P_2 })} \right] } \right] , \end{aligned}$$

(45)

and the first-order condition is given by:

$$\begin{aligned} \frac{dW^1}{d\Theta }&= \Delta \left( {X^1\frac{dP_1 }{d\Theta }+( {P_1 ( \Theta )-{C}'_1 })\left( {X_1^1 \frac{dP_1 }{d\Theta }+X_2^1 \frac{dP_2 }{d\Theta }}\right) }\right) \nonumber \\&+\mathrm{B}\left( {X^2\frac{dP_2 }{d\Theta }+( {P_2 ( \Theta )-{C}'_2 })\left( {X_1^2 \frac{dP_1 }{d\Theta }+X_2^2 \frac{dP_2 }{d\Theta }}\right) }\right) =0. \end{aligned}$$

(46)

We then derive the optimal level of state ownership:

$$\begin{aligned} \Theta ^*=1+\underbrace{\frac{\Delta ^2( {P_1 -{C}'_1 })X_2^1 \frac{\Pi _{21}^2 }{\Pi _{22}^2 }}{\Delta ( {1-\Delta })( {P_1 -{C}'_1 })X_2^1 \frac{\Pi _{21}^2 }{\Pi _{22}^2 }-\mathrm{B}( {P_2 -{C}'_2 })X_1^2 }}_{( +)}. \end{aligned}$$

(47)

Clearly, $\Theta ^*$ always attains a value of one in the case of international Bertrand rivalry.