Optimal Partial Privatization in an International Mixed Oligopoly Under Various Tax Principles

Abstract

This chapter considers optimal privatization policy in an international mixed oligopoly. Allowing for partial privatization and cost asymmetry, we analyze the optimal policies under various tax regimes: arbitrary taxation, origin principle, destination principle, import tariffs, and a combination of tax and import tariffs. Our main results are as follows. First, when the government can arbitrarily levy taxes on a public firm’s output, maximum welfare is independent of the degree of privatization as long as the public firm is at least partially privatized. Second, under tax schemes that restrict freedom of taxation, an optimal privatization policy depends on tax regimes and cost asymmetry. Third, the elimination of import tariffs and the privatization of public firms improve both domestic welfare and a foreign competitor’s profit if a production subsidy is introduced in exchange for tariff elimination. Our results suggest that fiscal incentives such as tax and subsidies are superior in maximizing welfare when compared with managerial incentives such as public ownership.

Appendix

1.1 A.6.1 Stability Condition

Consider a simple adjustment process according to the Nash conjecture: both domestic and foreign private firms respond to a change in the output of the public firm. Once the private firms have responded, the public firm adjusts its output subject to the optimal response function. First, regarding the change in the public firm’s output, the outputs of the private firms are adjusted as follows:

$$ \left[\begin{array}{c}\hfill d{q}_d/d{q}_0\hfill \\ {}\hfill d{q}_f/d{q}_0\hfill \end{array}\right]=\frac{1}{1-{\phi}_f^d{\phi}_d^f}\left[\begin{array}{c}\hfill {\phi}_0^d+{\phi}_f^d{\phi}_0^f\hfill \\ {}\hfill {\phi}_0^f+{\phi}_d^f{\phi}_0^d\hfill \end{array}\right]. $$

(A6.1)

Thus, stability requires $\left|{\phi}_f^d{\phi}_d^f\right|<1$. In such a case, the optimal response functions of the private firms are given by ${q}_d={\omega}^d\left({q}_0\right)$ and ${q}_f={\omega}^f\left({q}_0\right)$. Next, the public firm adjusts its outputs according to the following adjustment process :

$$ {q}_{0\left(t+1\right)}={\phi}^0\left[{\omega}^d\left({q}_{0(t)}\right),{\omega}^f\left({q}_{0(t)}\right)\right], $$

(A6.2)

where a subscript(s) denotes the time period. The approximation of the change in output around the equilibrium (denoted by q ^*₀ ) is represented by

$$ \Delta {q}_{0\left(t+1\right)}=\left({\phi}_d^0{\omega^d}^{\prime }+{\phi}_f^0{\omega^f}^{\prime}\right)\Delta {q}_{0(t)}, $$

(A6.3)

where $\Delta {q}_{0(s)}\equiv {q}_{0(s)}-{q}_0^{*}$. Thus, $\left|{\Delta}_2\right|=\left|{\phi}_d^0{\omega^d}^{\prime }+{\phi}_f^0{\omega^f}^{\prime}\right|<1$ is required for the stability.

1.2 A.6.2 Proof of Lemma 6.2

The effect of the change in the privatization ratio on welfare can be written as

$$ \begin{array}{ll}\frac{\partial W\left(0,\theta \right)}{\partial \theta }& ={\mathbf{w}}_q{\mathbf{D}}^{-1}\mathbf{b}\hfill \\ {}& =\frac{{\left({p}^{\prime}\right)}^2\left({q}_f+{q}_0\right)}{\left(1-{\Delta}_1\right)\left(1-{\Delta}_2\right){V}_{00}}\left[\left({q}_f+{q}_d\right)\left(1+{\phi}_0^f\right){\phi}_0^d+{q}_f\left(1+{\phi}_0^d\right){\phi}_0^f+\left({q}_f+z{q}_0\right)\left(1-{\phi}_f^d{\phi}_d^f\right)\right],\hfill \end{array} $$

(A6.4)

where w _q denotes the national marginal benefits of an increase in each firm’s output, which is represented as ${\left.{\mathbf{w}}_q\right|}_{\mathbf{t}=0}=-{p}^{\prime}\left[{q}_d+{q}_f, {q}_f, {q}_f+{q}_0z\right]$. Identical technologies among private firms imply that in the equilibrium, two private firms have the same level of output, denoted by $\tilde{q}\left(={q}_d={q}_f\right)$. Thus, the optimal response functions also have same slopes denoted as ${\tilde{\phi}}_q\left(={\phi}_0^d={\phi}_f^d={\phi}_0^f={\phi}_d^f\right)$. Solving $\partial W\left(0,\theta \right)/\partial \theta =0$, we have an implicit form of the optimal privatization ratio ${\theta}_{-t}^{*}$ as follows:

$$ {\theta}_{-t}^{*}=\left(\frac{-{\tilde{\phi}}_q}{1-{\tilde{\phi}}_q}\right)\left(\frac{3\tilde{q}}{q_0+\tilde{q}}\right). $$

(A6.5)

Under condition (i), because $\partial {q}_0/\partial \theta <0$ and $\partial {q}_i/\partial \theta >0$ for $i=d,f$ hold from (6.7), ${q}_0\ge \tilde{q}$ holds for $\theta \in \left[0,1\right]$. Thus, if condition (i) is met, ${\tilde{\phi}}_q/\left(1-{\tilde{\phi}}_q\right)\in \left(0,\ 1/2\right)$ and $3\tilde{q}/\left({q}_0+\tilde{q}\right)\in \Big(0,\ 3/2\Big]$. Therefore, ${\theta}_{-t}^{*}\in \left(0,\ 3/4\right)$ under condition (i). We turn to condition (ii). Linear demand function implies ${\tilde{\phi}}_q<-1/2$, where strict inequality follows from the convexity of the cost function. In this case, ${\tilde{\phi}}_q/\left(1-{\tilde{\phi}}_q\right)\in \left(0,\ 1/3\right)$ and $3\tilde{q}/\left({q}_0+\tilde{q}\right)\in \left(0,\ 3\right]$. Thus, ${\theta}_{-t}^{*}\in \left(0,\ 1\right)$ holds under condition (ii). From (A6.4), it is confirmed that $\partial W\left(0,0\right)/\partial \theta >0$ and $\partial W\left(0,1\right)/\partial \theta <0$ hold if either condition (i) or (ii) is met.

1.3 A.6.3 Proof of Proposition 6.3

Taking account of the feasibility set, we obtain the optimal tax for the given θ:

$$ {T}_O\left(\theta \right)=-2a{\left({\varLambda}_O\right)}^{-1}\left[3\left(3\rho +8\right){\theta}^2-\left(\rho +3\right)\theta +\left(5\rho +6\right)\rho \right], $$

(A6.6)

where ${\varLambda}_O\equiv 9\ \left(9\rho +8\right){\theta}^2-6\rho \theta +\left(17\rho +7\right)\rho +9>0$ for $\theta \in \left[0,1\right]$. Inserting (A6.6) into (6.16), we can write welfare as ${W}_O={W}_O\left(\theta \right)$ in which origin-based tax is optimally chosen. The first-order condition for θ is as follows:

$$ \frac{d{W}_O\left(\theta \right)}{d\theta }={a}^2{\left({\varLambda}_O\right)}^{-2}\left(7\theta \rho -8\theta +\rho \right)\left(6\theta \rho -7{\rho}^2+9\right)=0. $$

(A6.7)

Thus, we have two roots: ${\theta}_O^1=\rho /\left(8-7\rho \right)$ for $\rho \ne 8/7$ and ${\theta}_O^2=\left(7{\rho}^2-9\right)/\left(6\rho \right)$. However, the second-order condition implies that θ ² _O gives the minimum solution. Hence, ${\theta}_O^{*}={\theta}_O^1$ for $\rho \in \left(0,1\right]$. In (A6.7), it is confirmed that $d{W}_O/d\theta >0$ holds for $\theta \in \left[0,1\right]$ and $\rho \in \left(1,3/\sqrt{7}\right)$. That is, ${\theta}_O^{*}=1$ is optimal for $\rho \in \left(1,\ 3/\sqrt{7}\right)$. For $\rho \ge 3/\sqrt{7}$, θ ² _O becomes nonnegative and ${\theta}_O^1\notin \left[0,1\right]$. Thus, θ ^* _O is either zero or unity. A direct calculation yields ${W}_O(1)-{W}_O(0)>0$ for $\rho \in \left(1,\ 1.2956\right)$ and ${W}_O(1)-{W}_O(0)<0$ for $\rho \in \left(1.2957,\infty \right)$. Together with these facts described above, the claims are proved.

1.4 A.6.4 Proof of Proposition 6.6

For the given privatization ratio, the optimal taxes can be written as follows:

$$ {T}_C\left(\theta \right)=-2a{\left({\varLambda}_C\right)}^{-1}\left[3\left(\rho +4\right){\theta}^2-2\left(\rho +1\right)\theta +\left(2\rho +3\right)\rho \right], $$

(A6.8)

$$ {\tau}_C\left(\theta \right)=2a{\left({\varLambda}_C\right)}^{-1}\left[\left(9\rho +12\right){\theta}^2-\left(3\rho +1\right)\theta +\left(3\rho +4\right)\rho \right], $$

(A6.9)

where ${\varLambda}_C\equiv \left(45\rho +36\right){\theta}^2-6\rho \theta +\left(9\rho +15\right)\rho +5>0$ for $\theta \in \left[0,1\right]$. The second-order condition is satisfied because W(T _C , τ _C , θ) is concave in T _C and τ _C for $\theta \in \left[0,1\right]$. Inserting (A6.8) and (A6.9) into (6.10), we can write the welfare as ${W}_C={W}_C\left(\theta \right)$ in which the taxes are optimally chosen. The first-order condition for θ is as follows:

$$ \frac{d{W}_C\left(\theta \right)}{d\theta }={a}^2{\left({\varLambda}_C\right)}^{-2}\left[\left(3\theta \rho -4\theta +\rho \right)\left(6\theta \rho -3{\rho}^2+5\right)\right]=0. $$

(A6.10)

Although ${\theta}_C^1=\rho /\left(4-3\rho \right)$ for $\rho \ne 4/3$ and ${\theta}_C^2=\left(3{\rho}^2-5\right)/\left(6\rho \right)$ are two roots of (A6.10), θ ² _C gives a minimum value. Hence, ${\theta}_C^{*}={\theta}_C^1$ for $\rho \in \left(0,1\right]$. In contrast, $d{W}_C/d\theta >0$ holds for $\theta \in \left[0,1\right]$ and $\rho \in \left(1,\sqrt{5/3}\right)$. That is, ${\theta}_O^{*}=1$ is optimal for $\rho \in \left(1,\sqrt{5/3}\right)$. For $\rho \ge \sqrt{5/3}$, θ ² _C is nonnegative and ${\theta}_C^1\notin \left[0,1\right]$: θ ^* _C is either zero or unity. We obtain ${W}_C(1)-{W}_C(0)>0$ for $\rho \in \left(1,\ 1.7183\right]$, and ${W}_C(1)-{W}_C(0)<0$ for $\rho \in \left[1.7184,\infty \right)$.