The relationship between privatization and corporate taxation policies

Abstract

We investigate how the corporate (profit) tax rate affects the optimal degree of privatization in a mixed duopoly with a minimal profit constraint for the private firm. We show that the profit tax rate directly affects the behavior of the partially privatized firm, and therefore affects the behavior of the private firm through strategic interactions. Regardless of whether the constraint is binding, the optimal degree of privatization increases with the corporate tax rate. The reason is that an increase of corporate tax rate reduces the profits flowing to foreign investors, which mitigates the welfare losses of privatization. Furthermore, the optimal degree of privatization decreases (increases) with the foreign ownership share in the private firm if the constraint is ineffective (effective). This result suggests that a minimal profit constraint can be crucial in the optimal privatization policy.

Appendix

Proof of Proposition 1

First, we show $\partial X_6/\partial \alpha >0$ if $\alpha ^E<1$. Let

$$\begin{aligned} -\frac{1}{1-\alpha t}\frac{X_1p'}{X_2}:=X_7. \end{aligned}$$

From (11), we obtain

$$\begin{aligned} W^{S''}(\alpha )=\frac{\partial X_7}{\partial \alpha }X_6+X_7\frac{\partial X_6}{\partial \alpha }. \end{aligned}$$

Because $W^{S''}(\alpha )<0$, $X_6=0$ at the equilibrium point, and $X_7<0$, we obtain $\partial X_6/\partial \alpha >0$.

Next, we show $\partial X_6/\partial t<0$ if $\alpha ^E<1$. We obtain

$$\begin{aligned} \frac{\partial X_6}{\partial t}= & {} -\alpha \left\{ [q_1-q_0-\beta (1-t)q_1]p''q_1+[q_1-2q_0-2\beta (1-t)q_1]p'+[\beta (1-t)q_1+q_0]c_1''\right\} \\&-\alpha [-\beta (1-t)q_1(p''q_1+2p')+(1-t)\beta q_1c_1'']. \end{aligned}$$

As we show after (11), $\alpha ^E<1$ if and only if

$$\begin{aligned} {[}q_1-q_0-\beta (1-t)q_1]p''q_1+[q_1-2q_0-2\beta (1-t)q_1]p'+[\beta (1-t)q_1+q_0]c_1''>0. \end{aligned}$$

Thus, from the assumption that $p''q_1+p'<0$ and $c_1''\ge 0$, we obtain $\partial X_6/\partial t<0$ when $\alpha ^E<1$. These two facts and (13) imply Proposition 1. $\square$

Proof of Proposition 2

In the proof of Proposition 1, we have already shown that $\partial X_6/\partial \alpha >0$ if $\alpha ^E<1$. This and (14) imply Proposition 2. $\square$

Proof of Lemma 3

Suppose that t increases marginally, from $t_a$ to $t_b$. We will show that this change increases W.

Suppose that $\alpha ^E<1$ and $p-c_0'>0$ when $t=t_a$. Given $\alpha$, this change increases the resulting $q_0$ (Lemma 1(i)). Suppose that the government increases $\alpha$ to keep the resulting $q_0$ unchanged. Note that $q_1$ remains unchanged if $q_0$ remains unchanged because neither t nor $\alpha$ affects $q_1$ directly, and both affect $q_1$ through the change in $q_0$. Because $q_0^S(\alpha ,t)$ is decreasing in $\alpha$ (and thus $q_0^S(\alpha ,t)<q_0^S(1,t)$ for any $\alpha <1$ and $t \in [0,1)$) and $q_0^S(1,t)$ is independent of t, the government can choose such an $\alpha$ as long as $\alpha ^E<1$. Because Q, $q_0$, and $q_1$ remain unchanged, CS, $\pi _0$, and $\pi _1$ remain unchanged. Thus, W increases by $\beta (t_b-t_a)\pi _1$. Given $t_b$, the above $\alpha$ is not the optimal $\alpha$. Nevertheless, W increases with an increase in t, and much more if the government chooses the optimal $\alpha$.

Suppose that $\alpha ^E<1$ and $p-c_0' \le 0$ when $t=t_a$. Given $\alpha$, this change decreases the resulting $q_0$ (Lemma 1(i)). Suppose that the government decreases $\alpha$ to keep the resulting $q_0$ unchanged. Because $q_0^S(\alpha ,t)$ is decreasing in $\alpha$ and $\alpha ^E>0$, $q^S(0,t_a)>q_0^S(\alpha ^E,t_a)$. Due to the continuity of $q^S(\alpha ,t)$, there exists an $\alpha '$ such that $q_0^S(\alpha ',t_b)=q_0^S(\alpha ^E,t_a)$ if $t_b-t_a$ is sufficiently small. Because Q, $q_0$, and $q_1$ remain unchanged, CS, $\pi _0$, and $\pi _1$ remain unchanged. Thus, W increases by $\beta (t_b-t_a)\pi _1$. $\alpha '$ is not the optimal $\alpha$. Nevertheless, W increases with the increase in t, and much more if the government chooses the optimal $\alpha$.

Suppose that $\alpha ^E=1$ when $t=t_a$. Suppose that the government keeps $\alpha ^E=1$ after the change in t, which does not affect Q, $q_0$, and $q_1$ (Lemma 1(ii)). Thus, W increases by $\beta (t_b-t_a)\pi _1$ through the change in t. $\square$

Proof of Lemma 4

From (3), we find that an increase in $\alpha$ increases $q_1$ and reduces Q. Both increase $\pi _1^S$. No other effect on $\pi _1$ exists. Therefore, $\pi _1^S$ is increasing in $\alpha$. This implies Lemma 4(i).

We obtain

$$\begin{aligned} \frac{\partial [(1-t)\pi _1]}{\partial t}=-\pi _1+(1-t)p'q_1 \frac{\partial q_0}{\partial t}+(1-t)(p'q_1+p-c_1)\frac{\partial q_1}{\partial t}. \end{aligned}$$

(16)

The first term in (16) is negative, the second term is non-positive if $p \ge c_0'$ (Lemma 1) and the third term is zero from (2). These imply Lemma 4(ii).

Lemma 2 states that an increase in $\beta$ decreases $q_1$ and increases Q as long as $\alpha <1$. Both reduce $\pi _1^S$. No other effect on $\pi _1$ exists. Therefore, $\pi _1^S$ is decreasing in $\beta$. This implies Lemma 4(iii). $\square$

Proof of Proposition 4

Given that the constraint is binding when the government chooses the optimal $\alpha$ and t simultaneously, we take the total derivative of the constraint (i.e., $(1-t)\pi _1=F$).

$$\begin{aligned}&-(pq_1-c_1)dt+(1-t)p'q_1dQ+(1-t)(p-c_1')dq_1=0,\\&-(pq_1-c_1)dt+(1-t)p'q_1dq_0=0,\\&-\pi _1dt+(1-t)p'q_1\left[ -\frac{X_1(p''q_1+2p'-c_1'')}{X_2}\right] d\alpha \\&+(1-t)p'q_1 \left[ -\frac{X_4(p''q_1+2p'-c_1'')}{X_2}\right] dt \\&+(1-t)p'q_1\left[ -\frac{X_5(p''q_1+2p'-c_1'')}{X_2}\right] d\beta =0. \end{aligned}$$

From these, we obtain

$$\begin{aligned}&\frac{d \alpha ^E}{d \beta }=-\frac{X_5}{X_1}, \end{aligned}$$

(17)

$$\begin{aligned}&\frac{d t^E}{d \beta }=-\frac{(1-t)p'q_1X_5(p''q_1+2p'-c_1'')}{X_2\pi _1+(1-t)p'q_1X_4(p''q_1+2p'-c_1'')}. \end{aligned}$$

(18)

From (10), $X_5$ is non-negative and positive if $\alpha <1.$ Since $X_1<0$, we obtain Proposition 4(i).

Since $X_5\ge 0$ and the equality holds if and only if $\alpha =1$, the numerator in (18) is non-negative and strictly positive if $\alpha <1$. Then, we show that the denominator in (18) is positive, which implies Proposition 4(ii).

Because $X_2$, $\pi _1$ and $(1-t)p'q_1(p''q_1+2p'-c_1'')$ are positive, the denominator in (18) is positive if $X_4 \ge 0$. From the discussion in footnote 17 and $X_4$ in (8), we obtain

$$\begin{aligned} X_4=\frac{p'(1-\alpha )\cdot [-\alpha q_0+\beta (1-\alpha )q_1]}{1-\alpha t}=\frac{(p-c_0')(1-\alpha )}{1-t}. \end{aligned}$$

This is positive because we assume $p \ge c_0'$. $\square$