Environmental Regulation in a Mixed Economy

Abstract

Many developing countries are mixed economies in which public and private firms engage in Cournot competition. We show that some fundamental results in environmental economics fail to hold in these economies: more stringent environmental regulation does not necessarily reduce pollution levels, the equivalence between environmental taxes and standards breaks down, and not every emission level can be induced by emission taxes. These results are due to the endogeneity of the public firm CEO’s career choices. Instruments that can induce the CEO to choose a public career are most effective in reducing emissions and improving social welfare.

Appendix: Technical Details

1.1 Emission taxes

Given the linear quadratic setup, we are able to obtain closed form solutions of the equilibrium output and abatement levels. It is straightforward to show that

• (i) If $\tau \ge c$ , then $\alpha _{0} =\alpha _{1} =1$ and

  $$\begin{aligned} \left\{ {\begin{array}{l} q_{0} =\frac{a-k-c}{1+2\lambda } \\ q_{1} =\lambda \frac{a-k-c}{1+2\lambda } \\ \end{array}} \right. \end{aligned}$$

  (8)

• (ii) If $\tau <c$ and $\lambda <\lambda ^{I}$ , then $\alpha _{0} =1, \alpha _{1} =0$ and

  $$\begin{aligned} \left\{ {\begin{array}{l} q_{0} =\frac{\tau -2c+a-k}{1+2\lambda } \\ q_{1} =\frac{\lambda a-\lambda k-\lambda \tau -\tau +c}{1+2\lambda }\\ \end{array}} \right. \end{aligned}$$

  (9)

• (iii) If $\tau <c$ and $\lambda \ge \lambda ^{I}$ , then $\alpha _{0} =0, \alpha _{1} =0$ and

$$\begin{aligned} \left\{ {\begin{array}{l} q_{0} =\frac{2\lambda \delta -2\lambda \tau -2\delta +\tau +a-k}{1+2\lambda } \\ q_{1} =\frac{-\tau +\lambda a+\delta -\lambda \delta -\lambda k}{1+2\lambda } \\ \end{array}} \right. \end{aligned}$$

(10)

$\hbox {When }\lambda <1$, the public firm differs from the private firm in two aspects due to the former’s concern about social welfare: it might abate, and it tends to produce more so as to raise consumer surplus. The two firms also respond differently to changes in exogenous parameters. Proposition A1, which can be easily derived from (10), highlights the differences for the case of (iii) above.

Proposition A1

Consider case (iii), i.e., $\alpha _{0} =0, \alpha _{1} =0$ and $q_{0}$ and $q_{1}$ are given in (10).

• (i) Suppose $\lambda <1$ , i.e., the public firm cares about the social welfare. As pollution damage $\delta$ increases, the public firm’s output $q_{0}$ decreases but the private firm’s output $q_{1}$ increases.

• (ii) As pollution tax $\tau$ increases, the public firms’ output $q_{0}$ decreases if $\lambda \ge 1/2$ and the private firms’ output $q_{1}$ always decreases, and $q_{1}$ decreases more than $q_{0}$ does.

• (iii) As consumer demand parameter a increases, both firms’ outputs rise, but the public firm’s output rises more than that of the private firm.

• (iv) As the marginal cost k rises, both firms’ outputs decrease, but the public firm’s output decreases more than the private firm.

• (v) As $\lambda$ increases, i.e., as the public firm cares more about its profit, the public firm’s output $q_{0}$ decreases, and the private firm’s output $q_{1}$ increases.

We omit the proof, which is straightforward utilizing (10). $\hbox {When }\lambda <1$, the public firm cares about the social welfare. Thus, as the pollution damage $\delta$ increases, the public firm reduces its $\hbox {output }q_{0}$ but the private firm increases its output due to strategic substitution $\hbox {between }q_{0}\,\hbox {and }q_{1}$. In general, the private firm produces less as tax increases, but the public firm might produce more since the tax revenue is part of the social welfare. It will produce less if it cares more about its profit than about social welfare, i.e., $\hbox {if }\lambda \ge 1/2$. When demand increases, both firms produce more but since the marginal consumer surplus is higher, the public firm has incentive to increase its output more than the private firm. When cost k rises, the fact that social welfare includes both firms’ profits means that the public firm reduces its output more than the private firm. As $\lambda$ increases, the public firm cares more about its profit and thus reduces its $\hbox {output }q_{0}$, and the private firm increases its output $q_{1}\hbox {in }$ response, mimicking results in the standard mixed oligopoly literature.

Proof of Proposition 1

Given $\tau <c$, it is straightforward to show that (cf. Fig. 1)  $(\hat{{\pi }}_{0} (\lambda ,\tau )-\hat{{\pi }}_{1} (\lambda ,\tau ))|_{\alpha _{0} =0,\alpha _{1} =0} >(\hat{{\pi }}_{0} (\lambda ,\tau )-\hat{{\pi }}_{1} (\lambda ,\tau ))|_{\alpha _{0} =1,\alpha _{1} =0}$: at any chosen $\lambda$, the public firm obtains a higher profit when it does not abate. The definition of $\lambda _{1}^{P}$ and $\lambda _{2}^{P}$ then implies that $(\hat{{\pi }}_{0} (\lambda _{1}^{P} (\tau ),\tau )-\hat{{\pi }}_{1} (\lambda _{1}^{P} (\tau ),\tau ))|_{\alpha _{0} =0,\alpha _{1} =0} >(\hat{{\pi }}_{0} (\lambda _{2}^{P} (\tau ),\tau )-\hat{{\pi }}_{1} (\lambda _{2}^{P} (\tau ),\tau ))|_{\alpha _{0} =1,\alpha _{1} =0}$. Thus,

$$\begin{aligned} \hbox {f}(\hat{{\pi }}_{0} (\lambda _{1}^{P} (\tau ),\tau )-\hat{{\pi }}_{1} (\lambda _{1}^{P} (\tau ),\tau ))|_{\alpha _{0} =0,\alpha _{1} =0} \end{aligned}$$

(11)

is the highest payoff the private career CEO can obtain when choosing $\lambda$.

Under Case (i) of Proposition 1, $\lambda _{1}^{P} (\tau )>\lambda ^{I}(\tau )$. By $\hbox {setting }\lambda ^{P}=\lambda _{1}^{P}$, the CEO obtains the highest payoff in (11). Under Case (ii), the CEO has no incentive to choose $\lambda <\lambda ^{I}\,\hbox {since }$ it induces full abatement (cf. Fig. 1b) and since $\lambda ^{I} (\tau )<\lambda _{3}^{P}(\tau )\,\hbox {implies }$ that choosing $\lambda \in [\lambda ^{I},\lambda _{3}^{P})\,\hbox {generates }$ a payoff that exceeds the maximum payoff obtainable under full abatement (cf. the definition of $\lambda )$. Since $f(\cdot )$ is concave by assumption and $(\hat{{\pi }}_{0} (\lambda ,\tau )-\hat{{\pi }}_{1} (\lambda ,\tau ))|_{\alpha _{0} =0,\alpha _{1} =0}$ is concave in $\lambda$ (due to the linear quadratic setup), the optimal choice from $\lambda \in [\lambda ^{I},\lambda _{3}^{P})$ is $\lambda ^{P}=\lambda ^{I}$ . Under Case (iii), the definition of $\lambda$ and the condition that $\lambda ^{I} \ge \lambda _{3}^{P} \ge \lambda \,\hbox {imply }$ that the CEO can obtain a higher payoff by inducing full abatement and choosing $\lambda ^{P}=\lambda _{2}^{P}$ . $\square$

Layout of Fig. 2. The layout of the threshold profit weight functions in Fig. 2 has the following properties: (i)$\lambda _{1}^{P} (\tau )<\lambda _{2}^{P} (\tau )\le 1/4$ for all $\tau \le c,\lambda _{2}^{P} (\tau =c)=1/4$, and $\partial \lambda _{i}^{P} (\tau )/\partial \tau >0$, for $i=1, 2,$ and $\partial \lambda ^{I} (\tau )/\partial \tau >0$; (ii) $\lambda ^{I}(\tau )$ crosses $\lambda _{1}^{P} (\tau )$ only once and from below at point $\tau _{1}$, defined as $\tau _{1} \equiv \max \left\{ {\tau \ge 0,\hbox { s.t. }\lambda ^{I} (\tau )=\lambda _{1}^{P} (\tau )} \right\}$ if it exists and $\tau _{1} =0$ otherwise; (iii) $\lambda ^{I}(\tau )$ crosses $\lambda _{3}^{P} (\tau )$ only once and from below at point $\tau _{2}$, defined as $\tau _{2} \equiv \max \left\{ {\tau \ge 0,\hbox { s.t. }\lambda ^{I} (\tau )=\lambda _{3}^{P} (\tau )} \right\}$ if it exists and $\tau _{2} =0$ otherwise; and (iv) $\tau _{1} <\tau _{2} <c$.

Property (i) can be established directly from Eqs. (5), (6) and (7). From (5) and (6), $\lambda ^{I}(\tau )$ and $\lambda _{1}^{P} (\tau )$ can cross at most twice. Figure 2 depicts a situation when they cross once where $\lambda ^{I}(\tau )$ crosses from below. We can show that if $\lambda ^{I} (\tau )$ ever crosses $\lambda _{1}^{P} (\tau )$ from above, it will always cross $\lambda _{1}^{P} (\tau )$ again from below because $\frac{\partial \lambda _{1}^{P} (\tau )}{\partial \tau }=\frac{9(a-k-\delta )}{2(2a-2k-3\delta +\tau )^{2}}$ and is bounded above but $\frac{\partial \lambda ^{I} (\tau )}{\partial \tau }=\frac{\delta -c}{(\delta -\tau )^{2}}$ and goes to infinity as $\tau$ increases. That is, eventually the slope of $\lambda ^{I} (\tau )$ will exceed that of $\lambda _{1}^{P} (\tau )$ and they will cross again. A third possibility is when $\lambda ^{I} (\tau )$ lies entirely above $\lambda _{1}^{P} (\tau )$, in which case $\tau _{1} =0$. Although we only analyze the situation when they cross only once, the same methods can be used to analyze the other two situations (crossing twice or never crossing) and our main results still hold. Property (iii) can be established in a similar fashion to Property (ii). In Property (iv), $\tau _{1} <\tau _{2}$ follows from the fact $\hbox {that }\lambda ^{I} (\tau )$ is increasing, $\lambda ^{I} (\tau )\,\hbox {crosses } \lambda _{1}^{P} (\tau )$ from below, and $\lambda _{1}^{P} (\tau )<\lambda _{2}^{P} (\tau )$. Since $\lim _{\tau \rightarrow c} \lambda ^{I} (\tau )=1$ and $\lambda _{3}^{P} (c)<1$, we know $\tau _{2}<c$.

Proof of Proposition 2

When $\tau >c$, both firms fully abate and thus pay no pollution tax. Their profits and the social welfare are independent of $\tau$, establishing (i) of the Proposition. For (ii), we can establish that $\partial W^{*}/\partial \tau -\partial (\pi _{0}^{*} -\pi _{1}^{*} )/\partial \tau =(\delta -c)-(a-k-\tau )/2$, which is negative from Assumption 2(ii), $\tau <c$, and $\delta >\tau$(implied by Assumption 2(i)). For (iii) of the Proposition, we can show $\hbox {that }\partial W^{*}/\partial \tau -\partial (\pi _{0}^{*} -\pi _{1}^{*})/\partial \tau =(\delta -c)+(a-k-\tau )/6$, which is positive from Assumption 1 and because $\delta >c$(Assumption 2(i)). $\square$

1.2 Emission standards

Given $\hbox {standard }\bar{{\alpha }}$ and weight $\lambda$, it is straightforward to show that in Stage 3 Nash equilibrium, the private firm always abates up to the standard. Further,

• (i) If $\lambda <\tilde{\lambda }^{I}$ , the public firm fully abates ($\alpha _0 =1$ ) and the outputs are given by

  $$\begin{aligned} \left\{ {\begin{array}{l} q_{0} =\frac{c\overline{\alpha }-2c+a-k}{1+2\lambda } \\ q_{1} =\frac{\lambda a-\lambda k-\lambda c\overline{\alpha }-c\overline{\alpha }+c}{1+2\lambda } \\ \end{array}} \right. \end{aligned}$$

  (12)

• (ii) If $\lambda \ge \tilde{\lambda }^{I}, \alpha _{0} =\bar{{\alpha }}$ and the equilibrium outputs are

$$\begin{aligned} \left\{ {\begin{array}{l} q_{0} =\frac{-2\lambda \delta \overline{\alpha }-2\delta +2\delta \overline{\alpha }+2\lambda \delta +a-k-c\overline{\alpha }}{1+2\lambda } \\ q_{1} =\frac{\lambda \delta \overline{\alpha }+\lambda a-\lambda k-\lambda c\overline{\alpha }+\delta -\delta \overline{\alpha }-\lambda \delta }{1+2\lambda } \\ \end{array}} \right. \end{aligned}$$

(13)

Similar to the case of taxes, when the public firm fully abates, further tightening of the standard (increasing $\bar{{\alpha }})$ only hurts the private firm since it will have to abate more. Thus in (12), $q_{0}$ is increasing while $q_{1}$ is decreasing in $\hbox {standard }\bar{{\alpha }}$. When both firms are constrained by the standard, the private firm’s output is always decreasing in the standard, again since it cares only about its profit and abatement is costly. Even though the standard is binding for the public firm, the public firm’s output can still be increasing in the standard as long as its profit weight is not too high: a sufficient condition for this to be true $\hbox {is }\lambda \le 1/2$. Finally, we can show that in (13), $q_{0}$ is decreasing while $q_{1}$ is increasing in pollution damage $\delta$: $\partial q_{0} /\partial \delta \propto -(1-\bar{{\alpha }})(1-\lambda )\le 0$ and $\partial q_{1} /\partial \delta \propto (1-\bar{{\alpha }})(1-\lambda )\ge 0$, where the inequalities follow from $\lambda \le 1$ and $\bar{{\alpha }}\le 1$.