Profit-enhancing environmental policy: uninformed regulation in an entry-deterrence model

Abstract

This paper considers a polluting firm, subject to environmental policy, who seeks to deter the entry of potential competitors. We investigate under which conditions firm profits are enhanced by regulation. We show that, contrary to common belief, inefficient firms may support environmental regulation when their production is especially polluting. In addition, we evaluate how this result is affected by the regulator’s prior beliefs accuracy and the environmental damage from pollution.

Appendix

1.1 Appendix 1: Convex production costs

We examine how our equilibrium results are affected by the consideration of convex production costs, i.e., every firm’s costs become $c_{K}\cdot q^{2}$ where $K=\{H,L\}$. In particular, we first identify output levels under complete information and under the separating equilibrium. Next we evaluate the low-cost incumbent separating effort (as in Lemma 1), its profits under the separating equilibrium $\pi _{SE}^{L,R}(\rho )$ (as in Lemma 2) and, finally, the profit difference $\pi _{SE}^{L,R}(\rho )-\pi _{CI}^{L,R}$ (as in Proposition 1).

1.1.1 Complete information

When the incumbent maximizes its first-period profits, $(1-q)q-c_{K}q^{2}-t_{1}q$, it obtains an output function $q^{K}(t_{1})=\frac{1-t_{1}}{2(1+c_{K})}$. The socially optimal output in this context is $q_{K}^{SO}=\frac{1}{1+2d+2c_{K}}$. Hence, the regulator sets an emission fee $t_{1}$ that solves $\frac{1-t_{1}}{2(1+c_{K})}=\frac{1 }{1+2d+2c_{K}}$, i.e., $t_{1}^{K}=\frac{2d-1}{1+2d+2c_{K}}$, ultimately yielding an output level $q^{K}(t_{1}^{K})=q_{K}^{SO}$.

1.1.2 Separating equilibrium

Espinola-Arredondo and Munoz-Garcia (2015), Appendix 2, shows that under convex production costs the low-cost incumbent selects an output function $q^{A}(t_{1})=\frac{\left( 1-t_{1}\right) \left( 1+2d+2c_{inc}^{H}\right) +\left( 1+c_{inc}^{H}\right) \sqrt{3\delta }}{2\left( 1+c_{inc}^{H}\right) \left( 1+2d+2c_{inc}^{H}\right) }$, and that the regulator, anticipating such output, sets an emission fee $t_{1}^{*}$. The expression of the emission fee is, however, intractable and thus we next provide it evaluated at the same parameter values considered throughout the paper, $c_{H}=1/3$, $c_{L}=1/4$ and $\delta =1$.

$$\begin{aligned} t_{1}^{*}=\frac{12d\left[ A+4\sqrt{3}(1-\rho )\right] -7A+36\sqrt{3} (1-\rho )+A\rho }{A(9+12d+\rho )} \end{aligned}$$

where $A\equiv 5+6d$. At these parameter values, the low-cost incumbent is, thus, induced to produce

$$\begin{aligned} q^{A}\left( t_{1}^{*}\right) =\frac{3\sqrt{3}}{2A}+\frac{3\left[ 4A^{2}-9A\sqrt{3}(1-\rho )-12Ad\sqrt{3}(1-\rho )\right] }{2A^{2}(9+12d+\rho ) } \end{aligned}$$

in the separating equilibrium, and $q^{L}(t_{1}^{L})=\frac{2}{3+4d}$ in the complete information context. Hence, the separating effort is $q^{A}\left( t_{1}^{*}\right) -q^{L}(t_{1}^{L})$. Figure 6 evaluates the separating effort at the same environmental damages as Fig. 1. Hence, convex costs also yield a separating effort that increases in $\rho$, and experiences a downward shift as d increases. Unlike with linear costs, the separating effort is larger under convex costs, given that marginal costs are lower than in the linear case.

Fig. 6

Separating effort—convex cost

Equilibrium profits for the low-cost incumbent, $\pi _{SE}^{L,R}(\rho )$, are also intractable, but for the above parameter values become

$$\begin{aligned} \frac{25\left[ 9073134+189\left( 6325+189\sqrt{3}\right) \rho +25(11907\sqrt{ 3}-133160)\rho ^{2}\right] }{3969\left( 378+25\rho \right) ^{2}} \end{aligned}$$

when $d=0.51$, $\frac{1134+\rho \left[ 117+9\sqrt{3}\left( 9+3\rho \right) -302\rho \right] }{9\left( 18+\rho \right) ^{2}}$ when $d=3/4$, and $\frac{ 75816+432\rho (11+27\sqrt{3})+\rho ^{2}\left( 972\sqrt{3}-10883\right) }{ 324\left( 27+\rho \right) ^{2}}$ when $d=1.5$. Figure 7 depicts these profits as a function of $\rho$, showing that profits reach a maximum at $\rho _{1}\cong 0.01$ when $d=0.51$, but increases to $\rho _{1}\cong 0.25$ at $d=3/4$, and to $\rho _{1}\cong 1$ when $d=1.5$; thus reflecting a similar pattern as when production costs are linear.

Fig. 7

Profits in the SE—convex cost

We now compare equilibrium profits under the separating equilibrium and complete information, $\pi _{SE}^{L,R}(\rho )-\pi _{CI}^{L,R}$, where $\pi _{SE}^{L,R}(\rho )$ was obtained above and $\pi _{CI}^{L,R}\left( q_{L}^{SO}\right) =\frac{(1+\delta )\left[ 8(d+c_{L})-1\right] }{ 4(1+2d+2c_{L})^{2}}$, which becomes $\frac{3175}{7938}$ when $d=0.51$, $\frac{7}{18}$ when $d=3/4$, and $\frac{26}{81}$ when $d=1.5$. The profit difference as a function of $\rho$ is depicted in Fig. 8, illustrating that the value of $\rho$ for which $\pi _{SE}^{L,R}(\rho )=\pi _{CI}^{L,R}\left( q_{L}^{SO}\right)$, $\rho _{2}$, increases in d, as in the case of linear costs.

Fig. 8

Profit difference—convex cost

1.1.3 Pooling equilibrium

Let us next analyze the pooling equilibrium (PE) under convex production costs. Specifically, we first find the high-cost incumbent profits, $\pi _{PE}^{H,R}$ (as in Lemma 3), and compare them relative to complete-information profits, $\pi _{CI}^{H,R}$ (as in Proposition 2).

Under the PE, in the first period, the high-cost incumbent produces according to the low-cost output function $q^{L}(t_{1})=\frac{1-t_{1}}{ 2(1+c_{L})}$ and the regulator sets a fee $t_{1}^{L}=\frac{2d-1}{1+2d+2c_{L}}$ that helps the incumbent conceal its type. In the second period, since entry is deterred, the incumbent produces monopoly output and regulation is set accordingly. Hence, its profits are

$$\begin{aligned} \pi _{PE}^{H,R}=\frac{1-c_{H}+2c_{L}}{(1+2d+2c_{L})^{2}}+\frac{\delta (1+c_{H})}{(1+2d+2c_{H})^{2}} \end{aligned}$$

which are constant in $\rho$. In addition, for the above parameter values, $\pi _{PE}^{H,R}$ is monotonically decreasing in damage d. Intuitively, the negative effect of a more stringent fee dominates its positive effect (ameliorating the incumbent’s mimicking effort), as this effort is cheaper under convex than linear costs, i.e., marginal costs are lower for all inframarginal units.

Finally, comparing $\pi _{PE}^{H,R}$ against $\pi _{CI}^{H,R}$, where $\pi _{CI}^{H,R}=\frac{(4+\delta )(1+c_{H})}{4(1+2d+2c_{H})^{2}}$ we obtain that their difference is

$$\begin{aligned} \frac{(3\delta -4)(1+c_{H})}{4(1+2d+2c_{H})^{2}}-\frac{2d+c_{H}}{ (1+2d+2c_{L})^{2}}+\frac{1}{(1+2d+2c_{L})} \end{aligned}$$

which for our parameter values yields $\pi _{PE}^{H,R}>\pi _{CI}^{H,R}$ for all values of d.

Proof of Lemma 1

The output difference $q^{A}(t_{1}^{*})-q^{L}(t_{1}^{L})$ is positive if $c_{H}<\frac{\sqrt{3\delta }+2c_{L}}{\sqrt{3\delta }+2}\equiv \alpha _{A}$, a cutoff that originates at $c_{H}=\frac{\sqrt{3\delta }}{\sqrt{3\delta }+2}$ when $c_{L}=0$ and reaches $c_{H}=1$ when $c_{L}=1$. In addition, cutoff $\alpha _{A}$ satisfies $\alpha _{A}>\alpha _{1}\equiv \frac{\sqrt{3\delta } +(1+2d)c_{L}}{\sqrt{3\delta }+(1+2d)}$ since $\alpha _{1}$ originates at $c_{H}=\frac{\sqrt{3\delta }}{\sqrt{3\delta }+(1+2d)}$ and reaches $c_{H}=1$ when $c_{L}=1$, and their vertical intercepts satisfy $\frac{\sqrt{3\delta } }{\sqrt{3\delta }+(1+2d)}<\frac{\sqrt{3\delta }}{\sqrt{3\delta }+2}$ since $d>1/2$. Hence, for all parameter values in which the SE exists, $\alpha <\alpha _{1}$, the output difference $q^{A}(t_{1}^{*})-q^{L}(t_{1}^{L})$ is positive. Finally, the output difference increases in $\rho$, reaches its highest value, $\frac{\sqrt{3\delta }(1-c_{H})-2(c_{H}-c_{L})}{2(1+2d)}$ , when $\rho =1$; and collapses to zero when $\rho =0$. $\square$

Proof of Lemma 2

First-period profits in the SE are $\left( 1-q^{A}(t_{1}^{*})\right) q^{A}(t_{1}^{*})-c_{L}\cdot q^{A}(t_{1}^{*})$, and rearranging yields

$$\begin{aligned} \frac{(\lambda +2c_{L}(1-\rho )-(2+\sqrt{3\delta }\rho ))(\sqrt{3\delta } \rho -4d-\lambda +2c_{L}(2d+\rho )}{4(1+2d)^{2}} \end{aligned}$$

where $\lambda \equiv \rho c_{H}(2+\sqrt{3\delta })$. Differentiating with respect to $\rho$, yields $\frac{[\sqrt{3\delta }(1-c_{H})-2(c_{H}-c_{L})][ \lambda +c_{L}(1-2d-2\rho )-(1+\sqrt{3\delta }\rho -2d)]}{2(1+2d)^{2}}$ which is zero if $\rho =\frac{\left( 1-2d\right) \left( c_{L}-1\right) }{ \sqrt{3\delta }(1-c_{H})-2(c_{H}-c_{L})}\equiv \rho _{1}$. In addition, cutoff $\rho _{1}$ increases in d. That is, $\frac{\partial \rho _{1}}{ \partial d}=\frac{2(1-c_{L})}{\sqrt{3\delta }(1-c_{H})-2(c_{H}-c_{L})}>0$ if $\sqrt{3\delta }(1-c_{H})-2(c_{H}-c_{L})>0$ or $c_{H}<\frac{\sqrt{3\delta } +2c_{L}}{\sqrt{3\delta }+2}\equiv \overline{c_{H}}$ which always is satisfied since the cost parameter that supports the separating equilibrium is $c_{H}<\frac{\sqrt{3\delta }+(1+2d)c_{L}}{\sqrt{3\delta }+(1+2d)}< \overline{c_{H}}.$ $\square$

Proof of Proposition 1

Profits in the SE, $\pi _{SE}^{L,R}\left( \rho \right)$, are

$$\begin{aligned} \pi _{SE}^{L,R}\left( \rho \right)\equiv & {} \left( 1-q^{A}(t_{1}^{*})\right) q^{A}(t_{1}^{*})-c_{L}\cdot q^{A}(t_{1}^{*})\\&+\delta \left[ \left( 1-x_{inc}^{L}(t_{1}^{L})\right) x_{inc}^{L}(t_{1}^{L})-c_{L}\cdot x_{inc}^{L}(t_{1}^{L})\right] , \end{aligned}$$

since $x_{inc}^{L}(t_{1}^{L})=\frac{1-c_{L}}{1+2d}$, $\pi _{SE}^{L,R}\left( \rho \right)$ simplifies

$$\begin{aligned} \frac{ \begin{array}{c} 4d(\rho (\Gamma +\gamma )-2(1+\delta )(1-c_{L}))(c_{L}-1)+\rho (-3\rho \delta (1-c_{H})^{2} \\ -2\Gamma (c_{L}(1-2\rho )-1+2\rho c_{H})-2\gamma (\rho c_{H}-1+(1-\rho )c_{L})) \end{array} }{4(1+2d)^{2}} \end{aligned}$$

where $\Gamma \equiv -\sqrt{3\delta }(1-c_{H})$ and $\gamma \equiv 2(c_{H}-c_{L})$. Under CI, profits $\pi _{CI}^{L,R}=(1+\delta )\frac{ 2d(1-c_{L})^{2}}{(1+2d)^{2}}$. Hence, the profit difference $\pi _{SE}^{L,R}(\rho )-\pi _{CI}^{L,R}$ is

$$\begin{aligned} \frac{\rho \left[ \begin{array}{c} 2\sqrt{3\delta }(2d-1)-3\delta \rho -\rho (2\Gamma \gamma +2\gamma ^{2}-3\delta c_{H}(2-c_{H}))+ \\ 2(2d-1)(c_{L}(2+\sqrt{3\delta }-2c_{L})-(2+\sqrt{3\delta })c_{H}(1-c_{L})) \end{array} \right] }{4(1+2d)^{2}} \end{aligned}$$

which becomes zero for all $\rho \ge \rho _{2}$, where $\rho _{2}\equiv \frac{2(1-2d)(1-c_{L})(\Gamma +\gamma )}{3\delta (1-c_{H})^{2}+2\Gamma \gamma +\gamma ^{2}}$. In addition, cutoff $\rho _{2}$ is positive for all feasible values of d, and $\rho _{2}\le 1$ for all $d\le \frac{ (2+\sqrt{3\delta })(1-c_{H})}{4(1-c_{L})}$. $\square$

Proof of Lemma 3

Profits in the PE are

$$\begin{aligned} \pi _{PE}^{H,R}= & {} \left( 1-q^{L}(t_{1}^{L})\right) q^{L}(t_{1}^{L})-c_{H}\cdot q^{L}(t_{1}^{L}) \\&+\,\delta \left[ \left( 1-q^{H}(t_{1}^{H})\right) q^{H}(t_{1}^{H})-c_{H}\cdot q^{H}(t_{1}^{H})\right] \end{aligned}$$

Plugging output levels $q^{L}(t_{1}^{L})$ and $q^{H}(t_{1}^{H})$ into $\pi _{PE}^{H,R}$ yields

$$\begin{aligned} \pi _{PE}^{H,R}=\frac{2d(1+\delta )+2d\delta c_{H}^{2}-c_{L}(c_{L}+2d-1)+c_{H}\left[ (1-2d)c_{L}-1-2d(1+2\delta )\right] }{(1+2d)^{2}} \end{aligned}$$

Differentiating $\pi _{PE}^{H,R}$ with respect to d, and solving for d, we obtain s

$$\begin{aligned} d_{4}\equiv \frac{1+\delta (1+c_{H}^{2})-c_{H}(-1+2\delta +c_{L})+c_{L}(2c_{L}-3)}{2(-1+c_{H})(-1-\delta +\delta c_{H}+c_{L})}. \end{aligned}$$

$\square$

Proof of Proposition 2

Profit $\pi _{PE}^{H,R}$ is in the proof of Lemma 3, and

$$\begin{aligned} \pi _{CI}^{L,R}=(2+\delta )\frac{d(1-c_{H})^{2}}{(1+2d)^{2}}. \end{aligned}$$

Hence, $\pi _{PE}^{H,R}\ge \pi _{CI}^{H,R}$ for all $d<d_{5}$; where $d_{5}\equiv \frac{(c_{H}-c_{L})(1-c_{L})}{(1-c_{H})\left[ \delta +(2-\delta )c_{H}-2c_{L}\right] }$. In addition,

$$\begin{aligned} \pi _{PE}^{H,R}-\pi _{CI}^{H,R}=\frac{d\delta -d(2-\delta )c_{H}^{2}-c_{L}\left( 2d-1+c_{L}\right) +c_{H}(2d(1-\delta )-1+(1+2d)c_{L}) }{(1+2d)^{2}} \end{aligned}$$

Differentiating with respect to d, and solving for d yields

$$\begin{aligned} d=\frac{(2-\delta )c_{H}^{2}-\delta +6c_{L}-4c_{L}^{2}-2c_{H}(3-\delta -c_{L})}{2(1-c_{H})\left[ (\delta -2)c_{H}+2c_{L}-\delta \right] }\equiv d_{5}. \end{aligned}$$

$\square$

Proof of Corollary 1

Profit difference $\pi _{PE}^{H,R}-\pi _{CI}^{H,R}$ is given in the proof of Proposition 2. Since

$$\begin{aligned} \pi _{PE}^{H,NR}\equiv & {} \left( 1-q^{L}\right) q^{L}-c_{H}\cdot q^{L}+\delta \frac{\left( 1-c_{H}\right) ^{2}}{4} \\= & {} \frac{1+\delta (1+c_{H}^{2}-2c_{H}(1+\delta -c_{L})-c_{L}^{2})}{4} \end{aligned}$$

and $\pi _{CI}^{H,NR}=\frac{\left( 1-c_{H}\right) ^{2}}{4}+\delta \frac{ \left( 1-c_{H}\right) ^{2}}{9}$, then

$$\begin{aligned} \pi _{PE}^{H,NR}-\pi _{CI}^{H,NR}= & {} \left[ \frac{1+\delta (1+c_{H}^{2}-2c_{H}(1+\delta -c_{L})-c_{L}^{2})}{4}\right] \\&-\left[ \frac{ \left( 1-c_{H}\right) ^{2}}{4}+\delta \frac{\left( 1-c_{H}\right) ^{2}}{9} \right] \end{aligned}$$

Hence, $\pi _{PE}^{H,R}-\pi _{CI}^{H,R}\ge \pi _{PE}^{H,NR}-\pi _{CI}^{H,NR}$ for all $d\in \left[ d_{6},d_{6}^{\prime }\right]$, where

$$\begin{aligned} d_{6}\equiv \frac{D}{\varphi -\psi }+\frac{1}{\varphi -\psi }\left[ 4\delta +\left( 4\delta -9\right) (c_{H}-2)c_{H}+9(c_{L}-2)c_{L})^{2}-(\varphi -\psi )E\right] ^{1/2} \end{aligned}$$

and

$$\begin{aligned} d_{6}^{\prime }\equiv \frac{-D}{\varphi -\psi }+\frac{1}{\varphi -\psi } \left[ 4\delta +\left( 4\delta -9\right) (c_{H}-2)c_{H}+9(c_{L}-2)c_{L})^{2}-(\varphi -\psi )E\right] ^{1/2} \end{aligned}$$

where $\varphi =5\delta +(5\delta -9)c_{H}^{2}$, $\psi =2c_{H}(5\delta -9c_{L})+9c_{L}^{2}$, $D\equiv 9(c_{H}-c_{L})(c_{H}+c_{L}-2)-4\delta (1-c_{H})^{2}$, and $E\equiv \varphi +9c_{L}(3c_{L}-4)-2c_{H}(9c_{L}+5\delta -18)$. $\square$