Gresham’s law in environmental protection

Abstract

Under incomplete environmental enforcement, the high-cost (less efficient) firm may strategically violate the environmental standard, causing the low-cost (more efficient) firm to exit the market—a phenomenon similar to Gresham’s law in which bad money drives out good money. Tightening environmental regulation without increasing probability and penalties helps the high-cost firm to drive the low-cost firm out of the market. This explains why serious pollution and inefficient production co-exist in developing economies.

Appendices

Appendix 1

Substituting $a = 4,b = 1,c_{x} = 1,s = 1$ into the expected profit function in Table 1, we obtain the following expected (second-stage equilibrium) profits of all outcomes: $\begin{gathered} E\pi_{x}^{\text{CC}} = \frac{{(1 + c_{y} )^{2} }}{9},\,\,E\pi_{y}^{\text{CC}} = \frac{{(4 - 2c_{y} )^{2} }}{9},\,\,E\pi_{x}^{\text{CN}} = \frac{{c_{y}^{2} }}{9},\,\,E\pi_{y}^{\text{CN}} = \frac{{(6 + 2c_{y} )^{2} }}{9} - \,\theta f,\, \hfill \\ E\pi_{x}^{\text{NC}} = \frac{{(3 + c_{y} )^{2} }}{9} - \,\theta f,\,\,E\pi_{y}^{\text{NC}} = \frac{{(3 - 2c_{y} )^{2} }}{9},\,\,E\pi_{x}^{\text{NN}} = \frac{{(2 + c_{y} )^{2} }}{9} - \,\theta f,\, \hfill \\ E\pi_{y}^{\text{NN}} = \frac{{(5 - 2c_{y} )^{2} }}{9} - \,\theta f,\,E\pi_{x}^{\text{CO}} = \frac{4}{4},\,\,E\pi_{x}^{\text{NO}} = \frac{9}{4} - \,\theta f,\,\,E\pi_{y}^{\text{OC}} = \frac{{(3 - c_{y} )^{2} }}{4},\, {\text{and}}\, \hfill \\ E\pi_{y}^{\text{ON}} = \frac{{(4 - c_{y} )^{2} }}{4} - \,\theta f. \hfill \\ \end{gathered}$

According to the above expected profit, we find out the conditions for an outcome to be an SPNE and depict them in Fig. 1 as follows.

1.1 (C, C) as an SPNE

For (C, C) to be an SPNE, (1) and (2) must hold.

(1) Given firm x adopts C, firm y’s best response is adopting C.

$E\pi_{y}^{\text{CC}} = \frac{{(4 - 2c_{y} )^{2} }}{9} > E\pi_{y}^{\text{CN}} = \frac{{(6 - 2c_{y} )^{2} }}{9} - \theta f$ and $E\pi_{y}^{\text{CC}} = \frac{{(4 - 2c_{y} )^{2} }}{9} > E\pi_{y}^{\text{CO}} = 0$ must hold. By rearranging the former condition, we have $\theta f \ge \frac{{(6 - 2c_{y} )^{2} }}{9} - \frac{{(4 - 2c_{y} )^{2} }}{9} = \frac{{20 - 8c_{y} }}{9}$, such that the function of line $E\pi_{y}^{\text{CC}} = E\pi_{y}^{\text{CN}}$ in Fig. 1 is $\theta f = \frac{{20 - 8c_{y} }}{9}$ and the points on the right (left) side of it represent $\theta f > {(<)}\frac{{20 - 8c_{y} }}{9}$ and $E\pi_{y}^{\text{CC}} > {(<)}E\pi_{y}^{\text{CN}}$. Rearranging the latter condition, we get $c_{y} \le 2$, and so the function of line $E\pi_{y}^{CC} = 0$ is $c_{y} \le 2$ and the points above (under) it represent $c_{y} > {(<)}2$ and $E\pi_{y}^{CC} < {(>)}0$.

(2) Given firm y adopts C, firm x’s best response is adopting C.

$E\pi_{x}^{\text{CC}} = \frac{{(1 + c_{y} )^{2} }}{9} \ge E\pi_{x}^{\text{NC}} = \frac{{(3 + c_{y} )^{2} }}{9} - \theta f$ and $E\pi_{x}^{\text{CC}} = \frac{{(1 + c_{y} )^{2} }}{9} \ge E\pi_{x}^{\text{OC}} = 0.$ By rearranging the former condition, we have $\theta f \ge \frac{{8 + 4c_{y} }}{9}$, such that the function of line $E\pi_{x}^{\text{CC}} = E\pi_{x}^{\text{NC}}$ is $\theta f \ge \frac{{8 + 4c_{y} }}{9}$, and the points on the right (left) side of it represent $\theta f > ( < )\frac{{8 + 4c_{y} }}{9}$ and $E\pi_{x}^{\text{CC}} > ( < )E\pi_{x}^{\text{NC}}$. Moreover, the latter condition always holds.

To sum up, for (C, C) to be an SPNE, (1) and (2) must hold simultaneously, such that area (C, C) in Fig. 1 is surrounded by $c_{y} \le 2,\,\theta f \ge \frac{{20 - 8c_{y} }}{9}$, and $\theta f \ge \frac{{8 + 4c_{y} }}{9}$.

By the same way with (C, C), we also get the conditions for the other outcomes to be SPNE and depict them in Fig. 1.

1.2 (C, N) as an SPNE

All $\begin{gathered} E\pi_{y}^{\text{CN}} = \frac{{(6 - 2c_{y} )^{2} }}{9} - \theta f \ge E\pi_{y}^{\text{CC}} = \frac{{(4 - 2c_{y} )^{2} }}{9},\quad E\pi_{y}^{\text{CN}} = \frac{{(6 - 2c_{y} )^{2} }}{9} - \theta f \ge 0, \hfill \\ E\pi_{y}^{\text{CN}} = \frac{{c_{y}^{2} }}{9} \ge E\pi_{x}^{\text{NN}} = \frac{{(2 + c_{y} )^{2} }}{9} - \theta f, \hfill \\ \end{gathered}$ and $E\pi_{y}^{\text{CN}} = \frac{{c_{y}^{2} }}{9} \ge 0$ must hold. Rearranging them, we get $\frac{{4 + 4c_{y} }}{9} \le \theta f \le \frac{{20 - 8c_{y} }}{9}$ and $0 \le c_{y} \le \frac{4}{3}$. This is the area surrounded by line $E\pi_{y}^{\text{CC}} = E\pi_{y}^{\text{CN}}$ and line $E\pi_{x}^{\text{CN}} = E\pi_{x}^{\text{NN}}$ and the horizontal axis.

1.3 (N, C) as an SPNE

All $\begin{gathered} E\pi_{x}^{\text{NC}} = \frac{{(3 + c_{y} )^{2} }}{9} - \theta f \ge E\pi_{x}^{\text{CC}} = \frac{{(1 + c_{y} )^{2} }}{9},E\pi_{x}^{\text{CN}} = \frac{{c_{y}^{2} }}{9} \ge E\pi_{x}^{\text{NN}} = \frac{{(2 + c_{y} )^{2} }}{9} - \theta f, \hfill \\ E\pi_{y}^{\text{NC}} = \frac{{(3 - 2c_{y} )^{2} }}{9} \ge E\pi_{y}^{\text{NN}} = \frac{{(5 - 2c_{y} )^{2} }}{9} - \theta f, \hfill \\ \end{gathered}$ and $E\pi_{y}^{\text{NC}} \ge 0$must hold. By simplifying them, we have $\frac{{16 - 8c_{y} }}{9} \le \theta f \le \frac{{8 + 4c_{y} }}{9}$ and $\frac{2}{3} \le c_{y} \le \frac{3}{2}$. This is the area surrounded by line $E\pi_{x}^{\text{NC}} = E\pi_{x}^{\text{CC}} ,$ line $E\pi_{y}^{\text{NC}} = E\pi_{y}^{\text{NN}} ,$ and line $E\pi_{y}^{\text{NC}} = 0.$

1.4 (N, N) as an SPNE

All $\begin{gathered} E\pi_{x}^{\text{NN}} = \frac{{(2 + c_{y} )^{2} }}{9} - \theta f \ge E\pi_{x}^{\text{CN}} = \frac{{c_{y}^{2} }}{9},E\pi_{x}^{\text{NN}} = \frac{{(2 + c_{y} )^{2} }}{9} - \theta f \ge 0, \hfill \\ E\pi_{y}^{\text{NN}} = \frac{{(5 - 2c_{y} )^{2} }}{9} - \theta f \ge E\pi_{y}^{\text{NC}} = \frac{{(3 - 2c_{y} )^{2} }}{9},{\text{ and }}E\pi_{y}^{\text{NN}} = \frac{{(5 - 2c_{y} )^{2} }}{9} - \theta f \ge 0 \hfill \\ \end{gathered}$ must hold. By simplifying the above, we have: when $0 \le c_{y} \le 1,\,\theta f \le \frac{{4 + 4c_{y} }}{9}$; when $1 \le c_{y} \le \frac{3}{2},\,\theta f \le \frac{{16 - 8c_{y} }}{9}$; when $\frac{3}{2} \le c_{y} \le \frac{5}{2},\,\theta f \le \frac{{(5 - 2c_{y} )^{2} }}{9}$. This is the area surrounded by lines $E\pi_{x}^{\text{NN}} = E\pi_{x}^{\text{CN}} ,\,E\pi_{y}^{\text{NN}} = E\pi_{y}^{\text{NC}} ,E\pi_{y}^{\text{NN}} = 0,$ the vertical axis, and the horizontal axis.

1.5 (C, O) as an SPNE

All $E\pi_{y}^{\text{CC}} = \frac{{(4 - 2c_{y} )^{2} }}{9} \le 0,\,E\pi_{y}^{\text{CN}} = \frac{{(6 - 2c_{y} )^{2} }}{9} - \theta f \le 0$ and $E\pi_{x}^{\text{CO}} = \frac{4}{4} \le E\pi_{x}^{\text{NO}} = \frac{9}{4} - \theta f$ must hold. Simplifying them, we get $c_{y} \ge 2$ and $\theta f \ge \frac{5}{4}$. This is the area on the top right side of lines $E\pi_{x}^{\text{CO}} = E\pi_{x}^{\text{NO}}$ and $E\pi_{y}^{\text{CC}} = 0.$

1.6 (N, O) as an SPNE

All $E\pi_{y}^{\text{NC}} = \frac{{(3 - 2c_{y} )^{2} }}{9} < 0,\,E\pi_{y}^{\text{NN}} = \frac{{(5 - 2c_{y} )^{2} }}{9} - \theta f \le 0,$ and $E\pi_{x}^{\text{CO}} = \frac{4}{4} \ge \,E\pi_{x}^{\text{NO}} = \frac{9}{4} - \theta f$ must hold. By simplifying the above, we have $3 - 2c_{y} \le 0$; when $\frac{3}{2} \le c_{y} \le \frac{5}{2}$, $\frac{{(5 - 2c_{y} )^{2} }}{9} \le \theta f \le \frac{5}{4}$; and when $c_{y} \ge \frac{5}{2},\,\theta f \le \frac{5}{4}.$ This is the area surrounded by lines $E\pi_{y}^{\text{NC}} = 0,\,E\pi_{y}^{\text{NN}} = 0,\,$ and $E\pi_{x}^{\text{CO}} = E\pi_{x}^{\text{NO}} .$

1.7 (O, N) as an SPNE

All $E\pi_{x}^{\text{CN}} = \frac{{c_{y}^{2} }}{9} \le 0,E\pi_{x}^{\text{NN}} = \frac{{\left( {2 + c_{y} } \right)^{2} }}{9} - \theta f < 0,$ and $E\pi_{y}^{\text{ON}} = \frac{{\left( {4 - c_{y} } \right)^{2} }}{4} - \theta f \ge \,\,E\pi_{y}^{\text{OC}} = \frac{{\left( {3 - c_{y} } \right)^{2} }}{4}$ must hold. By simplifying the above, we have $c_{y} = 0$ and $\frac{4}{9} \le \theta f \le \frac{7}{4}$. This is the thick line on the horizontal axis. Furthermore, (O, C) as an SPNE is the dotted line on the horizontal axis, and (O, O) will not be an SPNE under these given parameters.

Appendix 2

By Table 1, when $s = 1.7$, for (O,N) to be a SPNE, the following conditions must hold: $E\pi_{x}^{ON} = 0 \ge E\pi_{x}^{NN} = \frac{{(4 - 2 \cdot 1 + c_{y} )^{2} }}{9} = \frac{{(2 + c_{y} )^{2} }}{9} - \theta f$ $E\pi_{x}^{ON} = 0 \ge E\pi_{x}^{CN} = \frac{{(4 - 2 \cdot (1 + 1.7) + c_{y} )^{2} }}{9} = \frac{{(c_{y} - 1.4)^{2} }}{9}$ $E\pi_{y}^{ON} = \frac{{(4 - c_{y} )^{2} }}{4} - \theta f \ge E\pi_{y}^{OC} = \frac{{(4 - c_{y} - 1.7)^{2} }}{4}$ and $E\pi_{y}^{ON} = \frac{{(4 - c_{y} )^{2} }}{4} - \theta f \ge E\pi_{y}^{OO} = 0$.

Rearranging them, we have $c_{y} \le 1.4,\,\theta f \ge \frac{{(2 + c_{y} )^{2} }}{9},\,\theta f \le \frac{{(4 - c_{y} )^{2} }}{4},$ and $\theta f \le \frac{{(6.3 - 2c_{y} ) \cdot 1.7}}{4}$. Because $c_{y} \le 1.4$ and $\theta f \le \frac{{(6.3 - 2c_{y} ) \cdot 1.7}}{4}$ imply $\theta f \le \frac{{(4 - c_{y} )^{2} }}{4}$, they are simplified to be $c_{y} \le 1.4,\,\theta f \ge \frac{{(2 + c_{y} )^{2} }}{9},$ and $\theta f \le \frac{{(6.3 - 2c_{y} ) \cdot 1.7}}{4}$. Moreover, according to Appendix 1, for (O, N) to be an SPNE when $s = 1$, both $\frac{{4 + 4c_{y} }}{9} \le \theta f \le \frac{{20 - 8c_{y} }}{9}$ and $0 \le c_{y} \le \frac{4}{3}$ must hold.

Based on the above, we overlap the area of (C, N) as an SPNE when $s = 1$ and that of (O, N) as an SPNE when $s = 1.7$ in Fig. 3. In Fig. 3, $tuv$ is the area of (C, N) as an SPNE when $s = 1$, whereas $qrst$ is the area of (O, N) as an SPNE when $s = 1.7$, and $lmn$ is their overlapping area. By the aforementioned conditions, we also find the equations of the lines surrounding $lmn$ are $\theta f = \frac{{(2 + c_{y} )^{2} }}{9},\,\theta f = \frac{{20 - 8c_{y} }}{9},$ and $c_{y} = 1$.