Environmental liability law and R&D subsidies: results on the screening of firms and the use of uniform policy

Abstract

This paper analyzes both R&D in pollution control technology and pollution abatement by firms that are subject to environmental liability law (either strict liability or negligence) and are granted R&D subsidies. Firms differ in their R&D costs (private information) and experience technology spillovers. Policy makers may induce first-best abatement and R&D levels despite asymmetric information by graduating policy instruments to screen firms. The chances of implementing first-best activity levels by such means differ under strict liability and negligence, and examples suggest that negligence performs better. The paper also studies the case in which uniform policy levels are imposed on heterogeneous firms, showing that strict liability tends to outperform negligence from a social welfare perspective in this scenario.

Appendices

Appendix 1

In Sect. 2, we propose that $x^{\rm FB}_{\rm L}>x^{\rm FB}_{\rm H}$ and $r^{\rm FB}_{\rm L}>r^{\rm FB}_{\rm H}$. This may be established by assuming otherwise and showing a contradiction.

• (i) Suppose that $x^{\rm FB}_{\rm L}<x^{\rm FB}_{\rm H}$, then $C_x(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L})=-D'(x^{\rm FB}_{\rm L})>-D'(x^{\rm FB}_{\rm H})=C_{x}(x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H})$ holds using (3). Given that $C_{xx}>0$, $C_x(x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H})> C_x(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm H})$. This leads to $C_x(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L})>C_x(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm H})$. This would imply that $T^{\rm FB}_{\rm L}<T^{\rm FB}_{\rm H}$ (i.e., that $r^{\rm FB}_{\rm H}>r^{\rm FB}_{\rm L}$) because $C_{xT}<0$. Social costs cannot be minimal at such a vector of individual behavior $(x_{\rm L},x_{\rm H},r_{\rm L},r_{\rm H})=(x^{\rm FB}_{\rm L},x^{\rm FB}_{\rm H},r^{\rm FB}_{\rm L},r^{\rm FB}_{\rm H})$ because costs would be lower when firms behave according to $(x_{\rm L},x_{\rm H},r_{\rm L},r_{\rm H})=(x^{\rm FB}_{\rm H},x^{\rm FB}_{\rm L},r^{\rm FB}_{\rm H},r^{\rm FB}_{\rm L})$. In detail, we obtain

  $$\begin{aligned}&C\left( x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L}\right) +D\left( x^{\rm FB}_{\rm L}\right) +Lr^{\rm FB}_{\rm L}+C\left( x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H}\right) +D\left( x^{\rm FB}_{\rm H}\right) +Hr^{\rm FB}_{\rm H} \nonumber \\&\qquad -\left( C\left( x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H}\right) +D\left( x^{\rm FB}_{\rm H}\right) +Lr^{\rm FB}_{\rm H}+C\left( x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L}\right) +D\left( x^{\rm FB}_{\rm L}\right) +Hr^{\rm FB}_{\rm L}\right) \nonumber \\&\quad =(H-L)\left( r^{\rm FB}_{\rm H}-r^{\rm FB}_{\rm L}\right) >0. \end{aligned}$$

  (30)

• (ii) Suppose that $x^{\rm FB}_{\rm L}=x^{\rm FB}_{\rm H}$, then (3) implies that $r^{\rm FB}_{\rm L}=r^{\rm FB}_{\rm H}$. Using these levels in (2) would imply $H=L$, which contradicts a central assumption of our framework. From (i) and (ii), we know that $x^{\rm FB}_{\rm L}>x^{\rm FB}_{\rm H}$ must hold. Since $C_{x}(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L})=-D'(x^{\rm FB}_{\rm L})<-D'(x^{\rm FB}_{\rm H})=C_{x}(x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H})< C_{x}(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm H})$, we conclude that $r^{\rm FB}_{\rm L}>r^{\rm FB}_{\rm H}$.

Appendix 2

The first-best level of the subsidy is given by $s^{\rm FB}_{i}=-\alpha C_{T}(x^{\rm FB}_{j},T^{\rm FB}_{j})$. Accordingly, the ranking $s^{\rm FB}_{\rm L}>s^{\rm FB}_{\rm H}$ follows when $-C_{T}(x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H})>-C_{T}(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L})$. Restating condition (2) gives

$$\begin{aligned} L&=-C_{T}\left( x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L}\right) -\alpha C_{T}\left( x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H}\right) =-C_{T}\left( x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L}\right) \nonumber \\&\quad \left[ 1 +\alpha C_{T}\left( x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H}\right) /C_{T}\left( x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L}\right) \right] \end{aligned}$$

(31)

$$\begin{aligned} H&=-C_{T}\left( x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H}\right) -\alpha C_{T}\left( x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L}\right) =-C_{T}\left( x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L}\right) \nonumber \\&\quad \left[ \alpha +C_{T}\left( x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H}\right) /C_{T}\left( x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L}\right) \right] . \end{aligned}$$

(32)

Using $L<H$ implies $1+\alpha C_{T}(x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H})/C_{T}(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L})<\alpha +C_{T}(x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H})/C_{T}(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L})$, which in turn explains $-C_{T}(x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H})>-C_{\rm T}(x^{\rm FB}_{\rm L},T^{\rm FB}_{\rm L})$.

Appendix 3

Under strict liability, firm L finds it privately optimal to select $(x^{\rm FB}_{\rm L},r^{\rm FB}_{\rm L})$ because $U_{\rm L}=\hbox {min}\{U_{\rm L},V_{\rm L},W_{\rm L}\}$. The ranking $U_{\rm L}<V_{\rm L}$ follows from

$$\begin{aligned} U_{\rm L}&=C\left( \bar{x}(T^{\rm FB}_{\rm L}),T^{\rm FB}_{\rm L}\right) +D\left( \bar{x}\left( T^{\rm FB}_{\rm L}\right) \right) +\left( L-s^{\rm FB}_{\rm L}\right) r^{\rm FB}_{\rm L} \nonumber \\&=\min _{r_{\rm L}}\left\{ C\left( \bar{x}\left( r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) ,r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) +D\left( \bar{x}\left( r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) \right) +\left( L-s^{\rm FB}_{\rm L}\right) r_{\rm L}\right\} \nonumber \\&<\min _{r_{\rm L}}\left\{ C\left( \bar{x}\left( r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) ,r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) +D\left( \bar{x}\left( r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) \right) +\left( L-s^{\rm FB}_{\rm H}\right) r_{\rm L}\right\} \le V_{\rm L}. \end{aligned}$$

(33)

The ranking $U_{\rm L}<W_{\rm L}$ follows from

$$\begin{aligned} U_{\rm L}&=\min _{r_{\rm L}}\left\{ C\left( \bar{x}\left( r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) ,r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) +D\left( \bar{x}\left( r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) \right) +\left( L-s^{\rm FB}_{\rm L}\right) r_{\rm L}\right\} \nonumber \\&<\min _{r_{\rm L}}\left\{ C\left( \bar{x}\left( r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) ,r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) +D\left( \bar{x}\left( r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) \right) +Lr_{\rm L}\right\} \le W_{\rm L}. \end{aligned}$$

(34)

Firm H optimizes given $r_{\rm L}=r^{\rm FB}_{\rm L}$. The ranking $U_{\rm H}<W_{\rm H}$ follows from

$$\begin{aligned} U_{\rm H}&=\min _{r_{\rm H}}\left\{ C\left( \bar{x}\left( r_{\rm H}+\alpha r^{\rm FB}_{\rm L}\right) ,r_{\rm H}+\alpha r^{\rm FB}_{\rm L}\right) +D\left( \bar{x}\left( r_{\rm H}+\alpha r^{\rm FB}_{\rm L}\right) \right) +\left( H-s^{\rm FB}_{\rm H}\right) r_{\rm H}\right\} \nonumber \\&<\min _{r_{\rm H}}\left\{ C\left( \bar{x}(r_{\rm H}+\alpha r^{\rm FB}_{\rm L}),r_{\rm H}+\alpha r^{\rm FB}_{\rm L}\right) +D\left( \bar{x}\left( r_{\rm H}+\alpha r^{\rm FB}_{\rm L}\right) \right) +Hr_{\rm H}\right\} \le W_{\rm H}. \end{aligned}$$

(35)

However, it may be that $V_{\rm H}<U_{\rm H}$, i.e., that firm H finds it privately optimal to mimic firm L. There are no incentives to mimic when

$$\begin{aligned} U_{\rm H}&=C\left( x^{\rm FB}_{\rm H},T^{\rm FB}_{\rm H}\right) +D\left( x^{\rm FB}_{\rm H}\right) +(H-s_{\rm H})r^{\rm FB}_{\rm H} \nonumber \\&<C\left( \bar{x}\left( (1+\alpha ) r^{\rm FB}_{\rm L}\right) ,(1+\alpha ) r^{\rm FB}_{\rm L}\right) +D\left( \bar{x}((1+\alpha ) r^{\rm FB}_{\rm L})\right) +\left( H-s^{\rm FB}_{\rm L}\right) r^{\rm FB}_{\rm L}=V_{\rm H}. \end{aligned}$$

(36)

Appendix 4

In the following analysis, we argue that firm L chooses $x_{\rm L}$ from the set $[x^{\rm FB}_{\rm H},x^{\rm FB}_{\rm L}]$ and $r_{\rm L}$ from the set $[r^{\rm FB}_{\rm H},r^{\rm FB}_{\rm L}]$. (i) Firm L chooses abatement at least as high as $x^{\rm FB}_{\rm H}$ because

$$\begin{aligned} \min _{x_{\rm L}<x^{\rm FB}_{\rm H}}\left\{ C\left( x_{\rm L},r_{\rm L}+\alpha r^{\rm FB}_{\rm H}\right) +D(x_{\rm L})+(L-s(r_{\rm L}))r_{\rm L}\right\} >U_{\rm L}>\hbox {PC}_{\rm L}^{\rm SN}\left( x^{\rm FB}_{\rm L},r^{\rm FB}_{\rm L}\right) . \end{aligned}$$

(37)

(ii) Firm L chooses R&D at least as high as $r^{\rm FB}_{\rm H}$ because firm H does so under strict liability (see Proposition 1). This ensures that $r_{\rm L}\ge r^{\rm FB}_{\rm H}$ because firm L enjoys a lower spillover than firm H ($\alpha r^{\rm FB}_{\rm H}<\alpha r^{\rm FB}_{\rm L}$), firm L bears lower R&D costs, and $r_{\rm L}$ falling short of $r^{\rm FB}_{\rm H}$ would mean that firm L would receive no subsidy at all. (iii) The privately optimal levels will not exceed $(x^{\rm FB}_{\rm L},r^{\rm FB}_{\rm L})$ because marginal benefits of choosing higher levels are smaller under the negligence regime than under strict liability. (iv) When firm L selects $x_{\rm L}=x^{\rm FB}_{\rm L}$, it will select $r^{\rm FB}_{\rm L}$. This follows from the analysis of strict liability.

Appendix 5

In the following analysis, we argue that firm H chooses $x_{\rm H}$ from the set $[x^{\rm FB}_{\rm H},x^{\rm FB}_{\rm L}]$ and $r_{\rm H}$ from the set $[r^{\rm FB}_{\rm H},r^{\rm FB}_{\rm L}]$. (i) Firm H chooses at least $(x^{\rm FB}_{\rm H},r^{\rm FB}_{\rm H})$ because

$$\begin{aligned}&\min _{x_{\rm H},r_{\rm H}}\left\{ C\left( x_{\rm H},r_{\rm H}+\alpha r^{\rm FB}_{\rm L}\right) +\hbox {LL}(x_{\rm H})+(H-s(r_{\rm H}))r_{\rm H}\right\} \nonumber \\&\quad <\min _{x_{\rm H},r_{\rm H}}\left\{ C\left( x_{\rm H},r_{\rm H}+\alpha r^{\rm FB}_{\rm L}\right) +D(x_{\rm H})+(H-s(r_{\rm H}))r_{\rm H}\right\} \nonumber \\&\quad =\min \{U_{\rm H},V_{\rm H},W_{\rm H}\}<W_{\rm H}, \end{aligned}$$

(38)

where $W_{\rm H}$ is implied by activity levels below $(x^{\rm FB}_{\rm H},r^{\rm FB}_{\rm H})$. (ii) Firm H chooses weakly less than $(x^{\rm FB}_{\rm L},r^{\rm FB}_{\rm L})$. This has been established for firm L above and clearly also holds for firm H. (iii) Firm H choosing $r^{\rm FB}_{\rm H}$ is inconsistent with firm H choosing $x_{\rm H}$ from the interval $(x^{\rm FB}_{\rm H},x^{\rm FB}_{\rm L})$. This results from

$$\begin{aligned} \hbox {arg min}_{x_{\rm H}\in X}\left\{ C\left( x_{\rm H},T^{\rm FB}_{\rm H}\right) +\delta D(x_{\rm H})\right\} \le \hbox {arg min}_{x_{\rm H}\in X}\left\{ C(x_{\rm H},T^{\rm FB}_{\rm H})+D(x_{\rm H})\right\} =x^{\rm FB}_{\rm H}. \end{aligned}$$

(39)

The firm will either remain at $x^{\rm FB}_{\rm H}$ or implement $x^{\rm FB}_{\rm L}$. (iv) When firm H chooses $x_{\rm H}$ from $X$ and $r_{\rm H}$ from $R$, it will always select the combination of activity levels $(x^{\rm FB}_{\rm H},r^{\rm FB}_{\rm H})$.

Appendix 6

When firms are subject to strict liability, the privately optimal R&D levels change with the level of the uniform subsidy according to

$$\begin{aligned} \frac{\hbox {d}r^*_{\rm L}}{\hbox {d}s}&=\frac{1}{Z} \left[ C_{T_{\rm H} x_{\rm H}}\frac{\hbox {d}x_{\rm H}}{\hbox {d}T_{\rm H}}+C_{T_{\rm H}T_{\rm H}}-\alpha C_{T_{\rm L} x_{\rm L}}\frac{\hbox {d}x_{\rm L}}{\hbox {d}T_{\rm L}}-\alpha C_{T_{\rm L}T_{\rm L}}\right] \end{aligned}$$

(40)

$$\begin{aligned} \frac{\hbox {d}r^*_{\rm H}}{\hbox {d}s}&=\frac{1}{Z} \left[ C_{T_{\rm L} x_{\rm L}}\frac{\hbox {d}x_{\rm L}}{\hbox {d}T_{\rm L}}+C_{T_{\rm L}T_{\rm L}}-\alpha C_{T_{\rm H} x_{\rm H}}\frac{\hbox {d}x_{\rm H}}{\hbox {d}T_{\rm H}}-\alpha C_{T_{\rm H}T_{\rm H}}\right] , \end{aligned}$$

(41)

where $Z>0$ is the determinant of the according Hessian matrix.

Appendix 7

When firms are subject to negligence, the privately optimal R&D levels change with the level of the uniform subsidy according to

$$\begin{aligned} \frac{\hbox {d}\bar{\bar{r}}_{\rm L}}{\hbox {d}s^N}&=\frac{1}{Z} \left[ C_{T_{\rm H} T_{\rm H}}-\alpha C_{T_{\rm L} T_{\rm L}}\right] \end{aligned}$$

(42)

$$\begin{aligned} \frac{\hbox {d}\bar{\bar{r}}_{\rm H}}{\hbox {d}s^N}&=\frac{1}{Z} \left[ C_{T_{\rm L} T_{\rm L}}-\alpha C_{T_{\rm H} T_{\rm H}}\right] , \end{aligned}$$

(43)

where $Z>0$ is the determinant of the according Hessian matrix.