Market distortions and optimal environmental policy instruments

Abstract

In practice, a market does not comprise only one type of firm, resulting in two distortions: negative externalities caused by pollution damage and pricing power enjoyed by dominant firms. This paper examines choice of environmental policy instruments (tax-centered, quota-centered, and mixed policy) in markets where multiple dominant firms are price makers and multiple fringe firms are price takers. Environmental policy is not necessarily applied to all firms or facilities. This study focuses on the situation where only dominant firms are objects of environmental policy because this situation best reflects actual policy instruments. Understanding whether abatement costs exceed the environmental damage is essential to determining the best policy. The major finding of the study is that deadweight loss is reduced if dominant firms adopt eco-friendly technology and the regulator increases the ratio of taxed dominant firms to all dominant firms. Additionally, mixed policy is efficient when market distortion as a result of pricing power decreases.

Appendix: Technical details

Derivation of the inverse residual demand function and MAC of each type of firm The marginal cost function of an individual fringe firm is given by $MC_i^F \left( q \right) \,=\,k+mq$. I obtain the aggregate marginal costs of fringe firms as $MC_{tot}^F \left( q \right) \,=\,k+\frac{mq}{\left( {N-n} \right) }$. The supply function for fringe firms is $q^{F}\left( p \right) \,=\,\frac{\left( {N-n} \right) \left( {p-k} \right) }{m}$. Suppose that individual dominant firms share the same cost functions, given by $C^{D}\left( {q_j } \right) \,=\,cq_j$. Then, the residual demand function that dominant firms face is $q^{R}\left( p \right) \,=\,q^{M}\left( p \right) -q^{F}\left( p \right)$ and I can obtain the inverse as $p^{R}\left( {q^{D},\theta } \right) \,=\,a-bq^{D}+\theta$ by replacing $a\,=\,\frac{mv+k\left( {N-n} \right) }{mx+\left( {N-n} \right) } \hbox {and}\,\, b\,=\,\frac{m}{mx+\left( {N-n} \right) }$.

The respective aggregate outputs of groups A and B are $q_A^D \,=\,r\sum _{j\,=\,1}^n q_j$ and $q_B^D \,=\,\left( {1-r} \right) \sum _{j\,=\,1}^n q_j$, and I define the aggregate output of the two groups together as $q_A^D +q_B^D \,=\,q^{D}$.

I now obtain the revenue function of dominant firms as follows:

$$\begin{aligned} R_j \left( {q_j ,\theta } \right)&\,=\,&p^{R}\left( {q^{D},\theta } \right) q_j\\&\,=\,&\left\{ {a-b\left( {q_1 +q_2 +\cdots +qn} \right) +\theta } \right\} q_j \end{aligned}$$

The marginal revenue function is expressed as $R_j \left( {q_j ,\theta } \right) \equiv \frac{dR_j \left( {q_j ,\theta } \right) }{dq_j }$; thus, the MAC is ${\textit{MAC}}_j^D \left( {q_j ,\theta } \right) \,=\,\beta -b\left( {n+1} \right) q_j +\theta$. The MAC of fringe firms is given by ${\textit{MAC}}^{F}\left( {q,\theta } \right) \,=\,p-MC_{tot}^F \left( q \right)$. Hence, the function is ${\textit{MAC}}^{F}\left( {q,\theta } \right) \,=\,\gamma -\frac{mq}{\left( {N-n} \right) }$.

Derivation of the respective deadweight loss under the regulations The expected efficiency losses of dominant firms under implementing a tax, a quota, and a mixed policy are given by the following equations.

$$\begin{aligned} E\left[ {{\textit{EL}}_\textit{tax}^D } \right]&\,=\,&nE\left[ {\mathop {\smallint }\limits _{q_\textit{tax}^D }^{q_j^*} \left( {{\textit{MAC}}^{D}\left( {q_j ,\theta } \right) -MD\left( {q,\varepsilon } \right) } \right) dq} \right] \nonumber \\&\,=\,&\frac{nf^{2}\sigma _\theta ^2 }{2b^{2}\left( {n+1} \right) ^{2}\left[ {b\left( {n+1} \right) +f} \right] }+\frac{n\sigma _\varepsilon ^2 }{2\left[ {b\left( {n+1} \right) +f} \right] }\end{aligned}$$

(A.1)

$$\begin{aligned} E\left[ {{\textit{EL}}_\textit{quota}^D } \right]&\,=\,&nE\left[ {\mathop {\smallint }\limits _{q_\textit{quota}^D }^{q_j^*} \left( {{\textit{MAC}}^{D}\left( {q_j ,\theta } \right) -MD\left( {q,\varepsilon } \right) } \right) dq} \right] \nonumber \\&\,=\,&\frac{n\sigma _\theta ^2 }{2\left[ {b\left( {n+1} \right) +f} \right] }+\frac{n\sigma _\varepsilon ^2 }{2\left[ {b\left( {n+1} \right) +f} \right] }\end{aligned}$$

(A.2)

$$\begin{aligned} E\left[ {{\textit{EL}}_{mix}^D } \right]&\,=\,&nE\left[ { \mathop {\smallint }\limits _{q_{mix}^D }^{q_j^*} \left( {{\textit{MAC}}^{D}\left( {q_j ,\theta } \right) -MD\left( {q,\varepsilon } \right) } \right) dq} \right] \nonumber \\&\,=\,&\frac{n\left[ {f^{2}r+b^{2}\left( {1-r} \right) \left( {n+1} \right) ^{2}} \right] \sigma _\theta ^2 }{2b^{2}\left( {n+1} \right) ^{2}\left[ {b\left( {n+1} \right) +f} \right] }\nonumber \\&+\frac{n\sigma _\varepsilon ^2 }{2\left[ {b\left( {n+1} \right) +f} \right] } \end{aligned}$$

(A.3)

From the above equation, we can see that the uncertainty due to parameter $\theta$, which is key in determining the better policy. Then, whether the marginal MAC is larger than the MD or not becomes particularly important for choosing the environmental policy, as shown in each slope level f and $b\left( {n+1} \right)$.

Similarly, the efficiency loss for fringe firms is given as follows:

$$\begin{aligned} \left[ {{\textit{EL}}_\textit{tax}^F } \right]&\,=\,&E\left[ {\mathop {\smallint }\limits _{q_\textit{tax}^F }^{q_\textit{tax}^{F*} } \left( {{\textit{MAC}}_\textit{tax}^F \left( {q,\theta } \right) -MD\left( {q,\varepsilon } \right) } \right) dq} \right] \nonumber \\&\,=\,&\frac{f^{2}\left( {N-n} \right) ^{3}\left[ {\left( {b\left( {n+1} \right) +f} \right) \left( {a-k} \right) -bn\beta } \right] ^{2}}{2m^{2}\left( {f\left( {N-n} \right) +m} \right) \left[ {b\left( {n+1} \right) +f} \right] ^{2}}\nonumber \\&+\frac{f^{2}\left( {N-n} \right) ^{3}\sigma _\theta ^2 }{2m^{2}\left( {n+1} \right) ^{2}\left( {f\left( {N-n} \right) +m} \right) }+\frac{\left( {N-n} \right) \sigma _\varepsilon ^2 }{2\left[ {f\left( {N-n} \right) +m} \right] }\end{aligned}$$

(A.4)

$$\begin{aligned} E\left[ {{\textit{EL}}_\textit{quota}^F } \right]&\,=\,&E\left[ {\mathop {\smallint } \limits _{q_\textit{quota}^F }^{q_\textit{quota}^{F*} } \left( {{\textit{MAC}}_\textit{quota}^F \left( {q,\theta } \right) -MD\left( {q,\varepsilon } \right) } \right) dq} \right] \nonumber \\&\,=\,&\frac{f^{2}\left( {N-n} \right) ^{3}\left[ {\left( {b\left( {n+1} \right) +f} \right) \left( {a-k} \right) -bn\beta } \right] ^{2}}{2m^{2}\left( {f\left( {N-n} \right) +m} \right) \left[ {b\left( {n+1} \right) +f} \right] ^{2}}\nonumber \\&+\frac{f^{2}\left( {N-n} \right) ^{3}\sigma _\theta ^2 }{2m^{2}\left( {f\left( {N-n} \right) +m} \right) }\nonumber \\&+\frac{\left( {N-n} \right) \sigma _\varepsilon ^2 }{2\left[ {f\left( {N-n} \right) +m} \right] }\end{aligned}$$

(A.5)

$$\begin{aligned} E\left[ {{\textit{EL}}_{mix}^F } \right]&\,=\,&E\left[ {\mathop {\smallint } \limits _{q_{mix}^F }^{q_{mix}^{F*} } \left( {{\textit{MAC}}_{mix}^F \left( {q,\theta } \right) -MD\left( {q,\varepsilon } \right) } \right) dq} \right] \nonumber \\&\,=\,&\frac{f^{2}\left( {N-n} \right) ^{3}\left[ {\left( {b\left( {n+1} \right) +f} \right) \left( {a-k} \right) -bn\beta } \right] ^{2}}{2m^{2}\left( {f\left( {N-n} \right) +m} \right) \left[ {b\left( {n+1} \right) +f} \right] ^{2}}\nonumber \\&+\frac{f^{2}\left( {N-n} \right) ^{3}\left[ {n\left( {1-r} \right) +1} \right] ^{2}\sigma _\theta ^2 }{2m^{2}\left( {n+1} \right) ^{2}\left( {f\left( {N-n} \right) +m} \right) }+\frac{\left( {N-n} \right) \sigma _\varepsilon ^2 }{2\left[ {f\left( {N-n} \right) +m} \right] } \end{aligned}$$

(A.6)

As is the case with the expected efficiency loss for dominant firms, the only uncertainty of the MAC influences the choice of policy regulation. I can obtain the expected deadweight loss under a tax policy by summing Eqs. (A.1) and (A.4). The deadweight loss under a quota policy is given by summing Eqs. (A.2) and (A.5). Similarly, the expected deadweight loss under a mixed policy is given by summing (A.3) and (A.6). Then, the uncertainty of MD (as shown in the second term in the above equations) does not have weight when the appropriate policy is determined.