Emission regulation of conventional energy-intensive industries

Abstract

Global climate change is closely related to conventional energy consumption. Taking the regulations for emissions into account, this article uses a game theory approach to identify industries depending on conventional energies to reduce emissions. This paper proposes to design a suitable supervisory system for emission regulation based on limited supervisor and asymmetric production efficiency. Two different supervision mechanism, random and selected supervisions, are employed. Some interesting conclusions are achieved. Firstly, the greater the level of competition, the smaller the number of firms with emission-reduction technology (ERT) are. Interestingly, the number of firms without ERT increases faster than does the number of firms with ERT. Secondly, under the asymmetric case, the threshold value for firms with low production costs that always employ emission-reduction technology is presented. Finally, this paper proves that firms with higher production costs have greater incentives to avoid emission restriction. Based on the above conclusions, the corresponding policy implications or regulation institutions to reduce climate changes are outlined. Random inspect is optimal if firms’ efficiency information, measured by production cost, is incomplete, while selected supervise is better if efficiency information is complete.

Appendix: Threshold values

$$\hat{k} = \frac{{(A - \bar{c})\sqrt {\frac{N - m}{N}} - [A - (N + 1)c + N\bar{c}]}}{{(c - \bar{c})(1 - \sqrt {\frac{N - m}{N}} )}},$$

(34)

$$\hat{h} = \frac{{(N + 1)c - (L + 1)\bar{c} - (N - L)\bar{c}}}{{(c - \bar{c})(1 - \sqrt {\frac{N - m}{N}} )}} - \frac{{[A + Lc_{L} - (L + 1)\bar{c}]}}{{(c - \bar{c})}},$$

(35)

$$\hat{J} = \frac{{\sqrt {\frac{N - m}{N}} (A - \bar{c}) - [A - (N + 1)c_{L} + N\bar{c}]}}{{(c_{L} - c)(1 - \sqrt {\frac{N - m}{N}} )}},$$

(36)

$$\hat{h}^{0} = \frac{{(N + 1)c - (L + 1)\bar{c} - (N - L)\bar{c}}}{{(c - \bar{c})(1 - \sqrt {1 - \frac{m}{N - L}} )}} - \frac{{[A + Lc_{L} - (L + 1)\bar{c}]}}{{(c - \bar{c})}}.$$

(37)

Proof of Proposition 1

By the above analysis, there exists $\hat{k} = \frac{{(A - \bar{c})\sqrt {\frac{N - m}{N}} - [A - (N + 1)c + N\bar{c}]}}{{(c - \bar{c})(1 - \sqrt {\frac{N - m}{N}} )}} = \frac{N + 1}{{1 - \sqrt {\frac{N - m}{N}} }} - \frac{{(A - \bar{c})}}{{(c - \bar{c})}}$, such that it has $\frac{{[A - (N - \hat{k} + 1)c + (N - \hat{k})\bar{c}]^{2} }}{{(N + 1)^{2} }} = \frac{N - m}{N}\frac{{[A + \hat{k}c - (\hat{k} + 1)\bar{c}]^{2} }}{{(N + 1)^{2} }}$ or $\frac{{A - (N - \hat{k} + 1)c + (N - \hat{k})\bar{c}}}{N + 1} = \sqrt {\frac{N - m}{N}} \frac{{A + \hat{k}c - (\hat{k} + 1)\bar{c}}}{N + 1}$. For $k < \hat{k}$, it has the following relationship:

$$\begin{aligned} \frac{{A - (N - k + 1)c + (N - k)\bar{c}}}{N + 1} = \frac{{A - (N - \hat{k} + 1)c + (N - \hat{k})\bar{c}}}{N + 1} - \frac{{(\hat{k} - k)(c - \bar{c})}}{N + 1} \hfill \\ \quad < \sqrt {\frac{N - m}{N}} [\frac{{A + \hat{k}c - (\hat{k} + 1)\bar{c}}}{N + 1} - \frac{{(\hat{k} - k)(c - \bar{c})}}{N + 1}] = \sqrt {\frac{N - m}{N}} \frac{{A + kc - (k + 1)\bar{c}}}{N + 1}. \hfill \\ \end{aligned}$$

Thus, the expected profits of firms without emission-reduction technology are higher than are the expected profits for firms with emission-reduction technology for $k < \hat{k}$. By the similar way, the opposite conclusions hold under $k > \hat{k}$.Conclusions are achieved, and the proof is complete. □

Proof of Proposition 2

Here, it addresses the effects of firm number in this industry on the equilibrium. For convenience, $\hat{k} = \frac{N + 1}{{1 - \sqrt {\frac{N - m}{N}} }} - \frac{{(A - \bar{c})}}{{(c - \bar{c})}}$ manifests the following relationship: $\frac{{\hat{k}}}{N} = \frac{{1 + \frac{1}{N}}}{{1 - \sqrt {\frac{N - m}{N}} }} - \frac{{(A - \bar{c})}}{{N(c - \bar{c})}} = \frac{{(N + 1)(1 + \sqrt {1 - \frac{m}{N}} )}}{m} - \frac{{(A - \bar{c})}}{{N(c - \bar{c})}}$. Apparently, $\frac{{\hat{k}}}{N}$ increases with the number of firms in this industry. That is, $\frac{{N - \hat{k}}}{{\hat{k}}}$ decreases with the number of firms in this industry. The ratios between firms with emission-reduction technology and firms without emission-reduction technology become smaller as the increase in total firms in this industry. On the other hand, $N - \hat{k} = \frac{{Nm - N(N + 1)(1 + \sqrt {1 - \frac{m}{N}} )}}{m} + \frac{{(A - \bar{c})}}{{(c - \bar{c})}}$. Obviously, $N - \hat{k}$ (or $k^{*,1}$) decreases with the number of firms in this industry.Conclusions are achieved, and the proof is complete. □

Proof of Proposition 5

The conclusions in proposition 5 can be obtained based on Eqs. (19) and (20). On the one hand, $c_{L} < c$ implies $\frac{{A - (N - L + 1)c_{L} + (N - L - h)\bar{c} + hc}}{N + 1} > \frac{{A + Lc_{L} + (N - L - h)\bar{c} - (N + 1 - h)c}}{N + 1}$. On the other hand, for all h, it has $A - (N + 1)c_{L} > \sqrt {\frac{N - m}{N}} [A - (N + 1)\bar{c}]$. And then the inequality $\frac{{A - (N - L + 1)c_{L} + (N - L - h)\bar{c} + hc}}{N + 1} > \sqrt {\frac{N - m}{N}} \frac{{A + Lc_{L} - (L + h + 1)\bar{c} + hc}}{N + 1}$ is always hold.

Conclusions are achieved, and the proof is complete. □