Effective ambient charges on non-point source pollution in a two-stage Bertrand duopoly

ABSTRACT

This paper analyses the effectiveness of the ambient charges for controlling emissions of non-point source pollutions. To this end, we construct a two-stage Bertrand duopoly game, in which optimal abatement technologies are chosen first and then the optimal prices as well as the optimal productions are determined. It is shown that the ambient charge is always effective at the second stage. Since the effect could be ambiguous at the first stage, this paper sheds light on the conditions under which the ambient charge becomes effective.

KEYWORDS:

• Ambient charge
• Bertrand duopoly competition
• non-point source pollution
• two-stage game

Acknowledgments

The authors greatly appreciate a referee for his/her constructive comments that surely improve the paper. They would like to thank participants at a seminar organized by the Institute of Economic Research of Chuo University and also thank Takaaki Ishida, Yusuke Kumahara and Hinata Shiratori for helpful discussions. The first author highly acknowledges the financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 20K01566) and Chuo University (Grant for Special Research). The usual disclaimers apply.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The referee casts doubt about the realistic validity for NPS pollution in the duopoly framework in which the government may calculate the value of individual pollutions. However, we believe with the following two reasons that the government might be left in the dark. The first reason concerns the stochastic diffusion process of emission that will change the size and properties of pollution emitted by a firm. The second relates to the definition of a duopoly firm. In our model, the duopoly firm is not a mom-and-pop store but a conglomerate, large cooperation made up of many smaller companies involved in a variety of different activities. It might cost a lot to identify individual responsibility with sufficient accuracy. As examples of a duopoly industry, we have in our mind, Boeing and Airbus, Pepsi and Coca-Cola, Canon and Nikon, etc.

2 The common parameters a and b are only for simplicity. It is possible to take $a_i\ne a_j$ and $b_i\ne b_j$ at the expense of tedious calculations.

3 There is another way to show this condition. Following (Singh and Vives 1984), we can derive the exact forms of the linear functions given in (2(2) $\begin{array}{cc}{p}_i& =\alpha -\text{β}{q}_i-\gamma q_j,\\ p_j& =\alpha -\gamma q_i-\text{β}{q}_j,\end{array}$(2) ) as the optimal solutions that solve a net utility-maximizing problem of the representative consumer,$U(q_i,q_j)-\left(p_i{q}_i+p_j{q}_j\right),$where U is the utility function,$U(q_i,q_j)=\alpha \left(q_i+q_j\right)-\frac{1}{2}\left(\text{β}{q}_i^2+2\gamma q_i{q}_j+\text{β}{q}_j^2\right),$with the parameter conditions, ${\text{β}}^2-{\gamma }^2>0$ and $\text{β}-\gamma >0$, both of which correspond to $1-b^2>0$ and 1−b > 0 in our framework.

4 In particular, under Assumption 2.2,$\begin{array}{rl}ab+b{\eta }_j-\theta b{\phi }_j& =b\left(a+{\eta }_j-\theta {\phi }_j\right)\\ & >b({\eta }_j+\theta {\phi }_j+{\eta }_j-\theta {\phi }_j)\\ & =2b{\eta }_j>0.\end{array}$With this inequality, we have $p_i^{\ast }>0$. In the same way, $p_j^{\ast }>0$ can be shown.

5 The results to be obtained below could hold for other specifications of the parameters.

6 This result depends on the parameter specificaion. If the values of ${\eta }_i$ and ${\eta }_i$ are interchanged, then the yellow curve can be located above the meshed curve.

Additional information

Funding

The first author highly acknowledges the financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 20K01566) and Chuo University (Grant for Special Research).