Environmental taxation and mergers in oligopoly markets with product differentiation

Abstract

We analyze the effect of mergers on optimal environmental taxation in a Cournot oligopoly market with product differentiation. Our result indicates that the adjustment in emission tax crucially depends on the post-merger output distortion and pollution intensities. Specifically, we find that the optimal emission tax increases post-merger as long as pollution intensity of firms is higher and output distortion smaller post-merger than pre-merger. Furthermore, our result suggests that there is no need to revise environmental policy in markets where pollution intensity of firms does not change post-merger and (i) products are completely differentiated, or (ii) there are many firms for any degree of product differentiation.

Appendix

We examine the comparative static effects of the tax, $t_{m}$, in the post-merger equilibrium. Differentiation of (13)–(16) yields the following system

$$\begin{aligned}&\left[ { \begin{array} [c]{cccc} -2(\beta +\gamma )-C_{q_{1}q_{1}} &{}\quad -\gamma (n-2) &{}\quad -C_{q_{1}e_{1}} &{}\quad 0\\ -C_{e_{1}q_{1}} &{}\quad 0 &{} \quad -C_{e_{1}e_{1}} &{}\quad 0\\ -2\gamma &{}\quad -2\beta -\gamma (n-3)-C_{\tilde{q}\tilde{q}} &{} \quad 0 &{}\quad -C_{\tilde{q}\tilde{e}}\\ 0 &{} \quad -C_{\tilde{e}\tilde{q}} &{} \quad 0 &{}\quad -C_{\tilde{e}\tilde{e}}\\ &{} &{} &{} \end{array} }\right] \left[ { \begin{array} [c]{c} dq_{1}\\ d\tilde{q}\\ de_{1}\\ d\tilde{e}\end{array} }\right] \\&\quad =\left[ { \begin{array} [c]{c} 0\\ dt_{m}\\ 0\\ dt_{m}\\ \end{array} }\right] \end{aligned}$$

where the determinant of the coefficient matrix $\eta <0$:

$$\begin{aligned} \eta= & {} -C_{\tilde{e}\tilde{e}}C_{e_{1}e_{1}}\left[ \left( 2\beta +\gamma (n-3) +\frac{C_{\tilde{q}\tilde{q}} C_{\tilde{e}\tilde{e}}-C^{2}_{\tilde{e}\tilde{q}}}{C_{\tilde{e}\tilde{e}}}\right) \right. \\&\left. \times \left( 2(\beta +\gamma ) +\frac{C_{q_{1}q_{1}}C_{e_{1}e_{1}} - C^{2}_{e_{1}q_{1}}}{C_{e_{1}e_{1}}} \right) - 2\gamma ^{2} (n-2)\right] \end{aligned}$$

We follow Lahiri and Symeonidis (2007) in the definition of pollution intensity, $-C_{qe}/C_{ee}>0$, where subscripts denote partial derivatives. When deriving the comparative statics analysis we assume identical pollution intensity within firms in the post-merger market; this is to facilitate the comparison with the pre-merger market. That is, $-C_{q_{1}e_{1}} /C_{e_{1}e_{1}}=-C_{\tilde{q}\tilde{e}}/C_{\tilde{e}\tilde{e}}>0$.

Using the above system the effect of the tax post-merger on output, $q_{1}$ and $\tilde{q}$ is given by

$$\begin{aligned} \frac{\eta }{C_{e_{1}e_{1}}C_{\tilde{e}\tilde{e}}} dq_{1}&= \frac{-C_{e_{1}q_{1}}}{C_{e_{1}e_{1}} }\left[ 2\beta -\gamma +\frac{\left( C_{\tilde{e}\tilde{e}}C_{\tilde{q}\tilde{q}}-C_{\tilde{e}\tilde{q}}^{2}\right) }{C_{\tilde{e}\tilde{e}} }\right] dt_{m}\end{aligned}$$

(32)

$$\begin{aligned} \frac{\eta }{C_{e_{1}e_{1}}C_{\tilde{e}\tilde{e}}} d\tilde{q}&= \frac{-C_{\tilde{e}\tilde{q}}}{C_{\tilde{e}\tilde{e}} }\left[ 2\beta +\frac{\left( C_{e_{1}e_{1}}C_{q_{1}q_{1}}-C_{e_{1}q_{1}}^{2}\right) }{C_{e_{1}e_{1}}}\right] dt_{m}\end{aligned}$$

(33)

where $2\beta -\gamma >0$, and $C_{qq}C_{ee}-C_{eq}^{2}>0$, $C_{ee}>0$ by the properties of the cost function. Hence, output falls with the tax, i.e., $\eta <0$ and $dq_{1}>0$, $d\tilde{q}>0$.

The effects of the tax post-merger on emissions are given by:

$$\begin{aligned} \eta de_{1}&=-\frac{C_{e_{1}q_{1}}}{C_{e_{1}e_{1}}} \left[ -C_{e_{1}e_{1}} C_{\tilde{e}\tilde{e}}\frac{C_{e_{1}q_{1}}}{C_{e_{1}e_{1}}}\left( 2\beta -\gamma +\frac{\left( C_{\tilde{e}\tilde{e}} C_{\tilde{q}\tilde{q}}-C_{\tilde{e}\tilde{q}}^{2}\right) }{C_{\tilde{e}\tilde{e}}}\right) + \eta /C_{e_{1}q_{1}} \right] dt_{m}\end{aligned}$$

(34)

$$\begin{aligned} \eta d\tilde{e}&=-\frac{C_{\tilde{e}\tilde{q}}}{C_{\tilde{e}\tilde{e}}} \left[ -C_{e_{1}e_{1}} C_{\tilde{e}\tilde{e}}\frac{C_{\tilde{e}\tilde{q}}}{C_{\tilde{e}\tilde{e}}}\left( 2\beta +\frac{\left( C_{e_{1}e_{1}} C_{q_{1}q_{1}}-C_{e_{1}q_{1}}^{2}\right) }{C_{e_{1}e_{1}}}\right) + \eta /C_{\tilde{e}\tilde{q}} \right] dt_{m}\end{aligned}$$

(35)

The expressions in (32)–(35) reduce to the following in the special case of the end-of-pipe-type cost function $C(q,e)=cq+(\delta q-e)^{2}/2$, where $C_{qq}C_{ee}-C_{eq}^{2}=0$, $C_{ee}=1$, $-C_{q_{1}e_{1}} /C_{e_{1}e_{1}}=-C_{\tilde{q}\tilde{e}}/C_{\tilde{e}\tilde{e}}=\delta$, where $\delta$ is a constant pollution intensity of firms:

See Table 1.