Strategic environmental policy in a differentiated duopoly with overlapping ownership: a welfare analysis

Abstract

This paper analyzes strategic environmental policy when polluting firms engage in overlapping ownership arrangements (OOAs) under differentiated duopoly with quantity competition. Specifically, we focus on two pollution control instruments: emission taxes and standards. The key findings are as follows: (i) Compared to the case without ownership, the optimal environmental tax rate and absolute standard are higher (lower) when the polluting firms’ products are complements (substitutes). (ii) Both the tax and standard policies are equally efficient in their effects on the firms’ output and abatement decisions, consumer surplus, environmental quality, and social welfare, regardless of whether the differentiated products are complements or substitutes. (iii) If the government sets equity share and emission tax (or standard) simultaneously to maximize social welfare, the optimal equity share may exceed 50% for OOA firms producing two complements and is inversely related to the degree of product complementarity. Our results have welfare implications for the choice of environmental regulation between taxes and standards, when equity share is exogenous, or when the government determines an optimal mix of equity share and emission tax (or standard).

Appendix A

A-1. Comparison between the equilibrium outcomes under OOAs $(\alpha > 0)$ and those in the absence of ownership $(\alpha = 0).$ The case of tax emission policy

$$t^{T} \left( {\alpha > 0} \right) - t^{T} \left( {\alpha = 0} \right) = \frac{2A(1 - \alpha )\,( - 13\alpha + 6\gamma - 3\alpha \gamma + 13)}{{ - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170}} - \frac{2A(6\gamma + 13)}{{27\gamma^{2} + 138\gamma + 170}}$$

$$= - \frac{{18A\alpha \gamma (120\gamma - 134\alpha + 27\gamma^{2} - 81\alpha \gamma - 9\alpha \gamma^{2} + 134)}}{{(27\gamma^{2} + 138\gamma + 170)(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)}}.$$

It follows that $t^{T} \left( {\alpha > 0} \right) - t^{T} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$q^{T} \left( {\alpha > 0} \right) - q^{T} \left( {\alpha = 0} \right) = \frac{9(1 - \alpha )\,[8(1 - \alpha ) + 3\gamma ]A}{{ - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170}} - \frac{9A(3\gamma + 8)}{{(27\gamma^{2} + 138\gamma + 170)}}$$

$$= - \frac{{\gamma 27A\alpha (138\gamma - 182\alpha + 27\gamma^{2} - 66\alpha \gamma + 182)}}{{\,(27\gamma^{2} + 138\gamma + 170)(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)}}.$$

We thus have $(\alpha > 0) - q^{T} (\alpha = 0) > 0$ when $\gamma < 0.$

$$e^{T} \left( {\alpha > 0} \right) - e^{T} \left( {\alpha = 0} \right) = \frac{3A(1 - \alpha )( - 11\alpha + 3\gamma + 3\alpha \gamma + 11)\,}{{ - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170}} - \frac{3A(3\gamma + 11)}{{27\gamma^{2} + 138\gamma + 170}}$$

$$= - \frac{{\gamma 81A\alpha (6\gamma - 16\alpha + 5\alpha \gamma + 3\alpha \gamma^{2} + 16)}}{{(27\gamma^{2} + 138\gamma + 170)(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)}}.$$

It follows that $e^{T} \left( {\alpha > 0} \right) - e^{T} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$a_{1}^{T} \left( {\alpha > 0} \right) - a_{1}^{T} \left( {\alpha = 0} \right) = \frac{{3(26A - 52A\alpha + 12A\gamma + 26A\alpha^{2} - 18A\alpha \gamma + 6A\alpha^{2} \gamma )}}{{2( - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170)}} - \frac{78A + 36A\gamma }{{54\gamma^{2} - 276\gamma + 340}}$$

$$= - \frac{{27A\alpha \gamma (120\gamma - 134\alpha + 27\gamma^{2} - 81\alpha \gamma - 9\alpha \gamma^{2} + 134)}}{{\,(27\gamma^{2} + 138\gamma + 170)(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)}}.$$

This result implies that $a_{1}^{T} (\alpha > 0) - a_{1}^{T} (\alpha = 0) > 0$ when $\gamma < 0.$

$$ED^{T} \left( {\alpha > 0} \right) - ED^{T} \left( {\alpha = 0} \right) = \frac{{12A^{2} (1 - \alpha )^{2} ( - 11\alpha + 3\gamma + 3\alpha \gamma + 11)^{2} }}{{( - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170)^{2} }} - \frac{{12A^{2} (3\gamma + 11)^{2} }}{{(27\gamma^{2} + 138\gamma + 170)^{2} }}.$$

It follows that $ED^{T} \left( {\alpha > 0} \right) - ED^{T} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$CS^{T} \left( {\alpha > 0} \right) - CS^{T} \left( {\alpha = 0} \right) = \frac{{81A^{2} (1 - \alpha )^{2} (1 + \gamma )\,(8 - 8\alpha + 3\gamma )^{2} }}{{( - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170)^{2} }} - \frac{{81A^{2} (1 + \gamma )(3\gamma + 8)^{2} }}{{(138\gamma + 27\gamma^{2} + 170)^{2} }}$$

It follows that $CS^{T} \left( {\alpha > 0} \right) - CS^{T} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$\begin{gathered} \pi_{i}^{T} \left( {\alpha > 0} \right) - \pi_{i}^{T} \left( {\alpha = 0} \right) \hfill \\ = \left( {\frac{\begin{gathered} 3A^{2} (1 - \alpha )\,( - 9\alpha^{3} \gamma^{2} + 1650\alpha^{3} \gamma - 1897\alpha^{3} - 1251\alpha^{2} \gamma^{2} \hfill \\ - 1848\alpha^{2} \gamma + 5691\alpha^{2} + 243\alpha \gamma^{3} + 981\alpha \gamma^{2} - 1254\alpha \gamma \hfill \\ - 5691\alpha + 279\gamma^{2} + 1452\gamma + 1897) \hfill \\ \end{gathered} }{{(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)^{2} }}} \right) - \frac{{3A^{2} (1452\gamma + 279\gamma^{2} + 1897)}}{{(138\gamma + 27\gamma^{2} + 170)^{2} }} \hfill \\ \end{gathered}$$

$$= \frac{{27A^{2} \alpha \gamma \left( \begin{gathered} 729\alpha^{3} \gamma^{5} - 126\,198\alpha^{3} \gamma^{4} - 1185\,435\alpha^{3} \gamma^{3} - 3625\,812\alpha^{3} \gamma^{2} - 3856\,716\alpha^{3} \gamma \hfill \\ - 501\,160\alpha^{3} + 100\,602\alpha^{2} \gamma^{5} + 1311\,714\alpha^{2} \gamma^{4} + 6230\,196\alpha^{2} \gamma^{3} + 13\,084\,848\alpha^{2} \gamma^{2} \hfill \\ + 10\,933\,416\alpha^{2} \gamma + 1503\,480\alpha^{2} - 19\,683\alpha \gamma^{6} - 381\,996\alpha \gamma^{5} - 2668\,302\alpha \gamma^{4} \hfill \\ - 8867\,286\alpha \gamma^{3} - 14\,570\,136\alpha \gamma^{2} - 10\,296\,684\alpha \gamma - 1503\,480\alpha + 19\,683\gamma^{6} \hfill \\ + 258\,066\gamma^{5} + 1365\,174\gamma^{4} + 3668\,868\gamma^{3} + 5111\,100\gamma^{2} + 3219\,984\gamma + 501\,160 \hfill \\ \end{gathered} \right)}}{{(27\gamma^{2} + 138\gamma + 170)^{2} (6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)^{2} }}$$

We thus have $\pi^{T} \left( {\alpha > 0} \right) - \pi^{T} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$\begin{gathered} SW^{T} \left( {\alpha > 0} \right) - SW^{T} \left( {\alpha = 0} \right) \hfill \\ = \frac{{9A^{2} \left( {1 - \alpha } \right)\,\left( { - 11\alpha + 3\gamma + 3\alpha \gamma + 11} \right)}}{{( - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170)}} - \frac{{9A^{2} \left( {3\gamma + 11} \right)}}{{27\gamma^{2} + 138\gamma + 170}} \hfill \\ \end{gathered}$$

$$= - \frac{{\gamma 243A^{2} \alpha \left( { - 16\alpha + 6\gamma + 5\alpha \gamma + 3\alpha \gamma^{2} + 16} \right)}}{{\left( {138\gamma + 27\gamma^{2} + 170} \right)\,\left( { - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170} \right)}}$$

It follows that $SW^{T} \left( {\alpha > 0} \right) - SW^{T} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

A-2. Comparison between the equilibrium outcomes under OOAs $(\alpha > 0)$ and those in the absence of ownership $(\alpha = 0).$ The case of emission standard policy

$$\begin{gathered} s^{s} \left( {\alpha > 0} \right) - s^{s} \left( {\alpha = 0} \right) \hfill \\ = \frac{{33A - 66A\alpha + 9A\gamma + 33A\alpha^{2} - 9A\alpha^{2} \gamma }}{{ - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170}} - \frac{33A + 9A\gamma }{{27\gamma^{2} + 138\gamma + 170}} \hfill \\ \end{gathered}$$

$$= - \frac{{81A\alpha \gamma (6\gamma - 16\alpha + 5\alpha \gamma + 3\alpha \gamma^{2} + 16)}}{{(27\gamma^{2} + 138\gamma + 170)(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)\,}}$$

This result implies that $s^{s} \left( {\alpha > 0.5} \right) - s^{s} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$\begin{gathered} q^{s} \left( {\alpha > 0} \right) - q^{s} \left( {\alpha = 0} \right) \hfill \\ = \frac{{ - 9A\left( {\alpha - 1} \right)\,\left( {3\gamma - 8\alpha + 8} \right)}}{{6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170}} - \frac{72A + 27A\gamma }{{27\gamma^{2} + 138\gamma + 170}} \hfill \\ \end{gathered}$$

$$= - \frac{{27A\alpha \gamma (138\gamma - 182\alpha + 27\gamma^{2} - 66\alpha \gamma + 182)}}{{(27\gamma^{2} + 138\gamma + 170)(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)\,}}$$

It follows that $q^{s} \left( {\alpha > 0} \right) - q^{s} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$\begin{gathered} \pi_{1}^{s} \left( {\alpha > 0} \right) - \pi_{1}^{s} \left( {\alpha = 0} \right) \hfill \\ = 3A^{2} \frac{\begin{gathered} (1 - \alpha )(9\alpha^{3} \gamma^{2} + 1662\alpha^{3} \gamma - 2183\alpha^{3} - 1287\alpha^{2} \gamma^{2} \hfill \\ - 1662\alpha^{2} \gamma + 6549\alpha^{2} + 243\alpha \gamma^{3} + 963\alpha \gamma^{2} \hfill \\ - 1662\alpha \gamma - 6549\alpha + 315\gamma^{2} + 1662\gamma + 2183) \hfill \\ \end{gathered} }{{(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)^{2} }} - \frac{{3A^{2} (315\gamma^{2} + 1662\gamma + 2183)}}{{(27\gamma^{2} + 138\gamma + 170)^{2} }}. \hfill \\ \end{gathered}$$

$$\begin{gathered} \pi_{i}^{s} (\alpha > 0) - \pi_{i}^{s} (\alpha = 0) \hfill \\ = \frac{\begin{gathered} 81A^{2} \alpha \gamma ( - 243\alpha^{3} \gamma^{5} - 47\,358\alpha^{3} \gamma^{4} - 409\,599\alpha^{3} \gamma^{3} - 1176\,492\alpha^{3} \gamma^{2} - 1081\,508\alpha^{3} \gamma \hfill \\ + 70\,720\alpha^{3} + 34\,992\alpha^{2} \gamma^{5} + 447\,444\alpha^{2} \gamma^{4} + 2056\,572\alpha^{2} \gamma^{3} + 4041\,984\alpha^{2} \gamma^{2} + 2789\,008\alpha^{2} \gamma \hfill \\ - 212\,160\alpha^{2} - 6561\alpha \gamma^{6} - 127\,818\alpha \gamma^{5} - 878\,796\alpha \gamma^{4} - 2808\,918\alpha \gamma^{3} - 4238\,280\alpha \gamma^{2} - 2333\,492\alpha \gamma \hfill \\ + 212\,160\alpha + 6561\gamma^{6} + 84\,564\gamma^{5} + 433\,836\gamma^{4} + 1103\,004\gamma^{3} + 1372\,788\gamma^{2} + 625\,992\gamma - 70\,720 \hfill \\ \end{gathered} }{{(27\gamma^{2} + 138\gamma + 170)^{2} (6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)^{2} }} \hfill \\ \end{gathered}$$

A numerical simulation shows that $\pi_{i}^{s} \left( {\alpha > 0} \right) - \pi_{i}^{s} \left( {\alpha = 0} \right) > 0$ for any $- 1 < \gamma < 1.$

$$\begin{gathered} a_{1}^{s} \left( {\alpha > 0} \right) - a_{1}^{s} \left( {\alpha = 0} \right) \hfill \\ = \frac{{3A\left( {1 - \alpha } \right)\,\left( {6\gamma - 13\alpha - 3\alpha \gamma + 13} \right)}}{{6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170}} - \frac{{3A\left( {6\gamma + 13} \right)}}{{27\gamma^{2} + 138\gamma + 170}} \hfill \\ \end{gathered}$$

$$= - \frac{{\gamma 27A\alpha (120\gamma - 134\alpha + 27\gamma^{2} - 81\alpha \gamma - 9\alpha \gamma^{2} + 134)}}{{27\gamma^{2} + 138\gamma + 170(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)}}.$$

This result implies that $a_{1}^{s} \left( {\alpha > 0} \right) - a_{1}^{s} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$\begin{gathered} ED^{s} \left( {\alpha > 0} \right) - ED^{s} \left( {\alpha = 0} \right) \hfill \\ = \frac{4}{3}\left( {\frac{{33A - 66A\alpha + 9A\gamma + 33A\alpha^{2} - 9A\alpha^{2} \gamma }}{{ - 340\alpha + 138\gamma + 170\alpha^{2} + 27\gamma^{2} - 144\alpha \gamma + 6\alpha^{2} \gamma + 170}}} \right)^{2} - \frac{{12\left( {11A + 3A\gamma } \right)^{2} }}{{\left( {27\gamma^{2} + 138\gamma + 170} \right)^{2} }} \hfill \\ \end{gathered}$$

It follows that $ED^{s} \left( {\alpha > 0} \right) - ED^{s} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$\begin{gathered} CS^{s} (\alpha > 0) - CS^{s} (\alpha = 0) \hfill \\ = (\gamma + 1)\left( {\frac{ - 9A(\alpha - 1)\,(3\gamma - 8\alpha + 8)}{{6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170}}} \right)^{2} - \frac{{81A^{2} (3\gamma + 8)^{2} (\gamma + 1)}}{{(138\gamma + 27\gamma^{2} + 170)^{2} }} \hfill \\ \end{gathered}$$

$$= - \frac{{243A^{2} \alpha \gamma (\gamma + 1)\left( \begin{gathered} - 15\,444\alpha^{3} \gamma^{3} - 152\,280\alpha^{3} \gamma^{2} - 482\,004\alpha^{3} \gamma - 495\,040\alpha^{3} + 11\,664\alpha^{2} \gamma^{4} \hfill \\ + 176\,256\alpha^{2} \gamma^{3} + 900\,720\alpha^{2} \gamma^{2} + 1926\,864\alpha^{2} \gamma + 1485\,120\alpha^{2} - 2187\alpha \gamma^{5} \hfill \\ - 56\,376\alpha \gamma^{4} - 435\,780\alpha \gamma^{3} - 1496\,664\alpha \gamma^{2} - 2407\,716\alpha \gamma - 1485\,120\alpha \hfill \\ + 4374\gamma^{5} + 56\,376\gamma^{4} + 290\,520\gamma^{3} + 748\,224\gamma^{2} + 962\,856\gamma + 495\,040 \hfill \\ \end{gathered} \right)}}{{(27\gamma^{2} + 138\gamma + 170)^{2} (6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)^{2} }}.$$

It follows that $CS^{s} \left( {\alpha > 0} \right) - CS^{s} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$

$$\begin{gathered} SW^{s} \left( {\alpha > 0} \right) - SW^{s} \left( {\alpha = 0} \right) \hfill \\ = \frac{{9A^{2} \left( {1 - \alpha } \right)\,\left( {3\gamma - 11\alpha + 3\alpha \gamma + 11} \right)}}{{6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170}} - \frac{{9A^{2} \left( {3\gamma + 11} \right)}}{{27\gamma^{2} + 138\gamma + 170}} \hfill \\ \end{gathered}$$

$$= - \frac{{\gamma 243A^{2} \alpha (6\gamma - 16\alpha + 5\alpha \gamma + 3\alpha \gamma^{2} + 16)}}{{(27\gamma^{2} + 138\gamma + 170)(6\alpha^{2} \gamma + 170\alpha^{2} - 144\alpha \gamma - 340\alpha + 27\gamma^{2} + 138\gamma + 170)}}$$

We thus have the result that $SW^{s} \left( {\alpha > 0} \right) - SW^{s} \left( {\alpha = 0} \right) > 0$ when $\gamma < 0.$