Environmental policy in a mixed market: abatement subsidies and emission taxes

Abstract

This paper investigates the optimal subsidy for activities abating emission in a mixed market, where the state-owned public firm competes against private firms. First, we consider the case, where the government uses only the abatement subsidy as a policy instrument. We derive a formula of the optimal abatement subsidy under general assumptions. We then examine the privatization policy and compare the abatement subsidy policy to the emission tax policy under a special case, which is analytically tractable. Next, we consider several combinations of tax and subsidy. It is shown that the first-best outcome is achieved under the combined policies.

Appendix

In this section, we consider the case where the government employs only an abatement subsidy. We show that in both regimes, (1) the output of each private firm is non-decreasing in s, and (2) the abatement of each private firm is increasing in s.

We assume that $P^{\prime}(Q) +qP^{\prime\prime}(Q)<0.$ This guarantees the stability of equilibrium: under the assumption, each firm’s best reply is downward sloping. We also assume that $D^{\prime\prime} \ge 0.$

First, we consider the case of the mixed oligopoly. Define

$$ \Uppi_{qq}^{0}:=P^{\prime} -C_{qq}^{0}- D^{\prime}e_{qq}^{0}-D^{\prime\prime}(e_{q}^{0})^{2} $$

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$$ \Uppi_{aq}^{0}:=- C_{aq}^{0}- D^{\prime}e_{aq}^{0}-D^{\prime\prime}e_{q}^{0} e_{a}^{0} $$

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$$ \Uppi_{aa}^{0}:=- C_{aa}^{0} - D^{\prime}e_{a}^{0}-D^{\prime\prime}(e_{a}^{0})^{2} $$

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$$ J:=nP^{\prime\prime}q_{1} +(n+1)P^{\prime}-C_{qq}^{1} $$

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where

$$ \begin{aligned} C_{qq}^{0}&=C_{qq}(q_0^{*},a_{0}^{*}),\quad C_{aq}^{0}=C_{aq}(q_0^{*},a_0^{*}), \quad C^0_{aa}=C_{aa}(q^*_0,a^*_0),\quad C_{qq}^{1}=C_{qq}(q_1^{*},a_1^{*}),\\ e_{q}^{0} &= {\frac{\partial e(q_{0}^{*},a_{0}^{*})}{\partial q}},\quad e_{q}^{0} = {\frac{\partial e(q_{0}^{*},a_{0}^{*})}{\partial a}},\quad e_{qq}^{0} = {\frac{\partial^{2} e(q_{0}^{*},a_{0}^{*})}{\partial q^{2}}},\quad e_{aq}^{0} = {\frac{\partial^{2} e(q_{0}^{*},a_{0}^{*})}{\partial a \partial q}}.\end{aligned} $$

We also define the following:

$$ \begin{aligned} C_{aq}^{1}&=C_{aq}(q_{1}^{*},a_{1}^{*}), \quad C_{aq}^{1}=C_{aq}(q_{1}^{*},a_{1}^{*}),\\ e_{q}^{1} &= {\frac{\partial e(q_{1}^{*},a_{1}^{*})}{\partial q}}, \quad e_{q}^{1} = {\frac{\partial e(q_{1}^{*},a_{1}^{*})}{\partial a}}.\end{aligned} $$

The second-order condition of the public firm implies

$$ \Uppi_{qq}^{0}<0, \Uppi_{aa}^{0}<0, \Uppi_{qq}^{0}\Uppi_{aa}^{0}-(\Uppi_{aq}^{0})^{2}>0. $$

We assume that $\Uppi_{aq}^{0}<0.$

From Eqs. 1–4, by using the implicit function theorem, we can obtain the following:

$$ K \left( \begin{array}{l} {\hbox{d}} q_{1}^{*}/{\hbox{d}} s \\ {\hbox{d}} a_{1}^{*}/{\hbox{d}} s \\ {\hbox{d}} q_{0}^{*}/{\hbox{d}} s \\ {\hbox{d}} a_{0}^{*}/ {\hbox{d}} s \\ \end{array}\right) = - \left( \begin{array}{l} 0\\ 1\\ 0\\ 0\end{array}\right), $$

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where

$$ K := \left( \begin{array}{llll} J &\quad -C_{aq}^{1} &\quad P^{\prime}+ q_{1} P^{\prime\prime} &\quad 0\\ -C_{aq}^{1}&\quad -C_{aa}^{1}&\quad 0 &\quad 0\\ nP^{\prime} - D^{\prime\prime} e_{q}^{0} e_{q}^{1}&\quad -D^{\prime\prime}e_{q}^{0}e_{a}^{1}&\quad \Uppi_{qq}^{0} &\quad\Uppi_{aq}^{0}\\ D^{\prime\prime}e_{a}^{0} e_{q}^{1}&\quad -D^{\prime\prime}e_{a}^{0}e_{a}^{1}&\quad \Uppi_{aq}^{0} & \quad\Uppi_{aa}^{0}\end{array}\right). $$

We can check that |K| < 0.

• (1)
  $$ {\frac{{\hbox{d}} q_{1}^{*}}{{\hbox{d}} s}}= {\frac{ -C_{aq}^{1}\{ \Uppi_{qq}^{0}\Uppi_{aa}^{0} -(\Uppi_{aq}^{0})^{2}\} + D^{\prime\prime} e_{q}^{0} e_{a}^{1}(P^{\prime}+q_{1}P^{\prime\prime})\Uppi_{aa}^{0} - D^{\prime\prime}e_{a}^{0} e_{a}^{1}(P^{\prime}+q_{1}P^{\prime\prime})\Uppi_{aq}^{0}}{|K|}} \le 0. $$

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  The equality holds if and only if C _aq  = 0 and $D^{\prime\prime}=0.$

• (2)
  $$ {\frac{{\hbox{d}} a_{1}^{*}}{{\hbox{d}} s}}={\frac{J \{\Uppi_{qq}^{0}\Uppi_{aa}^{0}-(\Uppi_{aq}^{0})^{2}\} + (P^{\prime}+q_{1}P^{\prime\prime})\{(nP^{\prime} - D^{\prime\prime} e_{q}^{0} e_{q}^{1})\Uppi_{aa}^{0} - (D^{\prime}e_{a}^{0} e_{q}^{1})\Uppi_{aq}^{0}\}}{|K|}}>0. $$

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Next, we consider the case of the private oligopoly. The equilibrium conditions are as follows:

$$ P(Q^{**}) + q_{1}^{**} P^{\prime}(Q^{**}) - {\frac{\partial C(q_{1}^{**},a_{1}^{**})}{\partial q_{1}}} =0, $$

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$$ - {\frac{\partial C(q_{1}^{**},a_{1}^{**})}{\partial a_{1}}}+s=0. $$

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By using the implicit function theorem, we obtain the following:

$$ L \left( \begin{array}{l} {\hbox{d}} q_{1}^{**}/{\hbox{d}} s\\ {\hbox{d}} a_{1}^{**}/{\hbox{d}} s\end{array}\right) = - \left( \begin{array}{l} 0\\ 1\end{array}\right), $$

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where

$$ L := \left( \begin{array}{ll} (n+1)P^{\prime\prime}q +(n+2)P^{\prime} -C_{qq} &\quad -C_{aq}\\ -C_{aq} &\quad -C_{aa}\end{array}\right). $$

We can check that |L| > 0.

• (1)
  $$ {\frac{{\hbox{d}} q_{1}^{*}}{{\hbox{d}} s}}=- {\frac{ C_{aq}}{|L|}} \le 0. $$

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  The equality holds if and only if C _aq  = 0.

• (2)
  $$ {\frac{{\hbox{d}} a_{1}^{*}}{{\hbox{d}} s}}=- {\frac{s \{(n+1)P^{\prime\prime}q +(n+2)P^{\prime} -C_{qq}\}}{|L|}}>0. $$

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