Emission quota versus emission tax in a mixed duopoly

Abstract

Emission taxes are compared with emission quotas in a mixed and, after privatization, a pure duopoly. Previous studies have shown that in a mixed duopoly, direct regulation is superior to indirect regulation, regardless of the degree of partial privatization. These findings were reached on the basis that direct regulation was regarded as an emission standard in which the government sets the uniform abatement effort of each firm, and indirect regulation was regarded as an emission tax. This study considers another indirect regulation: an emission quota that the regulator sets for each firm uniformly or differentially. We show that in a mixed duopoly, a differentiated emission quota does more to improve welfare than an emission tax. In a comparison of the emission tax with a uniform emission quota, the superiority of environmental regulations in terms of social welfare depends on the parameters of the cost functions.

Appendices

Appendix 1

Proof of Proposition 2. First, we compare welfare under the two emission quotas. By definition, the government can choose $\bar{e}_0=\bar{e}_1=\bar{e}^{\rm UQ}$ under the differentiated emission quota. From the results, we know that $\bar{e}_0 \neq \bar{e}_1 \neq \bar{e}^{\rm UQ}$. Therefore, $W^{\rm DQ}>W^{\rm UQ}$.

Second, we compare welfare under a differentiated emission quota and emission tax. We obtain

$$ \begin{aligned} W^{\rm DQ}-W^{\rm T}={\frac{k^2 (k+2)^2 c^2 \alpha^2}{2 \Updelta^{\rm T} \Updelta^{\rm DQ}}}>0. \end{aligned} $$

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Moreover, comparing $W^{\rm UQ}$ and $W^{\rm T}$ with $W^{\rm P}$, we obtain

$$ \begin{aligned} W^{\rm UQ}-W^{\rm P}&={\frac{(k+2)^2 \left[c^3+3(k+1)c^2+(3 k^2+6 k+1)c+k^3+3 k^2+k+1\right]\alpha^2}{\Updelta^{\rm UQ} \Updelta^{\rm P}}}>0, \\ W^{\rm T}-W^{\rm P}&={\frac{(k+2)^2 \left[(k+1)c^3+(k^2+5 k+3)c^2+(2 k^2+4 k+1)c+2 k+1 \right]\alpha^2}{2 \Updelta^{\rm T} \Updelta^{\rm P}}}>0. \\ \end{aligned} $$

Finally, we compare welfare under a uniform emission quota and emission tax. We obtain

$$ \begin{aligned} W^{\rm UQ}-W^{\rm T}={\frac{k (k+2)^2 \phi(c,k) \alpha^2}{2 \Updelta^{\rm T} \Updelta^{\rm UQ}}}, \\ \end{aligned} $$

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where $\phi(c,k)=-1-2c-c^2-3k-4ck+c^2k-2k^2+ck^2$. We examine whether ϕ(c, k) is positive. When k = 1, we get ϕ(c,1) < 0. Suppose k > 1. Then, we obtain

$$ \begin{aligned} \phi(c,k)>0 \quad {\hbox{if}\; \hbox{and}\; \hbox{only}\; \hbox{if}}\quad c>\bar{c}\; \hbox{or} \; c<\underline {c}, \\ \end{aligned} $$

where

$$ \begin{aligned} \bar{c} &= {\frac{2+4k-k^2+\sqrt{8k+16k^2+k^4}}{2(k-1)}}, \\ \underline {c} &= {\frac{2+4k-k^2-\sqrt{8k+16k^2+k^4}}{2(k-1)}}. \\ \end{aligned} $$

We can easily find that $\bar{c}>1$ and $\underline {c}<0$. Moreover, we check the sign of $d \bar{c}/ d k$. Calculating $d \bar{c}/ d k$, we obtain

$$ \begin{aligned} {\frac{d \bar{c}}{d k}} &= -{\frac{-k^4+2k^3+20k+4+\{(k-1)^2+5\} \sqrt{k(k^3+16k+8)}}{2(k-1)^2 \sqrt{k(k^3+16k+8)}}}, \end{aligned} $$

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We find that the sign of the numerator determines the sign of $d\bar{k}/dc$. With respect to the numerator, $\{(k-1)^2+5\}\sqrt{k(k^3+16k+8)}$ is positive, while $-k^4+2k^3+20k+4$ would be negative if k is slightly above 3.6. Thus, we calculate the value of the former squared minus the latter squared, and then we find

$$ \begin{aligned} 4(k-1)^2(7k^4+4k^3+48k^2+24k-4)>0. \end{aligned} $$

Therefore, $d\bar{c}/d k<0$.$\square$

Appendix 2

2.1 Emission standard

We consider the differentiated emission standard under the framework of the basic model in Naito and Ogawa (2009). The government sets the abatement effort of each firm given by $\bar{a}_i$. In this case, the firm only chooses its output level. The maximization problem of each firm is

$$ \begin{aligned} {\mathop{\max}\limits_{q_0}} \quad W(q_0, q_1, \bar{a}_0, \bar{a}_1), \end{aligned} $$

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$$ \begin{aligned} {\mathop{\max}\limits_{q_1}} \quad \pi_1(q_0, q_1, \bar{a}_1). \end{aligned} $$

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The first-order conditions of the above maximization problem are

$$ \begin{aligned} {\frac{\partial W}{\partial q_0}} &= \alpha-(2+c)q_0- 2q_1+\sum_{i=0}^1 \bar{a}_i =0,\\ {\frac{\partial \pi_1}{\partial q_1}} &= \alpha-q_0-(2+c) q_1 =0. \\ \end{aligned} $$

The government chooses $\bar{a}_0$ and $\bar{a}_1$ to maximize welfare, given the firms’ behavior. We obtain the following equilibrium outcome. The superscript ${\rm S}$ denotes the equilibrium outcome under the emission standard in a mixed duopoly.

$$ \begin{aligned} q_0^{\rm S}&={\frac{\left[(k+2)c^3+2(2k+5)c^2+2(k+7)c+4\right]\alpha}{\Updelta^{\rm S}}},\\ q_1^{\rm S}&={\frac{(c+1)\left[(k+2)c^2+2(2k+3)c+2k\right]\alpha}{\Updelta^{\rm S}}}, \\ a_i^{\rm S}&=\bar{a}_i^S={\frac{(c+1)(2c^2+7c+2)\alpha}{\Updelta^{\rm S}}},\\ e_0^{\rm S}&={\frac{\left[kc^3+(4k+1)c^2+(2k+5)c+2\right]\alpha}{\Updelta^{\rm S}}}, \\ e_1^{\rm S}&={\frac{(c+1)\left[k c^2+(4k-1)c+2(k-1)\right]\alpha}{\Updelta^{\rm S}}},\\ Q^{\rm S}&={\frac{\left[2(k+2)c^3+9(k+2)c^2+4(2k+5)c+2(k+2)\right]\alpha}{\Updelta^{\rm S}}},\\ A^{\rm S}&={\frac{2(c+1)(2c^2+7c+2)\alpha}{\Updelta^{\rm S}}}, \\ E^{\rm S}&={\frac{\left[2kc^3+9kc^2+2(4k+1)c+2k\right]\alpha}{\Updelta^{\rm S}}}, \\ W^{\rm S}&={\frac{\left[2(k+2)c^3+4(2k+5)c^2+(7k+22)c+2(k+2)\right]\alpha^2}{2\Updelta^{\rm S}}}, \\ \end{aligned} $$

where $\Updelta^{\rm S}=(k+2)c^4+2(4k+7)c^3+4(5k+8)c^2+2(8k+13)c+4(k+1)>0$.

Appendix 3

We calculate equilibrium outcomes under four environmental regulations—uniform emission taxes and quotas and differentiated emission taxes and quotas—where the public firm is a CSPS maximizer. The superscripts ${\rm UT'}$, ${\rm DT'}$, ${\rm DQ'}$, and ${\rm UQ'}$ denote the equilibrium outcomes under the uniform emission tax, the differentiated emission tax, the differentiated emission quota, and the uniform emission quota, respectively.

3.1 Uniform emission tax

The maximization problem of each firm is given by

$$ \begin{aligned} {\mathop{\max}\limits_{q_0, a_0}} \quad \bar{W}(q_0,a_0,q_1,a_1)-t(e_0+e_1), \end{aligned} $$

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$$ \begin{aligned} {\mathop{\max}\limits_{q_1, a_1}} \quad \pi_1(q_0, q_1, a_1)-te_1, \end{aligned} $$

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where $\bar{W}(q_0,a_0,q_1,a_1)=\int_0^{Q} (\alpha-s)ds-\sum_{i=0}^{1}cq_i^2/2-\sum_{i=0}^{1}ka_i^2/2$. Here, $\bar{W}^t(q_0,a_0,q_1,a_1)$ denotes $\bar{W}(q_0,a_0,q_1,a_1)-t(e_0+e_1)$. We note that the term related to the emission tax appears in Eq. 14, which is different from the case where the public firm is a welfare maximizer. The first-order conditions of the above maximization problem are as follows:

$$ \begin{aligned} {\frac{\partial \bar{W}^t}{\partial q_0}} &= \alpha-(1+c)q_0- q_1 -t =0,\\ {\frac{\partial \bar{W}^t}{\partial a_0}} &= - k a_0+t=0, \\ {\frac{\partial \pi_1^t}{\partial q_1}} &= \alpha-q_0-(2+c) q_1-t =0, \\ {\frac{\partial \pi_1^t}{\partial a_1}} &= -k a_1+t=0. \\ \end{aligned} $$

The government chooses t to maximize welfare, given the firms’ behavior. We obtain the following equilibrium outcome.

$$ \begin{aligned} t^{\rm UT'}&={\frac{\left[4 c^3+(3 k+14)c^2+2(2 k+5)c+k+2\right]\alpha}{\Updelta^{\rm UT'}}}, \\ q_0^{\rm UT'}&={\frac{(c+1)(k+2)\left[2 c^2+2(k+3)c+k+2\right]\alpha}{\Updelta^{\rm UT'}}}, \\ q_1^{\rm UT'}&={\frac{c(k+2)\left[2 c^2+2(k+3)c+k+2\right]\alpha}{\Updelta^{\rm UT'}}}, \\ a_i^{\rm UT'}&={\frac{\left[4 c^3+(3 k+14)c^2+2(2 k+5)c+k+2\right]\alpha}{\Updelta^{\rm UT'}}}, \\ e_0^{\rm UT'}&={\frac{\left[2 k c^3+ (2 k^2+9 k+2)c^2+(3 k^2+10 k+6)c+(k+1)(k+2)\right]\alpha}{\Updelta^{\rm UT'}}}, \\ e_1^{\rm UT'}&={\frac{\left[2 k c^3+(2 k^2+7 k-2)c^2+(k^2-6)c-(k+2)\right]\alpha}{\Updelta^{\rm UT'}}}, \\ Q^{\rm UT'}&={\frac{(k+2)(2 c+1)\left[2 c^2+2(k+3)c+k+2\right]\alpha}{\Updelta^{\rm UT'}}}, \\ A^{\rm UT'}&={\frac{2\left[4 c^3+(3 k+14)c^2+2(2 k+5)c+k+2\right]\alpha}{\Updelta^{\rm UT'}}}, \\ E^{\rm UT'}&={\frac{k\left[4 c^3+4(k+4)c^2+2(2 k+5)c+k+2\right]\alpha}{\Updelta^{\rm UT'}}}, \\ W^{\rm UT'}&={\frac{(k+2)\left[4 c^3+4(k+4)c^2+2(2 k+5)c+(k+2)\right]\alpha^2}{2\Updelta^{\rm UT'}}}, \\ \end{aligned} $$

where $\Updelta^{\rm UT'}=2(k+2)c^4+2(k^2+10k+12)c^3+2(5 k^2+25 k+22)c^2+(9k^2+32k+24)c+2(k+1)(k+2)>0$.

3.2 Differentiated emission tax and differentiated emission quota

First, we consider the equilibrium under the differentiated emission tax. The maximization problem of each firm is given by

$$ \begin{aligned} {\mathop{\max}\limits_{q_0, a_0}} \quad \bar{W}(q_0,a_0,q_1,a_1)-t_0 e_0-t_1 e_1, \end{aligned} $$

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$$ \begin{aligned} {\mathop{\max}\limits_{q_1, a_1}} \quad \pi_1(q_0, q_1, a_1)-t_1 e_1. \end{aligned} $$

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Here, $\hat{W}^t(q_0,a_0,q_1,a_1)$ denotes $\bar{W}(q_0,a_0,q_1,a_1)-t_0 e_0 - t_1 e_1$, and $\hat{\pi}_1^t(q_0,q_1,a_1)$ denotes $\pi_1(q_0,q_1,a_1)-t_1 e_1$. The first-order conditions of the above maximization problem are as follows:

$$ \begin{aligned} {\frac{\partial \hat{W}^t}{\partial q_0}} &= \alpha-(1+c)q_0- q_1 -t_0 =0,\\ {\frac{\partial \hat{W}^t}{\partial a_0}} &= - k a_0+t_0=0, \\ {\frac{\partial \hat{\pi}_1^t}{\partial q_1}} &= \alpha-{q}_{0}-(2+c) q_{1}-t_{1} =0,\\ {\frac{\partial \hat{\pi}_1^t}{\partial a_1}} &= -k a_1+t_1=0. \end{aligned} $$

The government chooses t ₀ and t ₁ to maximize welfare, given the firms’ behavior. We obtain the following equilibrium outcome.

$$ \begin{aligned} t_0^{\rm DT'}&={\frac{k\left[2 c^3+4(k+2)c^2+(2 k^2+9 k+7)c+k+1\right]\alpha}{\Updelta^{\rm DT'}}}, \\ t_1^{\rm DT'}&={\frac{k\left[2 c^3+3(k+2)c^2+(k^2+5 k+3)c+ k+1\right]\alpha} {\Updelta^{\rm DT'}}}, \\ q_0^{\rm DT'}&={\frac{(c+k+1)(k+2)\left[c^2+(k+3)c+1\right]\alpha}{\Updelta^{\rm DT'}}}, \\ q_1^{\rm DT'}&={\frac{c(k+2)\left[c^2+(2 k+3)c+k^2+3 k+1\right]\alpha}{\Updelta^{\rm DT'}}}, \\ a_0^{\rm DT'}&={\frac{\left[2 c^3+4(k+2)c^2+(2 k^2+9 k+7)c+k+1\right]\alpha}{\Updelta^{\rm DT'}}}, \\ a_1^{\rm DT'}&={\frac{\left[2 c^3+3(k+2)c^2+(k^2+5 k+3)c+k+1\right]\alpha}{\Updelta^{\rm DT'}}}, \\ e_0^{\rm DT'}&={\frac{\left[k c^3+ 2 k(k+2)c^2+(k^3+ 4 k^2+3 k+1)c+(k+1)^2\right]\alpha}{\Updelta^{\rm DT'}}}, \\ e_1^{\rm DT'}&={\frac{\left[k c^3+2k(k+2)c^2+(k+1)(k^2+3 k-1)c-(k+1)\right]\alpha}{\Updelta^{\rm DT'}}}, \\ Q^{\rm DT'}&={\frac{(k+2)\left[2 c^3+(4 k+7)c^2+(2 k^2+7 k+5)c+k+1\right]\alpha}{\Updelta^{\rm DT'}}}, \\ A^{\rm DT'}&={\frac{\left[4 c^3+7(k+2)c^2+(3 k^2+14 k+10)c+2(k+1)\right]\alpha} {\Updelta^{\rm DT'}}}, \\ E^{\rm DT'}&={\frac{k\left[2 c^3+4(k+2)c^2+(2 k^2+8 k+5)c+k+1\right]\alpha} {\Updelta^{\rm DT'}}}, \\ W^{\rm DT'}&={\frac{(k+2)\left[2 c^3+4(k+2)c^2+(2 k^2+8 k+5)c+k+1\right]\alpha^2} {2\Updelta^{\rm DT'}}}, \\ \end{aligned} $$

where $\Updelta^{\rm DT'}=(k+2)c^4+2(k^2+6 k+6)c^3+(k+1)(k+2)(k+11)c^2+(k+1)(4 k^2+17 k+12)c+2(k+1)^2>0$.

Next, we consider the equilibrium under the differentiated emission quota. The maximization problem of each firm is given by

$$ {\mathop{\max}\limits_{q_0, a_0}} \quad \bar{W}(q_0, q_1, a_0, a_1) \; \hbox{s.t.} \; \bar{e}_0 = e_0, $$

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$$ {\mathop{\max}\limits_{q_1, a_1}} \quad \pi_1(q_0, q_1, a_1) \; \hbox{s.t.} \; \bar{e}_1 = e_1. $$

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Using the Lagrange undetermined multiplier and calculating the first-order condition of the Lagrangian function of each firm, we find that

$$ \begin{aligned} {\frac{\partial {\rm LW}^{\rm DQ'}}{\partial q_0}} &= \alpha-(1+c)q_0- q_1-\lambda_{0}^{\rm DQ'} =0, \\ {\frac{\partial {\rm LW}^{\rm DQ'}}{\partial a_0}} &= -k a_0-a_1+\lambda_{0}^{\rm DQ'}=0, \\ {\frac{\partial {\rm LW}^{\rm DQ'}}{\partial \lambda_0^{\rm DQ'}}} &= \bar{e}_0-q_0+a_0=0, \\ {\frac{\partial L\pi_1^{\rm DQ'}}{\partial q_1}} &= \alpha-q_0-(2+c) q_1-\lambda_{1}^{\rm DQ'} =0, \\ {\frac{\partial L\pi_1^{\rm DQ'}}{\partial a_1}} &= -k a_1+\lambda_{1}^{\rm DQ'}=0, \\ {\frac{\partial L\pi_1^{\rm DQ'}}{\partial \lambda_1^{\rm DQ'}}} &= \bar{e}_1-q_1+ a_1=0, \\ \end{aligned} $$

where $\hbox{LW}^{\rm DQ'}=\bar{W}+\lambda_0^{\rm DQ'} \{\bar{e}_0-(q_0-a_0)\}$, $L\pi_1^{\rm DQ'}=\pi_1+\lambda_1^{\rm DQ'}\{\bar{e}_1-(q_1-a_1)\}$, and $\lambda_i^{\rm DQ'}$ denotes the shadow price of the emission constraint of firm i. Given the above conditions, the government chooses $\bar{e}_0$ and $\bar{e}_1$ to maximize W. Calculating the equilibrium outcome, we find that they are the same as those under the differentiated emission tax in a mixed duopoly when the public firm is a CSPS maximizer; that is, ${\rm DQ'}={\rm DT'}$ holds with regard to the equilibrium outcome.

3.3 Uniform emission quota

The maximization problems of firm 0 and firm 1 are given by

$$ {\mathop{\max}\limits_{q_0, a_0}} \quad \bar{W}(q_0, q_1, a_0, a_1) \; \hbox{s.t.} \; \bar{e} = e_0, \\ $$

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$$ {\mathop{\max}\limits_{q_1, a_1}} \quad \pi_1(q_0, q_1, a_1) \; \hbox{s.t.} \; \bar{e} = e_1. \\ $$

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Calculating the equilibrium outcome, we find that they are the same as those under the uniform emission quota in a mixed duopoly when the public firm is a welfare maximizer; that is, ${\rm UQ}={\rm UQ'}$ holds with regard to the equilibrium outcome.

Because the emission quota is binding on all firms, the environmental damage is constant, and therefore, the decision of the public firm is the same regardless of the public firm’s objective.

Appendix 4

Proof of$d\bar{k}/d c>0$in Proposition 3

$$ \begin{aligned} W^{\rm UT'}-W^{\rm P}=-{\frac{(k+2)^2\chi(c,k)\alpha^2} {2\Updelta^{\rm UT'}\Updelta^{\rm P}}}, \end{aligned} $$

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where χ(c, k) is defined as $c k^2-(c-1)^2 k-2(c^3+3 c^2+c+1)$.

If χ(c, k) < 0, we find that $W^{\rm UT'}-W^{\rm P}>0$. We examine whether χ(c, k) is positive.

$$ \chi(c,k)<0\ {\hbox{if}\ \hbox{and}\ \hbox{only}\ \hbox{if}}\ \underline {k} \le k < \bar{k}, $$

where

$$ \begin{aligned} \bar{k} &= {\frac{(1-c)^2+(1+c)\sqrt{9c^2+2c+1}}{2c}}, \\ \underline {k} &= {\frac{(1-c)^2+(1+c)\sqrt{9c^2+2c+1}}{2c}}. \\ \end{aligned} $$

We can easily find that $\bar{k}>1$ and $\underline {k}<0$. When k = 1, we get χ(c,1) < 0. Therefore, we obtain Proposition 3.

$$ {\frac{d \bar{k}}{d c}} = {\frac{9c^3+c^2-c-1+(c-1)(c+1) \sqrt{9c^2+2c+1}}{2c^2 \sqrt{9c^2+2c+1}}}, $$

We find that the sign of the numerator determines the sign of $d\bar{k}/d c$. With respect to the numerator, both $(c-1)(c+1) \sqrt{9c^2+2c+1}$ and $9c^3+c^2-c-1$ are positive. Therefore, $d\bar{k}/d c>0$.

4.1 Comparison of Q^UT′ and Q^P

$$ Q^{\rm UT'}-Q^{\rm P}=-{\frac{(k+2)\psi(c,k)\alpha}{\Updelta^{\rm UT'}\Updelta^{\rm P}}}, $$

where $\psi(c,k)=2(k+2)c^4+2(k^2+11 k+14)c^3+(13 k^2+66 k+64)c^2-(k^3-14 k^2-62 k-52)c+(k+2)(5 k+6)$. From the above, we find that Q^P could be larger than Q^UT' if k is extremely high and c is nearly equal to 1.