Policy analysis for emission-reduction with green technology investment in manufacturing

Abstract

Different environmental policies create different incentives for emission reduction. The paper examines the effect of two environmental policies, the emission abatement subsidy and emission tax policies, on a market with manufacturer investment in a green technology to reduce emission. Compared to environmental taxation, the results show that the subsidy policy offers a greater incentive to abate emission and yields higher industry profit. However, regarding social welfare, the subsidy policy leads to lower social welfare and environmental performance than the tax policy when emission is highly damaging to the environment and emission abatement is sufficiently costly. From the industrial perspective, increasing technological efficiency is not necessarily beneficial even if it is costless as the government will adjust the environmental policy accordingly for social welfare optimization, may at the manufacturer’s expense. Finally, extensions considering a combined policy (both subsidy and tax), a multiplicative emission cost function, and the problem in a supply chain context are performed to check the robustness of the results.

Appendix: Tables and Proofs

See Tables 1, 2, 3, 4, 5, 6 and 7.

Table 1 Optimal solutions under no environmental policy

Table 2 Equilibrium outcomes for the subsidy and tax policies

Table 3 Equilibrium outcomes for the subsidy and tax policies under Cournot competition

Table 4 Equilibrium outcomes for the subsidy and tax policies under Bertrand competition

Table 5 Equilibrium values with a combined policy

Table 6 Equilibria for the subsidy and tax policies with a multiplicative emission function

Table 7 Equilibrium outcomes for the subsidy and tax policies in a supply chain

Proof of Lemma 1

From the “Net emission” values under the subsidy and tax policies in Table 2, we have

$\frac{{\partial e^{S} }}{\partial d} = - \frac{{\lambda \left( {\alpha - c} \right)}}{{2\beta \left( {\lambda + d} \right)^{2} }} < 0$,

$\frac{{\partial e^{S} }}{\partial \lambda } = \frac{{d\left( {\alpha - c} \right)}}{{2\beta \left( {\lambda + d} \right)^{2} }} > 0$,

$\frac{{\partial e^{T} }}{\partial d} = - \frac{{\lambda \left( {2\beta + \lambda } \right)^{2} \left( {3\beta + \lambda } \right)\left( {\alpha - c} \right)}}{{\left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {4\beta + \lambda } \right)} \right]^{2} }} < 0$, and.

$\frac{{\partial e^{T} }}{\partial \lambda } = \frac{{\beta \left( {12\beta^{2} d + 8\beta d\lambda + \beta \lambda^{2} + d\lambda^{2} } \right)\left( {\alpha - c} \right)}}{{\left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {4\beta + \lambda } \right)} \right]^{2} }} > 0$.

Proof of Proposition 1

(i) Differentiating the “Manufacturer’s profit” value under the subsidy policy in Table 2 with respect to d, we have $\frac{{\partial \Pi_{M}^{S} }}{\partial d} = \frac{{\lambda^{2} d\left( {\alpha - c} \right)^{2} }}{{4\beta^{2} \left( {\lambda + d} \right)^{3} }} > 0$.

(ii) Differentiating the “Manufacturer’s profit” value under the subsidy policy in Table 2 with respect to $\lambda$, we obtain $\frac{{\partial \Pi_{M}^{S} }}{\partial \lambda } = \frac{{d^{2} \left( {d - \lambda } \right)\left( {\alpha - c} \right)^{2} }}{{8\beta^{2} \left( {\lambda + d} \right)^{3} }} \ge 0$ if $d \ge \lambda$, with equality holding at $d = \lambda$; otherwise, $\frac{{\partial \Pi_{M}^{S} }}{\partial \lambda } < 0$.

(iii) Differentiating the “Manufacturer’s profit” value under the tax policy in Table 2 with respect to d, we have $\frac{{\partial \Pi_{M}^{T} }}{\partial d} = - \frac{{2\beta \lambda^{3} \left( {2\beta + \lambda } \right)\left( {3\beta + \lambda } \right)^{2} \left( {\alpha - c} \right)^{2} }}{{\left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {4\beta + \lambda } \right)} \right]^{3} }} < 0$.

(iv) Differentiating the “Manufacturer’s profit” value under the tax policy in Table 2 with respect to $\lambda$, we obtain $\frac{{\partial \Pi_{M}^{T} }}{\partial \lambda } = \frac{{f_{1} \left( d \right)\left( {\alpha - c} \right)^{2} }}{{2\left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {4\beta + \lambda } \right)} \right]^{3} }}$, where.

$f_{1} \left( d \right) = \left[ \begin{gathered} - \left( {2\beta + \lambda } \right)^{4} d^{3} - 3\beta \lambda \left( {4\beta + \lambda } \right)\left( {2\beta + \lambda } \right)^{2} d^{2} \hfill \\ + \beta^{2} \lambda^{2} \left( {60\beta^{2} + 36\beta \lambda + 5\lambda^{2} } \right)d + \beta^{3} \lambda^{3} \left( {20\beta + 7\lambda } \right) \hfill \\ \end{gathered} \right]$.

Since ${\text{sgn}} \left( {\frac{{\partial \Pi_{M}^{T} }}{\partial \lambda }} \right) = {\text{sgn}} \left( {f_{1} \left( d \right)} \right)$, we only need to discuss ${\text{sgn}} \left( {f_{1} \left( d \right)} \right)$ below. Define $f_{1}^{{\prime}} \left( d \right)$ and $f_{1} \left( d \right)$ as follows:

$\left\{ \begin{gathered} f_{1}^{^{\prime}} \left( d \right) = \frac{{\partial f_{3} \left( d \right)}}{\partial d} = - 3\left( {2\beta + \lambda } \right)^{4} d^{2} - 6\beta \lambda \left( {4\beta + \lambda } \right)\left( {2\beta + \lambda } \right)^{2} d + \beta^{2} \lambda^{2} \left( {60\beta^{2} + 36\beta \lambda + 5\lambda^{2} } \right) \hfill \\ f_{1} \left( d \right) = \frac{{\partial^{2} f_{3} \left( d \right)}}{{\partial d^{2} }} = - 6\left( {2\beta + \lambda } \right)^{4} d - 6\beta \lambda \left( {4\beta + \lambda } \right)\left( {2\beta + \lambda } \right)^{2} \hfill \\ \end{gathered} \right.$.

If $d > d^{T}$, $f_{1} \left( d \right) < 0$, which means $f_{1}^{^{\prime}} \left( d \right)$ decreases in d. Furthermore, it can be verified that $f_{1}^{^{\prime}} \left( {d^{T} } \right) < 0$, which suggests $f_{1}^{^{\prime}} \left( {d^{T} } \right) < 0$ holds for all $d > d^{T}$. This in turn means $f_{1} \left( d \right)$ decreases with respect to d when $d > d^{T}$. Algebraic calculation shows that $f_{1} \left( {d^{T} } \right) > 0$. Since the cubic and quadratic terms are negative, $f_{1} \left( {d^{T} } \right) < 0$ must hold when d is sufficiently large. Thus, it can be proved that there exists a $d_{1} > d^{T}$ such that $f_{1} \left( d \right) \le 0$ for all $d \ge d_{1}$, with $f_{1} \left( {d_{1} } \right) = 0$ at $d = d_{1}$, where $d_{1}$ is the largest root of $f_{1} \left( {d_{1} } \right) = 0$. Thus, we can conclude the required.

Proof of Lemma 2

Comparing corresponding price and quantity decisions under the subsidy and tax policies in Table 2, we derive.

$p^{S} - p^{T} = - \frac{{\lambda \left( {4\beta d - 3\beta \lambda + d\lambda } \right)\left( {\alpha - c} \right)}}{{4\left[ {\left( {4\beta + \lambda } \right)^{2} d + \beta \lambda \left( {16\beta + \lambda } \right)} \right]}} < 0$, and.

$q^{S} - q^{T} = \frac{{\lambda \left( {4\beta d - 3\beta \lambda + d\lambda } \right)\left( {\alpha - c} \right)}}{{4\beta \left[ {\left( {4\beta + \lambda } \right)^{2} d + \beta \lambda \left( {16\beta + \lambda } \right)} \right]}} > 0$.

In addition, since there is no pass-through under the subsidy policy but it exists under the tax policy, we have $p^{N} = p^{S} < p^{T}$ and $q^{N} = q^{S} > q^{T}$.

Proof of Lemma 3

Comparing the “Abatement level” values under the subsidy and tax policies in Table 2, we have.

$a^{S} - a^{T} = \frac{{\lambda f_{2} \left( d \right)\left( {\alpha - c} \right)}}{{2\beta \left( {\lambda + d} \right)\left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {4\beta + \lambda } \right)} \right]}}$,

where $f_{2} \left( d \right) = \left( {2\beta + \lambda } \right)d^{2} + \beta \left( {2\beta - \lambda } \right)d + 2\beta^{2} \lambda$.

Since $f_{2} \left( d \right) > 0$ for $d > d^{T}$, we have $a^{S} - a^{T} > 0$.

Proof of Proposition 2

Comparing the “Net emission” values under the subsidy and tax policies in Table 2, we obtain $e^{S} - e^{T} = \frac{{\lambda \left[ {\left( { - 2\beta^{2} + 2\beta \lambda + \lambda^{2} } \right)d - \beta \lambda \left( {2\beta + \lambda } \right)} \right]\left( {\alpha - c} \right)}}{{2\beta \left( {\lambda + d} \right)\left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {4\beta + \lambda } \right)} \right]}}$. Algebraic calculation shows that $e^{S} \ge e^{T}$ holds only when $\lambda > \left( {\sqrt 3 - 1} \right)\beta$ and $d \ge d_{2} \left( { > d^{T} } \right)$, with equality holding at $d = d_{2}$, where $d_{2} = \frac{{ - \beta \lambda \left( {2\beta + \lambda } \right)}}{{ - \lambda^{2} - 2\beta \lambda + 2\beta^{2} }}$; otherwise, $e^{S} < e^{T}$. Since the manufacturer has no incentive to abate emission under no environmental policy. The net emission under no environmental policy is the highest.

Proof of Lemma 4

First, comparing the “Manufacturer’s profit” values under the subsidy and tax policies in Table 2 yields $\Pi_{M}^{S} - \Pi_{M}^{N} = \frac{{\lambda d^{2} \left( {\alpha - c} \right)^{2} }}{{8\beta^{2} \left( {\lambda + d} \right)^{2} }} > 0$. Next, we have.

$\Pi_{M}^{N} - \Pi_{M}^{T} = \frac{{\lambda \left( {\lambda d - \beta \lambda + 2d\beta } \right)\left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {10\beta + 3\lambda } \right)} \right]\left( {\alpha - c} \right)^{2} }}{{4\beta \left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {4\beta + \lambda } \right)} \right]^{2} }}$,

which is positive when $d > d^{T}$. Summarizing above results, we have $\Pi_{M}^{S} > \Pi_{M}^{N} > \Pi_{M}^{T}$.

Proof of Proposition 3

$SW^{S} - SW^{T} = \frac{{\lambda f_{3} \left( d \right)\left( {\alpha - c} \right)^{2} }}{{8\beta^{2} \left( {d + \lambda } \right)\left[ {\left( {2\beta + \lambda } \right)^{2} d + \beta \lambda \left( {4\beta + \lambda } \right)} \right]}}$,

where $f_{3} \left( d \right) = \left( {4\beta^{2} - \beta \lambda - \lambda^{2} } \right)d^{2} + \beta \lambda \left( {3\beta + 2\lambda } \right)d - \beta^{2} \lambda^{2}$.

Since ${\text{sgn}} \left( {SW^{S} - SW^{T} } \right) = {\text{sgn}} \left( {f_{3} \left( d \right)} \right)$, we only need to analyze ${\text{sgn}} \left( {f_{3} \left( d \right)} \right)$ below.

(a). If $\lambda = {{\left( {\sqrt {17} - 1} \right)\beta } \mathord{\left/ {\vphantom {{\left( {\sqrt {17} - 1} \right)\beta } 2}} \right. \kern-\nulldelimiterspace} 2}$, then $f_{3} \left( d \right) > 0$ for $d > d^{T}$. In this case, $SW^{S} - SW^{T} > 0$.

(b). If $\lambda \ne {{\left( {\sqrt {17} - 1} \right)\beta } \mathord{\left/ {\vphantom {{\left( {\sqrt {17} - 1} \right)\beta } 2}} \right. \kern-\nulldelimiterspace} 2}$, solving $f_{3} \left( d \right) = 0$, we obtain two real roots, $d_{3}$ and $d_{4}$, as follows:

$d_{3} = \frac{{\beta \lambda \left( { - 3\beta - 2\lambda + \sqrt {25\beta^{2} + 8\beta \lambda } } \right)}}{{2\left( {4\beta^{2} - \beta \lambda - \lambda^{2} } \right)}}$ and $d_{4} = \frac{{\beta \lambda \left( { - 3\beta - 2\lambda - \sqrt {25\beta^{2} + 8\beta \lambda } } \right)}}{{2\left( {4\beta^{2} - \beta \lambda - \lambda^{2} } \right)}}$.

If $\lambda < {{\left( {\sqrt {17} - 1} \right)\beta } \mathord{\left/ {\vphantom {{\left( {\sqrt {17} - 1} \right)\beta } 2}} \right. \kern-\nulldelimiterspace} 2}$, $f_{3} \left( d \right)$ is a quadratic function graphed by a parabola opening upward, with $d_{4} < d_{3} < d^{T}$. In this case, $f_{3} \left( d \right) > 0$ for $d > d^{T}$. Thus, $SW^{S} - SW^{T} > 0$. If $\lambda > {{\left( {\sqrt {17} - 1} \right)\beta } \mathord{\left/ {\vphantom {{\left( {\sqrt {17} - 1} \right)\beta } 2}} \right. \kern-\nulldelimiterspace} 2}$, $f_{3} \left( d \right)$ is a quadratic function graphed by a parabola opening downward, with $d_{3} < d^{T} < d_{4}$. In this case, $f_{3} \left( d \right) \ge \left( < \right)0$ for $d^{T} < d \le d_{4} \left( {d > d_{4} } \right)$. Thus, $SW^{S} - SW^{T} < 0$ for $d > d_{4}$ and $SW^{S} - SW^{T} \le 0$ for $d^{T} < d \le d_{4}$. Summarizing the results above, we can see that $SW^{S} < SW^{T}$ if and only if $d > d_{4}$ and $\lambda > {{\left( {\sqrt {17} - 1} \right)\beta } \mathord{\left/ {\vphantom {{\left( {\sqrt {17} - 1} \right)\beta } 2}} \right. \kern-\nulldelimiterspace} 2}$. Otherwise, $SW^{S} \ge SW^{T}$, with equality holding at $d = d_{4}$. From the government’s perspective, the case of no environmental policy is a special case of subsidy policy or tax policy. Thus, $SW^{N} < SW^{S}$ and $SW^{N} < SW^{T}$ always hold.

Comparing the “Social welfare” values under the subsidy and tax policies in Table 2 yields.

Proof of Lemma 5

Based the corresponding results in Table 5 for the combined policy, we have:

(i). $\frac{{\partial s^{C} }}{\partial d} = \frac{{ - \beta \lambda^{2} \left( {\alpha - c} \right)}}{{\left( {\beta d + \beta \lambda + d\lambda } \right)^{2} }} < 0$ and $\frac{{\partial s^{C} }}{\partial \lambda } = \frac{{ - \beta d^{2} \left( {\alpha - c} \right)}}{{\left( {\beta d + \beta \lambda + d\lambda } \right)^{2} }} < 0$;

(ii). $\frac{{\partial t^{C} }}{\partial d} = \frac{{2\beta \lambda^{2} \left( {\alpha - c} \right)}}{{\left( {\beta d + \beta \lambda + d\lambda } \right)^{2} }} > 0$ and $\frac{{\partial t^{C} }}{\partial \lambda } = \frac{{2\beta d^{2} \left( {\alpha - c} \right)}}{{\left( {\beta d + \beta \lambda + d\lambda } \right)^{2} }} > 0$.

(iii). $\frac{{\partial q^{C} }}{\partial d} = \frac{{ - \lambda^{2} \left( {\alpha - c} \right)}}{{\left( {\beta d + \beta \lambda + d\lambda } \right)^{2} }} < 0$ and $\frac{{\partial q^{C} }}{\partial \lambda } = \frac{{ - d^{2} \left( {\alpha - c} \right)}}{{\left( {\beta d + \beta \lambda + d\lambda } \right)^{2} }} < 0$.

Proof of Proposition 4

(i). $\frac{{\partial s^{SQ} }}{\partial \theta } = \frac{{ - 2\lambda d\left( {\alpha - c} \right)}}{{\beta \left( {\theta + 2} \right)^{2} \left( {\lambda + 2d} \right)}} < 0$;

(ii).$\frac{{\partial a_{i}^{SQ} }}{\partial \theta } = \frac{{ - 2d\left( {\alpha - c} \right)}}{{\beta \left( {\theta + 2} \right)^{2} \left( {\lambda + 2d} \right)}} < 0$.

(iii). $\frac{{\partial q_{i}^{SQ} }}{\partial \theta } = \frac{{ - \left( {\alpha - c} \right)}}{{\beta \left( {\theta + 2} \right)^{2} }} < 0$.

(iv).$\frac{{\partial e_{i}^{SQ} }}{\partial \theta } = \frac{{ - \lambda \left( {\alpha - c} \right)}}{{\beta \left( {\theta + 2} \right)^{2} \left( {\lambda + 2d} \right)}} < 0$;

(v).$\frac{{\partial p_{i}^{SQ} }}{\partial \theta } = \frac{{ - \left( {\alpha - c} \right)}}{{\left( {\theta + 2} \right)^{2} }} < 0$;

(vi).$\frac{{\partial \Pi_{Mi}^{SQ} }}{\partial \theta } = \frac{{2\left[ {2\left( {\lambda + 2\beta } \right)d^{2} + 4\beta \lambda d + \beta \lambda^{2} } \right]\left( {\alpha - c} \right)^{2} }}{{ - \beta^{2} \left( {2 + \theta } \right)^{3} \left( {\lambda + 2d} \right)^{2} }} < 0$;

(vii). $\frac{{\partial SW^{SQ} }}{\partial \theta } = \frac{{\left[ {2\left( {2\lambda - \beta \theta - 4\beta } \right)d - \beta \lambda \left( {4 + \theta } \right)} \right]\left( {\alpha - c} \right)^{2} }}{{ - \beta^{2} \left( {2 + \theta } \right)^{3} \left( {\lambda + 2d} \right)}} < 0$ holds in the applicable range of $\lambda$ and $d$ with positive social welfare, i.e., $\lambda < 3 + \beta$ or $d < \frac{{\left( {\beta + 3} \right)\lambda }}{{2\left( {3 + \beta - \lambda } \right)}}$.

Proof of Proposition 5

Differentiating the manufacturer’s profit in Table 5, we have:

$\frac{{\partial \Pi_{M}^{C} }}{\partial d} = \frac{{ - \beta \lambda^{2} \left( {d + 2\lambda } \right)\left( {\alpha - c} \right)^{2} }}{{\left( {\beta d + \beta \lambda + d\lambda } \right)^{3} }} < 0$, and.

$\frac{{\partial \Pi_{M}^{C} }}{\partial \lambda } = \frac{{ - d^{2} \left( {3\beta d + 5\beta \lambda + d\lambda } \right)\left( {\alpha - c} \right)^{2} }}{{2\left( {\beta d + \beta \lambda + d\lambda } \right)^{3} }} < 0$.

Fig. 1

Sequence of decisions

Fig. 2

Variation of the manufacturer’s profit with respect to $d$ and $\lambda$

Fig. 3

Comparisons of net emission under different cases

Fig. 4

Comparisons of social welfare under different cases