Environmental governance strategies in a two-echelon supply chain with tax and subsidy interactions

Abstract

Environmental impact is one of the major causes of tax and subsidy interactions between the government and firms. The government sets the tax of the upstream impact and designs the subsidy threshold value to promote the green effort of firms. This paper presents the optimal decision of the supply chain and the effective environmental governance strategy of the government. A Stackelberg game between the government and a two-echelon supply chain is constructed, where a supplier makes core parts for a manufacturer. First, we compared the effects of subsidy on the pricing decisions of firms and identified the profitable condition of the green effort of manufacturers. Results corroborate that a negotiated space for suppliers and manufacturers exists to reduce the environmental impact from the source, especially when the tax legislation is strict. Second, we obtained the optimal upstream contract of environmental impact in supply chains, which appears a phenomenon of the Matthew effect under any type of governance strategies. This finding affirms that subsidies make supply chains inclined to choose a greener upstream decision. The condition when subsidies make sense by effective environment improvement in the supply phase was further derived. Finally, with tax and subsidy governance strategies, the multi-perspective social welfare was simulated and analyzed. Accordingly, we verified that the growth boundary of the supply-chain profit with high subsidy threshold is better than that of the social welfare or the environment performance. It implies that setting the green subsidy standard originally high is a comprehensively sustainable choice for social welfare. In addition, the simulations of three tax structures were analyzed in extensions.

Appendices

Appendix 1: Proofs

Proof of Proposition 1

Taking the scenario of subsidizing manufacturer as an example, the proof of the other conditions can be alike. Taking the first order derivative of $\pi_{m}^{{S_{m} }} \left( p \right)$ regarding $p$, we obtained:

$$ \frac{{\partial \pi_{m}^{{S_{m} }} \left( p \right)}}{\partial p} = 1 - \left( {1 - \mu \left( {\theta - 1} \right)} \right)\left( {2p + s - w - \theta c_{m} } \right) = 0, $$

which implies that

$$ p = \frac{1}{2}\left( { - s + w + \theta c_{m} + \frac{1}{{1 - \mu \left( {\theta - 1} \right)}}} \right). $$

(A1)

Substituting $p$ with (A1) and taking the first order derivative of $\pi_{s}^{{S_{m} }} \left( w \right)$ regarding $w$, we acquired:

$$ \frac{{\partial \pi_{s}^{{S_{m} }} \left( w \right)}}{\partial w} = \frac{1}{2}\left( {1 + \left( {1 - \mu \left( {\theta - 1} \right)} \right)\left( {t\alpha + s + c_{s} - \theta c_{m} - 2w} \right)} \right) = 0, $$

which implies that

$$ w^{{S_{m} *}} = \frac{{1 - \left( {1 - \mu \left( {\theta - 1} \right)} \right)\left( {\theta c_{m} - c_{s} - \alpha t - s} \right)}}{{2\left( {1 - \mu \left( {\theta - 1} \right)} \right)}}. $$

(A2)

Substituting $w$ with the optimal expression (A2) in (A1), we obtained the results of Proposition 1. Reasonably supposing that consumers are always sensitive to the price, we obtained the following:

$$ 1 - \mu \left( {\theta - 1} \right) > 0 $$

(A3)

Thus, we acquired the second order derivative of $\pi_{s}^{{{\text{S}}_{\text{m}} }} \left( {\text{w}} \right)$ and $\pi_{m}^{{{\text{S}}_{\text{m}} }} \left( {\text{p}} \right),$ respectively, as follows:

$$ \begin{aligned} \frac{{\partial^{2} \pi_{m}^{{S_{m} }} \left( p \right)}}{{\partial p^{2} }} & = - 2\left( {1 - \mu \left( {\theta - 1} \right)} \right) < 0, \\ \frac{{\partial^{2} \pi_{s}^{{S_{m} }} \left( w \right)}}{{\partial w^{2} }} & = - \left( {1 - \mu \left( {\theta - 1} \right)} \right) < 0. \\ \end{aligned} $$

The solution satisfies the optimal conditions; thus, $\left( {w^{{S_{m} *}} ,p^{{S_{m} *}} } \right)$ is an optimal price decision.

Proof of Proposition 3

As

$$ \begin{aligned} \frac{{\partial w^{ *} }}{\partial \theta } & = \frac{\mu }{{\left( {1 - \mu \left( {\theta - 1} \right)} \right)^{2} }} - c_{m} \\ \frac{{\partial p^{ *} }}{\partial \theta } & = \frac{1}{4}\left( {\frac{3\mu }{{\left( {1 - \mu \left( {\theta - 1} \right)} \right)^{2} }} + c_{m} } \right) > 0, \\ \end{aligned} $$

we acquired the result (1) in Proposition 3.

As

$$ \begin{aligned} \frac{{\partial w^{*} }}{\partial \mu } & = \frac{\theta - 1}{{2\left( {1 - \mu \left( {\theta - 1} \right)} \right)^{2} }} \\ \frac{{\partial p^{*} }}{\partial \mu } & = \frac{{3\left( {\theta - 1} \right)}}{{4\left( {1 - \mu \left( {\theta - 1} \right)} \right),^{2} }} \\ \end{aligned} $$

we obtained the result (2) in Proposition 3.

As

$$ \frac{{\partial w^{*} }}{\partial \alpha } = \frac{t}{2},\frac{{\partial p^{*} }}{\partial \alpha } = \frac{t}{4}, $$

we acquired the result (3) in Proposition 3.

Proof of Proposition 4

With optimal results in Proposition 2, we obtained the profit as follows.

$$ \begin{aligned} \pi^{1} \left( \theta \right) & = \frac{{3\left( {1 - \left( {1 - \mu \left( {\theta - 1} \right)} \right)\left( {\theta c_{m} + c_{s} + \alpha t} \right)} \right)^{2} }}{{16\left( {1 - \mu \left( {\theta - 1} \right)} \right)}} \\ \pi^{*} \left( \theta \right) & = \frac{{3\left( {1 - \left( {1 - \mu \left( {\theta - 1} \right)} \right)\left( {\theta c_{m} + c_{s} + \alpha t - s} \right)} \right)^{2} }}{{16\left( {1 - \mu \left( {\theta - 1} \right)} \right).}} \\ \end{aligned} $$

Condition (1) in Proposition 4 can be obtained by simplifying the following:

$$ \pi^{1} \left( \theta \right) > \pi^{1} \left( {\theta = 1} \right), $$

i.e.,

$$ \frac{{\left( {1 - \left( {1 - \mu \left( {\theta - 1} \right)} \right)\left( {\theta c_{m} + c_{s} + \alpha t} \right)} \right)^{2} }}{{\left( {1 - \mu \left( {\theta - 1} \right)} \right)}} > \left( {1 - \left( {c_{m} + c_{s} + \alpha t} \right)} \right).^{2} $$

Condition (2) in Proposition 4 can be obtained by the simplifying the following:

$$ \pi^{*} \left( \theta \right) > \pi^{1} \left( {\theta = 1} \right) $$

i.e.,

$$ \frac{{\left( {1 - \left( {1 - \mu \left( {\theta - 1} \right)} \right)\left( {\theta c_{m} + c_{s} + \alpha t - s} \right)} \right)^{2} }}{{\left( {1 - \mu \left( {\theta - 1} \right)} \right)}} > \left( {1 - \left( {c_{m} + c_{s} + \alpha t - s} \right)} \right).^{2} $$

Proof of Proposition 5

Taking Model R as an example. Given that the first and second order derivative of $\pi^{1} \left( \alpha \right)$ satisfy

$$ \frac{{\partial \pi^{1} \left( \alpha \right)}}{\partial \alpha } = - \frac{3}{8}t\left( {1 - \left( {1 - \mu \left( {\theta - 1} \right)} \right)\left( {\theta c_{m} + c_{s} + t\alpha } \right)} \right) $$

and

$$ \frac{{\partial \pi^{1} \left( \alpha \right)}}{{\partial \alpha^{2} }} = \frac{{3\left( {1 - \mu \left( {\theta - 1} \right)} \right)t^{2} }}{8} > 0. $$

$\pi^{1} \left( \alpha \right)$ is on the convexity of $\alpha$. $\hat{\alpha }^{R}$ stands for the level of environment impact in supply phase, where profit of the manufacturer is minimum, and the quadratic function graph are right-and-left symmetric. The minimum value of $\pi^{1} \left( \alpha \right)$ is obtained when $\alpha = \hat{\alpha }^{R}$. Therefore, the supplier and the manufacturer will choose the value farthest from $\hat{\alpha }^{R}$. Similar result of $\pi^{*} \left( \alpha \right)$ is derived in Model S under the tax & subsidy governance.

Proof of Proposition 6

Given that

$$ \begin{aligned} \hat{\alpha }^{S} - \hat{\alpha }^{R} = \frac{s}{{\left( {1 - \mu \left( {\theta - 1} \right)} \right)t}} > 0, \hfill \\ \hat{\alpha }^{S} > \hat{\alpha }^{R} . \hfill \\ \end{aligned} $$

When $\hat{\alpha }^{R} \ge \frac{{\alpha_{H} + \alpha_{L} }}{2}$, $\alpha_{L}$ is the farthest point from both the symmetry axis $\hat{\alpha }^{R}$ and $\hat{\alpha }^{S}$. Therefore, $\pi^{1} \left( \alpha \right)$ and $\pi^{*} \left( \alpha \right)$ reach the maximum at $\alpha_{L}$. Afterward, we obtained the result (1) in Proposition 6.

When $\hat{\alpha }^{S} < \frac{{\alpha_{H} + \alpha_{L} }}{2}$, $\alpha_{H}$ is the farthest point from the symmetry axis $\hat{\alpha }^{R}$ and $\hat{\alpha }^{S}$. Therefore, $\pi^{1} \left( \alpha \right)$ and $\pi^{*} \left( \alpha \right)$ reach the maximum at $\alpha_{L}$. Afterward, we acquired the result (2) in Proposition 6.

When $\hat{\alpha }^{R} < \frac{{\alpha_{L} + \alpha_{H} }}{2} \le \hat{\alpha }^{S}$,$\pi^{1} \left( \alpha \right)$ and $\pi^{*} \left( \alpha \right)$ appear as different trends. $\alpha_{H}$ is the farthest point from $\hat{\alpha }^{R}$. On the contrary, $\alpha_{L}$ is the farthest point from $\hat{\alpha }^{S}$. Afterward, we obtained the result (3) in Proposition 6.

Appendix 2: Simulations

See Tables 3 and 4.