# Tax or Subsidy? Optimal Carbon Emission Policy: A Supply Chain Perspective
## Modeling
### Demand Expansion Effect and Carbon Emission Policy Exist Together  
We model government policy in a similar way to CAFE standards. In order to further study the existence of government carbon emission policies, we consider penalty and subsidy policies within the scope of government regulations in this model. Therefore, in addition to the previous parameters, \(t\) denotes the penalty or subsidy levied per unit in greening standards for products produced by the enterprise. From a practical point of view, we assume \(t<\lambda\). For simplicity and generality, it is assumed that the environmental standards set by the government is 0 (unless otherwise stated). Thus, the objective function of the firm is:  
\[ \begin{array}{ll}Max\pi_{(t|s)}(\theta)&=[p-(c+\lambda\theta)+t\theta]d\\ &=[p-(c+\lambda\theta)+t\theta](a+b\theta-\gamma p)\end{array}  \tag{6}\]  
The index \((t|s)\) denotes the tax or subsidy. The expression above articulates an interesting phenomenon. The government levies a linear tax for every unit of greening that the enterprise falls short of, which is multiplied by total production quantity \(t\theta d(\theta)\). On the contrary, when the greening level of products is higher than normal, the government subsidizes a proportion of the greening difference \(t\theta d(\theta)\). The goal of firms is to determine the greening level of products according to carbon emission policies and production costs so as to maximize the objective function (6). Therefore, decision-making follows this sequence: firstly, the firm determines the level of greening improvement of their products and then realizes demand by setting greening level. By solving this problem, we draw the following conclusions (refer to the Appendix A for proof):  
**Proposition 1**. : _The optimal level of greening achieved by the firm is:_  
\[\theta_{(\mathrm{fjs})}=\frac{b(p-c)-(\lambda-t)(a-\gamma p)}{2b(\lambda-t)} \Bigg{\{} \begin{array}{l}<0,\lambda>\Delta_{1}\\ \geq 0,t<\lambda\leq\Delta_{1}\end{array}  \tag{7}\]  
_where \(\Delta_{1}=t+\frac{b(p-c)}{(a-\gamma p)}\)._  
It can be inferred from the above proposition that when the greening cost \(\lambda\) is higher than the limit denoted by \(\Delta_{1}\), the greening level of products does not meet the standard set by the government. Note that \(\Delta_{1}\) is increasing in \(t\), indicating that with the increase of government taxes, firms struggle to achieve a higher level of greening (\(\frac{\partial\Delta_{1}}{\partial t}>0\)). Obviously, in the process of determining the best greening level of products, costs and taxes remain the most prominent challenges faced by firms. Under the given assumptions, substituting \(\theta_{(\mathrm{fjs})}\) for Equations (1) and (6) can express market demand and the profit of the firm as follows  
\[\left\{ \begin{array}{l}d_{(\mathrm{fjs})}=\frac{b(p-c)+(\lambda-t)(a-\gamma p )}{2(\lambda-t)}\\ \pi_{(\mathrm{fjs})}=\frac{\left[b(p-c)+(\lambda-t)(a-\gamma p)\right]^{2}}{ 4b(\lambda-t)}\end{array} \right. \tag{8}\]  
**Lemma 1**. : _(i) The optimal greening level \(\theta_{(\mathrm{fjs})}\) and the optimal total output \(d_{(\mathrm{fjs})}\) of the firm decreases in greening cost \(\lambda\) and increases in demand sensitivity to greening \(b\); (ii) the optimal profit of the firm decreases first when \(t<\lambda<\Delta_{1}\) then increases in \(\lambda\) and \(b\)._  
When the cost of greening rises, the firm will take the initiative to reduce the greening level of their products to reduce costs (see Figure 1a). At the same time, due to the existence of government-guided price, greening level of the product declines, resulting in lower product demand, so the production of the firm is also reduced (see Figure 1c). When \(t<\lambda<\Delta_{1}\), that is, when the greening level of the firm's product is higher than the standard set by the government, the government must subsidize the firm.