# Tax or Subsidy? Optimal Carbon Emission Policy: A Supply Chain Perspective
## Modeling
### The Impact of Government Policy
\begin{array}{ll}SW&=\pi_{(\text{fls})}+CS-t0d+E0d\\ &=\frac{\left[b(p-c)+(\lambda-t)(a-\gamma p)\right]}{8\gamma(\lambda-t)^{2}} \big{\{}[2\gamma(\lambda+E-2t)+b]b(p-c)+[2\gamma(\lambda-E)+b](\lambda-t)(a- \gamma p)\big{\}}\end{array}  \tag{10}\]  
where \(E\) is environmental damage per unit of greening level for each product produced. It is exogenously determined. We suppose that environmental damage is linear and is a function of the difference in greening levels and production. **Proposition 4**. : _When \(\lambda<E+\frac{b}{\gamma}+\frac{b(p-c)}{a-\gamma p}\), the government adopts a policy of subsidies; when \(\lambda>E+\frac{b}{\gamma}+\frac{b(p-c)}{a-\gamma p}\), the government adopts a tax policy._  
From Proposition 4, we can see that when the cost of greening \(\lambda\) is low, the cost of improving greening levels is also low. The greening level of products is relatively high. The firm has a strong willingness to produce more and higher greening level products. According to [PERSON]'s nudge theory [28], the government should subsidize firms that produce green products, and guide and encourage them to meet market demand (see Figure 3). When the cost of greening \(\lambda\) is relatively high, the cost of improving greening levels is also high, so the greening level of products is low and market demand is limited. This creates a problem for the government. Therefore, the government should punish firms by implementing tax policy, warn them and guide them to increase investment in green technology research, carry out technological innovation and strive to reduce greening costs. **Corollary 1**. : _When \(b>\frac{(a-\gamma p)+\gamma(p-c)}{\gamma(a-\gamma p)}(\lambda-E)\), the government adopts a subsidy policy; when \(b<\frac{(a-\gamma p)+\gamma(p-c)}{\gamma(a-\gamma p)}(\lambda-E)\), the government adopts a tax policy._  
This conclusion can be deduced from the condition of Proposition 4. It can be seen from Corollary 1 that when consumers' green-sensitivity \(b\) is relatively limited, the greening level of products has little influence on demand, so firms will produce products with lower greening levels, because, as Lemma 1 shows, improving the greening level of products will not increase profits. The government should adopt tax policies to punish firms for adopting low greening levels, make them aware of the importance of improving the greening level of products and guide the firm to produce greener products. When consumers are more sensitive to greening levels, there is a greater impact on demand, so the firm will produce products with higher greening levels to achieve higher profits. In this case, the government should reward firms, subsidize the cost of improving the greening level of products, develop positive models and encourage firms to produce greener products. Figure 3: Optimal value of social welfare \(SW\) versus \(\lambda\). **Corollary 2**. : _When \(E<\lambda-\frac{b}{\gamma}-\frac{b(p-c)}{a-\gamma p}\), the government adopts a subsidy policy; when \(E>\lambda-\frac{b}{\gamma}-\frac{b(p-c)}{a-\gamma p}\), the government adopts a tax policy._  
This conclusion can also be deduced from the condition of Proposition 4. It can be concluded from Corollary 2 that when environmental sensitivity is low, the difference in greening levels per product will cause less damage to the environment. If the firm produces products with lower greening levels, the impact on the environment will be relatively small. The government should implement tax policies to prevent firms from profiteering by producing products with low greening levels. When environmental sensitivity is relatively large, the difference in greening levels per product will cause greater environmental harm. Therefore, from an environmental protection perspective, the government hopes firms will produce greener products. In this case, the government should implement policies to subsidize investment in greening to encourage firms to produce greener products.