# Tax or Subsidy? Optimal Carbon Emission Policy: A Supply Chain Perspective
## Modeling
### The Impact of Government Policy  
In this section, we assess and analyze the optimal outcome for the firm in two situations to analyze the role of government policy, that is, how to implement government policy when demand expansion effect exists. By calculating this, we have obtained the conditions for the government to impose taxes or subsidies on the firm. By analyzing the impact of government policy on consumer surplus and social welfare, we study the influence of government policy in detail and then analyze optimal government policy. **Proposition 2**. : _The greenness of products in the presence of government policy is higher than that in the absence of government policy, that is, \(\theta_{(\text{tg})}\geq\theta_{B}\)._Comparing \(\theta_{\left(\mathrm{fjs}\right)}\) and \(\theta_{B}\), we can come to the conclusion above. Government policy (taxes or subsidies) plays a necessary role in forcing manufacturers to offer greener products than when government policy does not exist (notice that \(\theta_{\left(\mathrm{fjs}\right)}\) is increasing in \(t\)). The above analysis identifies that with the increase of \(t\), the greening level of products increases. This means that only manufacturers will benefit from green-sensitive markets in the absence of government policy, but additional government policy will result in increased greening levels of products, thereby improving the environment. Therefore, from an environmental perspective, the existence of government policies is necessary. Voluntary measures taken by firms to maximize profits and green conscious consumers needs may promote greening measures, but the existence of government policy can further enhance this. In the past, government policy has been thoroughly analyzed in associated literature. We next analyze the impact of government policy on consumer surplus and social welfare. Consumer surplus is represented by CS [27]:  
\[CS_{\left(\mathrm{fjs}\right)}=\int_{0}^{d_{\left(\mathrm{fjs}\right)}}p\! \left(x,\theta_{\left(\mathrm{fjs}\right)}\right)\!dx-pd_{\left(\mathrm{fjs} \right)}=\frac{\left[b(p-c)+(\lambda-t)(a-\gamma p)\right]^{2}}{8\gamma( \lambda-t)^{2}} \tag{9}\]  
where \(p\!\left(x,\theta_{\left(\mathrm{fjs}\right)}\right)\) is the inverse demand function, expressed by \(\frac{a+b\theta-x}{\gamma}\) and \(x\) denotes the production output. **Proposition 3**. : _Consumer surplus in the presence of government policy is higher than that in the absence of government policy, that is, \(CS_{\left(\mathrm{fjs}\right)}>CS_{B}\)._  
From the analysis above, although manufacturers may not benefit from government policy, consumer surplus increases after the implementation of government policy. Therefore, leaving aside the environmental perspective, from a consumer perspective, government policy is also necessary. **Lemma 2**. : _Consumer surplus \(CS_{\left(\mathrm{fjs}\right)}\) decreases in greening cost \(\lambda\) and increases in consumers' green sensitivity \(b\)._  
Figure 2 shows that the increase in greening costs reduces consumer surplus, which poses a major challenge to consumers and the government (see Figure 1(a)). However, consumers with higher green-sensitivity can alleviate the impact of cost (see Figure 1(b)). Thus, the government is now focusing on fostering consumers' green awareness. Previous analysis shows that in the green market, the existence of government carbon emission policies enables firms to provide greener products. Although the government can influence strategic decisions by formulating carbon emission policies to improve greening levels, at the same time, the government should also support firms by cultivating green consumer markets. Figure 2: Optimal value of consumer surplus \(CS_{\left(\mathrm{fjs}\right)}\) versus \(\lambda\), \(b\). The government is concerned with the overall maximization of social welfare (SW). Social welfare comes from a comprehensive consideration of profit, consumer surplus, government revenue and environmental externality [20] expressed as:  
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