# Tax or Subsidy? Optimal Carbon Emission Policy: A Supply Chain Perspective
## 4 Model Extensions  
We have analyzed and concluded on a series of situations in which the firm must decide the greening level of their products where price is exogenous. When the price of green products is endogenous, that is, when the firm makes decisions on greening and price, does the conclusion drawn above still hold?  
Now the objective of the firm is  
\[Max\ \pi(p,\theta)=[p-(c+\lambda\theta)+t\theta]d=[p-(c+\lambda\theta)+t \theta](a+b\theta-\gamma p) \tag{11}\]  
**Proposition 5**.: _There is no optimal analytic solution to the case when the enterprise makes a decision on price and greening levels._  
As evidenced in this calculation, price and greening level decision-making optimization cannot be modelled together for a number of reasons. For instance, numerical conditions relating to government decisions are unavailable, thus, we use simulations to analyze the problem. The structure is \(Max\ \pi(p,\theta\big{|}t)\), given \(p\), \(t\), then we can get \(\theta^{*}(p,t)\). Substituting \(\theta^{*}(p,t)\) to the objective function, \(Max\ \pi(p,\theta\big{|}t)=Max\ \pi(p,\theta^{*}(p,t)\big{|}t)\). The optimal solution is \(p^{*}(t)\), \(\theta^{*}(p^{*}(t),t)\). Similarly, substituting \(p^{*}(t),\theta^{*}(p^{*}(t),t)\) to the \(SW\), we can get the optimal \(t^{*}\) from \(Max\ SW(t)\), thus \(p^{*}=p^{*}(t^{*})\), \(\theta^{*}=\theta^{*}(p^{*}(t^{*}),t^{*})\). So \(\pi^{*}\), the amount of tax/subsidy, \(SW^{*}\) is obtained accordingly. The results are shown in Figure 4.  
As can be seen from Figure 4, with the increase of \(\lambda\), the cost of improving the greening of products also increases, meaning enterprises will lower the greening level of their products to reduce costs. At the same time, the effect of product prices on profit is more obvious; businesses set prices as high as possible within the scope of government regulations. These two factors lead to reduced demand (see Figure 4a). When greening costs rise, the government will increase \(t\). When the greening level of the enterprise meets government standards, subsidies will be increased to encourage continuous improvement to the level of greening of products. When the greening level of products does not comply with standards, government policy is to impose taxes. The government will raise taxes, \(t\), to increase the cost of lowering greening levels to the enterprise, thus preventing it from lowering the greening level of its products (see Figure 4b).  
When greening cost increases, the enterprise will reduce the greening level of its products. When it drops by a certain amount, it becomes lower than the standard set by the government, meaning subsidy is replaced with tax. When government policy is to subsidize and greening cost increases, the enterprise will reduce greening and set higher prices for higher profit, resulting in lower demand and production. Government subsidies are reduced and profit decreases. Simultaneously, the greening level of the product is reduced, resulting in a lower consumer surplus and environmental externalities, leading to lower social welfare (see Figure 4c). When government policy is to tax and greening cost increases, similar to subsidy policy, the manufacturer will lower the greening level of its products so it will cost less and lead to high profits. In this case, the consumer surplus and environmental externality decreases while the manufacturer's profits and the tax revenues increase, leading to an increase in overall social welfare.