# Tax or Subsidy? Optimal Carbon Emission Policy: A Supply Chain Perspective
## 5 Conclusions
\(\Box\)  
The derivatives of \(\theta_{(\text{fig})}\) w.r.t \(\lambda\) and \(b\) are \(\frac{\partial\theta_{(\text{fig})}}{\partial\lambda}=-\frac{p-c}{2(\lambda-t )^{2}}<0\) and \(\frac{\partial\theta_{(\text{fig})}}{\partial b}=\frac{a-\gamma p}{2b^{2}}>0\), respectively. Similarly, the derivatives of \(d_{(\text{fig})}\) w.r.t \(\lambda\) and \(b\) are \(\frac{\partial d_{(\text{fig})}}{\partial\lambda}=-\frac{p-c}{2(\lambda-t)^{2 }}<0\), \(\frac{\partial d_{(\text{fig})}}{\partial b}=\frac{p-c}{2(\lambda-t)}>0\). Further, taking the partial derivatives of \(\pi_{(\text{fig})}\) w.r.t \(\lambda\) and \(b\), we have \(\frac{\partial\pi_{(\text{fig})}}{\partial\lambda}=\frac{[(a-\gamma p)(\lambda -t)+b(p-c)][(a-\gamma p)(\lambda-t)-b(p-c)]}{4b(\lambda-t)^{2}}\bigg{\{} \begin{array}{l}>0,\lambda>\Delta_{1}\\ <0,t<\lambda\leq\Delta_{1}\end{array} ,\quad\frac{\partial\pi_{(\text{fig})}}{ \partial b}=\frac{[(a-\gamma p)(\lambda-t)+b(p-c)][b(p-c)-(a-\gamma p)(\lambda -t)]}{4b(\lambda-t)^{2}}\bigg{\{} \begin{array}{l}<0,\lambda>\Delta_{1}\\ >0,t<\lambda\leq\Delta_{1}\end{array} .\Box\)  
\[CS_{(\text{fig})}=\int_{0}^{d_{(\text{fig})}}p\big{(}x,\theta_{(\text{fig}) }\big{)}dx-pd_{(\text{fig})}\quad CS_{B}=\int_{0}^{d_{B}}p(x,\theta_{B})dx-pd_ {B}\\ =\frac{[b(p-c)+(\lambda-t)(a-\gamma p)]^{2}}{8\gamma(\lambda-t)^{2}}\quad, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad  
**Proof of Lemma 2. ** The derivative of \(CS_{(t|\text{s})}\) w.r.t \(\lambda\) gives \(\frac{\partial CS_{(t|\text{s})}}{\partial\lambda}=-\frac{[b(p-c)+(\lambda-t)(a- \gamma p)]b(p-c)}{4\gamma(\lambda-t)^{3}}<0\). Similarly, the derivative of \(CS_{(t|\text{s})}\) w.r.t \(b\) gives \(\frac{\partial CS_{(t|\text{s})}}{\partial b}=\frac{[b(p-c)+(\lambda-t)(a- \gamma p)](p-c)}{4\gamma(\lambda-t)^{3}}>0\). \(\Box\)  
**Proof of Proposition 4. ** Let \(\frac{dSW}{dt}=\frac{[-4\gamma(p-c)+2(a-\gamma p)]t+(4\gamma E+2b)(p-c)+2\lambda (a-\gamma p)]b^{2}(p-c)}{8\gamma(\lambda-t)^{3}}=0\). Thus, we have \(t^{*}=\frac{(2\gamma E+b)(p-c)+\lambda(a-\gamma p)}{2\gamma(p-c)+(a-\gamma p)}\). When \(2b(2\gamma\lambda-2\gamma E-b)<0\), that is \(\lambda<E+\frac{b}{2\gamma}\), we have \(\frac{dSW}{dt}>0\), which implies \(SW\) is increasing in \(t\). As a result, the government should implement the subsidy policy and put all the subsidy budgets into the firm to achieve the overall maximization of social welfare. \(\Box\)  
In contrast, if \(\lambda>E+\frac{b}{2\gamma}\), \(SW\) increases first and then decreases with respect to \(t\). Therefore, there exists \(t=t^{*}\) so that \(SW\) reaches its maximum value \(SW(t^{*})=\frac{\gamma[(\lambda-E)(a-\gamma p)+b(p-c)]^{2}}{2b[2\gamma(\lambda -E)-b]}\). If \(\theta_{(t|\text{s})}>0\) we have \(E+\frac{b}{2\gamma}<\lambda<E+\frac{b}{\gamma}+\frac{b(p-c)}{a-\gamma p}\), which means the government should adopt subsidy policy; In contrast, if \(\theta_{(t|\text{s})}<0\), we have \(\lambda>E+\frac{b}{\gamma}+\frac{b(p-c)}{a-\gamma p}\), which implies the government should adopt tax policy. To sum up, the government should adopt subsidy policy when \(\lambda<E+\frac{b}{\gamma}+\frac{b(p-c)}{a-\gamma p}\) and tax policy when \(\lambda>E+\frac{b}{\gamma}+\frac{b(p-c)}{a-\gamma p}\). **Proof of Proposition 5. ** Taking the second order derivatives of the profit function, we have \(\frac{\partial^{2}\pi(p,\theta)}{\partial\theta^{2}}=-2b\lambda\), \(\frac{\partial^{2}\pi(p,\theta)}{\partial\theta\partial p}=\frac{\partial^{2 }\pi(p,\theta)}{\partial\theta\partial\theta}=b+\gamma\lambda\), \(\frac{\partial^{2}\pi(p,\theta)}{\partial p^{2}}=-2\gamma\). As a result, the determinant of Hessian matrix is \(\left| \begin{array}{cc}-2b\lambda&b+\gamma\lambda\\ b+\gamma\lambda&-2\gamma\end{array} \right|=-(b-\gamma\lambda)^{2}<0\), and hence the optimal value condition is not satisfied. \(\Box\)