## 4 Model Formulation and Solution
### Stackelberg Equilibrium  
In this study, the aim was to evaluate the manufacturing, replenishment, and improvement in the carbon emissions produced by the manufacturer and the retailer under a carbon tax policy by applying the Stackelberg game decision-making strategy where the manufacturer is the leader and the retailer is the follower. Under Stackelberg equilibrium, the manufacturer (leader) first decides the optimal shipping and investment strategies, and the retailer (follower) then decides its ordering and pricing decisions based on the leader's decisions. To achieve a Stackelberg equilibrium, the retailer (i.e., the follower) firstly finds its optimal value of \((p,T_{b})\) to maximize its own profit for the value of \((n,\omega)\) determined by the manufacturer (i.e., the leader). As to the retailer, the necessary conditions for \(\mathit{TP}_{CTb}(p,T_{b})\) to be maximum are \(\partial\mathit{TP}_{CTb}(p,\ T_{b})/\partial p=0\) and \(\partial\mathit{TP}_{CTb}(p,\ T_{b})/\partial T_{b}=0\), which give  
\[ \begin{split} D(p)+pD^{\prime}(p)&-\frac{D^{\prime}(p)}{T _{b}}\bigg{\{}\frac{(\underline{C}_{t}+v)+[1-m(\omega)]\big{(}\hat{C}_{t}+ \underline{\phi})C}{\theta_{2}}\big{(}e^{\theta_{2}T_{b}}-1\big{)}\\ &+\frac{h_{b}+[1-m(\omega)]\hat{h}_{v}C}{\theta_{2}^{2}}\Big{(}e^ {\theta_{2}T_{b}}-\theta_{2}T_{b}-1\big{)}\bigg{\}}=0\\ \end{split}  \tag{19}\]\[(A_{R}+C_{T})+[1 -m(\omega)]\big{(}\hat{A}_{R}+\hat{C}_{T}\big{)}C\] \[-\frac{\big{\{}(C_{t}+\upsilon)+[1-m(\omega)]\big{(}\hat{C}_{t}+ \vartheta)C\big{\}}\mathcal{D}(p)}{\theta_{2}^{2}}\Big{(}\theta_{2}T_{b}e^{ \theta_{2}T_{b}}-e^{\theta_{2}T_{b}}+1\Big{)}\] \[-\frac{h_{b}+[1-m(\omega)]\hat{h}_{b}C\mathcal{D}(p)}{\theta_{2} ^{2}}\Big{(}\theta_{2}T_{b}e^{\theta_{2}T_{b}}-e^{\theta_{2}T_{b}}+1\Big{)}=0\]  
However, finding the closed-form solution of \((p,\ T_{b})\) from Equations (19) and (20) is not easy. Instead, the retailer's problem can be solved using the following search procedure: For any given \(p\), it should be proven that the optimal value of \(T_{b}\) not only exists but is also unique. Conversely, for any given value of \(T_{b}\), a unique \(p\) that maximizes the retailer's total profit per unit time with carbon tax regulation exists. **Theorem 1**. : _For any given \(p\), the retailer's total profit per unit time of \(TP_{CTb}(p,T_{b})\) has a global maximum value at the point \(T_{b}=T_{b^{\prime}}^{*}\), where \(T_{b}^{*}\in(0,\infty)\) and satisfies Equation (20)._  
Proof. : Let  
\[F(T_{b})=(A_{R}+C_{T})+[1-m(\omega)]\big{(}\hat{A}_{R}+\hat{C}_{T}\big{)}C\] \[-\frac{\big{\{}(C_{t}+\upsilon)+[1-m(\omega)]\big{(}\hat{C}_{t}+ \vartheta)C\big{\}}\mathcal{D}(p)}{\theta_{2}}\big{(}\theta_{2}T_{b}e^{ \theta_{2}T_{b}}-e^{\theta_{2}T_{b}}+1\big{)}\] \[-\frac{h_{b}+[1-m(\omega)]\hat{h}_{b}C\mathcal{D}(p)}{\theta_{2} ^{2}}\Big{(}\theta_{2}T_{b}e^{\theta_{2}T_{b}}-e^{\theta_{2}T_{b}}+1\Big{)}\] \[T_{b}\in(0,\infty)\]  
Firstly, by taking the first derivative of \(F(T_{b})\) with respect to \(T_{b}\),  
\[\frac{dF(T_{b})}{dT_{b}}=-\Bigg{\{}\frac{(C_{t}+\upsilon)+[1-m(\omega)]\big{(} \hat{C}_{t}+\vartheta\big{)}C}{\theta_{2}}+\frac{h_{b}+[1-m(\omega)]\hat{h}_{ b}C}{\theta_{2}^{2}}\Bigg{\}}D(p)\theta_{2}^{2}T_{b}e^{\theta_{2}T_{b}}<0\]  
implying that \(F(T_{b})\) is a strictly decreasing function of \(T_{b}\in(0,\infty)\). Furthermore, \(\lim\limits_{T_{b}\to 0}F(T_{b})=(A_{R}+C_{T})+[1-m(\omega)]\big{(}\hat{A}_{R}+ \hat{C}_{T}\big{)}C>0\) and \(\lim\limits_{T_{b}\to\infty}F(T_{b})=-\infty\). By applying the Intermediate Value Theorem, a unique \(T_{b}\in(0,\infty)\) exists, such that \(F(T_{b})=0\).