## 4 Model Construction
### The Manufacturer Shall Bear the Advertising Expenses Alone (MI)  
In this investment mode, the manufacturer bears the advertising cost alone. According to the above questions and symbols, the profit functions of TPLSP, the manufacturer and the retailer are expressed in this section. The TPLSP profit function is as follows:  
\[ \begin{array}{ll}\pi_{T}&=(p_{t}-c_{t})D\\ &=a_{1}\eta^{2}*\mu*\sigma*p_{r}^{-k}-a_{2}\eta^{3}*\mu*\sigma*p_{r}^{-k}\\ \end{array}  \tag{2}\]  
Manufacturer's profit function is as follows:  
\[ \begin{array}{ll}\pi_{M}&=(w-c_{m}-p_{t}-a_{3}\sigma^{2})*D-a_{4}\mu^{2}\\ &=(w-c_{m}-a_{1}\eta-a_{3}\sigma^{2})*y_{0}*\eta*\mu*\sigma*p_{r}^{-k}-a_{4} \mu^{2}\\ \end{array}  \tag{3}\]  
Retailer profit function is as follows:  
\[ \begin{array}{ll}\pi_{R}&=(p_{r}-w-c_{r})*D\\ &=(p_{r}-w-c_{r})*y_{0}*\eta*\mu*\sigma*p_{r}^{-k}\\ \end{array}  \tag{4}\]  
According to the profit functions of all parties in the supply chain system and the Stark game order, Theorem 1 and Inference 1 can be obtained. **Theorem 1**. : _Under the condition that the manufacturer alone bears the advertising costs, the manufacturer chooses the optimal advertising level as \(\mu_{1}^{*}=\frac{(w-c_{m})*d*\sqrt{\frac{(w-c_{m})}{543}}*\frac{2(w-c_{m})}{54 4}*p_{r}^{-k}}{544}\), the optimal carbon emission reduction level as \(\sigma^{*}=\sqrt{\frac{(w-c_{m})}{543}}\), and the optimal fresh-keeping service level purchased as \(\eta*=\frac{2(w-c_{m})}{541}\). The retailer's optimal selling price is: \(p_{r}^{*}=\frac{2k(w+c_{r})}{(2k-1)}\)._  
It is proven that the second order partial derivative of product freshness \(\eta\), advertising level \(\mu\) and low carbon level \(\sigma\) can be obtained through the manufacturer's profit function:  
\[\frac{\partial^{2}\pi_{M}}{\partial\mu^{2}}=-2a_{4}<0 \tag{5}\]  
\[\frac{\partial^{2}\pi_{M}}{\partial\sigma^{2}}=-6a_{3}\sigma*y_{0}*\eta*\mu* p_{r}^{-k}<0 \tag{6}\]  
\[\frac{\partial^{2}\pi_{M}}{\partial\eta^{2}}=-2a_{1}*y_{0}*\mu*\sigma*p_{r}^{ -k}<0 \tag{7}\]  
Then, by setting \(\frac{\partial\pi_{M}}{\partial\mu}\), \(\frac{\pi_{M}}{\partial\sigma}\), and \(\frac{\partial\pi_{M}}{\partial\eta}\) equal to 0, respectively, we can derive:  
\[\left\{ \begin{array}{ll}(w-c_{m}-a_{1}\eta-a_{3}\sigma^{2})*y_{0}*\eta* \sigma*p_{r}^{-k}-2a_{4}\mu=0\\ (w-c_{m}-a_{1}\eta-3a_{3}\sigma^{2})*y_{0}*\eta*\mu*p_{r}^{-k}=0\\ (w-c_{m}-2a_{1}\eta-a_{3}\sigma^{2})*y_{0}*\mu*\sigma*p_{r}^{-k}=0\\ \end{array} \right. \tag{8}\]  
By solving the equations above, it can be calculated that the manufacturer's optimal advertising level is \(\mu_{1}^{*}=\frac{(w-c_{m})*d*\sqrt{\frac{5(w-c_{m})}{543}}*\frac{2(w-c_{m})} {541}*p_{r}^{-k}}{544}\), the optimal carbon emission reduction level is \(\sigma^{*}=\sqrt{\frac{(w-c_{m})}{543}}\), and the optimal fresh-keeping service level purchased is \(\eta*=\frac{2(w-c_{m})}{541}\). \[\text{Order}\ \ A=(w-c_{m})*d*\sqrt{\frac{5(w-c_{m})}{5q_{3}}}*\frac{2(w-c_{m})}{5q_{1}} /5q_{4}\ \ \ \text{and substitute}\ \mu_{1}^{*},\sigma^{*},\eta*\ \text{into}\ \pi_{r}\]  
to obtain:  
\[ \begin{array}{l}\frac{\partial\pi_{R}}{\partial p_{r}}=\frac{p_{r}^{2k}-2k(p_ {r}-c_{r}-w)p_{r}^{2k-1}}{(p_{r}^{2k})^{2}}*y_{0}*\eta*\eta^{*}*A*\sigma^{*}\\ =\frac{(p_{r}+2k*w+2k*c_{r}-2k*p_{r})p_{r}^{2k-1}}{(p_{r}^{2k})}*y_{0}*\eta* \eta*A*\sigma^{*}\\ =\frac{[2k(w+c_{r})-(2k-1)p_{r}]p_{r}^{2k}}{(p_{r}^{2k})}*y_{0}*\eta*\eta^{*}A *\sigma^{*}\end{array}  \tag{9}\]  
\[ \begin{array}{l}\text{Order}\ \ \frac{\partial\pi_{R}}{\partial p_{r}}=0,\ \text{available}\text{:}\ p_{r}^{*}=\frac{2k(w+c_{r})}{(2k-1)}=\frac{(w+c_{r})}{(1 -\frac{1}{2k})}.\end{array} \]  
\[\text{Certificate completion}.\]  
According to the results of the optimal decisions of manufacturers and retailers, the impact of the main parameters on their optimal decisions can be analyzed, as shown in Inference 1.