## 4 Model Construction
### Retailers Bear Advertising Expenses Alone (RI)  
At this point, the retailer bears the advertising cost alone. The profit functions of TPLSP, the manufacturer and the retailer are listed are presented in this section. The TPLSP profit function is as follows:  
\[ \begin{array}{l}\pi_{T}=(p_{t}-c_{t})q\\ =a_{1}\eta^{2}*\mu*\sigma*p_{r}^{-k}-a_{2}\eta^{3}*\mu*\sigma*p_{r}^{-k}\end{array}  \tag{10}\]  
The manufacturer's profit function is as follows:  
\[ \begin{array}{l}\pi_{M}=(w-c_{m}-p_{t}-a_{3}\sigma^{2})*q\\ =(w-c_{m}-a_{1}\eta-a_{3}\sigma^{2})*d*\eta*\mu*\sigma*p_{r}^{-k}\end{array}  \tag{11}\]  
The retailer profit function is as follows:  
\[ \begin{array}{l}\pi_{R}=(p_{r}-w-c_{r})*q-a_{4}\mu^{2}\\ =(p_{r}-w-c_{r})*d*\eta*\mu*\sigma*p_{r}^{-k}-a_{4}\mu^{2}\end{array}  \tag{12}\]  
According to the profit functions of all parties in the supply chain system and the Stark game order, Theorem 2 and Inference 2 can be obtained. **Theorem 2**: _Under the condition that retailers bear advertising costs alone, the optimal carbon emission reduction level \(\sigma^{*}=\sqrt{\frac{(w-c_{m})}{5q_{3}}}\) selected by manufacturers is the optimal fresh-keepingservice level purchased \(q*=\frac{2(w-c_{m})}{5q_{1}}\). The optimal selling price \(p_{r}^{*}=\frac{k(w+c_{r})}{(k-1)}\) and advertising level of retailers is \(\mu_{2}^{*}=\frac{d*\sqrt{\frac{(w-c_{m})}{5q_{3}}}*\frac{2(w-c_{m})}{5q_{1}}* \frac{k(w+c_{r})}{(k-1)}]^{1-k}}{2a_{4}k}\)._  
In the process of proving reference Theorem 1, similarly to Theorem 2, the influence of main parameters on the optimal decision of manufacturers and retailers can be analyzed. Inference 2  
(1) With the increase in the manufacturer's wholesale price or the decrease in the production cost, the optimal product freshness and the optimal low carbon level selected by the manufacturer have shown an upward trend. The fresh-keeping price coefficient is negatively related to the optimal fresh-keeping level selected by the manufacturer, and the low carbon cost coefficient is negatively related to the optimal low carbon level selected by the manufacturer. (2) With the increase in the manufacturer's wholesale price and retailer's selling cost, the retailer's optimal retail price will increase. The increase in the price elasticity coefficient will reduce the optimal selling price of retailers. (3) With the increase in potential market size, wholesale price and sales cost, the optimal advertising level selected by retailers will increase. If the preservation price coefficient, low carbon cost coefficient, advertising cost coefficient and price elasticity coefficient increase, the optimal advertising level selected by retailers will decrease. To prove the influence of main parameters on the optimal decision, we can observe the expression in Theorem 2. For the influence of price elasticity coefficient on the retailer's optimal advertising level, the specific solution process is listed in this section. First simplify \(\mu*\),  
\[ \begin{array}{l}\mu*=\frac{d*\sqrt{\frac{(w-c_{m})}{5q_{3}}}*\frac{2(w-c_{m} )}{5q_{1}}*\frac{k(w+c_{r})}{(k-1)}]^{1-k}}{2a_{4}k}\\ =\frac{d*\sqrt{\frac{(w-c_{m})}{5q_{3}}}*\frac{2(w-c_{m})}{5q_{1}}*(p_{r}^{*})} {2a_{4}k(p_{r}^{*})^{k}}\end{array}  \tag{13}\]  
The above formula \(\frac{d*\sqrt{\frac{(w-c_{m})}{5q_{3}}}*\frac{2(w-c_{m})}{5q_{1}}}{2a_{4}}=A\) can be changed into:  
\[\mu*=\frac{\mathrm{A}*(p_{r}^{*})}{k(p_{r}^{*})^{k}} \tag{14}\]  
The partial derivative of the above formula \(\mu*\) to the price elasticity coefficient \(k\) is obtained with:  
\[ \begin{array}{l}\frac{\partial*\mu*}{\partial k}=A*\frac{\frac{\partial p_{r }^{*}}{\partial k}*(p_{r}^{*})^{k}k-\left(p_{r}^{*}\right)[(p_{r}^{*})^{k}+k*k( p_{r}^{*})^{k-1}\frac{\partial p_{r}^{*}}{\partial k}]}{[k(p_{r}^{*})^{k}]^{2}}\\ =A*\frac{\frac{\partial p_{r}^{*}}{\partial k}*(p_{r}^{*})^{k}k(1-k)-(p_{r}^{* })^{k+1}}{[k(p_{r}^{*})^{k}]^{2}}\end{array}  \tag{15}\]  
Because \(k>1\), it is \(\frac{\partial*\mu*}{\partial k}<0\).