## 4 Model Construction
### Shared Advertising Expenses (CI)  
When the manufacturer and retailer share the advertising cost, assuming that the proportion of advertising cost borne by the manufacturer is \(\theta\), the profit functions of TPLSP, the manufacturer and the retailer are listed in this section. TPLSP profit function is as follows:  
\[ \begin{array}{l}\pi_{T}=(p_{t}-c_{t})q\\ =a_{1}\eta^{2}*\mu*\sigma*p_{r}^{-k}-a_{2}\eta^{3}*\mu*\sigma*p_{r}^{-k}\end{array}  \tag{16}\]Manufacturer's profit function is as follows:  
\[ \begin{array}{l}\pi_{T}=(w-c_{m}-p_{t}-a_{3}\sigma^{2})*q-\theta a_{4}\mu^{2}\\ =(w-c_{m}-a_{1}\eta-a_{3}\sigma^{2})*d*\eta*\mu*\sigma*p_{r}^{-k}-\theta a_{4}\mu ^{2}\end{array}  \tag{17}\]  
Retailer profit function is as follows:  
\[ \begin{array}{l}\pi_{R}=(p_{r}-w-c_{r})*q-(1-\theta)a_{4}\mu^{2}\\ =(p_{r}-w-c_{r})*d*\eta*\mu*\sigma*p_{r}^{-k}-(1-\theta)a_{4}\mu^{2}\end{array}  \tag{18}\]  
According to the above profit functions of all parties in the supply chain system and the Stark game order, Theorem 3 and Inference 3 can be obtained. **Theorem 3**. : _Under the condition that the manufacturer and retailer jointly bear the advertising expenses, the manufacturer selects the optimal advertising level as \(\mu_{3}^{*}=\frac{(w-c_{m})*d*\sqrt{\frac{5(w-c_{m})}{5\delta_{3}}}\frac{2(w- c_{m})}{5\delta_{1}}sp_{r}^{-k}}{5\theta a_{4}}\), the optimal carbon emission reduction level as \(\sigma^{*}=\sqrt{\frac{(w-c_{m})}{5\delta_{3}}}\), and the optimal fresh-keeping service level as \(\eta*=\frac{2(w-c_{m})}{5\delta_{1}}\). The retailer's optimal selling price is \(p_{r}^{*}=\frac{2\mathds{k}(w+c_{r})}{(2k-1)}\)._  
The process of proving reference Theorem 1. By comparing MI contract (manufacturer as advertising investor), RI contract (retailer as advertising investor) and CI contract (manufacturer and retailer cooperate in advertising investment), we can draw the following conclusions:  
Inference 3  
(1) By the comparison of \(\mu_{1}^{*}\) and \(\mu_{3}^{*}\), it can be seen that the joint assumption of advertising costs by manufacturers and retailers will promote manufacturers to improve their advertising level. (2) By the comparison of \(\mu_{1}^{*}\) and \(\mu_{2}^{*}\), we can see that at that time of \(\frac{2}{5}(k-1)>\frac{w+c_{r}}{w-c_{m}}\), the advertising level under the MI investment mode was higher than that under the RI investment mode. At the time of \(\frac{2}{5}(k-1)<\frac{w+c_{r}}{w-c_{m}}\), the advertising level under the MI investment mode was lower than that under the RI investment mode. At that time of \(\frac{2}{5}(k-1)=\frac{w+c_{r}}{w-c_{m}}\), the MI investment model and the RI investment model had the same level of advertising. (3) Under the MI investment mode, the optimal selling price of retailers is less than that under the RI investment mode. Prove that subtracting \(\mu_{3}^{*}\) and \(\mu_{1}^{*}\) yields:  
\[\begin{array}{l}\mu_{3}^{*}-\mu_{1}^{*}=\frac{(w-c_{m})*d*\sqrt{\frac{5(w-c_ {m})}{5\delta_{3}}}\frac{2(w-c_{m})}{5\delta_{1}}*p_{r}^{-k}}{5\delta_{4}}- \frac{(w-c_{s})*d*\sqrt{\frac{5(w-c_{m})}{5\delta_{3}}}*\frac{2(w-c_{m})}  
According to the parameter settings in the above table, the following data analysis diagram can be obtained. Figure 3 shows the impact of the fresh-keeping service price coefficient and the wholesale price on the optimal fresh-keeping service level selected by manufacturers. According to Figure 3, when the price coefficient of the fresh-keeping service is lower or the wholesale price of products is higher, the optimal fresh-keeping service level selected by the manufacturer is higher; on the contrary, the fresh-keeping service level selected by the manufacturer is lower. This is because the increase in the price coefficient of fresh-keeping services will increase the production costs of manufacturers and reduce their investment in fresh-keeping services. The increase in wholesale prices will help manufacturers invest more money in fresh-keeping services, and better fresh-keeping services will bring more product sales, forming a virtuous circle.