## 2 Materials and Methods
### Models and Solutions
#### 2.3.2 Operating Subsidy Model  
To improve the supply capacity of ecological products, the financial subsidy is paid directly to the supplier and distributor in proportion to their ecosystem protection investment, which is an effective way to reduce the supply chain cost and encourage sustainable operations. The function of the operating subsidy is described as:  
\[E(f)=f\times\frac{\mu\theta^{2}}{2} \tag{12}\]  
where \(f\) is the government subsidy coefficient, and the subsidy distribution proportion of the supplier and distributor is consistent with their payments \(r\) and \(1\,-\,r\) separately, and the utility functions of the supplier, distributor, and supply chain are described as follows:  
\[U_{1}(\tau_{S})=(w-c)\ (a-bp+k\theta)-r(1-f)\frac{\mu\theta^{2}}{2} \tag{13}\]  
\[U_{1}(\tau_{R})=(p-w)\ (a-bp+k\theta)-(1-r)(1-f)\frac{\mu\theta^{2}}{2} \tag{14}\]  
\[U_{1}(\tau_{T})=(p-c)(a-bp+k\theta)-(1-f)\frac{\mu\theta^{2}}{2} \tag{15}\]  
As with the subside-free model solution, the optimal market price is solved by the derivative of Equation (14) as follows:  
\[p^{*}=\frac{a+bw+k\theta}{2b} \tag{16}\]  
Let \(\frac{\partial^{2}U_{1}(\tau_{S})}{\partial w^{2}}=-b=A\), \(\frac{\partial^{2}U_{1}(\tau_{S})}{\partial w^{2}}=\frac{k}{2}=B\), and \(\frac{\partial^{2}U_{1}(\tau_{S})}{\partial\theta^{2}}=-r(1-f)\mu=C\), then \(A<0\) and \(AC-B^{2}=br(1-f)\mu-\frac{k^{2}}{4}\) can be calculated. When \(br(1-f)\mu>\frac{k^{2}}{4}\), the unique optimal \(w\) and \(\theta\) for the distributor's maximum benefit can be proposed as follows:  
\[ \begin{cases}\frac{\partial U_{1}(\tau_{S})}{\partial w}=0\\ \frac{\partial U_{1}(\tau_{S})}{\partial\theta}=0\end{cases} \Longrightarrow \begin{cases}w^{*}=\frac{\alpha k^{2}-2\mu r(1-f)(a-bc)}{k^{2}-4b\mu r(1-f)\mu }\\ \theta^{*}=\frac{k(bc-a)}{k^{2}-4b\mu r(1-f)\mu}\end{cases}  \tag{17}\]  
So \(p^{*}=\frac{\alpha k^{2}-\mu(1-f)(3a-bc)}{k^{2}-4b\mu r(1-f)\mu}\) is obtained by Equation (16), and the utility functions of the supplier, distributor, and supply chain are calculated as follows:  
\[U_{1}^{*}(\tau_{S})=-\frac{k^{2}\mu r(1-f)(-a+bc)^{2}}{2(-4b\mu r(1-f)+k^{2}) ^{2}}+\left(-c+\frac{\alpha k^{2}-2\mu r(1-f)(a-bc)}{-4b\mu r(1-f)+k^{2}} \right)\left(a-b\frac{\alpha k^{2}-\mu r(1-f)(3a-bc)}{-4b\mu r(1-f)+k^{2}}+ \frac{k^{2}(-a+bc)}{-4b\mu r(1-f)+k^{2}}\right) \tag{18}\]  
\[ \begin{split} U_{1}^{*}(\tau_{R})=-\frac{k^{2}\mu(1-f)(1-f)(-a+ bc)^{2}}{2(-4b\mu r(1-f)+k^{2})^{2}}\\ +\left(\frac{-\alpha k^{2}+2\mu r(1-f)(a-bc)}{-4b\mu r(1-f)+k^{2}} \right)\end{split}  \tag{19}\]\[ \begin{array}{c}U_{1}^{*}(\pi_{T})=\frac{4 bp(1-f)\left(ck^{2}-\mu r(1-f)(3a-bc) \right)}{\left(-4b\mu r(1-f)+k^{2}\right)^{2}}-\frac{k^{2}\mu(1-f)(-a+bc)^{2}}{ 2(-4b\mu r(1-f)+k^{2})^{2}}-\frac{\mu(1-f)(3a-bc)}{-4b\mu r(1-f)+k^{2}}\\ \hskip 14.226378 pt+\left(-c+\frac{ck^{2}-\mu r(1-f)(3a-bc)}{-4b\mu r(1-f)+k^{2} }\right)\left(a-b\frac{ck^{2}-\mu r(1-f)(3a-bc)}{-4b\mu r(1-f)+k^{2}}+\frac{k^{2 }(-a+bc)}{-4b\mu r(1-f)+k^{2}}\right)\end{array}  \tag{20}\]  
Proposition 2: _Under the operating subsidy, the \(\theta\), \(p\), \(w\), and the supplier's utility are directly proportional to the government subsidy coefficient, but inversely proportional to the cost-sharing contract coefficient; under specified conditions, the utilities of the distributor, and supply chain are all in inverse proportion to the government subsidy coefficient, but in direct proportion to the cost-sharing contract coefficient._  
Proof: According to the given condition \(\theta>0\) and \(br(1-f)\mu>\frac{k^{2}}{4}\), it can be inferred that \(\theta^{*}=\frac{k(bc-a)}{k^{2}-4b\mu r(1-f)\mu}>0\), \(bc-a<0\), \(\frac{\partial\theta}{\partial f}=-\frac{4 brk(bc-a)}{(k^{2}-4b\mu r(1-f))^{2} }>0\), and \(\frac{\partial\theta}{\partial t}=\frac{4 bk\mu(bc-a)(1-f)}{(k^{2}-4b\mu r(1-f ))^{2}}<0\).