## 2 Materials and Methods
### Models and Solutions  
#### 2.3.1 Subsidy-Free Model  
The eco-product supply chain is a sequential game where the supplier has a pricing priority based on product costs, and the distributor can determine the selling price after referencing the wholesale price. The utility functions of the supplier, distributor, and supply chain are described as follows:  
\[U_{0}(\pi_{S})=(w-c)(a-bp+k\theta)-\frac{\mu\theta^{2}}{2}r \tag{1}\]  
\[U_{0}(\pi_{R})=(p-w)(a-bp+k\theta)-\frac{\mu\theta^{2}}{2}(1-r) \tag{2}\]  
\[U_{0}(\pi_{T})=(p-c)(a-bp+k\theta)-\frac{\mu\theta^{2}}{2} \tag{3}\]  
The hysteron-proteron method is used to solve the first derivative and the second derivative of the market price \(p\) according to Equation (2), and the reaction function of the distributor and the market price can be obtained. For \(\frac{\partial U_{0}(\pi_{R})}{\partial p^{2}}=-2b<0\), by equating \(\frac{\partial U_{0}(\pi_{R})}{\partial p}\) to zero, the optimal pricing of the market price can be obtained:  
\[\frac{\partial U_{0}(\pi_{R})}{\partial p}=0\Rightarrow p_{0}^{*}=\frac{a+bw +k\theta}{2b} \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Name** & **Symbol** & **Name** \\ \hline \(\pi_{S}\) & supplier’s profit & \(U(\pi_{S})\) & supplier utility \\ \(\pi_{R}\) & distributor’s profit & \(U(\pi_{R})\) & distributor utility \\ \(\pi_{T}\) & total profit of supply chain & \(U(\pi_{T})\) & supply chain utility \\ \(q\) & market demand for ecological products & \(\mu\) & cost coefficient of ecological conservation \\  & & & and restoration \\ \(a\) & potential market size, \(a>0\) & \(r\) & ratio of suppliers sharing the ecological \\  & & & conservation and restoration cost \\ \(b\) & coefficient of sensitivity of price, \(b>0\) & \(1-r\) & ratio of distributors sharing the ecological \\  & & & conservation and restoration cost \\ \(k\) & consumer preferences of eco-products, \(k>0\) & \(\delta\) & the cost-sharing contract coefficient between \\ \(\theta\) & ecological level of supply chain, \(\theta>0\) & \(E(f)\) & supplier and distributor \\  & & & rate of subsidization based on ecological \\ \(w\) & wholesale price of supplier & \(f\) & conservation and restoration cost of supplier \\  & & & and distributor, \(f\in\) (0,1) \\ \(p\) & market price of distributor & \(E(\varphi)\) & the total amount of the consumption subsidy \\ \(c\) & cost of eco-product production, \(p\geq w\geq c\) & \(\varphi\) & subsidized price for lowering each eco-product \\  & & & price in the market, \(\varphi>0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variable declaration of the model. With simultaneous Equations (2) and (4), the revenue of the supplier can be calculated as:  
\[U_{0}(\pi_{S})=\frac{(w-c)(a-bw+k\theta)}{2}-\frac{\mu\theta^{2}}{2}r \tag{5}\]  
Then, derivations of Equation (5) regarding \(w\) and \(\theta\) are taken separately, and produce the following results:  
\[\frac{\partial U_{0}(\pi_{S})}{\partial w}=\frac{a-2 bw+k\theta+bc}{2} \tag{6}\]  
\[\frac{\partial U_{0}(\pi_{S})}{\partial\theta}=\frac{(w-c)k}{2}-\mu\theta r \tag{7}\]  
After setting \(\frac{\partial^{2}U_{0}(\pi_{S})}{\partial w^{2}}=-b=A\), \(\frac{\partial^{2}U_{0}(\pi_{S})}{\partial w^{2}\partial\theta^{2}}=\frac{k}{ 2}=B\), \(\frac{\partial^{2}U_{0}(\pi_{S})}{\partial\theta^{2}}=-\mu r=C\), it can obtain \(A<0\), \(AC-B^{2}=b\mu r-\frac{k^{2}}{4}\), when \(b\mu r>\frac{k^{2}}{4}\), the Hessian matrix of \(U_{0}(\pi_{S})\) is negative, which means \(U_{0}(\pi_{S})\) is a concave function, and it has unique optimal \(w\) and \(\theta\) making the distributor receive the maximum benefit. Hence, let:  
\[ \begin{cases}\frac{\partial U_{0}(\pi_{S})}{\partial w}=0\\ \frac{\partial U_{0}(\pi_{S})}{\partial\theta}=0\end{cases} \Longrightarrow \begin{cases}w^{*}=\frac{ck^{2}-2\mu r(a+k\zeta)}{k^{2}-4b\mu r}\\ \theta^{*}=\frac{k(bc-a)}{k^{2}-4b\mu r}\end{cases}