## 2 Materials and Methods
### Models and Solutions
#### 2.3.3 Consumption Subsidy Model  
To stimulate consumption and expand eco-product demand, the government provides purchase subsidies to consumers, reflected in the reduction inmarket price. The subsidized price becomes \(p-\varphi\), and the utility functions also change accordingly as follows:  
\[U_{2}(\pi_{S})=(w-c)[a-b\times(p-\varphi)+k\theta]-\frac{\mu\theta^{2}}{2}r \tag{21}\]  
\[U_{2}(\pi_{R})=(p-w)[a-b\times(p-\varphi)+k\theta]-\frac{\mu\theta^{2}}{2}(1-r) \tag{22}\]  
\[U_{2}(\pi_{T})=(p-c)(a-b\times(p-\varphi)+k\theta)-\frac{\mu\theta^{2}}{2} \tag{23}\]  
As with the subside-free model solution, the equations are solved by the hysteronproton method as follows:  
Similarly, let \(\frac{\partial^{2}U_{2}(\pi_{S})}{\partial w^{2}}=-b=A\), \(\frac{\partial^{2}U_{2}(\pi_{S})}{\partial w\partial\theta}=\frac{k}{2}=B\), and \(\frac{\partial^{2}U_{2}(\pi_{S})}{\partial\theta^{2}}=-\mu r=C\), it can be calculated that \(A<0\) and \(AC-B^{2}=b\mu r-\frac{k^{2}}{4}\). When \(b\mu r>\frac{k^{2}}{4}\), the results are as follows:  
\[p^{*}=\frac{a+b(w+\varphi)+k\theta}{2b}=\frac{\left(b^{2}-4b\mu r\right)\left( 3a+bc+3b\varphi\right)-3k^{2}(a-bc+b\varphi)}{4b(b^{2}-4b\mu r)} \tag{24}\]  
\[w^{*}=\frac{a+b(c+\varphi)+k\theta}{2b}=\frac{2b\left(ck^{2}-2b\mu r\right)-4 b\mu r(a+b\varphi)}{2b(k^{2}-4b\mu r)} \tag{25}\]  
\[\theta^{*}=\frac{k(bc-b\varphi-a)}{k^{2}-4b\mu r} \tag{26}\]  
\[\begin{array}{c}U_{2}^{*}(\pi_{S})=-\frac{k^{2}\mu r(-a+bc-b\varphi)^{2}}{2( -4b\mu r+k^{2})^{2}}\\ \hskip 14.226378 pt+\left\{a-b\left[-\frac{3 bk^{2}\left(b^{2}-4b\mu r\right)(a- bc+b\phi)}{4}-\phi+\left(b^{2}-4b\mu r\right)(3a+bc+3b\phi\right)\right]\\ \hskip 14.226378 pt+\frac{k^{2}(-a+bc-b\phi)}{-4b\mu r+k^{2}}\right\}\left[-2b^{2 }\mu r(a+b\phi)\left(-4b\mu r+k^{2}\right)+2b\left(-2b\mu r+ck^{2}\right)-c \right]\end{array} \tag{27}\]\[U_{2}^{s}(\pi_{R})=-\frac{k^{2}\mu(1-f)((1-\tau)-(a+bc-b\varphi)^{2} }{2(-4b\mu\tau+k^{2})^{2}} \tag{28}\] \[+\bigg{\{}a-b\bigg{[}\frac{-34b^{2}\left(b^{2}-4b\mu\tau\right)(a- bc+b\varphi)}{4}+\left(b^{2}-4b\mu\tau\right)(3a+bc+3b\varphi)\bigg{]}+\frac{k^{2} (-a+bc-b\varphi)}{-4b\mu\tau+k^{2}}\bigg{\}}\] \[\times\bigg{[}2b^{2}\mu(a+b\varphi)\big{(}-4b\mu\tau+k^{2}\big{)} -\frac{3b^{2}\left(b^{2}-4b\mu\tau\right)(a-bc+b\varphi)}{4}-2b\big{(}-2b\mu \tau+ck^{2}\big{)}+\left(b^{2}-4b\mu\tau\right)(3a+bc+3b\varphi)\bigg{]}\]  
\[U_{2}^{s}(\pi_{T})=-\frac{k^{2}\mu(1-f)(-a+bc-b\varphi)^{2}}{2(-4b\mu\tau+k^{ 2})^{2}}\delta \tag{29}\] \[+\bigg{\{}a-b\bigg{[}\frac{-34b^{2}\left(b^{2}-4b\mu\tau\right)( a-bc+b\varphi)}{4}+\left(b^{2}-4b\mu\tau\right)(3a+bc+3b\varphi)\bigg{]}+ \frac{k^{2}(-a+bc-b\varphi)}{-4b\mu\tau+k^{2}}\bigg{\}}\] \[\times\bigg{[}-\frac{3b^{2}\left(b^{2}-4b\mu\tau\right)(a-bc+b \varphi)}{4}-c+\left(b^{2}-4b\mu\tau\right)(3a+bc+3b\varphi)\bigg{]}\]  
**Proposition 3**: _Under the consumption subsidy, all the decision variables are proportionate to the increased amount of subsidy \(\varphi\)._  
Due to \(\theta>0\) and \(b\mu\tau>\frac{k^{2}}{4}\), \(k^{2}-4b\mu\tau<0\), it can be inferred that \(\theta^{*}=\frac{k(bc-b\varphi-a)}{k^{2}-4b\mu\tau}>0\), \(bc-b\varphi-a<0\), and \(\frac{3\theta}{\theta\tau}=-\frac{4b\mu(bc-b\varphi-a)}{\left(k^{2}-4b\mu\tau \right)^{2}}>0\); furthermore, \(\frac{\partial p}{\partial\mu}>0\), \(\frac{3\theta\tau}{\partial\tau}>0\), \(\frac{\partial U_{2}(\pi_{R})}{\partial\tau}>0\), \(\frac{\partial U_{2}(\pi_{R})}{\partial\tau}>0\) and \(\frac{\partial U_{2}(\pi_{T})}{\partial\tau}>0\). \(\square\)