## 5 Analysis of Contract Characteristics  
High recovery and dismantling capacity. The recycler is willing to pay for a higher level of effort, and the high recovery and dismantling ability of the agent's efforts is higher than the low recovery and dismantling ability. At this time, if the entrusting party increases the revenue sharing ratio, it will positively motivate the agent's recycling efforts. **Proof. ** Solving the first-order partial derivatives of Formulae (14) and (15) with regard to recycling and dismantling ability and revenue sharing ratio:  
\[\frac{\partial e_{HH}^{*}}{\partial r_{H}}=\frac{f(v)\beta_{L}-p_{m}+p_{t}}{k}>0,\frac{\partial e_{LL}^{*}}{\partial r_{L}}=\frac{f(v)\beta_{H}-p_{m}+p_{t}}{k}>0 \tag{41}\]  
\[\frac{\partial e_{LL}^{*}}{\partial\beta_{H}}=\frac{f(v)r_{H}}{k},\frac{ \partial e_{LL}^{*}}{\partial\beta_{L}}=\frac{f(v)r_{L}}{k} \tag{42}\]  
\[\frac{\partial e_{HH}^{*}}{\partial r_{H}}-\frac{\partial e_{LL}^{*}}{ \partial r_{L}}=\frac{f(v)\left(\beta_{H}-\beta_{L}\right)}{k}>0,\frac{ \partial e_{HH}^{*}}{\partial r_{H}}>\frac{\partial e_{LL}^{*}}{\partial r_{L}} \tag{43}\]  
\(\square\)  
**Characteristic 2**. : _The revenue sharing ratio \(\beta_{j}\) is positively correlated with the recycling effort level \(e_{ij}\), with market uncertainty \(\sigma^{2}\), risk aversion degree \(\rho\), and effort cost factor \(k\), which shows a negative correlation._  
**Proof. ** This can be obtained from Equations (18) and (19):  
\[\frac{\partial\beta_{H}^{*}}{\partial r_{H}}=\frac{2k\rho\sigma^{2}r_{H}\left[ f(v)p_{n}-f(v)c_{r}-p_{t}\right]}{f(v)\left(r_{H}^{2}+k\rho\sigma^{2}\right) ^{2}}>0,\frac{\partial\beta_{H}^{*}}{\partial k}=-\frac{\rho\sigma^{2}r_{H}^{ 2}\left[f(v)p_{n}-f(v)c_{r}-p_{t}\right]}{f(v)\left(r_{H}^{2}+k\rho\sigma^{2} \right)^{2}}<0 \tag{44}\]  
\[\frac{\partial\beta_{H}^{*}}{\partial\rho}=-\frac{k\sigma^{2}r_{H}^{2}\left[ f(v)p_{n}-f(v)c_{r}-p_{t}\right]}{f(v)\left(r_{H}^{2}+k\rho\sigma^{2}\right) ^{2}}<0,\frac{\partial\beta_{H}^{*}}{\partial\sigma^{2}}=-\frac{k\rho r_{H}^{ 2}\left[f(v)p_{n}-f(v)c_{r}-p_{t}\right]}{f(v)\left(r_{H}^{2}+k\rho\sigma^{2} \right)^{2}}<0 \tag{45}\]  
\[\frac{\partial\beta_{L}^{*}}{\partial r_{L}}>0,\frac{\partial\beta_{L}^{*}}{ \partial k}<0,\frac{\partial\beta_{L}^{*}}{\partial\rho}<0,\frac{\partial \beta_{L}^{*}}{\partial\sigma^{2}}<0 \tag{46}\]  
This also shows that with the increase of market uncertainty \(\sigma^{2}\), degree of risk avoidance \(\rho\), and effort cost coefficient \(k\), the agent is willing to accept the risk reduction. \(\square\)  
**Characteristic 3**. : _With the increase in the market's proportion \(\delta\) of high recovery and dismantling capacity recyclers, with dismantling ability \(r_{H}\), the ratio of recyclers with low recovery and dismantling ability revenue sharing is reduced by \(\beta_{L}^{*}\). That is, the greater the difference between \(r_{H}\) and \(r_{L}\), the smaller the value of \(\beta_{L}^{*}\). The low recycling ability of the agent is not affected by the high capacity side, reflecting the high-end distortion in economics. At this point, the information screening contract triggered the market encroachment effect, and the income share of the agent with low capacity was distorted downward._  
**Proof. **  
\[\frac{\partial\beta_{H}^{*}}{\partial r_{L}}=0 \tag{47}\]  
\[\frac{\partial\beta_{L}^{*}}{\partial r_{H}}=\frac{2(\delta-1)\delta r_{H} \left[k\rho\sigma^{2}\left(p_{t}-p_{m}\right)-r_{L}^{2}\left(f(v)c_{r}+p_{m}-f (v)p_{n}\right)\right]}{f(v)\left(\delta r_{H}^{2}+(\delta-1)(-k)\rho\sigma^{2} +(1-2\delta)r_{L}^{2}\right)^{2}}<0 \tag{48}\]  
When the recycler is risk neutral, i.e., \(\rho\to 0\):  
\[\beta_{H}^{*}-\beta_{L}^{*}=\frac{\delta\left(r_{H}^{2}-r_{L}^{2}\right)\left[ f(v)\left(p_{n}-c_{r}\right)-p_{m}\right]}{f(v)\left(\delta r_{H}^{2}+(1-2 \delta)r_{L}^{2}\right)}>0 \tag{49}\]  
\(\square\)  
**Characteristic 4**. : _Reverne sharing ratio. The premise that the contract is effective is contingent on the fact that the recycler has risk-taking ability.