# Competing Reverse Channels' Performance with Sustainable Recycle Innovation Input
## Appendix A Proofs for Theorems, Lemmas and Propositions
### Proof for Theorem 2  
In model \(BE\), we solve for the collector's profit function first.  
\[\Pi_{\text{C}}^{B}(\tau,e)=b\tau D(p)-A\tau D(p)-\frac{H}{2}\tau^{2}-\frac{R} {2}e^{2}\]  
The first order condition of \(\Pi_{\text{C}}(\tau,e)\) on \(\tau\) and \(e\) are  
\[\frac{\partial\Pi_{\text{C}}(\tau,e)}{\partial\tau}= (b-A)(\alpha-\beta p)-H\tau\] \[\frac{\partial\Pi_{\text{C}}(\tau,e)}{\partial e}= K(b-A)\tau-Re\]  
The second order condition of \(\Pi_{\text{C}}(\tau,e)\) on \(\tau\) and \(e\) are  
\[\frac{\partial^{2}\Pi_{\text{C}}(\tau,e)}{\partial e^{2}}= -R<0\] \[\frac{\partial^{2}\Pi_{\text{C}}(\tau,e)}{\partial\tau^{2}}= -H<0\] \[\frac{\partial^{2}\Pi_{\text{C}}(\tau,e)}{\partial\tau\partial e}= K(b-A)\]  
So the determinant is \(RH-K^{2}(b-A)^{2}\).  
We assume that \(RH-K^{2}(b-A)^{2}>0\). Thus the collector's profit function is jointly strictly concave in \(\tau\) and \(e\).  
We assume the first order condition equations above are 0, then we get  
\[\tau(b,p)= \frac{R(b-A)(\alpha-\beta p)}{RH-K^{2}(b-A)^{2}}\] \[e(b,p)= \frac{K(b-A)^{2}(\alpha-\beta p)}{RH-K^{2}(b-A)^{2}}\]  
Solving for the manufacturer's profit function  
\[\Pi_{M}^{BE}(p,b) =\left[p-c_{m}+(c_{m}-c_{r}-b)\,\tau\right]D(p,e)\] \[=\left[p-c_{m}+(\Delta-b)\tau\right]D(p,e)\]  
We substitute the values of \(\tau\) and \(e\) into the above equation and derive  
\[\Pi_{M}(p,b)=\left[p-c_{m}+(\Delta-b)\frac{R(b-A)(\alpha-\beta p)}{RH-K^{2}(b- A)^{2}}\right]\frac{RH(\alpha-\beta p)}{RH-K^{2}(b-A)^{2}}\]  
The first order condition of \(\Pi_{M}(p,b)\) on \(p\) and \(b\) are  
\[\frac{\partial\Pi_{M}(p,b)}{\partial b}= \frac{R(\Delta-2b+A)\left[RH-K^{2}(b-A)^{2}\right]}{2K^{2}(b-A)}+ 2R(\Delta-b)(b-A)+\frac{(p-cm)\left[RH-K^{2}(b-A)^{2}\right]}{(\alpha-\beta p)}\] \[\frac{\partial\Pi_{M}(p,b)}{\partial p}= \frac{RH\left(\alpha-2\beta p+\beta c_{m}\right)}{RH-K^{2}(b-A)^{ 2}}+2(\Delta-b)\frac{-\beta R(b-A)}{RH-K^{2}(b-A)^{2}}\frac{RH(\alpha-\beta p )}{RH-K^{2}(b-A)^{2}}\]  
The second order condition of \(\Pi_{M}(p,b)\) on \(p\) and \(b\) are  
\[\frac{\partial^{2}\Pi_{M}(p,b)}{\partial b_{1}^{2}}= -\frac{2 RH\left(\alpha-\beta c_{m}\right)^{2}}{[2M-2\beta RN]^{2} }\left\{\frac{\beta R-K^{2}}{\beta}+\frac{K^{4}}{M^{2}}\frac{(\Delta-A)^{2}( \beta R)^{2}}{4\left(\beta R-K^{2}\right)^{2}}\left[\frac{4M-4\beta RN}{\beta }\right]\right\}<0\] \[\frac{\partial^{2}\Pi_{M}(p,b)}{\partial p^{2}}= -2\beta[M-\beta RN]\frac{RH}{M^{2}}<0\] \[\frac{\partial^{2}\Pi_{M}(p,b)}{\partial p\partial b_{1}}= \left(\alpha-\beta c_{m}\right)K^{2}\frac{(\Delta-A)\beta R}{( \beta R-K^{2})}\frac{RH}{M^{2}}\]  
So the determinant is  
\[\left(\beta R-K^{2}\right)\frac{\left(RH\right)^{2}\left(\alpha-\beta c_{m} \right)^{2}}{M^{2}[M-\beta RN]}>0\]  
So we assume \(M-\beta RN>0\) and \(\beta R-K^{2}>0\).  
Because the Hessian H is negative definite, thus manufacturer's profit function is jointly concave in \(b\) and \(p\).  
We assume the first order condition equations above are 0, then we get  
\[b^{BE*}= \frac{\beta R(\Delta+A)-2K^{2}A}{2\beta R-2K^{2}}\] \[p^{BE*}= \frac{\left(\alpha+\beta c_{m}\right)M-2\alpha\beta RN}{2\beta M- 2\beta^{2}RN}\]  
where \(M=RH-\frac{K^{2}\beta^{2}R^{2}(\Delta-A)^{2}}{4(\beta R-K^{2})^{2}}\), \(N=\frac{\beta R\left(\beta R-2K^{2}\right)(\Delta-A)^{2}}{4(\beta R-K^{2})^{2}}\).  
The optimal collection rate and innovation input are obtained as  
\[\tau^{BE*}= \frac{\beta R}{\beta R-K^{2}}*\frac{R(\Delta-A)\left(\alpha-\beta c_ {m}\right)}{4M-4\beta RN}\] \[e^{BE*}= \frac{K}{2}\frac{\left[\beta R\right]^{2}}{\left[\beta R-K^{2} \right]^{2}}\frac{\left(\Delta-A\right)^{2}\left(\alpha-\beta c_{m}\right)}{4M -4\beta RN}\]  
where \(M=RH-\frac{K^{2}\beta^{2}R^{2}\left(\Delta-A\right)^{2}}{4\left(\beta R-K^{2} \right)^{2}}\), \(N=\frac{\beta R\left(\beta R-2K^{2}\right)\left(\Delta-A\right)^{2}}{4\left( \beta R-K^{2}\right)^{2}}\).  
From the above equilibrium values we derive the collector's profit, manufacturer's profit and the supply chain's profit.