## 4 Equilibrium Analysis and Inference
### The Impacts on the Profit  
The comparison of the overall profit of merchants between the CD and DD can be articulated as follows. \[\pi^{c}-\pi^{d}=\frac{\delta_{2}[(4\delta_{1}^{2}+\delta_{1}\delta_{2}-\delta _{2}^{2})\Delta\delta+2\beta K_{2}\Delta\delta\Delta s+\beta^{2}K_{3}(\Delta s )^{2}]}{16\alpha K_{1}^{2}\Delta\delta} \tag{18}\]  
Given that \(\delta_{1}>\delta_{2}>0\) and \(\alpha,\beta,\Delta s,K_{1}\sim K_{3}\) are all greater than 0, it follows that \(\pi^{c}>\pi^{d}\). This indicates that the overall profit of merchants in the CD scenario is greater than in the DD scenario. In other words, when merchants collaborate closely to jointly determine product prices and service levels, it can yield greater benefits for the entire supply chain. A detailed analysis of the impact of various variables on the overall profit leads to the following inference. The overall profit \(\pi\) is monotonically related to the consumers' sensitivity to product price a and service level \(\beta\). The first-order derivatives of \(\pi^{c}\) and \(\pi^{d}\) with respect to \(\alpha\) and \(\beta\) are denoted as follows. \[\left\{ \begin{array}{c}\frac{\partial\pi^{c}}{\partial\alpha}=\frac{-[( \beta\Delta s+\Delta\delta)^{2}+\delta_{2}\Delta\delta]}{4\alpha^{2}\Delta \delta}\\ \frac{\partial\pi^{d}}{\partial\alpha}=\frac{-[K_{6}(\beta\Delta s)^{2}+2 \beta K_{4}\Delta\delta\Delta s+(K_{2}^{2}+2\beta K_{1})(\Delta\delta)^{2}]} {16\alpha^{2}K_{1}^{2}\Delta\delta}\\ \frac{\partial\pi^{c}}{\partial\beta}=\frac{(\beta\Delta s+\Delta\delta) \Delta s}{2\alpha\Delta\delta}\\ \frac{\partial\pi^{d}}{\partial\beta}=\frac{\Delta s(K_{5}\Delta\delta+K_{6} \beta\Delta s)}{8\alpha K_{1}^{2}\Delta\delta}\end{array} \right. \tag{19}\]  
Since \(\alpha,\beta,\delta_{i},\Delta\delta,\Delta s,K_{1}\sim K_{8}\) are all greater than 0, it follows that \(\partial\pi^{c}/\partial\alpha<0\), \(\partial\Delta\pi^{d}/\partial\alpha<0\), \(\Delta\pi^{c}/\partial\beta>0\), and \(\partial\pi^{d}/\partial\beta>0\). This indicates that the overall profit \(\pi\) is negatively correlated with consumers' price sensitivity \(\alpha\) and positively correlated with consumers' service sensitivity \(\beta\). QED. \(\square\)  
The overall profit \(\pi\) is negatively correlated with the service cost coefficient \(\eta\). The first-order derivatives of \(\pi^{c}\) and \(\pi^{d}\) with respect to \(\eta\) are denoted as follows. \[\frac{\partial\pi^{c}}{\partial\eta}=\frac{\partial\pi^{d}}{\partial\eta}=-\frac{( \Delta s)^{2}}{2} \tag{20}\]  
It can be observed that \(\partial\pi^{c}/\partial\eta<0\) and \(\partial\Delta\pi^{d}/\partial\eta<0\). This indicates that the profit of the CBEC supply chain \(\pi\) decreases as the service cost coefficient \(\eta\) increases. QED. \(\square\)  
**Theorem 6**: _The overall profit \(\pi\) is non-monotonically related to the service differentiation \(\Delta s\). The relationship between them is influenced by comprehensive factors._  
Proof of Theorem 6.: The first-order derivatives of \(\pi^{c}\) and \(\pi^{d}\) with respect to \(\Delta s\) are denoted as follows. \[\left\{ \begin{array}{c}\frac{\partial\pi^{c}}{\partial\Delta s}=\frac{ \partial\Delta s-K_{7}\Delta s}{\partial\Delta s}\\ \frac{\partial\pi^{d}}{\partial\Delta s}=\frac{\partial\Delta s-K_{8}\Delta s }{\partial\Delta s^{2}K_{1}^{2}\Delta s}\end{array} \right. \tag{21}\]  
In the CD scenario, when \(\Delta s<\beta\Delta/K_{7}\), \(\partial\pi^{c}/\partial\Delta s>0\), meaning that \(\Delta s\) is a monotonically increasing function of \(\pi^{c}\); otherwise, it is a monotonically decreasing function. In the DD scenario, when \(\Delta s<\beta K_{5}\Delta\delta/K_{8}\), \(\partial\pi^{d}/\partial\Delta s>0\), meaning that \(\Delta s\) is a monotonically increasing function of \(\pi^{d}\); otherwise, it is a monotonically decreasing function.