Will the community O2O service supply channel benefit the elderly healthcare service supply chain?

Abstract

This paper studies the coordination of the elderly healthcare service supply chain (EHSSC) within a dual-channel setting. A game-theoretic framework is introduced to analyse how best the elderly service integrator (ESI) and elderly service provider (ESP) would make decisions and interact amidst cooperation and competition. Three cases including the centralised case, the Bertrand game case and the Stackelberg game case are examined to obtain the optimal relationship structure suitable for the EHSSC coordination. The associated loss due to customer dissatisfaction and the value increment due to the ESI’s service enhancement effort are incorporated in the model. Also, by considering customer’s health status, optimal pricing strategies are investigated for both the ESI and ESP under two different service demand classes. The main results are summarised as follows. First, the value increment due to the ESI’s service enhancement effort may have different impacts on the ESP’s prices in different service demand classes. Second, the whole supply chain would achieve effective coordination when the ESP acts as the Stackelberg leader with an appropriate setting of the wholesale price. Third, a lower wholesale price set by the ESP could benefit both the ESI and ESP in achieving greater demand and profit.

Appendix

Proposition 1

Taking Eqs. (1) and (2) into $\pi_{ip}$, $\pi_{ip} = \mathop \sum \limits_{k = I,II} \left\{ {\left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{i}^{k} - c_{p}^{k} } \right) - \gamma e^{2} } \right]D_{i}^{k} + \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{p}^{k} - c_{p}^{k} } \right)D_{p}^{k} } \right\} - \lambda s^{2} ,$ then,

$$\frac{{\partial^{2} \pi_{ip} }}{{\partial p_{i}^{2} }} = - \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {b_{k} + \delta } \right) < 0,$$

$$\frac{{\partial^{2} \pi_{ip} }}{{\partial p_{p}^{2} }} = - 2\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {b_{k} + \delta } \right) < 0,$$

where $k = I,II$.

$$\frac{{\partial \pi_{ip} }}{{\partial p_{i} }} = \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left[ {\left( {1 - {{\varphi }}} \right)a_{k} - b_{k} p_{i}^{k} + \theta s + ce + \delta \left( {p_{p}^{k} - p_{i}^{k} + e} \right)} \right] - \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{i}^{k} - c_{p}^{k} } \right) - \gamma e^{2} } \right]\left( {b_{k} + \delta } \right) + \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{p}^{k} - c_{p}^{k} } \right)\delta$$

$$\frac{{\partial \pi_{ip} }}{{\partial p_{p} }} = \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{i}^{k} - c_{p}^{k} } \right) - \gamma e^{2} } \right]\delta + \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left[ {{{\varphi }}a_{k} - b_{k} p_{p}^{k} + \theta s + \delta \left( {p_{i}^{k} - p_{p}^{k} - e} \right)} \right] - \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{p}^{k} - c_{p}^{k} } \right)\left( {b_{k} + \delta } \right)$$

$$\frac{{\partial \pi_{ip} }}{{\partial p_{i} }} = 0$$

$$\frac{{\partial \pi_{ip} }}{{\partial p_{p} }} = 0$$

$$p_{i}^{k*} = \frac{{\left( {b_{k} + \delta - \varphi b_{k} } \right)a_{k} + \left( {b_{k} c + \delta c + b_{k} \delta } \right)e}}{{2\left( {b_{k}^{2} + 2\delta b_{k} } \right)}} + \frac{{c_{p}^{k} }}{2} + \frac{{\gamma e^{2} }}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]}} + \frac{\theta s}{{2b_{k} }}$$

$$p_{p}^{k*} = \frac{{\left( {\delta + \varphi b_{k} } \right)a_{k} + \left( {c - b_{k} } \right)\delta e}}{{2\left( {b_{k}^{2} + 2\delta b_{k} } \right)}} + \frac{{c_{p}^{k} }}{2} + \frac{\theta s}{{2b_{k} }}$$

where $k = I,II$.

Corollary 1

According to results from Proposition 1,

$$\frac{{\partial p_{p}^{k*} }}{{\partial {\text{s}}}} = \frac{\theta }{{2b_{k} }} > 0,$$

$$\frac{{\partial p_{i}^{k*} }}{{\partial {\text{s}}}} = \frac{\theta }{{2b_{k} }} - \frac{{\gamma e^{2} \varepsilon }}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]^{2} }}.$$

Then, If $\theta < \varepsilon \gamma e^{2} b_{k} ,$ then $\sqrt {\frac{{\gamma e^{2} b_{k} }}{\varepsilon \theta }} + 1 - \frac{1}{\varepsilon } > 1$. Since $s \in \left[ {0,1} \right] ,$ we can get $s < \sqrt {\frac{{\gamma e^{2} b_{k} }}{\varepsilon \theta }} + 1 - \frac{1}{\varepsilon }$. Thus, $\frac{{\partial p_{i}^{k*} }}{{\partial {\text{s}}}} < 0$. If $\varepsilon \gamma e^{2} b_{k} < \theta < \frac{{\varepsilon \gamma e^{2} b_{k} }}{{\left( {1 - \varepsilon } \right)^{2} }} ,$ then $0 < \sqrt {\frac{{\gamma e^{2} b_{k} }}{\varepsilon \theta }} + 1 - \frac{1}{\varepsilon } < 1$. Thus, when $0 \le s < \sqrt {\frac{{\gamma e^{2} b_{k} }}{\varepsilon \theta }} + 1 - \frac{1}{\varepsilon } ,$ then $\frac{{\partial p_{i}^{k*} }}{{\partial {\text{s}}}} < 0$; when $\sqrt {\frac{{\gamma e^{2} b_{k} }}{\varepsilon \theta }} + 1 - \frac{1}{\varepsilon } < s < 1,$ then $\frac{{\partial p_{i}^{k*} }}{{\partial {\text{s}}}} > 0$. If $\theta > \frac{{\varepsilon \gamma e^{2} b_{k} }}{{\left( {1 - \varepsilon } \right)^{2} }} ,$ then $\sqrt {\frac{{\gamma e^{2} b_{k} }}{\varepsilon \theta }} + 1 - \frac{1}{\varepsilon } < 0 ,$ Since $s \in \left[ {0,1} \right] ,$ we can get $s > \sqrt {\frac{{\gamma e^{2} b_{k} }}{\varepsilon \theta }} + 1 - \frac{1}{\varepsilon }$. Thus, $\frac{{\partial p_{i}^{k*} }}{{\partial {\text{s}}}} > 0.$

Corollary 2

According to results from Proposition 1,

$$\frac{{\partial p_{i}^{k*} }}{\partial e} = \frac{{\left( {b_{k} c + \delta c + b_{k} \delta } \right)}}{{2\left( {b_{k}^{2} + 2\delta b_{k} } \right)}} + \frac{\gamma e}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]}} > 0$$

$$\frac{{\partial p_{p}^{k*} }}{\partial e} = \frac{{\left( {c - b_{k} } \right)\delta }}{{2\left( {b_{k}^{2} + 2\delta b_{k} } \right)}}$$

Here, since $b_{I} > b_{II} ,$ if $c > b_{I} > b_{II} ,$ $\frac{{\partial p_{p}^{I*} }}{{\partial {\text{e}}}} > 0$ and $\frac{{\partial p_{p}^{II*} }}{{\partial {\text{e}}}} > 0 ,$ if $b_{I} > c > b_{II} ,$ $\frac{{\partial p_{p}^{I*} }}{{\partial {\text{e}}}} < 0$ and $\frac{{\partial p_{p}^{II*} }}{{\partial {\text{e}}}} > 0$; if $b_{I} > b_{II} > c$, $\frac{{\partial p_{p}^{I*} }}{{\partial {\text{e}}}} < 0$ and $\frac{{\partial p_{p}^{II*} }}{{\partial {\text{e}}}} < 0.$

Proposition 2

Since $\frac{{\partial^{2} \pi_{i} }}{{\partial p_{i}^{2} }} = - 2\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {b_{k} + \delta } \right) < 0 ,$ $\frac{{\partial^{2} \pi_{p} }}{{\partial p_{p}^{2} }} = - 2\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {b_{k} + \delta } \right) < 0,$ where $k = I,II.$

Then, the optimal selling prices for the ESI and ESP can be derived by the following equations:

$$\frac{{\partial \pi_{i} }}{{\partial p_{i} }} = \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left[ {\left( {1 - {{\varphi }}} \right)a_{k} - b_{k} p_{i}^{k} + \theta s + ce + \delta \left( {p_{p}^{k} - p_{i}^{k} + e} \right)} \right] - \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{i}^{k} - w_{p}^{k} } \right) - \gamma e^{2} } \right]\left( {b_{k} + \delta } \right) = 0$$

$$\frac{{\partial \pi_{p} }}{{\partial p_{p} }} = \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left[ {{{\varphi }}a_{k} - b_{k} p_{p}^{k} + \theta s + \delta \left( {p_{i}^{k} - p_{p}^{k} - e} \right)} \right] - \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{p}^{k} - c_{p}^{k} } \right)\left( {b_{k} + \delta } \right) + \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {w_{p}^{k} - c_{p}^{k} } \right)\delta = 0$$

From above equations, the optimal selling prices ($p_{iB}^{k}$ and $p_{pB}^{k}$) for the ESI and ESP are obtained as follows:

$$p_{iB}^{k} = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{1}^{k} + 2\left( {\delta + b_{k} } \right)^{2} \gamma e^{2} }}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} + \frac{\theta s}{{2b_{k} + \delta }},$$

$$p_{pB}^{k} = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{2}^{k} + \delta \left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} + \frac{\theta s}{{2b_{k} + \delta }},$$

where $f_{1}^{k} = \left( {2b_{k} + 2\delta - \delta \varphi - 2\varphi b_{k} } \right)a_{k} + \left( {2cb_{k} + 2c\delta + 2\delta b_{k} + \delta^{2} } \right)e + \left[ {2\left( {\delta + b_{k} } \right)^{2} + \delta^{2} } \right]w_{p}^{k} + \delta b_{k} c_{p}^{k} ,$ $f_{2}^{k} = \left( {\delta + \delta \varphi + 2\varphi b_{k} } \right)a_{k} + \left( {\delta c - 2\delta b_{k} - \delta^{2} } \right)e + 3\delta \left( {\delta + b_{k} } \right)w_{p}^{k} + 2b_{k} \left( {\delta + b_{k} } \right)c_{p}^{k} ,$ $k = I,II.$

Corollary 3

According to results from Proposition 2,

$$\frac{{\partial p_{pB}^{k*} }}{{\partial {\text{s}}}} = \frac{\theta }{{2b_{k} + \delta }} - \frac{{\delta \left( {\delta + b_{k} } \right)\gamma e^{2} \varepsilon }}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]^{2} \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}},$$

$$\frac{{\partial p_{iB}^{k*} }}{{\partial {\text{s}}}} = \frac{\theta }{{2b_{k} + \delta }} - \frac{{2\left( {\delta + b_{k} } \right)^{2} \gamma e^{2} \varepsilon }}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]^{2} \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}.$$

If $\theta < \frac{{\delta \varepsilon \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} ,$ $\sqrt {\frac{{\delta \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon } > 1 ,$ Since $s \in \left[ {0,1} \right] ,$ thus $s < \sqrt {\frac{{\delta \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon }$. Then, we can get $\frac{{\partial p_{pB}^{k*} }}{{\partial {\text{s}}}} < 0.$

If $\frac{{\delta \varepsilon \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} < \theta < \frac{{\delta \varepsilon \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]\left( {1 - \varepsilon } \right)^{2} }},$ then $0 < \sqrt {\frac{{\delta \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon } < 1$. Since $s \in \left[ {0,1} \right] ,$ when $0 \le s < \sqrt {\frac{{\delta \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon } ,$ then $\frac{{\partial p_{pB}^{k*} }}{{\partial {\text{s}}}} < 0$; when $\sqrt {\frac{{\delta \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon } < s < 1 ,$ then $\frac{{\partial p_{pB}^{k*} }}{{\partial {\text{s}}}} > 0.$

If $\theta > \frac{{\delta \varepsilon \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]\left( {1 - \varepsilon } \right)^{2} }} ,$ then $\sqrt {\frac{{\delta \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon } < 0 ,$ Since $s \in \left[ {0,1} \right] ,$ thus $s > \sqrt {\frac{{\delta \left( {2b_{k} + \delta } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon }$. Then, we can get $\frac{{\partial p_{pB}^{k*} }}{{\partial {\text{s}}}} > 0.$

Similarly, for the ESI, we can get that if $\theta < \frac{{2\left( {\delta + b_{k} } \right)^{2} \left( {2b_{k} + \delta } \right)\varepsilon \gamma e^{2} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} ,$ then $\frac{{\partial p_{iB}^{k*} }}{{\partial {\text{s}}}} < 0.$

If $\frac{{2\left( {\delta + b_{k} } \right)^{2} \left( {2b_{k} + \delta } \right)\varepsilon \gamma e^{2} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} < \theta < \frac{{2\left( {\delta + b_{k} } \right)^{2} \left( {2b_{k} + \delta } \right)\varepsilon \gamma e^{2} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]\left( {1 - \varepsilon } \right)^{2} }} ,$ when $0 \le s < \sqrt {\frac{{2\left( {\delta + b_{k} } \right)^{2} \left( {2b_{k} + \delta } \right)\varepsilon \gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon } ,$ then $\frac{{\partial p_{iB}^{k*} }}{{\partial {\text{s}}}} < 0$; when $\sqrt {\frac{{2\left( {\delta + b_{k} } \right)^{2} \left( {2b_{k} + \delta } \right)\varepsilon \gamma e^{2} }}{{\varepsilon \theta \left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}} + 1 - \frac{1}{\varepsilon } < s < 1 ,$ then $\frac{{\partial p_{iB}^{k*} }}{{\partial {\text{s}}}} > 0.$

If $\theta > \frac{{2\left( {\delta + b_{k} } \right)^{2} \left( {2b_{k} + \delta } \right)\varepsilon \gamma e^{2} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]\left( {1 - \varepsilon } \right)^{2} }} ,$ then $\frac{{\partial p_{iB}^{k*} }}{{\partial {\text{s}}}} > 0.$

Corollary 4

According to Proposition 2,

$$\frac{{\partial p_{iB}^{k*} }}{\partial e} = \frac{{2cb_{k} + 2c\delta + 2\delta b_{k} + \delta^{2} }}{{4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} }} + \frac{{4\left( {\delta + b_{k} } \right)^{2} \gamma e}}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} > 0,$$

$$\frac{{\partial p_{pB}^{k*} }}{\partial e} = \frac{{\delta \left( {c - 2b_{k} - \delta } \right)}}{{4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} }} + \frac{{2\delta \left( {\delta + b_{k} } \right)\gamma e}}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}}.$$

When $\frac{{\partial p_{pB}^{k*} }}{\partial e} = 0 ,$ $e = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {2b_{k} + \delta - c} \right)}}{{2\gamma \left( {\delta + b_{k} } \right)}} ,$ namely, $e_{I} = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {2b_{I} + \delta - c} \right)}}{{2\gamma \left( {\delta + b_{I} } \right)}} ,$ $e_{II} = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {2b_{II} + \delta - c} \right)}}{{2\gamma \left( {\delta + b_{II} } \right)}} ,$ then $e_{I} > e_{II} .$

Here, if $e > e_{I}$, $\frac{{\partial p_{pB}^{I*} }}{{\partial {\text{e}}}} > 0$ and $\frac{{\partial p_{pB}^{II*} }}{{\partial {\text{e}}}} > 0$; if $e_{I} > e > e_{II}$, $\frac{{\partial p_{pB}^{I*} }}{{\partial {\text{e}}}} < 0$ and $\frac{{\partial p_{pB}^{II*} }}{{\partial {\text{e}}}} > 0$; if $e_{I} > e_{II} > e$, $\frac{{\partial p_{pB}^{I*} }}{{\partial {\text{e}}}} < 0$ and $\frac{{\partial p_{pB}^{II*} }}{{\partial {\text{e}}}} < 0.$

Proposition 3

According to the results from Proposition 1and 2, when $p_{i}^{k*} = p_{iB}^{k} ,$ namely, $\frac{{\left( {b_{k} + \delta - \varphi b_{k} } \right)a_{k} + \left( {b_{k} c + \delta c + b_{k} \delta } \right)e}}{{2\left( {b_{k}^{2} + 2\delta b_{k} } \right)}} + \frac{{c_{p}^{k} }}{2} + \frac{{\gamma e^{2} }}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]}} + \frac{\theta s}{{2b_{k} }} = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{1}^{k} + 2\left( {\delta + b_{k} } \right)^{2} \gamma e^{2} }}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} + \frac{\theta s}{{2b_{k} + \delta }},$ then,

$$w_{p1}^{k} = \frac{{\delta \left\{ {\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{3}^{k} - \delta \left( {b_{k}^{2} + 2\delta b_{k} } \right)\gamma e^{2} } \right\}}}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {b_{k}^{2} + 2\delta b_{k} } \right)\left[ {2\left( {\delta + b_{k} } \right)^{2} + \delta^{2} } \right]}} + \frac{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right] - 2\delta b_{k} }}{{2\left[ {2\left( {\delta + b_{k} } \right)^{2} + \delta^{2} } \right]}}c_{p}^{k} + \frac{{\left( {2b_{k} + 3\delta } \right)\delta \theta s}}{{2b_{k} \left[ {2\left( {\delta + b_{k} } \right)^{2} + \delta^{2} } \right]}}.$$

Moreover, if $w_{p}^{k} > w_{p1}^{k}$, $p_{i}^{k*} < p_{iB}^{k*}$; if $w_{p}^{k} < w_{p1}^{k}$, $p_{i}^{k*} > p_{iB}^{k*} .$

When $p_{p}^{k*} = p_{pB}^{k*}$, namely, $\frac{{\left( {\delta + \varphi b_{k} } \right)a_{k} + \left( {c - b_{k} } \right)\delta e}}{{2\left( {b_{k}^{2} + 2\delta b_{k} } \right)}} + \frac{{c_{p}^{k} }}{2} + \frac{\theta s}{{2b_{k} }} = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{2}^{k} + \delta \left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} + \frac{\theta s}{{2b_{k} + \delta }},$ then,

$$w_{p2}^{k} = \frac{{f_{4}^{k} - 2\left( {b_{k}^{2} + 2\delta b_{k} } \right)\left( {\delta + b_{k} } \right)\gamma e^{2} }}{{6\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {b_{k}^{2} + 2\delta b_{k} } \right)\left( {\delta + b_{k} } \right)}} + \frac{{3\delta + 4b_{k} }}{{6\left( {\delta + b_{k} } \right)}}c_{p}^{k} + \frac{{\left( {2b_{k} + 3\delta } \right)\theta s}}{{6b_{k} \left( {\delta + b_{k} } \right)}}.$$

Moreover, if $w_{p}^{k} > w_{p2}^{k} ,$ $p_{p}^{k*} < p_{pB}^{k*}$; if $w_{p}^{k} < w_{p2}^{k} ,$ $p_{p}^{k*} > p_{pB}^{k*} ,$ where

$$f_{1}^{k} = \left( {2b_{k} + 2\delta - \delta \varphi - 2\varphi b_{k} } \right)a_{k} + \left( {2cb_{k} + 2c\delta + 2\delta b_{k} + \delta^{2} } \right)e + \left[ {2\left( {\delta + b_{k} } \right)^{2} + \delta^{2} } \right]w_{p}^{k} + \delta b_{k} c_{p}^{k} ,$$

$$f_{2}^{k} = \left( {\delta + \delta \varphi + 2\varphi b_{k} } \right)a_{k} + \left( {\delta c - 2\delta b_{k} - \delta^{2} } \right)e + 3\delta \left( {\delta + b_{k} } \right)w_{p}^{k} + 2b_{k} \left( {\delta + b_{k} } \right)c_{p}^{k} ,$$

$$f_{3}^{k} = \left( {3\delta b_{k} + 3\delta^{2} + 2\varphi b_{k}^{2} + \varphi \delta b_{k} } \right)a_{k} + \left( {3c\delta^{2} - \delta^{2} b_{k} - 2\delta b_{k}^{2} + 3c\delta b_{k} } \right)e,$$

$$f_{4}^{k} = \left( {4\delta b_{k} + 3\delta^{2} - 2\varphi b_{k}^{2} - \varphi \delta b_{k} + 2b_{k}^{2} } \right)a_{k} + \left( {4c\delta b_{k} + \delta^{2} b_{k} + 2\delta b_{k}^{2} + 3c\delta^{2} + 2cb_{k} } \right)e,\;k = I,II.$$

Proposition 4

Set $D_{iB}^{k*} = D_{i}^{k*} \left( {p_{iB}^{k*} } \right)$ and $D_{pB}^{k*} = D_{i}^{k*} \left( {p_{pB}^{k*} } \right) ,$ taking $p_{i}^{k*}$ and $p_{p}^{k*} ,$ $p_{iB}^{k*}$ and $p_{pB}^{k*}$ into $\left( {D_{i}^{k*} + D_{p}^{k*} } \right)$ and $\left( {D_{iB}^{k*} + D_{pB}^{k*} } \right),$

when $\left( {D_{i}^{k*} + D_{p}^{k*} } \right) - \left( {D_{iB}^{k*} + D_{pB}^{k*} } \right) = b_{k} p_{iB}^{k} + b_{k} p_{pB}^{k} - b_{k} p_{i}^{k} - b_{k} p_{p}^{k} > 0 ,$ there exist threshold wholesale prices $w_{p3}^{k} ,$ namely,

$$w_{p3}^{k} = \frac{{\delta \left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {a_{k} + ce} \right) - \delta b_{k} \gamma e^{2} }}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {b_{k}^{2} + 2\delta b_{k} } \right)}} + \frac{{b_{k} + \delta }}{{b_{k} + 2\delta }}c_{p}^{k} + \frac{\delta \theta s}{{b_{k} \left( {b_{k} + 2\delta } \right)}}.$$

Proposition 5

Since $\frac{{\partial^{2} \pi_{i} }}{{\partial p_{i}^{2} }} = - 2\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {b_{k} + \delta } \right) < 0 ,$ where $k = I,II.$

$$\frac{{\partial \pi_{i} }}{{\partial p_{i} }} = \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left[ {\left( {1 - {{\varphi }}} \right)a_{k} - b_{k} p_{i}^{k} + \theta s + ce + \delta \left( {p_{p}^{k} - p_{i}^{k} + e} \right)} \right] - \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{i}^{k} - w_{p}^{k} } \right) - \gamma e^{2} } \right]\left( {b_{k} + \delta } \right) = 0$$

$$p_{iS}^{k} = \frac{{\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left[ {\left( {1 - {{\varphi }}} \right)a_{k} + \theta s + ce + \delta \left( {p_{p}^{k} + e} \right)} \right] + \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)w_{p}^{k} + \gamma e^{2} } \right]\left( {b_{k} + \delta } \right)}}{{2\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {b_{k} + \delta } \right)}}$$

Taking $p_{iS}^{k}$ into $\pi_{p} ,$

$$\frac{{\partial \pi_{p} }}{{\partial p_{p} }} = \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left[ {{{\varphi }}a_{k} - b_{k} p_{p}^{k} + \theta s + \delta \left( {p_{iS}^{k} - p_{p}^{k} - e} \right)} \right] + \left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{p}^{k} - c_{p}^{k} } \right)\left( {\frac{{\delta^{2} }}{{2\left( {b_{k} + \delta } \right)}} - b_{k} - \delta } \right) + \frac{\delta }{2}\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {w_{p}^{k} - c_{p}^{k} } \right) = 0$$

Then,

$$p_{pS}^{k*} = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {f_{2}^{k} - \left( {\delta^{2} + \delta b_{k} } \right)w_{p}^{k} } \right] + \gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}} + \frac{{2b_{k}^{2} + 3\delta b_{k} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}}c_{p}^{k} + \frac{{\left( {2b_{k} + 3\delta } \right)\theta s}}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}},$$

where $f_{2}^{k} = \left( {\delta + \delta \varphi + 2\varphi b_{k} } \right)a_{k} + \left( {\delta c - 2\delta b_{k} - \delta^{2} } \right)e + 3\delta \left( {\delta + b_{k} } \right)w_{p}^{k} + 2b_{k} \left( {\delta + b_{k} } \right)c_{p}^{k} .$

$$p_{iS}^{k*} = \frac{{\delta p_{pS}^{k*} + \left( {1 - \varphi } \right)a_{k} + \left( {\delta + c} \right)e + \left( {\delta + b_{k} } \right)w_{p}^{k} }}{{2\left( {\delta + b_{k} } \right)}} + \frac{{\gamma e^{2} }}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]}} + \frac{\theta s}{{2\left( {\delta + b_{k} } \right)}}$$

When $p_{p}^{k*} = p_{pS}^{k*} ,$

$$\frac{{\left( {\delta + \varphi b_{k} } \right)a_{k} + \left( {c - b_{k} } \right)\delta e}}{{2\left( {b_{k}^{2} + 2\delta b_{k} } \right)}} + \frac{{c_{p}^{k} }}{2} + \frac{\theta s}{{2b_{k} }} = \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {f_{2}^{k} - \left( {\delta^{2} + \delta b_{k} } \right)w_{p}^{k} } \right] + \gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}} + \frac{{2b_{k}^{2} + 3\delta b_{k} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}}c_{p}^{k} + \frac{{\left( {2b_{k} + 3\delta } \right)\theta s}}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}},$$

Then, $w_{p4}^{k} = \frac{{\left( {b_{k} + \delta - \varphi b_{k} } \right)a_{k} + \left( {cb_{k} + \delta b_{k} + c\delta } \right)e}}{{2\left( {b_{k}^{2} + 2\delta b_{k} } \right)}} - \frac{{\gamma e^{2} }}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]}} + \frac{1}{2}c_{p}^{k} + \frac{\theta s}{{2b_{k} }}.$

Moreover, if $w_{p}^{k} > w_{p4}^{k} ,$ $p_{p}^{k*} < p_{pS}^{k*}$; if $w_{p}^{k} < w_{p4}^{k} ,$ $p_{p}^{k*} > p_{pS}^{k*}$.

Corollary 5

According to results from Proposition 5,

$$\frac{{\partial p_{pS}^{k*} }}{{\partial {\text{s}}}} = \frac{{\left( {2b_{k} + 3\delta } \right)\theta }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}} - \frac{{\gamma e^{2} \varepsilon \left( {\delta^{2} + \delta b_{k} } \right)}}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]^{2} \left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}}.$$

Since $\frac{{\partial p_{iS}^{k*} }}{{\partial {\text{s}}}} = \frac{\delta }{{2\left( {\delta + b_{k} } \right)}}\frac{{\partial p_{pS}^{k*} }}{{\partial {\text{s}}}} - \frac{{\gamma e^{2} \varepsilon }}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]^{2} }} + \frac{\theta }{{2\left( {\delta + b_{k} } \right)}},$ thus

$$\frac{{\partial p_{iS}^{k*} }}{{\partial {\text{s}}}} = \frac{{\left( {4b_{k}^{2} + 10b_{k} \delta + 5\delta^{2} } \right)\theta }}{{4\left( {\delta + b_{k} } \right)\left( {2b_{k}^{2} + 4b_{k} \delta + \delta^{2} } \right)}} - \frac{{\left( {4b_{k}^{2} + 8b_{k} \delta + 3\delta^{2} } \right)\gamma e^{2} \varepsilon }}{{4\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]^{2} \left( {2b_{k}^{2} + 4b_{k} \delta + \delta^{2} } \right)}}.$$

If $\theta < \frac{{\varepsilon \gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\left( {2b_{k} + 3\delta } \right)}} ,$ $\sqrt {\frac{{\gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\varepsilon \theta \left( {2b_{k} + 3\delta } \right)}}} + 1 - \frac{1}{\varepsilon } > 1 ,$ Since $s \in \left[ {0,1} \right] ,$ thus $s < \sqrt {\frac{{\gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\varepsilon \theta \left( {2b_{k} + 3\delta } \right)}}} + 1 - \frac{1}{\varepsilon }$. Then, we can get $\frac{{\partial p_{pS}^{k*} }}{{\partial {\text{s}}}} < 0$.

If $\frac{{\varepsilon \gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\left( {2b_{k} + 3\delta } \right)}} < \theta < \frac{{\varepsilon \gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\left( {1 - \varepsilon } \right)^{2} \left( {2b_{k} + 3\delta } \right)}} ,$ then $0 < \sqrt {\frac{{\gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\varepsilon \theta \left( {2b_{k} + 3\delta } \right)}}} + 1 - \frac{1}{\varepsilon } < 1$. Since $s \in \left[ {0,1} \right] ,$ when $0 \le s < \sqrt {\frac{{\gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\varepsilon \theta \left( {2b_{k} + 3\delta } \right)}}} + 1 - \frac{1}{\varepsilon } ,$ then $\frac{{\partial p_{pS}^{k*} }}{{\partial {\text{s}}}} < 0$; when $\sqrt {\frac{{\gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\varepsilon \theta \left( {2b_{k} + 3\delta } \right)}}} + 1 - \frac{1}{\varepsilon } < s < 1 ,$ then $\frac{{\partial p_{pS}^{k*} }}{{\partial {\text{s}}}} > 0.$

if $\theta > \frac{{\varepsilon \gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\left( {1 - \varepsilon } \right)^{2} \left( {2b_{k} + 3\delta } \right)}} ,$ then $\sqrt {\frac{{\gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\varepsilon \theta \left( {2b_{k} + 3\delta } \right)}}} + 1 - \frac{1}{\varepsilon } < 0 ,$ Since $s \in \left[ {0,1} \right] ,$ thus $s > \sqrt {\frac{{\gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\varepsilon \theta \left( {2b_{k} + 3\delta } \right)}}} + 1 - \frac{1}{\varepsilon }$. Then, we can get $\frac{{\partial p_{pS}^{k*} }}{{\partial {\text{s}}}} > 0.$

Similarly, for the ESI, we can get that if $\theta < \frac{{\gamma e^{2} \varepsilon \left( {\delta + b_{k} } \right)\left( {4b_{k}^{2} + 8b_{k} \delta + 3\delta^{2} } \right)}}{{\left( {4b_{k}^{2} + 10b_{k} \delta + 5\delta^{2} } \right)}} ,$ then $\frac{{\partial p_{iS}^{k*} }}{{\partial {\text{s}}}} < 0.$

If $\frac{{\gamma e^{2} \varepsilon \left( {\delta + b_{k} } \right)\left( {4b_{k}^{2} + 8b_{k} \delta + 3\delta^{2} } \right)}}{{\left( {4b_{k}^{2} + 10b_{k} \delta + 5\delta^{2} } \right)}} < \theta < \frac{{\gamma e^{2} \varepsilon \left( {\delta + b_{k} } \right)\left( {4b_{k}^{2} + 8b_{k} \delta + 3\delta^{2} } \right)}}{{\left( {1 - \varepsilon } \right)^{2} \left( {4b_{k}^{2} + 10b_{k} \delta + 5\delta^{2} } \right)}} ,$ when $0 \le s < \sqrt {\frac{{\gamma e^{2} \left( {\delta + b_{k} } \right)\left( {4b_{k}^{2} + 8b_{k} \delta + 3\delta^{2} } \right)}}{{\varepsilon \theta \left( {4b_{k}^{2} + 10b_{k} \delta + 5\delta^{2} } \right)}}} + 1 - \frac{1}{\varepsilon } ,$ then $\frac{{\partial p_{iS}^{k*} }}{{\partial {\text{s}}}} < 0$; when $\sqrt {\frac{{\gamma e^{2} \left( {\delta + b_{k} } \right)\left( {4b_{k}^{2} + 8b_{k} \delta + 3\delta^{2} } \right)}}{{\varepsilon \theta \left( {4b_{k}^{2} + 10b_{k} \delta + 5\delta^{2} } \right)}}} + 1 - \frac{1}{\varepsilon } < s < 1 ,$ then $\frac{{\partial p_{iS}^{k*} }}{{\partial {\text{s}}}} > 0$.

If $\theta > \frac{{\gamma e^{2} \varepsilon \left( {\delta + b_{k} } \right)\left( {4b_{k}^{2} + 8b_{k} \delta + 3\delta^{2} } \right)}}{{\left( {1 - \varepsilon } \right)^{2} \left( {4b_{k}^{2} + 10b_{k} \delta + 5\delta^{2} } \right)}} ,$ then $\frac{{\partial p_{iS}^{k*} }}{{\partial {\text{s}}}} > 0$.

Proposition 6

Set $p_{iB}^{I*} > p_{iS}^{I*}$,

then $\frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{1}^{I} + 2\left( {\delta + b_{I} } \right)^{2} \gamma e^{2} }}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{I} } \right)^{2} - \delta^{2} } \right]}} + \frac{\theta s}{{2b_{k} + \delta }} > \frac{{\delta p_{pS}^{I*} + \left( {1 - \varphi } \right)a_{I} + \left( {\delta + c} \right)e + \left( {\delta + b_{I} } \right)w_{p}^{I} }}{{2\left( {\delta + b_{I} } \right)}} + \frac{{\gamma e^{2} }}{{2\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]}} + \frac{\theta s}{{2\left( {\delta + b_{k} } \right)}},$

where $f_{1}^{I} = \left( {2b_{I} + 2\delta - \delta \varphi - 2\varphi b_{I} } \right)a_{I} + \left( {2cb_{I} + 2c\delta + 2\delta b_{I} + \delta^{2} } \right)e + \left[ {2\left( {\delta + b_{I} } \right)^{2} + \delta^{2} } \right]w_{p}^{I} + \delta b_{I} c_{p}^{I} .$

It can be derived that $w_{p}^{I} > \frac{{\delta \left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{5}^{I} + \delta^{2} \left( {\delta + b_{I} } \right)\gamma e^{2} }}{{4\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {b_{I}^{2} + 2\delta b_{I} } \right)\left( {\delta + b_{I} } \right)}} + \frac{{10\delta b_{I} + 4b_{I}^{2} + 5\delta^{2} }}{{4\left( {\delta + b_{I} } \right)\left( {b_{I} + 2\delta } \right)}}c_{p}^{I} + \frac{{\left( {3\delta + 2b_{I} } \right)\delta \theta s}}{{4b_{I} \left( {\delta + b_{I} } \right)\left( {b_{I} + 2\delta } \right)}},$

where $f_{5}^{I} = \left( {\delta + \varphi \delta + 2\varphi b_{I} } \right)a_{I} + \delta \left( {c\delta - \delta - 2b_{I} } \right)e.$

Similarly, if $p_{iB}^{I*} \le p_{iS}^{I*} ,$ there is $w_{p}^{I} \le \frac{{\delta \left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{5}^{I} + \delta^{2} \left( {\delta + b_{I} } \right)\gamma e^{2} }}{{4\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {b_{I}^{2} + 2\delta b_{I} } \right)\left( {\delta + b_{I} } \right)}} + \frac{{10\delta b_{I} + 4b_{I}^{2} + 5\delta^{2} }}{{4\left( {\delta + b_{I} } \right)\left( {b_{I} + 2\delta } \right)}}c_{p}^{I} + \frac{{\left( {3\delta + 2b_{I} } \right)\delta \theta s}}{{4b_{I} \left( {\delta + b_{I} } \right)\left( {b_{I} + 2\delta } \right)}}$

Therefore, set $w_{p5}^{k} = \frac{{\delta \left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{5}^{k} + \delta^{2} \left( {\delta + b_{k} } \right)\gamma e^{2} }}{{4\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left( {b_{k}^{2} + 2\delta b_{k} } \right)\left( {\delta + b_{k} } \right)}} + \frac{{10\delta b_{I} + 4b_{I}^{2} + 5\delta^{2} }}{{4\left( {\delta + b_{I} } \right)\left( {b_{I} + 2\delta } \right)}}c_{p}^{I} + \frac{{\left( {3\delta + 2b_{I} } \right)\delta \theta s}}{{4b_{I} \left( {\delta + b_{I} } \right)\left( {b_{I} + 2\delta } \right)}},$

if $w_{p}^{k} > w_{p5}^{k} ,$ then $p_{iB}^{k*} > p_{iS}^{k*} ,$ $p_{pB}^{k*} > p_{pS}^{k*} ,$

if $w_{p}^{k} = w_{p5}^{k} ,$ then $p_{iB}^{k*} = p_{iS}^{k*} ,$ $p_{pB}^{k*} = p_{pS}^{k*} ,$

if $w_{p}^{k} < w_{p5}^{k} ,$ $p_{iB}^{k*} < p_{iS}^{k*} ,$ $p_{pB}^{k*} < p_{pS}^{k*}$.

Proposition 7

If $p_{pS}^{k*} > p_{pB}^{k*} ,$ then $\frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {f_{2}^{k} - \left( {\delta^{2} + \delta b_{k} } \right)w_{p}^{k} } \right] + \gamma e^{2} \left( {\delta^{2} + \delta b_{k} } \right)}}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}} + \frac{{2b_{k}^{2} + 3\delta b_{k} }}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}}c_{p}^{k} + \frac{{\left( {2b_{k} + 3\delta } \right)\theta s}}{{\left[ {4\left( {\delta + b_{k} } \right)^{2} - 2\delta^{2} } \right]}} - \frac{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]f_{2}^{k} + \delta \left( {\delta + b_{k} } \right)\gamma e^{2} }}{{\left[ {1 - \varepsilon \left( {1 - s} \right)} \right]\left[ {4\left( {\delta + b_{k} } \right)^{2} - \delta^{2} } \right]}} - \frac{\theta s}{{2b_{k} + \delta }} > 0.$$\pi_{pS}^{*} - \pi_{pB}^{*} > 0$.

Similarly, if $p_{pS}^{k*} < p_{pB}^{k*}$, then $\pi_{pS}^{*} - \pi_{pB}^{*} > 0$.

Further, if $w_{p}^{k} > w_{p5}^{k}$, $\pi_{iS}^{*} < \pi_{iB}^{*}$; if $w_{p}^{k} = w_{p5}^{k}$, $\pi_{iS}^{*} = \pi_{iB}^{*}$; if $w_{p}^{k} < w_{p5}^{k}$, $\pi_{iS}^{*} > \pi_{iB}^{*}$.

Proposition 8

Let $\pi_{o} - \pi_{t} > 0$, where $\pi_{o}$ is the ESP’s revenue from the online channel $\pi_{o} = \mathop \sum \limits_{k = I,II} \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{p}^{k} - c_{p}^{k} } \right)} \right]D_{p}^{k} ,$ and $\pi_{t}$ is the ESP’s revenue from the traditional channel $\pi_{t} = \mathop \sum \limits_{k = I,II} \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {w_{p}^{k} - c_{p}^{k} } \right)} \right]D_{i}^{k}$.

When $\mathop \sum \limits_{k = I,II} \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {p_{p}^{k} - c_{p}^{k} } \right)} \right]D_{p}^{k} > \mathop \sum \limits_{k = I,II} \left[ {\left( {1 - \varepsilon \left( {1 - s} \right)} \right)\left( {w_{p}^{k} - c_{p}^{k} } \right)} \right]D_{i}^{k} ,$

Hence, $\varphi_{0}$ is obtained: $\varphi_{0} = \frac{{\mathop \sum \nolimits_{k = I,II} \left[ {\left( {a_{k} - b_{k} p_{i}^{k} } \right)w_{p}^{k} + \left( {\delta + b_{k} } \right)\left( {p_{p}^{k} } \right)^{2} + \delta \left( {p_{p}^{k} - p_{i}^{k} } \right)w_{p}^{k} + \left( {\delta + c} \right)ew_{p}^{k} + \delta ep_{p}^{k} - \delta p_{i}^{k} p_{p}^{k} + \left( {w_{p}^{k} - p_{p}^{k} } \right)\theta s} \right]}}{{\mathop \sum \nolimits_{k = I,II} a_{k} \left[ {\left( {p_{p}^{k} - c_{p}^{k} } \right) + \left( {w_{p}^{k} - c_{p}^{k} } \right)} \right]}}.$