Optimal pricing decision in a multi-channel supply chain with a revenue-sharing contract

Abstract

As a significant component of supply chain management, multi-channel pricing decision has received extensive attention in recent years. Many studies have focused on single-channel pricing decision, while limited research has been done on multi-channel pricing decision with a revenue-sharing contract. This paper establishes a multi-channel optimal pricing decision model with a revenue-sharing contract (entailing a revenue apportionment and an additional reward) in the context of a cross-channel effect, consumers trust utility, and after-sales service utility, all of which play roles in increasing or reducing supply chain members’ expected profits. The results indicate that, in a bid to obtain maximum profit, manufacturer and reseller will take different measures for varying levels of differences between cross-channel effects of direct seller (D-seller) and reseller (R-seller), for different levels of consumer trust utility, and for different levels of after-sales service utility. Manufacturer and reseller both try their best to decrease the impact of sales format differences on electronic channel when the differences are small, but the action is opposite when the differences are large. In addition, manufacturer should not blindly improve the additional trust of direct-sale stores relative to reseller, but instead should increase additional reward to R-seller when manufacturer decides to improve the after-sales service of products through D-seller.

Appendices

Appendix

Proof of Theorem 1

From Eqs. (6) and (8), the first order and second order conditions of $\Pi_{R}$ and $\Pi_{M}$ with respect to $p_{R}$ and $p_{D}$ give the equations:

$$ \frac{{\partial \Pi_{R} }}{{\partial p_{R} }} = \frac{{\delta \left( {p_{D} - 2p_{R} + m} \right) - d_{R} }}{2m}, $$

(25)

$$ \frac{{\partial \Pi_{M} }}{{\partial p_{D} }} = \frac{1}{2} + \frac{{C_{D} - d_{R} - \lambda_{D} + \lambda_{R} + \left( {2 - \delta } \right)p_{R} - 2p_{D} }}{2m}; $$

(26)

$$ \frac{{\partial^{2} \Pi_{R} }}{{\partial p_{R}^{2} }} = - \frac{\delta }{m} < 0,\,\frac{{\partial^{2} \Pi_{M} }}{{\partial p_{D}^{2} }} = - \frac{1}{m} < 0. $$

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It is easy to recognize that $\Pi_{R}$ and $\Pi_{M}$ are convex functions with respective to $p_{R}$ and $p_{D}$, respectively. Thus, optimal solutions for each both exist and are unique. Letting $\frac{{\partial \Pi_{R} }}{{\partial p_{R} }} = 0$ and $\frac{{\partial \Pi_{M} }}{{\partial p_{D} }} = 0$, the optimal pricing strategies will be obtained as follows:

$$ p_{R}^{*} = \frac{\eta }{\delta + 2} - \frac{{d_{R} }}{\delta }, $$

(28)

$$ p_{D}^{*} = \frac{2\eta }{{\delta + 2}} - m - \frac{{d_{R} }}{\delta }. $$

(29)

where $\eta = C_{D} - \lambda_{D} + \lambda_{R} + 3m$. Under the condition of Assumption 1, we can derive:

$$ 0 \le d_{R} \le \frac{\delta }{\delta + 2}{\kern 1pt} \cdot \min \left\{ {\begin{array}{*{20}c} {\eta ,} & {2\eta - \left( {\delta + 2} \right)m} \\ \end{array} } \right\}. $$

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This ensures that the optimal retail prices are both positive values, i.e., $p_{R}^{*} \ge 0$ and $p_{D}^{*} \ge 0$. Thus, Theorem 1 is proven.

Proof of Proposition 1

From Eq. (10), the partial derivative of $p_{D}^{*}$ is solved with respect to $\lambda_{D}$ and $\lambda_{R}$,

$$ \frac{{\partial p_{D}^{*} }}{{\partial \lambda_{D} }} = - \frac{2}{{\delta { + 2}}} < 0,\,\frac{{\partial p_{D}^{*} }}{{\partial \lambda_{R} }} = \frac{2}{{\delta { + 2}}} > 0. $$

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Thus, the manufacturer’s optimal price $p_{D}^{*}$ decreases with increase in $\lambda_{D}$ and increases with increase in $\lambda_{R}$. Taking a partial derivative of $p_{R}^{*}$ with respect to $\lambda_{D}$ and $\lambda_{R}$ from Eq. (9), we get:

$$ \frac{{\partial p_{R}^{*} }}{{\partial \lambda_{D} }} = - \frac{1}{{\delta { + 2}}} < 0,\,\frac{{\partial p_{R}^{*} }}{{\partial \lambda_{R} }} = \frac{1}{{\delta { + 2}}} > 0. $$

(32)

Therefore $p_{R}^{*}$ also decreases with $\lambda_{D}$ and increases with $\lambda_{R}$. In the same way:

$$ \frac{{\partial p_{D}^{*} }}{\partial \delta } = \frac{{d_{R} }}{{\delta^{2} }} - \frac{2\eta }{{\left( {\delta + 2} \right)^{2} }}, $$

(33)

$$ \frac{{\partial p_{R}^{*} }}{\partial \delta } = \frac{{d_{R} }}{{\delta^{2} }} - \frac{\eta }{{\left( {\delta + 2} \right)^{2} }}. $$

(34)

According to Assumption 1, $d_{R} < \frac{{\left( {\delta + 2} \right)^{2} }}{{\delta^{2} }}\eta$, and the inequalities $\frac{{\partial p_{D}^{*} }}{\partial \delta } < 0$ and $\frac{{\partial p_{R}^{*} }}{\partial \delta } < 0$ are obtained. Thus, the optimal retail prices for the manufacturer and R-seller both decrease with increase in the revenue apportionment $\delta$. Proposition 1 is therefore confirmed.

Proof of Proposition 2

From Eqs. (9) and (10), we obtain:

$$ p_{D}^{*} - p_{R}^{*} = \frac{{C_{D} - \lambda_{D} + \lambda_{R} + 3m}}{\delta + 2} - m = \frac{{C_{D} - \lambda_{D} + \lambda_{R} + \left( {1 - \delta } \right)m}}{\delta + 2}, $$

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where $\lambda_{D} - \lambda_{R} < C_{D} + \left( {1 - \delta } \right)m$, the inequality $p_{D}^{*} > p_{R}^{*}$ is obtained. Thus, in this case, D-seller will set a higher optimal retail price than R-seller; where $\lambda_{D} - \lambda_{R} > C_{D} + \left( {1 - \delta } \right)m$, the inequality $p_{D}^{*} < p_{R}^{*}$ is obtained; where $\lambda_{D} - \lambda_{R} = C_{D} + \left( {1 - \delta } \right)m$, the equality $p_{D}^{*} = p_{R}^{*}$ is obtained.

Proof of Proposition 3

Solving the partial derivative of Eq. (11) with respect to $\lambda_{D}$ and $\lambda_{R}$, we find:

$$ \frac{{\partial \Pi_{R}^{*} }}{{\partial \lambda_{D} }} = - \frac{\delta }{{m\left( {\delta + 2} \right)^{2} }}\left( {C_{D} - \lambda_{D} + \lambda_{R} + 3m} \right), $$

$$ \frac{{\partial \Pi_{R}^{*} }}{{\partial \lambda_{R} }} = \frac{\delta }{{m\left( {\delta + 2} \right)^{2} }}\left( {C_{D} - \lambda_{D} + \lambda_{R} + 3m} \right). $$

It is easy to identify that $\frac{{\partial \Pi_{R}^{*} }}{{\partial \lambda_{D} }} = - \frac{{\partial \Pi_{R}^{*} }}{{\partial \lambda_{R} }}$; $\Pi_{R}^{*}$ has a contrary monotonicity with respect to $\lambda_{D}$ and $\lambda_{R}$. According to Assumption 1, $C_{D} - \lambda_{D} + \lambda_{R} + 3m \ge \frac{\delta + 2}{2}m > 0$. Thus, the inequalities $\frac{{\partial \Pi_{R}^{*} }}{{\partial \lambda_{D} }} < 0$ and $\frac{{\partial \Pi_{R}^{*} }}{{\partial \lambda_{R} }} > 0$ always hold. The result is that $\Pi_{R}^{*}$ decreases with increase in $\lambda_{D}$ and increases with increase in $\lambda_{R}$. Because R-seller cannot affect the value of $\lambda_{D}$ (the cross-channel effect of D-seller’s sales volumes on the E-channel), R-seller should make some effort to maximize the effect of his sales volumes (which occur in the T-channel) on sales volumes in the E-channel, in order to obtain maximum profit. Considering the difference in the cross-channel effect of sales by D-seller and R-seller, the following relationship is derived:

$$ \frac{{\partial \Pi_{R}^{*} }}{{\partial \left( {\lambda_{D} - \lambda_{R} } \right)}} = - \frac{\delta }{{2m\left( {2 + \delta } \right)^{2} }}\left( {C_{D} - \lambda_{D} + \lambda_{R} + 3m} \right) < 0. $$

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Thus, the optimal profit of R-seller decreases with increase in $\lambda_{D} - \lambda_{R}$. Because $\lambda_{R} \le \lambda_{D}$, when $\lambda_{R} = \lambda_{D}$, R-seller will obtain maximum profit:

$$ \Pi_{R}^{*} = \frac{{\delta \left( {C_{D} + 3m} \right)^{2} }}{{2m\left( {2 + \delta } \right)^{2} }}. $$

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Proposition 3 therefore holds.

Proof of Proposition 4

Solving the partial derivative of Eq. (12) with respect to $\lambda_{D}$ and $\lambda_{R}$, the following formulae are obtained:

$$ \frac{{\partial \Pi_{M}^{*} }}{{\partial \lambda_{D} }} = \frac{{m\left( {\delta^{2} + 3\delta - 1} \right) - C_{D} + \lambda_{D} - \lambda_{R} }}{{m\left( {\delta + 2} \right)^{2} }}, $$

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$$ \frac{{\partial \Pi_{M}^{*} }}{{\partial \lambda_{R} }} = \frac{{C_{D} - \lambda_{D} + \lambda_{R} + 5m + \delta m}}{{m\left( {\delta + 2} \right)^{2} }} = \frac{{\eta + \left( {\delta + 2} \right)m}}{{m\left( {\delta + 2} \right)^{2} }} > 0. $$

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Assuming the first order condition ${{\partial \Pi_{M}^{*} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{*} } {\partial \lambda_{D} }}} \right. \kern-\nulldelimiterspace} {\partial \lambda_{D} }} \ge 0$ to hold, the inequality $m\left( {\delta^{2} + 3\delta - 1} \right) - C_{D} + \lambda_{D} - \lambda_{R} \ge 0$ should be satisfied, i.e., when $\lambda_{D} - \lambda_{R} \ge C_{D} - m\left( {\delta^{2} + 3\delta - 1} \right)$, the manufacturer’s optimal profit $\Pi_{M}^{*}$ increases with increase in $\lambda_{1}$; if ${{\partial \Pi_{M}^{*} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{*} } {\partial \lambda_{D} }}} \right. \kern-\nulldelimiterspace} {\partial \lambda_{D} }} < 0$, the inequality $\lambda_{D} - \lambda_{R} < C_{D} - m\left( {\delta^{2} + 3\delta - 1} \right)$ is obtained, so that the manufacturer’s optimal profit $\Pi_{M}^{*}$ decreases with increase in $\lambda_{D}$. Because the inequality ${{\partial \Pi_{M}^{*} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{*} } {\partial \lambda_{R} }}} \right. \kern-\nulldelimiterspace} {\partial \lambda_{R} }} > 0$ always holds, the manufacturer’s optimal profit $\Pi_{M}^{*}$ always increases with increase in $\lambda_{R}$.

Firstly, we consider the case where the difference between D-seller and R-seller’s cross-channel effects is large (i.e., $\lambda_{D} - \lambda_{R} \ge C_{D} - m\left( {\delta^{2} + 3\delta - 1} \right)$). In order to obtain maximum profit, the manufacturer will attempt to maximize the effect of D-seller’s product sales volumes on the E-channel. According to Proposition 3, R-seller will also attempt to minimize the difference between D-seller and R-seller’s cross-channel effect. In accordance with the inequality relation $\lambda_{D} - \lambda_{R} \ge C_{D} - m\left( {\delta^{2} + 3\delta - 1} \right)$, R-seller should ensure $\lambda_{R} = \lambda_{D} - C_{D} + m\left( {\delta^{2} + 3\delta - 1} \right)$ in order to obtain maximum profit. Thus, the manufacturer’s optimal cross-channel effect is $\lambda_{D} = \lambda_{R}^{*} + C_{D} - m\left( {\delta^{2} + 3\delta - 1} \right)$.

Secondly, when the difference between D-seller and R-seller’s cross-channel effects is small (i.e., $\lambda_{D} - \lambda_{R} < C_{D} - m\left( {\delta^{2} + 3\delta - 1} \right)$), the manufacturer will seek a minimum effect of D-seller’s product sales volumes on the E-channel, and will hope for a maximum effect of R-seller’s product sales volumes on the E-channel. Thus, D-seller’s cross-channel effects parameter $\lambda_{D} = \lambda_{R}$ represents the manufacturer’s optimal outcome. Moreover, R-seller also obtains a maximum profit in this case.

Proof of Proposition 5

Solving the partial derivative of the formula for R-seller’s optimal profit $\Pi_{R}^{*}$ (Eq. 11) and the manufacturer’s optimal profit $\Pi_{M}^{*}$ (Eq. 12) with respect to the revenue apportionment parameter $\delta$:

$$ \frac{{\partial \Pi_{R}^{*} }}{\partial \delta } = \frac{2 - \delta }{{2m\left( {\delta + 2} \right)^{2} }}\eta^{2} > 0, $$

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$$ \frac{{\partial \Pi_{M}^{*} }}{\partial \delta } = \frac{{d_{R} }}{{\delta^{2} }} - \frac{\eta }{{\left( {\delta + 2} \right)^{2} }} - \frac{{\eta^{2} }}{{m\left( {\delta + 2} \right)^{3} }}, $$

(41)

Because $\frac{{\partial \Pi_{R}^{*} }}{\partial \delta } > 0$, R-seller’s optimal profit increases with increase in the revenue apportionment $\delta$. Considering the relationship between $\Pi_{M}^{*}$ and $\delta$, and the inequality:

$$ \frac{{d_{R} }}{{\delta^{2} }} - \frac{\eta }{{\left( {\delta + 2} \right)^{2} }} - \frac{{\eta^{2} }}{{m\left( {\delta + 2} \right)^{3} }} \ge 0, $$

(42)

the equality holds only when:

$$ \delta_{1} = \frac{\beta }{3\gamma } + \frac{3\gamma }{\beta } - \frac{{108d_{R} m\gamma^{2} }}{{\beta^{3} }}{ + }\kappa^{{\frac{{1}}{{3}}}} { + }\left( {\kappa { + }\frac{{\beta^{3} }}{{{27}\gamma^{{3}} }}} \right)^{{\frac{{1}}{{3}}}} , $$

(43)

and the parameters are

$$ \kappa = \sqrt[2]{{\frac{{4d_{R} m}}{\gamma } - \frac{{\beta^{3} }}{{27\gamma^{3} }} + \frac{{2d_{R} m\beta }}{{\gamma^{2} }} - \left( {\frac{{\beta^{2} }}{{9\gamma^{2} }} - \frac{{4d_{R} m}}{\gamma }} \right)^{3} }} - \frac{{4d_{R} m}}{\gamma } - \frac{{2d_{R} m\beta }}{{\gamma^{2} }}, $$

$$ \beta = \eta^{2} + 2m\eta - 6d_{R} m, $$

$$ \gamma = \left( {d_{R} - \eta } \right)m. $$

For convenience, this denotes $\delta_{1} = \frac{\beta }{3\gamma } + \frac{3\gamma }{\beta } - \frac{{108d_{R} m\gamma^{2} }}{{\beta^{3} }}{ + }\kappa^{{\frac{{1}}{{3}}}} { + }\left( {\kappa { + }\frac{{\beta^{3} }}{{{27}\gamma^{{3}} }}} \right)^{{\frac{{1}}{{3}}}}$.When $\delta \le \delta_{1}$, the inequality (A-3) holds, and the optimal profit of the manufacturer increases with increase in the revenue apportionment. Otherwise, the optimal profit of the manufacturer decreases with increase in the revenue apportionment. Therefore, from the manufacturer’s perspective, the optimal revenue apportionment is $\delta = \delta_{1} = \frac{\beta }{3\gamma } + \frac{3\gamma }{\beta } - \frac{{108d_{R} m\gamma^{2} }}{{\beta^{3} }}{ + }\kappa^{{\frac{{1}}{{3}}}} { + }\left( {\kappa { + }\frac{{\beta^{3} }}{{{27}\gamma^{{3}} }}} \right)^{{\frac{{1}}{{3}}}}$. Proposition 5 is therefore proved.

Proof of Proposition 6

From Eqs. (9), (10), (17) and (18), the following inequality relationships are obtained:

$$ \frac{\eta }{\delta + 2} \ge \frac{\eta - t}{{\delta + 2}}, $$

(44)

$$ \frac{2\eta }{{\delta + 2}} \le \frac{2\eta + \delta t}{{\delta + 2}}, $$

(45)

where the equality holds only when $t = 0$. Note that in such a scenario, where no value is assigned to the additional trust utility of the D-seller relative to the R-seller, the result of the basic model in Sect. 3 is obtained. Here, we recognize an additional trust utility $t > 0$, thus, $\hat{p}_{R}^{*} < p_{R}^{*}$ and $\hat{p}_{D}^{*} > p_{D}^{*}$. Proposition 6 is therefore proved.

Proof of Proposition 7

From Eqs. (7), (8), (17) and (18), for the extended model incorporating consumer’s additional trust, optimal profits of R-seller and the manufacturer are solved as follows:

$$ \hat{\Pi }_{R}^{*} = \frac{{\delta \left( {\eta - t} \right)^{2} }}{{2m\left( {\delta + 2} \right)^{2} }}, $$

(46)

$$ \hat{\Pi }_{M}^{*} = q_{E} + \lambda_{D} + t - C - m - C_{D} - \frac{{d_{R} }}{\delta } + \frac{\eta - t}{{\delta + 2}} + \frac{{\left( {\eta - t} \right)^{2} }}{{2m\left( {\delta + 2} \right)^{2} }}. $$

(47)

Applying the expressions of optimal profits in Eqs. (11), (12), (46) and (A-624), we can obtain:

$$ \hat{\Pi }_{R}^{*} - \Pi_{R}^{*} = \frac{\delta }{{2m\left( {\delta + 2} \right)^{2} }}\left[ {\left( {\eta - t} \right)^{2} - \eta^{2} } \right] = \frac{\delta }{{2m\left( {\delta + 2} \right)^{2} }}\left[ {\left( {\eta - t} \right)^{2} - \eta^{2} } \right] \le 0, $$

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$$ \begin{aligned} \hat{\Pi }_{M}^{*} - \Pi_{M}^{*} & = t{ + }\frac{\eta - t}{{\delta + 2}} + \frac{{\left( {\eta - t} \right)^{2} }}{{2m\left( {\delta + 2} \right)^{2} }} - \frac{\eta }{\delta + 2} - \frac{{\eta^{2} }}{{2m\left( {\delta + 2} \right)^{2} }} \\ & = \frac{t}{{2m\left( {\delta + 2} \right)^{2} }}\left[ {2m\left( {\delta + 1} \right)\left( {\delta + 2} \right) - 2\eta + t} \right] \\ \end{aligned} $$

(49)

When $t \ge 2\left[ {\eta - m\left( {\delta + 1} \right)\left( {\delta + 2} \right)} \right]$, the inequality $\hat{\Pi }_{M}^{*} \ge \Pi_{M}^{*}$ holds; otherwise, $\hat{\Pi }_{M}^{*} < \Pi_{M}^{*}$. Thus, a large additional trust utility will increase the manufacturer’s optimal profit; otherwise, the additional trust utility will decrease the manufacturer’s optimal profit. R-seller’ optimal profit is smaller in the model incorporating the additional trust utility than in that without regard to the additional trust utility.

Below, we analyze the change in the revenue apportionment $\delta$ when the parameter $\eta$ is changed to $\eta - t$. Solving R-seller and D-seller’s optimal profit with respect to revenue apportionment:

$$ \frac{{\partial \hat{\Pi }_{R}^{*} }}{\partial \delta } = \frac{{\left( {\eta - t} \right)^{2} \left( {2 - \delta } \right)}}{{2m\left( {\delta + 2} \right)^{3} }} \ge 0, $$

(50)

$$ \frac{{\partial \hat{\Pi }_{M}^{*} }}{\partial \delta } = \frac{{d_{R} }}{{\delta^{2} }} - \frac{\eta - t}{{\left( {\delta + 2} \right)^{2} }} - \frac{{\left( {\eta - t} \right)^{2} }}{{2m\left( {\delta + 2} \right)^{3} }}. $$

(51)

It is clear that R-seller’s optimal profit $\hat{\Pi }_{R}^{*}$ increases with its proportion of revenue $\delta$. By solving the equality ${{\partial \hat{\Pi }_{M}^{*} } \mathord{\left/ {\vphantom {{\partial \hat{\Pi }_{M}^{*} } {\partial \delta }}} \right. \kern-\nulldelimiterspace} {\partial \delta }} = 0$, the following equality is found:

$$ \frac{{d_{R} }}{{\delta^{2} }} = \frac{\eta - t}{{\left( {\delta + 2} \right)^{2} }} + \frac{{\left( {\eta - t} \right)^{2} }}{{2m\left( {\delta + 2} \right)^{3} }}. $$

(52)

For convenience, we introduce the functions $g_{1} \left( \delta \right) = \frac{{d_{R} }}{{\delta^{2} }}$ and $g_{2} \left( {\delta ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta } \right) = \frac{\eta - t}{{\left( {\delta + 2} \right)^{2} }} + \frac{{\left( {\eta - t} \right)^{2} }}{{2m\left( {\delta + 2} \right)^{3} }}$, i.e., Eq. (52) is equivalent to the following equality:

$$ g_{1} \left( \delta \right) = g_{2} \left( {\delta ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta } \right). $$

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It is clear that $g_{2} \left( {\delta ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta } \right)$ is an increasing function of $\eta$; $g_{1} \left( \delta \right)$ and $g_{2} \left( {\delta ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta } \right)$ are both decreasing functions of $\delta$, but:

$$ \frac{{{\text{d}}{\kern 1pt} {\kern 1pt} g_{1} \left( \delta \right)}}{{{\text{d}}{\kern 1pt} \delta }} > \frac{{\partial {\kern 1pt} g_{2} \left( {\delta ,{\kern 1pt} {\kern 1pt} \eta } \right)}}{{\partial {\kern 1pt} {\kern 1pt} \delta }}. $$

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This shows that $g_{1} \left( \delta \right)$ has a larger rate of change than $g_{2} \left( {\delta ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta } \right)$ with respect to $\delta$. This condition ensures that there exists a value of $\delta$ which allows Equation (A-8) to hold where the value of $\eta$ has been changed. Where the value $\eta$ decreases, the value $g_{2} \left( {\delta ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \eta } \right)$ will decrease. In order to ensure that Equation (A-8) holds, the value $\delta$ will increase. The optimal revenue apportionment $\delta$ is obtained by solving the Equation (A-8), so the value of the optimal revenue apportionment $\delta$ increases with the reduction from $\eta$ to $\eta - t$.

With the condition $\delta \in (0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1]$, Equation (A-8) is solved thus:

$$ \delta_{2} = \frac{{\beta_{2} }}{{3\gamma_{2} }} + \frac{{3\gamma_{2} }}{{\beta_{2} }} - \frac{{108d_{R} m\gamma_{2}^{2} }}{{\beta_{2}^{3} }}{ + }\kappa_{2}^{\frac{1}{3}} { + }\left( {\kappa_{2} { + }\frac{{\beta_{2}^{3} }}{{{27}\gamma_{2}^{3} }}} \right)^{{\frac{{1}}{{3}}}} , $$

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where the parameters are $\kappa_{2} = \sqrt[2]{{\frac{{4d_{R} m}}{{\gamma_{2} }} - \frac{{\beta^{3} }}{{27\gamma_{2}^{3} }} + \frac{{2d_{R} m\beta_{2} }}{{\gamma_{2}^{2} }} - \left( {\frac{{\beta_{2}^{2} }}{{9\gamma_{2}^{2} }} - \frac{{4d_{R} m}}{{\gamma_{2} }}} \right)^{3} }} - \frac{{4d_{R} m}}{{\gamma_{2} }} - \frac{{2d_{R} m\beta_{2} }}{{\gamma_{2}^{2} }}$, $\beta_{2} = \eta_{2}^{2} + 2m\eta_{2} - 6d_{R} m$, $\gamma_{2} = \left( {d_{R} - \eta_{2} } \right)m$, and $\eta_{2} = C_{D} - \lambda_{D} + \lambda_{R} + 3m - t$.

Therefore, $\delta_{2} \ge \delta_{1}$, so the equality holds only where $t = 0$ (i.e., where no value is assigned to consumer trust, and so this has no effect on pricing strategies). Proposition 7 is therefore confirmed.

Proof of Proposition 8

• (1) The derivative of R-seller and D-seller’s optimal retail prices (i.e., Eqs. 23 and 24) with respect to the additional after-sales service are, respectively:

$$ \frac{{\partial {\kern 1pt} \overline{p}_{R}^{*} }}{\partial f} = \frac{ - 1}{{\delta + 2}} < 0, $$

(56)

$$ \frac{{\partial {\kern 1pt} \overline{p}_{D}^{*} }}{\partial f} = = \frac{\delta }{\delta + 2} > 0. $$

(57)

Thus, ${\kern 1pt} \overline{p}_{R}^{*}$ decreases with increase in $f$, and $\overline{p}_{D}^{*}$ increases with increase in $f$. Because $\left| {\frac{{\partial {\kern 1pt} \overline{p}_{R}^{*} }}{\partial f}} \right| \ge \left| {\frac{{\partial {\kern 1pt} \overline{p}_{D}^{*} }}{\partial f}} \right|$, the rate of decrease in $\overline{p}_{R}^{*}$ is larger than the rate of increase in $\overline{p}_{D}^{*}$.

• (2) The derivative of R-seller and D-seller’s optimal retail prices (i.e., Eqs. (23) and (24)) with respective to the effect of after-sales service on the E-channel are, respectively:

$$ \frac{{\partial {\kern 1pt} \overline{p}_{R}^{*} }}{\partial \Delta f} = \frac{\mu - 1}{{\delta + 2}} < 0, $$

(58)

$$ \frac{{\partial {\kern 1pt} \overline{p}_{D}^{*} }}{\partial \Delta f} = = \frac{2\mu - 2}{{\delta + 2}} < 0. $$

(59)

Thus, $\overline{p}_{R}^{*}$ and $\overline{p}_{D}^{*}$ both decrease with increase in $\Delta f$. As $\Delta f$ increases, $\overline{p}_{D}^{*}$ has a greater rate of decrease than $\overline{p}_{R}^{*}$.

• (3) Comparing the three models’ optimal retail prices from Eqs. (9–10), (17–18) and (23–24), we see that the relationship ${\kern 1pt} {\kern 1pt} \hat{p}_{R}^{*} \le {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p_{R}^{*}$ always holds under Proposition 6. If ${\kern 1pt} \overline{p}_{R}^{*} \le {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat{p}_{R}^{*} \le {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p_{R}^{*}$, i.e.,

$$ \frac{{\eta - \left( {1 - \mu } \right)\Delta f - f - t + \mu \left( {\lambda_{1} - \lambda_{2} } \right)}}{\delta + 2} - \frac{{d_{R} }}{\delta } \le \frac{\eta - t}{{\delta + 2}} - \frac{{d_{R} }}{\delta }, $$

(60)

the inequality $\mu \le \frac{f + \Delta f}{{\lambda_{D} - \lambda_{R} + \Delta f}}$ should be satisfied; if ${\kern 1pt} {\kern 1pt} \hat{p}_{R}^{*} \le {\kern 1pt} {\kern 1pt} \overline{p}_{R}^{*} \le {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p_{R}^{*}$, i.e.,

$$ \frac{\eta - t}{{\delta + 2}} - \frac{{d_{R} }}{\delta } \le \frac{{\eta - \left( {1 - \mu } \right)\Delta f - f - t + \mu \left( {\lambda_{D} - \lambda_{R} } \right)}}{\delta + 2} - \frac{{d_{R} }}{\delta }, $$

(61)

$$ \frac{{\eta - \left( {1 - \mu } \right)\Delta f - f - t + \mu \left( {\lambda_{D} - \lambda_{R} } \right)}}{\delta + 2} - \frac{{d_{R} }}{\delta } \le \frac{\eta }{\delta + 2} - \frac{{d_{R} }}{\delta }, $$

(62)

the inequality $\frac{f + \Delta f}{{\lambda_{D} - \lambda_{R} + \Delta f}} \le \mu \le \frac{f + \Delta f + t}{{\lambda_{D} - \lambda_{R} + \Delta f}}$ should be satisfied; if ${\kern 1pt} \hat{p}_{R}^{*} \le {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p_{R}^{*} \le {\kern 1pt} {\kern 1pt} \overline{p}_{R}^{*}$, i.e.,

$$ \frac{\eta }{\delta + 2} - \frac{{d_{R} }}{\delta } \le \frac{{\eta - \left( {1 - \mu } \right)\Delta f - f - t + \mu \left( {\lambda_{D} - \lambda_{R} } \right)}}{\delta + 2} - \frac{{d_{R} }}{\delta }, $$

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the inequality $\mu \ge \frac{f + \Delta f + t}{{\lambda_{D} - \lambda_{R} + \Delta f}}$ should be satisfied.

We further consider the relationship between D-seller’s optimal retail prices under each of the three models. From Proposition 6, it is known that the inequality $p_{D}^{*} < \hat{p}_{D}^{*}$ always holds. If $\overline{p}_{D}^{*} < p_{D}^{*} < \hat{p}_{D}^{*}$, i.e.,

$$ \frac{{2\eta - 2\left( {1 - \mu } \right)\Delta f{ + }\delta \left( {f + t} \right) + 2\mu \left( {\lambda_{D} - \lambda_{R} } \right)}}{\delta + 2} - \frac{{d_{R} }}{\delta } - m \le \frac{2\eta }{{\delta + 2}} - m - \frac{{d_{R} }}{\delta }, $$

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the inequality $\mu \le \frac{{2\Delta f - \delta \left( {f + t} \right)}}{{2\left( {\lambda_{D} - \lambda_{R} + \Delta f} \right)}}$ will be obtained. If $p_{D}^{*} < \overline{p}_{D}^{*} < \hat{p}_{D}^{*}$, i.e.,

$$ \frac{2\eta }{{\delta + 2}} - m - \frac{{d_{R} }}{\delta } \le \frac{{2\eta - 2\left( {1 - \mu } \right)\Delta f{ + }\delta \left( {f + t} \right) + 2\mu \left( {\lambda_{D} - \lambda_{R} } \right)}}{\delta + 2} - \frac{{d_{R} }}{\delta } - m, $$

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$$ \frac{{2\eta - 2\left( {1 - \mu } \right)\Delta f{ + }\delta \left( {f + t} \right) + 2\mu \left( {\lambda_{D} - \lambda_{R} } \right)}}{\delta + 2} - \frac{{d_{R} }}{\delta } - m \le \frac{2\eta + \delta t}{{\delta + 2}} - m - \frac{{d_{R} }}{\delta }, $$

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the inequality $\frac{{2\Delta f - \delta \left( {f + t} \right)}}{{2\left( {\lambda_{D} - \lambda_{R} + \Delta f} \right)}} \le \mu \le \frac{2\Delta f - \delta f}{{2\left( {\lambda_{D} - \lambda_{R} + \Delta f} \right)}}$ will be obtained. Otherwise, $p_{D}^{*} < \hat{p}_{D}^{*} < \overline{p}_{D}^{*}$, and the inequality $\mu \ge \frac{2\Delta f - \delta f}{{2\left( {\lambda_{D} - \lambda_{R} + \Delta f} \right)}}$ will be obtained. The relationship $\frac{2\Delta f - \delta f}{{2\left( {\lambda_{D} - \lambda_{R} + \Delta f} \right)}} \le \frac{f + \Delta f}{{\lambda_{D} - \lambda_{R} + \Delta f}}$ holds in all scenarios, thus Proposition 8 always holds.

Proof of Proposition 9

From Eqs. (21–24), the manufacturer and R-seller’s optimal profit functions can be obtained:

$$ \overline{\Pi }_{R}^{*} = \frac{{\delta \left( {\eta_{1} - f - \Delta f + \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)} \right)^{2} }}{{2m\left( {\delta + 2} \right)^{2} }}, $$

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$$ \overline{\Pi }_{M}^{*} = f - C - C_{D} - m + t - \mu + \left( {1 - \mu } \right)\left( {\Delta f + \lambda_{D} + q_{E} } \right) - \frac{{d_{R} }}{\delta } + \frac{\rho }{\delta + 2} + \frac{{\rho^{2} }}{{2m\left( {\delta + 2} \right)^{2} }}, $$

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where $\rho = \eta_{1} - f - \Delta f + \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)$ and $\eta_{1} = C_{D} - \lambda_{D} + \lambda_{R} + 3m - t$. Comparing $\overline{\Pi }_{R}^{*}$ and $\hat{\Pi }_{R}^{*}$, we observe:

$$ \overline{\Pi }_{R}^{*} - \hat{\Pi }_{R}^{*} = - \frac{{\delta \left[ {f + \Delta f - \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)} \right]\left[ {2\eta_{1} - f - \Delta f + \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)} \right]}}{{2m\left( {\delta + 2} \right)^{2} }}. $$

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Because $\overline{p}_{R}^{*} \ge {0}$, the inequality $\rho > 0$ holds, when $f + \Delta f - \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right) \le 0$, i.e., $\mu \ge \frac{f + \Delta f}{{\Delta f + \lambda_{D} - \lambda_{R} }}$, the condition $\overline{\Pi }_{R}^{*} - \hat{\Pi }_{R}^{*} \ge 0$ always holds. Thus, if the additional after-sales service $\mu \ge \frac{f + \Delta f}{{\Delta f + \lambda_{D} - \lambda_{R} }}$, R-seller will obtain more profit by providing additional after-sales service than without providing it.

We then compare manufacturer’s optimal profit, $\overline{\Pi }_{M}^{*}$ and $\hat{\Pi }_{M}^{*}$,

$$ \overline{\Pi }_{M}^{*} - \hat{\Pi }_{M}^{*} = \frac{1}{\delta + 2}\left\{ \begin{gathered} \frac{{\left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)^{2} }}{{2m\left( {\delta + 2} \right)}}\mu^{2} - \left[ {\delta + 2 + \lambda_{D} + \lambda_{R} + 2q_{E} + \delta \left( {\Delta f + \lambda_{D} + q_{E} } \right) + \frac{{\left( {\Delta f + } \right.\lambda_{D} \left. { - \lambda_{R} } \right)\left( {f + \Delta f - \eta_{1} } \right)}}{{m\left( {\delta + 2} \right)}}} \right]\mu \hfill \\ + \frac{{\left( {\Delta f + f} \right)\left( {\Delta f + f - 2\eta_{1} } \right)}}{{2m\left( {\delta + 2} \right)}} + \left( {\delta + 1} \right)\left( {\Delta f + f} \right) \hfill \\ \end{gathered} \right\} $$

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Solving the above quadratic function with respect to $\mu$, we obtain $\mu \le \frac{{a_{2} - \sqrt {a_{2}^{2} - 4a_{1} a_{3} } }}{{a_{3} }}$, and the parameters are $a_{1} = \left( {\delta + 1} \right)\left( {f + \Delta f} \right) + \frac{{\left( {f + \Delta f} \right)\left( {f + \Delta f - 2\eta_{1} } \right)}}{{2m\left( {\delta + 2} \right)}}$, $a_{3} = \frac{{\left( {\Delta f + f} \right)\left( {\Delta f + f - 2\eta_{1} } \right)}}{{2m\left( {\delta + 2} \right)}} + \left( {\delta + 1} \right)\left( {\Delta f + f} \right)$, and $a_{2} = \delta + 2 + \lambda_{D} + \lambda_{R} + 2q_{E} + \delta \left( {\Delta f + \lambda_{D} + q_{E} } \right) + \frac{{\left( {\Delta f + } \right.\lambda_{D} \left. { - \lambda_{R} } \right)\left( {f + \Delta f - \eta_{1} } \right)}}{{m\left( {\delta + 2} \right)}}$. The manufacturer’s optimal profit is therefore increased in the scenario where additional after-sales service is provided if its marginal cost is low. Thus, the Proposition 9 is confirmed.

Proof of Proposition 10

From Eqs. (21–24), R-seller and the manufacturer’s optimal profit functions can be obtained, respectively:

$$ \overline{\Pi }_{R}^{*} = \frac{{\delta \left( {\eta_{1} - f - \Delta f + \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)} \right)}}{{2m\left( {\delta + 2} \right)^{2} }}, $$

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$$ \overline{\Pi }_{M}^{*} = f - C - C_{D} - m + t - a + \left( {1 - a} \right)\left( {\Delta f + \lambda_{D} + q_{E} } \right) - \frac{{d_{R} }}{\delta } + \frac{\rho }{\delta + 2} + \frac{{\rho^{2} }}{{2m\left( {\delta + 2} \right)^{2} }}, $$

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where $\rho = \eta_{1} - f - \Delta f + \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)$ and $\eta_{1} = C_{D} - \lambda_{D} + \lambda_{R} + 3m - t$. Solving for $\overline{\Pi }_{R}^{*}$ and $\overline{\Pi }_{M}^{*}$ with respect to $f$, we find:

$$ \frac{{\partial \overline{\Pi }_{R}^{*} }}{{\partial {\kern 1pt} f}} = \frac{{ - \delta \left[ {\eta_{1} - f - \Delta f + \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)} \right]}}{{m\left( {\delta + 2} \right)^{2} }} < 0, $$

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$$ \frac{{\partial \overline{\Pi }_{M}^{*} }}{{\partial {\kern 1pt} f}} = 1 - \frac{1}{\delta + 2} - \frac{{\eta_{1} - f - \Delta f + \mu \left( {\Delta f + \lambda_{D} - \lambda_{R} } \right)}}{{m\left( {\delta + 2} \right)^{2} }}. $$

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Because $\frac{{\partial \overline{\Pi }_{R}^{*} }}{{\partial {\kern 1pt} f}} < 0$, we know that $\overline{\Pi }_{R}^{*}$ is a decreasing function of $f$. If $\frac{{\partial \overline{\Pi }_{M}^{*} }}{{\partial {\kern 1pt} f}} \ge 0$, the inequality $\mu \le \frac{{m\left( {\delta + 2} \right)\left( {\delta + 1} \right) - \eta_{1} + f + \Delta f}}{{\Delta f + \lambda_{D} - \lambda_{R} }}$ will be obtained. Thus, $\overline{\Pi }_{M}^{*}$ is an increasing function of $f$ when the marginal cost satisfies $\mu \le \frac{{m\left( {\delta + 2} \right)\left( {\delta + 1} \right) - \eta_{1} + f + \Delta f}}{{\Delta f + \lambda_{D} - \lambda_{R} }}$. Otherwise, $\overline{\Pi }_{M}^{*}$ is a decreasing function of $f$.