Reselling or drop shipping: Strategic analysis of E-commerce dual-channel structures

Abstract

Practical and anecdotal evidence indicates that the drop-shipping policy has been extensively adopted in electronic commerce (E-commerce) practice. However, how the different structures of dual channels affect the drop-shipping strategy of a retailer and supply chain efficiency has not been adequately investigated. To fill this research gap, this study performs a game-theoretic analysis of the drop-shipping strategy of a retailer under two practical dual-channel supply chain structures, namely, manufacturer-owned and retailer-owned online channel structures (MOS and ROS). Under each structure, the optimal pricing and the variations in profits are analytically presented with reselling or drop shipping in the corresponding channel. We show that under MOS, drop shipping can lower both wholesale and retail prices. By contrast, under ROS, drop shipping can decrease the offline retail price and the wholesale price but lower the online retail price only when the ratio of the unit fulfillment fee to overstock inventory is relatively low. The retailer prefers to adopt the drop-shipping policy under both MOS and ROS. Counterintuitively, the profits of both retailer and manufacturer are independent of the unit order fulfillment fee under MOS; however, under ROS, the profit of the manufacturer indicates a decreasing trend, whereas that of the retailer continuously increases with the unit order fulfillment fee. In terms of the performance of the entire supply chain, ROS dominates MOS when the proportion of brick-and-mortar (BM) shoppers is sufficiently high. In specific, beyond a certain service level, drop shipping in the retailer-owned online channel is preferred; otherwise, reselling in such channel is dominant. When the proportion of BM shoppers is moderate, MOS with drop shipping in the BM channel is the best choice for the entire supply chain. When the proportion of BM shoppers is low, MOS with reselling in the BM channel becomes the dominant option.

Appendix

1.1 Proof of Lemma 1

For the given $z_{r}$ and $z_{e}$, we solve the first-order partial derivative of $p_{r}$ and $p_{e}$ from (6). We obtain

$$\frac{{\partial E(\pi_{t} )}}{{\partial p_{r} }} = \rho A{ - }2\alpha p_{r} { + }2\beta p_{e} { + }\mu_{r} { + (}\alpha { - }\beta )c - \varTheta (z_{r} ),$$

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$$\frac{{\partial E(\pi_{t} )}}{{\partial p_{e} }} = (1 - \rho )A - 2\alpha p_{e} + 2\beta p_{r} + \mu_{e} + (\alpha - \beta )c - \varPsi (z_{e} ).$$

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Obtaining the Hessian matrix of (6) at point $(p_{r} ,p_{e} )$ is easy:

$${\rm H} = \left( {\begin{array}{*{20}c} { - 2\alpha } & {2\beta } \\ {2\beta } & { - 2\alpha } \\ \end{array} } \right) = 4(\alpha^{2} - \beta^{2} ).$$

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Its first-order principal minor is $- 2\alpha < 0$, and the second-order principal minor is $4(\alpha^{2} - \beta^{2} ) > 0$. The total profit of the dual-channel supply chain is about the joint concave function of $p_{r}$ and $p_{e}$. Therefore, the only optimal solution can be obtained from the first-order condition of $(p_{r} ,p_{e} )$ by (6). $\partial E(\pi_{t} )/\partial p_{r} = 0$, $\partial E(\pi_{t} )/\partial p_{e} = 0$, we can obtain,

$$\begin{aligned} p_{rc}^{ * } & = \frac{{[\alpha \rho + \beta (1 - \rho )]A{ + }\alpha \mu_{r} + \beta \mu_{e} + (\alpha^{2} - \beta^{2} )c - \alpha \varTheta (z_{r} ) - \beta \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}, \\ p_{ec}^{ * } & = \frac{{[\alpha (1 - \rho ) + \beta \rho ]A{ + }\beta \mu_{r} + \alpha \mu_{e} + (\alpha^{2} - \beta^{2} )c - \beta \varTheta (z_{r} ) - \alpha \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}. \\ \end{aligned}$$

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1.2 Proof of Lemma 2

The decision-making questions of a retailer include the traditional retail channel pricing when her profit is maximized. By solving the first-order partial derivative of $p_{r}$ and making it equal to 0, namely, $\frac{{\partial E(\pi_{r} )}}{{\partial p_{r} }} = 0$, we can obtain

$$p_{r} (w,p_{e} ) = \frac{{\rho A + \beta p_{e} + \mu_{r} + \alpha w - \varTheta (z_{r} )}}{2\alpha }.$$

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We put the upper substitution into (10), and then find the partial derivatives of $w$ and $p_{e}$, respectively:

$$\begin{aligned} \frac{{\partial E(\pi_{m} )}}{\partial w} & = \rho A + 2\beta p_{e} - 2\alpha w + \mu_{r} + (\alpha - \beta )c + 2\varLambda (z_{r} ) - \varTheta (z_{r} ), \\ \frac{{\partial E(\pi_{m} )}}{{\partial p_{e} }} & = [2\alpha (1 - \rho ) + \beta \rho ]A - (4\alpha^{2} - 2\beta^{2} )p_{e} + 2\alpha \beta w + \beta \mu_{r} + 2\alpha \mu_{e} + (2\alpha^{2} - \beta^{2} - \alpha \beta )c - \beta \varTheta (z_{r} ) - 2\alpha \varPsi (z_{e} ). \\ \end{aligned}$$

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We can easily obtain the Hessian matrix of (10) at point $(w,p_{e} )$,

$${\rm H} = \left( {\begin{array}{*{20}c} { - 2\alpha } & {2\beta } \\ {2\alpha \beta } & { - (4\alpha^{2} - 2\beta^{2} )} \\ \end{array} } \right) = 8\alpha (\alpha^{2} - \beta^{2} ).$$

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Its first-order principal minor is $- 2\alpha < 0$, and its second-order principal minor is $8\alpha (\alpha^{2} - \beta^{2} ) > 0$. Thus, $E(\pi_{m} )$ is a concave function associated with $(w,p_{e} )$. Thus, the optimal solution $w_{s}^{ * } ,p_{es}^{ * }$ can be obtained by solving the first-order condition of (10) about $(w,p_{e} )$. $\frac{{\partial E(\pi_{m} )}}{{\partial p_{e} }} = 0$, $\frac{{\partial E(\pi_{m} )}}{\partial w} = 0$, we can obtain

$$\begin{aligned} w_{s - MS}^{*} & = \frac{{[\alpha \rho + \beta (1 - \rho )]A + \alpha \mu_{r} + \beta \mu_{e} + (\alpha^{2} - \beta^{2} )c + (\frac{{2\alpha^{2} - \beta^{2} }}{\alpha })\varLambda (z_{r} ) - \alpha \varTheta (z_{r} ) - \beta \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}, \\ p_{es - MS}^{*} & = \frac{{[\alpha (1 - \rho ) + \beta \rho ]A + \beta \mu_{r} + \alpha \mu_{e} + (\alpha^{2} - \beta^{2} )c + \beta \varLambda (z_{r} ) - \beta \varTheta (z_{r} ) - \alpha \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}. \\ \end{aligned}$$

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We can obtain the optimal pricing of $p_{rs}^{ * }$ by taking $w_{s - M}^{ * } ,p_{es - M}^{ * }$ into $p_{r} (w,p_{e} )$.

$$p_{rs - MS}^{*} = \frac{\begin{aligned} & [(3\alpha^{2} - \beta^{2} )\rho + 2\alpha \beta (1 - \rho )]A{ + (3}\alpha^{2} - \beta^{2} )\mu_{r} + 2\alpha \beta \mu_{e} + (\alpha + \beta )(\alpha^{2} - \beta^{2} )c \\ & \quad + \;2\alpha^{2} \varLambda (z_{r} ) - (3\alpha^{2} - \beta^{2} )\varTheta (z_{r} ) - 2\alpha \beta \varPsi (z_{e} ) \\ \end{aligned} }{{4\alpha (\alpha^{2} - \beta^{2} )}}.$$

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1.3 Proof of Lemma 3

The decision-making questions of a retailer include the traditional retail channel pricing when the profit of the retailer is maximized. We solve the first-order partial derivative of $p_{r}$ about (13) and make it equal to 0, namely, $\partial E(\pi_{r} )/\partial p_{r} = 0$. We can obtain

$$p_{r} (w,p_{e} ) = \frac{{\rho A + \beta p_{e} + \mu_{r} + \alpha w + \alpha T - \varTheta (z_{r} )}}{2\alpha }.$$

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We place the upper substitution (14) and then find the partial derivatives of $w$ and $p_{e}$ about $E(\pi_{m} )$ respectively,

$$\begin{aligned} \frac{{\partial E(\pi_{m} )}}{\partial w} & = \rho A + 2\beta p_{e} - 2\alpha w + \mu_{r} + (\alpha - \beta )c{ - }2\alpha T - \varTheta (z_{r} ), \\ \frac{{\partial E(\pi_{m} )}}{{\partial p_{e} }} & = [2\alpha (1 - \rho ) + \beta \rho ]A - (4\alpha^{2} - 2\beta^{2} )p_{e} + 2\alpha \beta w + \beta \mu_{r} + 2\alpha \mu_{e} + (2\alpha^{2} - \beta^{2} - \alpha \beta )c + 2\alpha \beta T - \beta \varTheta (z_{r} ) - 2\alpha \varPsi (z_{e} ). \\ \end{aligned}$$

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We can easily obtain the Hessian matrix of $E(\pi_{m} )$ at point $(w,p_{e} )$,

$${\rm H} = \left( {\begin{array}{*{20}c} { - 2\alpha } & {2\beta } \\ {2\alpha \beta } & { - (4\alpha^{2} - 2\beta^{2} )} \\ \end{array} } \right) = 8\alpha (\alpha^{2} - \beta^{2} ).$$

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Its first-order principal minor is $- 2\alpha < 0$, and its second-order principal minor is $8\alpha (\alpha^{2} - \beta^{2} ) > 0$. Thus, $E(\pi_{m} )$ is a concave function associated with $(w,p_{e} )$. Thus, the optimal solution can be obtained by solving the first-order condition of (14) about $(w,p_{e} )$, which is equal to 0.$\frac{{\partial E(\pi_{m} )}}{{\partial p_{e} }} = 0$, $\frac{{\partial E(\pi_{m} )}}{\partial w} = 0$, we can obtain

$$\begin{aligned} w_{{^{s - DMS} }}^{*} & = \frac{{[\alpha \rho + \beta (1 - \rho )]A{ + }\alpha \mu_{r} + \beta \mu_{e} + (\alpha^{2} - \beta^{2} )c - 2(\alpha^{2} - \beta^{2} )T - \alpha \varTheta (z_{r} ) - \beta \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}, \\ p_{{_{es - DMS} }}^{*} & = \frac{{[\alpha (1 - \rho ) + \beta \rho ]A{ + }\beta \mu_{r} + \alpha \mu_{e} + (\alpha^{2} - \beta^{2} )c - \beta \varTheta (z_{r} ) - \alpha \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}. \\ \end{aligned}$$

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The optimal pricing of $p_{rs - DMS}^{ * }$ can be obtained by taking $w_{{^{s - DMS} }}^{*}$, $p_{{_{es - DMS} }}^{*}$ into $p_{r} (w,p_{e} ).$

$$p_{rs - DMS}^{*} = \frac{\begin{aligned} & [(3\alpha^{2} - \beta^{2} )\rho + 2\alpha \beta (1 - \rho )]A{ + (3}\alpha^{2} - \beta^{2} )\mu_{r} + 2\alpha \beta \mu_{e} + (\alpha + \beta )(\alpha^{2} - \beta^{2} )c \\ & \quad - \;(3\alpha^{2} - \beta^{2} )\varTheta (z_{r} ) - 2\alpha \beta \varPsi (z_{e} ) \\ \end{aligned} }{{4\alpha (\alpha^{2} - \beta^{2} )}}.$$

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1.4 Proof of Lemma 4

The decision-making issues of a retailer are the pricing of online and offline channels when maximizing profits. By solving the first-order partial derivatives of $p_{r}$ and $p_{e}$, and making them equal to 0, namely, $\frac{{\partial E(\pi_{r} )}}{{\partial p_{r} }} = 0$ and $\frac{{\partial E(\pi_{r} )}}{{\partial p_{e} }} = 0$, we can obtain the expression about the wholesale price.

$$\begin{aligned} p_{r} (w) & = \frac{{[\alpha \rho + \beta (1 - \rho )]A{ + }\alpha \mu_{r} + \beta \mu_{e} + (\alpha^{2} - \beta^{2} )w - \alpha \varTheta (z_{r} ) - \beta \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}, \\ p_{e} (w) & = \frac{{[\alpha (1 - \rho ) + \beta \rho ]A{ + }\beta \mu_{r} + \alpha \mu_{e} + (\alpha^{2} - \beta^{2} )w - \beta \varTheta (z_{r} ) - \alpha \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}. \\ \end{aligned}$$

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We can easily obtain the Hessian matrix of (17) at point $(p_{r} ,p_{e} )$:

$${\rm H} = \left( {\begin{array}{*{20}c} { - 2\alpha } & {2\beta } \\ {2\beta } & { - 2\alpha } \\ \end{array} } \right) = 4(\alpha^{2} - \beta^{2} ).$$

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Its first-order principal minor is $- 2\alpha < 0$, and second-order principal minor is $4(\alpha^{2} - \beta^{2} ) > 0$. Thus, $E(\pi_{r} )$ is the joint concave function associated with $(p_{r} ,p_{e} )$.

We put the upper substitution (18) and then make $\frac{{\partial E(\pi_{m} )}}{\partial w} = 0$ to obtain the optimal wholesale price of the manufacturer:

$$w_{{^{s - RS} }}^{ * } = \frac{{A{ + }\mu_{r} + \mu_{e} + 2(\alpha - \beta )c + 2[\varLambda (z_{r} ) + \varPhi (z_{e} )] - [\varTheta (z_{r} ) + \varPsi (z_{e} )]}}{4(\alpha - \beta )}.$$

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By putting $w_{{^{s - RS} }}^{ * }$ into the $p_{r} (w)$, $p_{e} (w)$, we can obtain the retailer’s optimal pricing of the traditional retail and online channels.

$$p_{rs - RS}^{*} = \frac{\begin{aligned} & (4\alpha \rho + 5\beta - 4\beta \rho + \alpha )A{ + (5}\alpha + \beta )\mu_{r} + (5\beta + \alpha )\mu_{e} + 2(\alpha^{2} - \beta^{2} )c \\ & \quad + \;2(\alpha + \beta )[\varLambda (z_{r} ) + \varPhi (z_{e} )] - (5\alpha + \beta )\varTheta (z_{r} ) - ( 5\beta + \alpha )\varPsi (z_{e} ) \\ \end{aligned} }{{8(\alpha^{2} - \beta^{2} )}}.$$

$$p_{es - RS}^{*} = \frac{\begin{aligned} & (4\beta \rho + 5\alpha - 4\alpha \rho + \beta )A{ + (5}\beta + \alpha )\mu_{r} + (5\alpha + \beta )\mu_{e} + 2(\alpha^{2} - \beta^{2} )c \\ & \quad + \;2(\alpha + \beta )[\varLambda (z_{r} ) + \varPhi (z_{e} )] - (\alpha + 5\beta )\varTheta (z_{r} ) - ( 5\alpha + \beta )\varPsi (z_{e} ) \\ \end{aligned} }{{8(\alpha^{2} - \beta^{2} )}}{\kern 1pt} {\kern 1pt} .$$

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1.5 Proof of Lemma 5

The decision-making issues of the retailer include the pricing of online and offline channels when maximizing profits. By solving the first-order partial derivative of $p_{r}$ and $p_{e}$ for (21) and making them equal to 0, namely, $\frac{{\partial E(\pi_{r} )}}{{\partial p_{r} }} = 0$ and $\frac{{\partial E(\pi_{r} )}}{{\partial p_{e} }} = 0$, we can obtain the expression about wholesale price as follows:

$$\begin{aligned} p_{r} (w) & = \frac{{[\alpha \rho + \beta (1 - \rho )]A{ + }\alpha \mu_{r} + \beta \mu_{e} + (\alpha^{2} - \beta^{2} )w - \alpha \varTheta (z_{r} ) - \beta \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}, \\ p_{e} (w) & = \frac{{[\alpha (1 - \rho ) + \beta \rho ]A{ + }\beta \mu_{r} + \alpha \mu_{e} + (\alpha^{2} - \beta^{2} )w{ + }(\alpha^{2} - \beta^{2} )T - \beta \varTheta (z_{r} ) - \alpha \varPsi (z_{e} )}}{{2(\alpha^{2} - \beta^{2} )}}. \\ \end{aligned}$$

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We can easily obtain the Hessian matrix of (21) at point $(p_{r} ,p_{e} )$,

$${\rm H} = \left( {\begin{array}{*{20}c} { - 2\alpha } & {2\beta } \\ {2\beta } & { - 2\alpha } \\ \end{array} } \right) = 4(\alpha^{2} - \beta^{2} ).$$

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Thus, $E(\pi_{r} )$ is the joint concave function associated with $(p_{r} ,p_{e} )$.

We put the upper substitution (22) and then make $\frac{{\partial E(\pi_{m} )}}{\partial w} = 0$ to obtain the optimal wholesale price of the manufacturer,

$$w_{s - DRS}^{*} = \frac{{A{ + }\mu_{r} + \mu_{e} + 2(\alpha - \beta )c - 2(\alpha - \beta )T + 2\varLambda (z_{r} ) - \varTheta (z_{r} ) - \varPsi (z_{e} )}}{4(\alpha - \beta )}.$$

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By putting $w_{s - DRS}^{*}$ into $p_{r} (w)$, $p_{e} (w)$ we can obtain the retailer’s optimal pricing of the traditional retail and online channels.

$$\begin{aligned} p_{rs - DRS}^{*} & = \frac{\begin{aligned} & (4\alpha \rho + 5\beta - 4\beta \rho + \alpha )A{ + (5}\alpha + \beta )\mu_{r} + (5\beta + \alpha )\mu_{e} + 2(\alpha^{2} - \beta^{2} )c - 2(\alpha^{2} - \beta^{2} )T \\ & \quad + \;2(\alpha + \beta )\varLambda (z_{r} ) - (5\alpha + \beta )\varTheta (z_{r} ) - (\alpha + 5\beta )\varPsi (z_{e} ) \\ \end{aligned} }{{8(\alpha^{2} - \beta^{2} )}}, \\ p_{es - DRS}^{*} & = \frac{\begin{aligned} & (4\beta \rho + 5\alpha - 4\alpha \rho + \beta )A{ + (5}\beta + \alpha )\mu_{r} + (5\alpha + \beta )\mu_{e} + 2(\alpha^{2} - \beta^{2} )c - 2(\alpha^{2} - \beta^{2} )T \\ & \quad + \;2(\alpha + \beta )\varLambda (z_{r} ) - (\alpha + 5\beta )\varTheta (z_{r} ) - (5\alpha + \beta )\varPsi (z_{e} ) \\ \end{aligned} }{{8(\alpha^{2} - \beta^{2} )}}. \\ \end{aligned}$$

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