Who should pay for return freight in the online retailing? Retailers or consumers

Abstract

This paper studies the online shopping situation where retailer faces uncertain demand and uncertain consumer valuations. We compare the suitability and effectiveness of two return freight policies, consumer affording return freight (C-Policy) and retailer paying return freight (R-Policy). Moreover, we explore the effect of non-defective returns, and the retailer’s optimal decisions on retail price and order quantity. Our results suggest that return freight policy is related to the actual quantity of returns and the proportion of non-defective returns. In general, R-Policy is reasonable when the actual returns is low, C-Policy is reasonable when the actual returns is high. But when the actual returns is not too low or too high, the return freight policy is closely related to the proportion of non-defective returns. Our study shows C-Policy is better if the proportion of non-defective returns is lower, otherwise R-Policy is better. In addition, we find the optimal retail price and the optimal order quantity decisions are also related to the actual returns and the proportion of non-defective returns. Usually, the higher the actual returns, the lower the optimal retail price and the more optimal order quantity. The higher the proportion of non-defective returns, the higher the optimal retail price and the less the optimal order quantity. At last, we find high returns are harmful to retailers, which erode the profitability of online retailers. However, an interesting observation is that the damage of high returns can be alleviated when most of returns are non-defective returns. That indicates the risk of high returns is not as terrible as we intuitively think.

Appendices

Appendix 1

Proof of Proposition 3

The partial derivatives of $\Pi_{1} \left( {p_{1} ,Q_{1} } \right)$ are

$$ \begin{aligned} \frac{{\partial {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial p_{1} }} & = \left\{ {\left( {\gamma - \beta } \right)F\left[ {Q_{1} - \alpha - \left( {\gamma - \beta } \right)p_{1} + t\gamma } \right]} \right\}\left\{ {p_{1} - s - \phi {\text{U}}\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right]} \right\} \\ & \quad + Emin\left( {Q_{1} ,D_{1} } \right)\left\{ {1 - \phi u\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right] - \phi U\left( {p_{1} - t} \right)\left( {1 - {{\uptheta }}} \right)} \right\} \\ \end{aligned} $$

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$$ \frac{{\partial {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial Q_{1} }} = \left\{ {1 - F\left[ {Q_{1} - \alpha - \left( {\gamma - \beta } \right)p_{1} + t\gamma } \right]} \right\}\left\{ {p_{1} - s - \phi {\text{U}}\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right]} \right\} - \left( {c - s} \right) $$

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We can get:

$$ \begin{aligned} \frac{{\partial^{2} {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial p_{1}^{2} }} & = - \left( {\beta - \gamma } \right)^{2} f\left[ {Q_{1} - \alpha - \left( {\gamma - \beta } \right)p_{1} + t\gamma } \right]\left\{ {p_{1} - s - \phi U\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right]} \right\} \\ & \quad + 2\left( {\gamma - \beta } \right)F\left[ {Q_{1} - \alpha - \left( {\gamma - \beta } \right)p_{1} + t\gamma } \right]\left\{ {1 - \phi u\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right] - \phi U\left( {p_{1} - t} \right)\left( {1 - {{\uptheta }}} \right)} \right\} \\ & \quad - 2\phi \left( {1 - \theta } \right)u\left( {p_{1} - t} \right)Emin\left( {Q_{1} ,D_{1} } \right) \\ \end{aligned} $$

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$$ \frac{{\partial^{2} {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial Q_{1}^{2} }} = - f\left[ {Q_{1} - \alpha - \left( {\gamma - \beta } \right)p_{1} + t\gamma } \right]\left\{ {p_{1} - s - \phi U\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right]} \right\} $$

(26)

There are optimal solutions for retail price and order quantity. It can be proved that $\frac{{\partial^{2} {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial p_{1}^{2} }} < 0$ and $\frac{{\partial^{2} {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial Q_{1}^{2} }} < 0$, when p₁ − s − ϕU(p₁ − t)[(1 − θ)(p₁ − s) + c_r] > 0 and 1 − ϕU(p₁ − t)[(1 − θ)(p₁ − s) + c_r] − ϕU(p₁ − t)(1 − θ) > 0.

Meanwhile, the value of Hessian Matrix is greater than 0, which can be proved as follows.

$$ \begin{aligned} H_{1} & = \left| {\begin{array}{*{20}c} {\frac{{\partial^{2} {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial p_{1}^{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial p_{1} \partial Q_{1} }}} \\ {\frac{{\partial^{2} {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial Q_{1} \partial p_{1} }}} & {\frac{{\partial^{2} {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial Q_{1}^{2} }}} \\ \end{array} } \right| \\ & = 2\left( {\beta - \gamma } \right)f\left[ {Q_{1} - \alpha - \left( {\gamma - \beta } \right)p_{1} + t\gamma } \right]\left\{ {p_{1} - s - \phi {\text{U}}\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right]} \right\}\left\{ {1 - \phi u\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right] - \phi {\text{U}}\left( {p_{1} - t} \right)\left( {1 - {{\uptheta }}} \right)} \right\} \\ & \quad + 2\phi \left( {1 - {{\uptheta }}} \right)u\left( {p_{1} - t} \right)Emin\left( {Q_{1} ,D_{1} } \right)f\left[ {Q_{1} - \alpha - \left( {\gamma - \beta } \right)p_{1} + t\gamma } \right]\left\{ {p_{1} - s - \phi {\text{U}}\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right]} \right\} \\ & \quad - \left\{ {\left\{ {1 - F\left[ {Q_{1} - \alpha - \left( {\gamma - \beta } \right)p_{1} + t\gamma } \right]} \right\}\left\{ {1 - \phi u\left( {p_{1} - t} \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{1} - s} \right) + c_{r} } \right] - \phi U\left( {p_{1} - t} \right)\left( {1 - {{\uptheta }}} \right)} \right\}} \right\}^{2} > 0 \\ \end{aligned} $$

Therefore, the solution of optimal retail price $p_{1}^{*}$ and order quantity $Q_{1}^{*}$ is unique. Furthermore, according to $\frac{{\partial {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial p_{1} }} = 0$ and $\frac{{\partial {{\Pi }}_{1} \left( {p_{1} ,Q_{1} } \right)}}{{\partial Q_{1} }} = 0$, we derive the Proposition 3.

Appendix 2

Proof of Proposition 4

The partial derivatives of $\Pi_{2} \left( {p_{2} ,Q_{2} } \right)$ and the second derivatives are:

$$ \begin{aligned} \frac{{\partial {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial p_{2} }} & = \left( {\gamma - \beta } \right) F\left[ {Q_{2} - \alpha - \left( {\gamma - \beta } \right)p_{2} } \right]\left\{ {p_{2} - s - \phi U\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right]} \right\} \\ & \quad + Emin\left( {Q_{2} ,D_{2} } \right)\left\{ {1 - \phi u\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right] - \phi U\left( {p_{2} } \right)\left( {1 - {{\uptheta }}} \right)} \right\} \\ \end{aligned} $$

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$$ \frac{{\partial {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial Q_{2} }} = \left\{ {1 - F\left[ {Q_{2} - \alpha - \left( {\gamma - \beta } \right)p_{2} } \right]} \right\}\left\{ {p_{2} - s - \phi U\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right]} \right\} - \left( {c - s} \right) $$

(28)

$$ \begin{aligned} \frac{{\partial^{2} {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial p_{2}^{2} }} & = - \left( {\gamma - \beta } \right)^{2} f\left[ {Q_{2} - \alpha - \left( {\gamma - \beta } \right)p_{2} } \right]\left\{ {p_{2} - s - \phi U\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right]} \right\} \\ & \quad + 2\left( {\gamma - \beta } \right)F\left[ {Q_{2} - \alpha - \left( {\gamma - \beta } \right)p_{2} } \right]\left\{ {1 - \phi u\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right] - \phi U\left( {p_{2} } \right)\left( {1 - {{\uptheta }}} \right)} \right\} \\ & \quad - 2\phi \left( {1 - {{\uptheta }}} \right)u\left( {p_{2} } \right)Emin\left( {Q_{2} ,D_{2} } \right) \\ \end{aligned} $$

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$$ \frac{{\partial^{2} {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial Q_{2}^{2} }} = - f\left[ {Q_{2} - \alpha - \left( {\gamma - \beta } \right)p_{2} } \right]\left\{ {p_{2} - s - \phi U\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right]} \right\} $$

(30)

There are optimal retail price $p_{2}^{*}$ and order quantity $Q_{2}^{*}$. It can be proved that $\frac{{\partial^{2} {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial p_{2}^{2} }} < 0$ and $\frac{{\partial^{2} {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial Q_{2}^{2} }} < 0$, if p₂ − s − ϕU(p₂)[(1 − θ)(p₂ − s) + c_r + t] > 0 and 1 − ϕU(p₂)[(1 − θ)(p₂ − s) + c_r + t] − ϕU(p₂)(1 − θ) > 0.

Meanwhile, the value of Hessian Matrix is greater than 0, which can be proved as follows

$$ \begin{aligned} H_{2} & = \left| {\begin{array}{*{20}c} {\frac{{\partial^{2} {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial p_{2}^{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial p_{2} \partial Q_{2} }}} \\ {\frac{{\partial^{2} {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial Q_{2} \partial p_{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial Q_{2}^{2} }}} \\ \end{array} } \right| \\ & = 2\left( {\beta - \gamma } \right)f\left[ {Q_{2} - \alpha - \left( {\gamma - \beta } \right)p_{2} } \right]\left\{ {p_{2} - s - \phi U\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right]} \right\}\left\{ {1 - \phi u\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right] - \phi U\left( {p_{2} } \right)\left( {1 - {{\uptheta }}} \right)} \right\} \\ & \quad + 2\phi \left( {1 - {{\uptheta }}} \right)u\left( {p_{2} } \right)Emin\left( {Q_{2} ,D_{2} } \right)f\left[ {Q_{2} - \alpha - \left( {\gamma - \beta } \right)p_{2} } \right]\left\{ {p_{2} - s - \phi U\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right]} \right\} \\ & \quad - \left\{ {\left\{ {1 - F\left[ {Q_{2} - \alpha - \left( {\gamma - \beta } \right)p_{2} } \right]} \right\}\left\{ {1 - \phi u\left( {p_{2} } \right)\left[ {\left( {1 - {{\uptheta }}} \right)\left( {p_{2} - s} \right) + c_{r} + t} \right] - \phi U\left( {p_{2} } \right)\left( {1 - {{\uptheta }}} \right)} \right\}} \right\}^{2} > 0. \\ \end{aligned} $$

Therefore, the solution of optimal retail price $p_{2}^{*}$ and order quantity $Q_{2}^{*}$ is unique. Furthermore, according to $\frac{{\partial {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial p_{2} }} = 0$ and $\frac{{\partial {{\Pi }}_{2} \left( {p_{2} ,Q_{2} } \right)}}{{\partial Q_{2} }} = 0$, we derive the Proposition 4.