“Product + logistics” bundling sale and co-delivery in cross-border e-commerce

Abstract

When customers purchase products via cross-border e-commerce, they care about both the product quality and the logistics service quality. Actually, retailers are selling “product + logistics” to customers, although their contracted logistics service provider (LSP) might not be preferred by customers. In practice, it is observed that a retailer selling high-quality products tends to contract with multiple LSPs to ensure higher customer volumes and the overall high quality of “product + logistics”. However, interestingly, we find that the LSP’s profits might be negatively affected by serving two competing retailers, and preferences of the LSP and the retailer selling high-quality products through logistical cooperation result in two “prisoner’s dilemma” regions. We also identify the size of the system’s profit pie and the allocation rules among the competing LSPs and retailers. We show that it is possible to observe competing retailers’ co-delivery, which benefits both the LSP and the retailer selling high-quality products.

Appendix

Proof of Lemma 1

• (a) Comparing the logistics fees and product prices in the two models, we have $t_{A}^{D} - t_{A}^{B} = \frac{m\theta }{1 + r} - \frac{5m - \Delta }{{5\left( {1 + r} \right)}} = \frac{{\Delta + 5m\left( {\theta - 1} \right)}}{{5\left( {1 + r} \right)}}$. Note that $0 < \Delta < 3m\left( {1 - \theta } \right)$, $t_{A}^{D} - t_{A}^{B} < \frac{{2m\left( {\theta - 1} \right)}}{{5\left( {1 + r} \right)}} < 0$

  $$\begin{aligned} t_{B}^{D} - t_{B}^{B} & = \frac{m\theta }{1 + r} - \frac{5m + \Delta }{{5\left( {1 + r} \right)}} = \frac{{ - \Delta + 5m\left( {\theta - 1} \right)}}{{5\left( {1 + r} \right)}} < 0 \\ p_{A}^{D} - p_{A}^{B} & = \frac{{3m\left( {1 - \theta } \right) - \Delta }}{{3\left( {1 + r} \right)}} - \frac{5m - \Delta }{{5\left( {1 + r} \right)}} = \frac{ - 2\Delta - 15m\theta }{{15\left( {1 + r} \right)}} < 0 \\ p_{B}^{D} - p_{B}^{B} & = \frac{{3m\left( {1 - \theta } \right) + \Delta }}{{3\left( {1 + r} \right)}} - \frac{5m + \Delta }{{5\left( {1 + r} \right)}} = \frac{2\Delta - 15m\theta }{{15\left( {1 + r} \right)}}. \\ \end{aligned}$$

  Calculating $2\Delta - 15m\theta = 0$, we have $\Delta_{1} = \frac{15m\theta }{2}$. Then we show that, $p_{B}^{D} > p_{B}^{B}$ if $\Delta > \Delta_{1} ;\, p_{B}^{D} < p_{B}^{B}$ otherwise.

• (b) Comparing the demand of four participants in the two models, we have the demand of R_A,$q_{AA}^{D} - q_{AA}^{B} = \left( {\frac{1}{2} - \frac{\Delta }{{6m\left( {1 - \theta } \right)}}} \right) - \left( {\frac{1}{2} - \frac{\Delta }{10m}} \right) = \frac{{\Delta \left( {3\theta + 2} \right)}}{{30m\left( {\theta - 1} \right)}} < 0$ the demand of R_B,$q_{BA}^{D} + q_{BB}^{D} - q_{BB}^{B} = \left( {\frac{\Delta }{{6m\left( {1 - \theta } \right)}} + \frac{1}{2}} \right) - \left( {\frac{1}{2} + \frac{\Delta }{10m}} \right) = - \frac{{\Delta \left( {3\theta + 2} \right)}}{{30m\left( {\theta - 1} \right)}} > 0$ the demand of L_A, $q_{BA}^{D} + q_{AA}^{D} - q_{AA}^{B} = \left( {\frac{\Delta }{{6m\left( {1 - \theta } \right)}} + \frac{1}{2} - \frac{\Delta }{{6m\left( {1 - \theta } \right)}}} \right) - \left( {\frac{1}{2} - \frac{\Delta }{10m}} \right) = \frac{\Delta }{10m} > 0$ the demand of L_B,$q_{BB}^{D} - q_{BB}^{B} = \frac{1}{2} - \left( {\frac{1}{2} + \frac{\Delta }{10m}} \right) = - \frac{\Delta }{10m} < 0$

• (c) Conducting a sensitivity analysis of price difference with respect to θ and Δ, we have

  $$\begin{aligned} \frac{{\partial \left( {p_{A}^{D} - p_{A}^{B} } \right)}}{\partial \theta } & = - \frac{m}{1 + r} < 0,\quad \frac{{\partial \left( {p_{A}^{D} - p_{A}^{B} } \right)}}{\partial \Delta } = - \frac{2}{{15\left( {1 + r} \right)}} < 0 \\ \frac{{\partial \left( {p_{B}^{D} - p_{B}^{B} } \right)}}{\partial \theta } & = - \frac{m}{1 + r} < 0,\quad \frac{{\partial \left( {p_{B}^{D} - p_{B}^{B} } \right)}}{\partial \Delta } = \frac{2}{{15\left( {1 + r} \right)}} > 0 \\ \frac{{\partial \left( {t_{A}^{D} - t_{A}^{B} } \right)}}{\partial \theta } & = \frac{m}{1 + r} > 0,\quad \frac{{\partial \left( {t_{A}^{D} - t_{A}^{B} } \right)}}{\partial \Delta } = \frac{1}{{5\left( {1 + r} \right)}} > 0 \\ \frac{{\partial \left( {t_{B}^{D} - t_{B}^{B} } \right)}}{\partial \theta } & = \frac{m}{1 + r} > 0,\quad \frac{{\partial \left( {t_{B}^{D} - t_{B}^{B} } \right)}}{\partial \Delta } = - \frac{1}{{5\left( {1 + r} \right)}} < 0 \\ \end{aligned}$$

Proof of Lemma 2

In the dual-LSP model, R_B makes profits from customer group BA and customer group BB. While in the base model, R_B makes profits from customer group BB.

• (a) Comparing the customer groups’ marginal profit and conducting a sensitivity analysis with respect to Δ, we have

  $$\begin{aligned} \left( { p_{B}^{D} + t_{i}^{D} } \right) - \left( {p_{B}^{B} + t_{B}^{B} } \right) & = - \frac{\Delta + 15m}{{15\left( {1 + r} \right)}} < 0 \\ \frac{{\partial \left[ {\left( {p_{B}^{D} + t_{i}^{D} } \right) - \left( {p_{B}^{B} + t_{B}^{B} } \right)} \right]}}{\partial \Delta } & = - \frac{1}{{15\left( {1 + r} \right)}} < 0 \\ \end{aligned}$$

• (b) Comparing the proportion of R_B in the customer groups’ marginal profit, we have

  $$\frac{{p_{B}^{D} }}{{p_{B}^{D} + t_{i}^{D} }} - \frac{{p_{B}^{B} }}{{p_{B}^{B} + t_{B}^{B} }} = \frac{1}{2} - \frac{3m\theta }{3m + \Delta }\left\{ {\begin{array}{ll} { > 0,} \hfill & {\quad if \theta \in \left( {0,\frac{1}{2}} \right)\quad and\quad \Delta \in \left( {0,\bar{\Delta }} \right);\,\theta \in \left( {\frac{1}{2},\frac{2}{3}} \right)\quad and\quad \Delta \in \left( {3m\left( { - 1 + 2\theta } \right),\bar{\Delta }} \right)} \hfill \\ { < 0,} \hfill & {\quad otherwise} \hfill \\ \end{array} } \right.$$

• (c) Conducting a sensitivity analysis of the difference between p_B’s proportion with respect to θ and Δ, we have

  $$\begin{aligned} \frac{{\partial \left( {\frac{{p_{B}^{D} }}{{p_{B}^{D} + t_{i}^{D} }} - \frac{{p_{B}^{B} }}{{p_{B}^{B} + t_{B}^{B} }}} \right)}}{\partial \theta } & = - \frac{3m}{3m + \Delta } < 0 \\ \frac{{\partial \left( {\frac{{p_{B}^{D} }}{{p_{B}^{D} + t_{i}^{D} }} - \frac{{p_{B}^{B} }}{{p_{B}^{B} + t_{B}^{B} }}} \right)}}{\partial \Delta } & = \frac{3m\theta }{{\left( {3m + \Delta } \right)^{2} }} > 0 \\ \end{aligned}$$

Proof of Proposition 1

We next examine the total profit pie of R_B’s customer groups and the conditions under which R_B can earn a larger share of the pie.

• (a) Comparing the total profit pie of R_B’s customer groups in the two models, we have

  $$\begin{aligned} \left( {\pi_{BA}^{D} + \pi_{BB}^{D} } \right) - \pi_{BB}^{B} & = \left( {\pi_{{R_{B} }}^{D} + \pi_{{L_{B} }}^{D} + t_{A}^{D} q_{BA}^{D} } \right) - \left( {\pi_{{R_{B} }}^{B} + \pi_{{L_{B} }}^{B} } \right) \\ & = \left( {\frac{{\left[ {\Delta + 3m\left( {1 - \theta } \right)} \right]^{2} }}{{18m\left( {1 - \theta } \right)\left( {1 + r} \right)}} + \frac{m\theta }{{2\left( {1 + r} \right)}} + \frac{m\theta }{1 + r}.\frac{\Delta }{{6m\left( {1 - \theta } \right)}}} \right) \\ & \quad - \left( {\frac{{\left( {5m + \Delta } \right)^{2} }}{{50m\left( {1 + r} \right)}} + \frac{{\left( {5m + \Delta } \right)^{2} }}{{50m\left( {1 + r} \right)}}} \right) \\ & = \frac{{225m^{2} \left( { - 1 + \theta } \right) + 15m\Delta \left( { - 2 + 7\theta } \right) + \Delta^{2} \left( {7 + 18\theta } \right)}}{{450m\left( {1 + r} \right)\left( {1 - \theta } \right)}} < 0 \\ \end{aligned}$$

  for $0 < \theta < 1 and 0 < \Delta < 3m - 3m\theta$

• (b) Comparing the proportion of R_B in the total profit pie of R_B’s customer groups, we have

  $$\frac{{\pi_{{R_{B} }}^{D} }}{{\pi_{BA}^{D} + \pi_{BB}^{D} }} - \frac{{\pi_{{R_{B} }}^{B} }}{{\pi_{BB}^{B} }} = \frac{1}{2} - \frac{3m\theta }{3m + \Delta }\left\{ {\begin{array}{ll} { > 0,} \hfill & {\quad if\,\, \theta \in \left( {0,\frac{1}{2}} \right) \quad and \quad \Delta \in \left( {0,\bar{\Delta }} \right);} \hfill \\ {\theta \in \left( {\frac{1}{2},\frac{2}{3}} \right) \quad and\quad \Delta \in \left( {3m\left( { - 1 + 2\theta } \right),\bar{\Delta }} \right)} \hfill & {} \hfill \\ { < 0,} \hfill & {\quad otherwise} \hfill \\ \end{array} } \right.$$

Proof of Lemma 3

• (a) We use $\frac{{{\text{q}}_{{\text{BA}}}^{{\text{D}}} }}{{{\text{q}}_{{\text{AA}}}^{{\text{D}}} }}$ to characterize the transfer effect in the dual-LSP model. Conducting a sensitivity analysis of the transfer effect with respect to θ and Δ, we have

  $$\begin{aligned} \frac{{\partial \left( {\frac{{q_{BA}^{D} }}{{q_{AA}^{D} }}} \right)}}{\partial \theta } & = \frac{{\partial \left( {\frac{\Delta }{{3m\left( {1 - \theta } \right) - \Delta }}} \right)}}{\partial \theta } = \frac{3m\Delta }{{\left[ {3m\left( {1 - \theta } \right) - \Delta } \right]^{2} }} > 0 \\ \frac{{\partial \left( {\frac{{q_{BA}^{D} }}{{q_{AA}^{D} }}} \right)}}{\partial \Delta } & = \frac{{3m\left( {1 - \theta } \right)}}{{\left[ {3m\left( {1 - \theta } \right) - \Delta } \right]^{2} }} > 0 \\ \end{aligned}$$

• (b) In the dual-LSP model, L_A makes profits from customer group AA and customer group BA. While in the base model, L_A makes profits from customer group AA. Comparing the customer groups’ marginal profit and conducting a sensitivity analysis with respect to θ and Δ, we have

  $$\begin{aligned} \frac{{\partial \left[ {\left( {p_{A}^{D} + t_{A}^{D} } \right) - \left( {p_{A}^{B} + t_{A}^{B} } \right)} \right]}}{\partial \Delta } & = \frac{{\partial \left[ {\frac{ - 15m + \Delta }{{15\left( {1 + r} \right)}}} \right]}}{\partial \Delta } = \frac{1}{{15\left( {1 + r} \right)}} > 0 \\ \frac{{\partial \left[ {\left( {p_{B}^{D} + t_{A}^{D} } \right) - \left( {p_{A}^{B} + t_{A}^{B} } \right)} \right]}}{\partial \Delta } & = \frac{{\partial \left[ {\frac{ - 15m + 11\Delta }{{15\left( {1 + r} \right)}}} \right]}}{\partial \Delta } = \frac{11}{{15\left( {1 + r} \right)}} > 0 \\ \end{aligned}$$

• (c) Conducting a sensitivity analysis of the difference between t_A’s proportion with respect to θ and Δ, we have

  $$\begin{aligned} \frac{{\partial \left( {\frac{{t_{A}^{D} }}{{p_{A}^{D} + t_{A}^{D} }} - \frac{{t_{A}^{B} }}{{p_{A}^{B} + t_{A}^{B} }}} \right)}}{\partial \theta } & = \frac{{\partial \left( { - \frac{1}{2} + \frac{3m\theta }{3m - \Delta }} \right)}}{\partial \theta } = \frac{3m}{3m - \Delta } > 0 \\ \frac{{\partial \left( {\frac{{t_{A}^{D} }}{{p_{B}^{D} + t_{A}^{D} }} - \frac{{t_{A}^{B} }}{{p_{A}^{B} + t_{A}^{B} }}} \right)}}{\partial \theta } & = \frac{{\partial \left( { - \frac{1}{2} + \frac{3m\theta }{3m + \Delta }} \right)}}{\partial \theta } = \frac{3m}{3m + \Delta } > 0 \\ \frac{{\partial \left( {\frac{{t_{A}^{D} }}{{p_{A}^{D} + t_{A}^{D} }} - \frac{{t_{A}^{B} }}{{p_{A}^{B} + t_{A}^{B} }}} \right)}}{\partial \Delta } & = \frac{{\partial \left( { - \frac{1}{2} + \frac{3m\theta }{3m - \Delta }} \right)}}{\partial \Delta } = \frac{3m\theta }{{\left( {3m - \Delta } \right)^{2} }} > 0 \\ \frac{{\partial \left( {\frac{{t_{A}^{D} }}{{p_{B}^{D} + t_{A}^{D} }} - \frac{{t_{A}^{B} }}{{p_{A}^{B} + t_{A}^{B} }}} \right)}}{\partial \Delta } & = \frac{{\partial \left( { - \frac{1}{2} + \frac{3m\theta }{3m + \Delta }} \right)}}{\partial \Delta } = - \frac{3m\theta }{{\left( {3m + \Delta } \right)^{2} }} < 0 \\ \end{aligned}$$

Proof of Proposition 2

We next examine the total profit pie of L_A’s customer groups and the conditions under which L_A can earn a larger share of the pie.

• (a) Comparing the total profit pie of L_A’s customer groups in the two models, we have

  $$\begin{aligned} \left( {\pi_{AA}^{D} + \pi_{BA}^{D} } \right) - \pi_{AA}^{B} & = \left( {\pi_{{R_{A} }}^{D} + \pi_{{L_{A} }}^{D} + p_{B}^{D} q_{BA}^{D} } \right) - \left( {\pi_{{R_{A} }}^{B} + \pi_{{L_{A} }}^{B} } \right) \\ & = \frac{{ - 225m^{2} \left( { - 1 + \theta } \right) + 105m\Delta \left( { - 1 + \theta } \right) - 2\Delta^{2} \left( {16 + 9\theta } \right)}}{{450m\left( {1 + r} \right)\left( { - 1 + \theta } \right)}} \\ & \quad \left\{ {\begin{array}{l} { > 0, \quad if \theta \in \left( {0,\frac{{ - 49 + 5\sqrt {217} }}{36}} \right)\quad and\quad \Delta \in \left( {\frac{{15m\left[ { - 7\left( {1 - \theta } \right) + \sqrt {\left( {1 - \theta } \right)\left( {177 + 23\theta } \right)} } \right]}}{{4\left( {16 + 9\theta } \right)}},\bar{\Delta }} \right)} \\ { < 0,\quad otherwise} \\ \end{array} } \right. \\ \end{aligned}$$

• (b) Comparing the proportion of L_A in the total profit pie of L_A’s customer groups, we have

  $$\begin{aligned} \frac{{\pi_{{L_{A} }}^{D} }}{{\pi_{AA}^{D} + \pi_{BA}^{D} }} - \frac{{\pi_{{L_{A} }}^{B} }}{{\pi_{AA}^{B} }} & = \frac{{2\Delta^{2} + 3m\Delta \left( { - 1 + \theta } \right) + 9m^{2} \left( { - 1 + \theta } \right)\left( { - 1 + 2\theta } \right)}}{{2\left( { - 2\Delta^{2} + 9m^{2} \left( { - 1 + \theta } \right) - 3m\Delta \left( { - 1 + \theta } \right)} \right)}} \\ & \quad \left\{ {\begin{array}{l} { > 0, \quad if\quad \theta \in \left( {\frac{7}{15},\frac{1}{2}} \right) \quad and\quad \Delta \in \left( {\frac{{3m\left[ {\left( {1 - \theta } \right) - \sqrt {\left( {1 - \theta } \right)\left( { - 7 + 15\theta } \right)} } \right]}}{4},\quad \frac{{3m\left[ {\left( {1 - \theta } \right) + \sqrt {\left( {1 - \theta } \right)\left( { - 7 + 15\theta } \right)} } \right]}}{4}} \right);} \hfill \\ {\theta \in \left( {\frac{1}{2},\frac{2}{3}} \right) \quad and\quad \Delta \in \left( {0,\frac{{3m\left[ {\left( {1 - \theta } \right) + \sqrt {\left( {1 - \theta } \right)\left( { - 7 + 15\theta } \right)} } \right]}}{4}} \right)} \hfill \\ {\theta \in \left( {\frac{2}{3},1} \right)\quad and\quad \Delta \in \left( {0,\bar{\Delta }} \right)} \hfill \\ { < 0,\quad otherwise} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

Proof of Corollary 1

We next examine the conditions under which R_B and L_A have incentives to cooperate with each other.

• (a) Comparing the profits of R_B in the two models, we have

  $$\begin{aligned} \pi_{{R_{B} }}^{D} - \pi_{{R_{B} }}^{B} & = - \frac{{ - 60m\Delta \left( { - 1 + \theta } \right) + 225m^{2} \left( { - 1 + \theta } \right)\theta + \Delta^{2} \left( {16 + 9\theta } \right)}}{{450m\left( {1 + r} \right)\left( { - 1 + \theta } \right)}} \\ & \quad \left\{ {\begin{array}{ll} > 0, &\quad if \theta \in \left( {0,\frac{{ - 26 + 10\sqrt {10} }}{9}} \right) \quad and\quad \Delta \in \left( {\frac{{15m\left[ { - 2\left( {1 - \theta } \right) + \left( {2 + 3\theta } \right)\sqrt {1 - \theta } } \right]}}{16 + 9\theta },\bar{\Delta }} \right) \\ < 0, &\quad otherwise \\ \end{array} } \right. \\ \end{aligned}$$

• (b) Comparing the profits of L_A in the two models, we have

  $$\left\{ {\begin{array}{ll} > 0,&\quad if \theta \in \left( {\frac{4}{9},1} \right)\quad and \quad \Delta \in \left( {5m\left( {1 - \sqrt \theta } \right),\bar{\Delta }} \right) \\ < 0, &\quad otherwise \\ \end{array} } \right.$$