Implications of e-tailers’ transition from reselling to the combined reselling and agency selling

Abstract

Whether or not online retailers (“e-tailers”) should choose a reselling or agency selling business mode has been well studied by academics. In practice, many major e-tailers, including Amazon and JD.com, are moving toward a combined approach, but not toward the pure agency selling mode. We adopt a game theoretical model to examine the implications of this transition by considering a supplier that distributes its products through a bricks-and-mortar retailer and an e-tailer who has the flexibility of using reselling or combined reselling and agency selling. Conventional wisdom suggests that the agency selling would benefit the supply chain members due to the effect of eliminating double-marginalization. Our results, however, reveal that, comparing with the reselling mode, the additional agency selling is not always beneficial to the supplier nor always helpful to the bricks-and-mortar retailer. These findings not only complement the emerging literature regarding the strategic mode choice of e-tailers but also, from a practical perspective, urge caution in whether the supplier should participate in the e-tailer’s marketplace through agency selling.

Appendix

Proofs for “Implications of e-tailers’ transition from reselling to. the combined reselling and agency selling”. We use backward induction to solve for the equilibrium channel configuration. Firstly, conditional on the channel structure (R, RA), we derive the equilibrium outcomes like the demand, wholesale price, commission fees and service level with using backward induction for a specific subgame. These outcomes are presented in Table 2 of Sect. 4. Secondly, we notice that all parameters and variables must satisfy non-negativity constraints. Thus, we solve the parameter scope of these nonlinear conditions in model R and RA:$\Delta {*} = \frac{{{8}c_{r} k\gamma^{2} + {4}k\gamma^{3} - {33}c_{r} k\gamma - {29}k\gamma^{2} + {33}k\gamma + {9}c_{r} - {9}}}{{k\gamma {(4}\gamma { - 21)}}} < c_{e} < \frac{{\gamma (c_{r} + 1)}}{2},\frac{8}{{\gamma {(}\gamma { - 4)}^{{2}} }} < k$. In the range of the threshold $\Delta {*}$ ( referred to as $\underline{{c_{e} }} = \frac{{{8}c_{r} k\gamma^{2} + {4}k\gamma^{3} - {33}c_{r} k\gamma - {29}k\gamma^{2} + {33}k\gamma + {9}c_{r} - {9}}}{{k\gamma {(4}\gamma { - 21)}}},\overline{{c_{e} }} = \frac{{\gamma (c_{r} + 1)}}{2},\underline {k} = \frac{8}{{\gamma {(}\gamma { - 4)}^{{2}} }}$),for the rest of the paper), we compare the outcomes between Model R and Model RA and draw the Proposition 1–7.

Appendix A1. Proof of Proposition 1

Under model RA, in order to show the variation in the commission fee charged by the e-tailer w.r.t. the order-fulfillment costs, we take the derivative of the fee $f$ with respect to the e-tailer's order-fulfillment costs $c_{e}$, which yields:

$${\raise0.7ex\hbox{${\partial f}$} \!\mathord{\left/ {\vphantom {{\partial f} {\partial c}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c}$}}_{e} = \frac{{ - k\gamma (\gamma^{2} - 14\gamma + 30)}}{{4k\gamma^{3} - 36k\gamma^{2} + 66k\gamma + \gamma - 18}}$$

Clearly the numerator of the above expression is lower than zero under our assumption that $0 < \gamma < 1$ and $\underline{k} < k$. Thus, to prove ${\raise0.7ex\hbox{${\partial f}$} \!\mathord{\left/ {\vphantom {{\partial f} {\partial c}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c}$}}_{e} < 0$, we have to show that the denominator of the above expression is larger than zero, i.e., $4k\gamma^{3} - 36k\gamma^{2} + 66k\gamma + \gamma - 18 > 0$, after simplification, this reduces to $k > \frac{18 - \gamma }{{2\gamma {(2}\gamma^{2} - {18}\gamma { + 33)}}}$, which is true in the range of the threshold $\Delta {*}$. That is to say, for any $0 < \gamma < 1$,$k > \underline{k}$, $4k\gamma^{3} - 36k\gamma^{2} + 66k\gamma + \gamma - 18 > 0$ and ${\raise0.7ex\hbox{${\partial f}$} \!\mathord{\left/ {\vphantom {{\partial f} {\partial c_{e} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e} }$}} < 0$. Further we can conclude that the commission fee $f$ is decreasing in the value of $c_{e}$.

Similarly, we take the derivative of the fee $f$ with respect to the bricks-and-mortar retailer's order-fulfillment costs $c_{r}$, which yields: ${\raise0.7ex\hbox{${\partial f}$} \!\mathord{\left/ {\vphantom {{\partial f} {\partial c}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c}$}}_{r} = \frac{{( - 3k\gamma^{3} + 8k\gamma^{2} + \gamma )}}{{4k\gamma^{3} - 36k\gamma^{2} + 66k\gamma + \gamma - 18}}$. It is obvious that the value is always positive given the condition $0 < \gamma < 1$,$k > \underline{k}$. Therefore, the commission fee $f$ is increasing in the value of $c_{r}$, i.e., ${\raise0.7ex\hbox{${\partial f}$} \!\mathord{\left/ {\vphantom {{\partial f} {\partial c}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c}$}}_{r} > 0$.

Appendix A2. Proof of Proposition 2

We subtract the bricks-and-mortar retailer’s wholesales price from the e-tailer’s wholesales price in model R to obtain the following:

$$W_{Difference}^{R} = w_{e}^{R} - w_{r}^{R} = \frac{{k\gamma (\gamma - 4)^{2} }}{{8 - 2k\gamma (\gamma - 4)^{2} }}c_{e} + \frac{{k\gamma^{2} (\gamma - 4)^{2} + 2c_{r} \gamma - 2\gamma }}{{2k\gamma (\gamma - 4)^{2} - 8}} + \frac{{c_{r} - 1}}{2}$$

Notice the function $W_{Difference}^{R} (c_{e} )$ is linear, we can obtain $W_{Difference}^{R} (\overline{{c_{e} }} ) < 0$ and $W_{Difference}^{R} (\underline{{c_{e} }} ) > 0$ in the range of the threshold $\Delta {*}$, and $0 < \gamma < 1$.Hence, it follows that there exists a $c_{e1}^{R} \in (\underline{{c_{e} }} ,\overline{{c_{e} }} )$ such that $W_{Difference}^{R} (c_{e1}^{R} ) = 0$. Thus, that is to true for $c_{e} \in (c_{e1}^{R} ,\overline{{c_{e} }} )$, $W_{Difference}^{R} < 0$, i.e., $w_{e}^{R} < w_{r}^{R}$ and $c_{e} \in (\underline{{c_{e} }} ,c_{e1}^{R} )$, $W_{Difference}^{R} > 0$, i.e.,$w_{e}^{R} > w_{r}^{R}$.

Next we compare the e-tailer’s wholesales price with the bricks-and-mortar retailer’s wholesales price in model RA. Following a similar procedure, we find $\begin{gathered} W_{Difference}^{RA} = w_{e}^{RA} - w_{r}^{RA} \hfill \\ = \frac{{k\gamma ( - 2\gamma^{2} { + }18\gamma - 33)}}{{2k\gamma (2\gamma^{2} - 18\gamma + 33) + \gamma - 18}}c_{e} + \frac{{2k\gamma^{4} - 20k\gamma^{3} + 51k\gamma^{2} - 33k\gamma + 4c_{r} \gamma - 4\gamma - 9c_{r} + 33c_{r} k\gamma - 18c_{r} k\gamma^{2} + 2c_{r} k\gamma^{3} + 9}}{{2k\gamma (2\gamma^{2} - 18\gamma + 33) + \gamma - 18}} \hfill \\ \end{gathered}$ The function $W_{Difference}^{RA} (c_{e} )$ is still linear. Further, we obtain $W_{Difference}^{RA} (\overline{{c_{e} }} ) < 0$ and $W_{Difference}^{RA} (\underline{{c_{e} }} ) > 0$ in the range of the threshold $\Delta {*}$, and $0 < \gamma < 1$, which shows that there exists a $c_{e1}^{RA} \in (\underline{{c_{e} }} ,\overline{{c_{e} }} )$ such that $W_{Difference}^{RA} (c_{e1}^{RA} ) = 0$. Hence, for $c_{e} \in (c_{e1}^{RA} ,\overline{{c_{e} }} )$,$W_{Difference}^{RA} < 0$,i.e.,$w_{e}^{RA} < w_{r}^{RA}$ and for $c_{e} \in (\underline{{c_{e} }} ,c_{e1}^{RA} )$, $W_{Difference}^{RA} > 0$, i.e.,$w_{e}^{RA} > w_{r}^{RA}$.

Appendix A3. Proof of Proposition 3

The difference between the e-tailer’s profits under the combined reselling and agency selling mode and the e-tailer’s profits under the reselling mode is as follows:

$$\pi_{e}^{RA} - \pi_{e}^{R} = \frac{{k( - 2c_{e} + \gamma + c_{r} \gamma )^{2} ( - 8 + 16k\gamma - 8k\gamma^{2} + k\gamma^{3} )}}{{4( - 4 + 16k\gamma - 8k\gamma^{2} + k\gamma^{3} )^{2} }} + \frac{{\left[ \begin{gathered} - 648c_{e}^{2} k + 5\gamma - 10c_{r} \gamma + 5c_{r}^{2} \gamma + 912c_{e} k\gamma + 72c_{e}^{2} k\gamma + 384c_{e} c_{r} k\gamma + 2016c_{e}^{2} k^{2} \gamma - 352k\gamma^{2} - 174c_{e} k\gamma^{2} - 2c_{e}^{2} k\gamma^{2} \hfill \\ - 208c_{r} k\gamma^{2} + 30c_{e} c_{r} k\gamma^{2} - 88c_{r}^{2} k\gamma^{2} - 2904c_{e} k^{2} \gamma^{2} - 1488c_{e}^{2} k^{2} \gamma^{2} - 1128c_{e} c_{r} k^{2} \gamma^{2} + 100k\gamma^{3} + 10c_{e} k\gamma^{3} - 26c_{r} k\gamma^{3} \hfill \\ - 6c_{e} c_{r} k\gamma^{3} - 2c_{r}^{2} k\gamma^{3} + 1089k^{2} \gamma^{3} + 2334c_{e} k^{2} \gamma^{3} + 381c_{e}^{2} k^{2} \gamma^{3} + 726c_{r} k^{2} \gamma^{3} + 642c_{e} c_{r} k^{2} \gamma^{3} + 201c_{r}^{2} k^{2} \gamma^{{3}} - 8k\gamma^{4} \hfill \\ + 6c_{r} k\gamma^{4} - 946k^{2} \gamma^{4} - 666c_{e} k^{2} \gamma^{4} - 44c_{e}^{2} k^{2} \gamma^{4} - 442c_{r} k^{2} \gamma^{4} - 96c_{e} c_{r} k^{2} \gamma^{4} - 100c_{r}^{2} k^{2} \gamma^{4} + 295k^{2} \gamma^{5} + 84c_{e} k^{2} \gamma^{5} \hfill \\ + 2c_{e}^{2} k^{2} \gamma^{5} + 76c_{r} k^{2} \gamma^{5} + 4c_{e} c_{r} k^{2} \gamma^{5} + 10c_{r}^{2} k^{2} \gamma^{5} - 40k^{2} \gamma^{6} - 4c_{e} k^{2} \gamma^{6} - 4c_{r} k^{2} \gamma^{6} + {2}k^{2} \gamma^{7} \hfill \\ \end{gathered} \right]}}{{4( - 18 + \gamma + 66k\gamma - 36k\gamma^{2} + 4k\gamma^{3} )^{2} }}$$

Let $f(c_{e} ) = \pi_{e}^{RA} - \pi_{e}^{R}$, to prove $\pi_{e}^{RA} > \pi_{e}^{R}$, we have to show that $f(c_{e} ) > 0$. The function $f(c_{e} )$ is convex, i.e., ${\raise0.7ex\hbox{${\partial^{2} f(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial^{2} f(c_{e} )} {\partial c_{e}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e}^{2} }$}} > 0$. Further, we obtain that ${\raise0.7ex\hbox{${\partial f(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial f(c_{e} )} {\partial c_{e} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e} }$}} < 0$ and $f(\overline{c}_{e} ) > 0$ for the threshold $\Delta {*}$, and $0 < \gamma < 1$. Thus the function $f(c_{e} )$ is positive, that is to say, $\pi_{e}^{RA} > \pi_{e}^{R}$, Hence, the e-tailer’s profits under the coexistence of reselling and agency selling is always better than the e-tailer’s profits under the reselling mode.

Appendix A4. Proof of Proposition 4

We subtract the supplier’s profits under the reselling mode from the supplier’s profits under the combined reselling and agency selling mode to obtain the following:

$$\pi_{m}^{RA} - \pi_{m}^{R} = \begin{gathered} \frac{{k\gamma (\gamma - 4)( - 2c_{e} + \gamma + c_{r} \gamma )( - 2 + 2c_{r} - 16c_{e} k + 16k\gamma + 8c_{e} k\gamma - 8k\gamma^{2} - c_{e} k\gamma^{2} + k\gamma^{3} )}}{{4( - 4 + 16k\gamma - 8k\gamma^{2} + k\gamma^{3} )^{2} }} + \frac{{(c_{r} - 1)( - 2 + 2c_{r} + 8k\gamma + 4c_{e} k\gamma - 8c_{r} k\gamma - 6k\gamma^{2} - c_{e} k\gamma^{2} + 2c_{r} k\gamma^{2} + k\gamma^{3} )}}{{2( - 8 + 32k\gamma - 16k\gamma^{2} + 2k\gamma^{3} )}} \hfill \\ + \frac{{\left[ \begin{gathered} - 162 + 324c_{r} - 162c_{r}^{2} + 28\gamma - 56c_{r} \gamma + 28c_{r}^{2} \gamma + 1188k\gamma + 276c_{e} k\gamma - 2376c_{r} k\gamma - 276c_{e} c_{r} k\gamma + 1188c_{r}^{2} k\gamma - 2016c_{e}^{2} k^{2} \gamma - 956k\gamma^{2} - 160c_{e} k\gamma^{2} \hfill \\ + 1636c_{r} k\gamma^{2} + 160c_{e} c_{r} k\gamma^{2} - 680c_{r}^{2} k\gamma^{2} - 2178k^{2} \gamma^{2} + 1452c_{e} k^{2} \gamma^{2} + 1374c_{e}^{2} k^{2} \gamma^{2} + 43{56}c_{r} k^{2} \gamma^{2} + 2580c_{e} c_{r} k^{2} \gamma^{2} - 2178c_{r}^{2} k^{2} \gamma^{2} + 244k\gamma^{3} \hfill \\ + 20c_{e} k\gamma^{3} - 328c_{r} k\gamma^{3} - 20c_{e} c_{r} k\gamma^{3} + 84c_{r}^{2} k\gamma^{3} + 2376k^{2} \gamma^{3} - 684c_{e} k^{2} \gamma^{3} - 216c_{e}^{2} k^{2} \gamma^{3} - 6204c_{r} k^{2} \gamma^{3} - 2064c_{e} c_{r} k^{2} \gamma^{3} + 1812c_{r}^{2} k^{2} \gamma^{3} - 20k\gamma^{4} \hfill \\ + 20c_{r} k\gamma^{4} - 1154k^{2} \gamma^{4} - 76c_{e} k^{2} \gamma^{4} - 12c_{e}^{2} k^{2} \gamma^{4} + 2992c_{r} k^{2} \gamma^{4} + 508c_{e} c_{r} k^{2} \gamma^{4} - 464c_{r}^{2} k^{2} \gamma^{4} + 327k^{2} \gamma^{5} + 62c_{e} k^{2} \gamma^{5} + 3c_{e}^{2} k^{2} \gamma^{5} - 578c_{r} k^{2} \gamma^{5} \hfill \\ - 38c_{e} c_{r} k^{2} \gamma^{5} + 35c_{r}^{2} k^{2} \gamma^{5} - 50k^{2} \gamma^{6} - 6c_{e} k^{2} \gamma^{6} + 38c_{r} k^{2} \gamma^{6} + 3k^{2} \gamma^{7} \hfill \\ \end{gathered} \right]}}{{4( - 18 + \gamma + 66k\gamma - 36k\gamma^{2} + 4k\gamma^{3} )^{2} }} \hfill \\ \end{gathered}$$

Let $g(c_{e} ) = \pi_{m}^{RA} - \pi_{m}^{R}$, we can find that the function $g(c_{e} )$ is a convex function of $c_{e}$, i.e., ${\raise0.7ex\hbox{${\partial^{2} g(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial^{2} g(c_{e} )} {\partial c_{e}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e}^{2} }$}} > 0$. Further, we obtain that ${\raise0.7ex\hbox{${\partial g(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial g(c_{e} )} {\partial c_{e} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e} }$}} < 0$,$g(\underline{{c_{e} }} ) > 0$ and $g(\overline{{c_{e} }} ) < 0$ for the threshold $\Delta {*}$, and $0 < \gamma < 1$. Thus it follows that there exists a $c_{e2} \in (\underline{{c_{e} }} ,\overline{{c_{e} }} )$ such that $g(c_{e2} ) = 0$.Hence,$c_{e} \in (\underline{{c_{e} }} ,c_{e2} )$, $g(c_{e} ) > 0$, i.e.,$\pi_{m}^{RA} > \pi_{m}^{R}$, and for $c_{e} \in (c_{e2} ,\overline{{c_{e} }} )$,$g(c_{e} ) < 0$,i.e.,$\pi_{m}^{RA} < \pi_{m}^{R}$.

Appendix A5. Proof of Proposition 5

We subtract the bricks-and-mortar retailer’s profits under the reselling mode from the bricks-and-mortar retailer’s profits under the combined reselling and agency selling mode to obtain the following:

$$\pi_{r}^{RA} - \pi_{r}^{R} = \frac{{( - 2 + 2c_{r} + 8k\gamma + 4c_{e} k\gamma - 8c_{r} k\gamma - 6k\gamma^{2} - c_{e} k\gamma^{2} + 2c_{r} k\gamma^{2} + k\gamma^{3} )^{2} }}{{4( - 4 + 16k\gamma - 8k\gamma^{2} + k\gamma^{3} )^{2} }} + \frac{{( - 9 + 9c_{r} + 33k\gamma + 21c_{e} k\gamma - 33c_{r} k\gamma - 29k\gamma^{2} - 4c_{e} k\gamma^{2} + 8c_{r} k\gamma^{2} + 4k\gamma^{3} )^{2} }}{{4( - 18 + \gamma + 66k\gamma - 36k\gamma^{2} + 4k\gamma^{3} )^{2} }}$$

Let $h(c_{e} ) = \pi_{r}^{RA} - \pi_{r}^{R}$, to prove $\pi_{r}^{RA} < \pi_{r}^{R}$, we have to show that $h(c_{e} ) < 0$. The function $h(c)$ is convex, i.e., ${\raise0.7ex\hbox{${\partial^{2} h(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial^{2} h(c_{e} )} {\partial c_{e}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e}^{2} }$}} > 0$. Further, we obtain that $h(\overline{c}_{e} ) < 0$ and $h(\underline{{c_{e} }} ) < 0$ for the threshold $\Delta {*}$, and $0 < \gamma < 1$. Thus for any $c_{e} \in (\underline{{c_{e} }} ,\overline{c}_{e} )$,$k > \underline{k}$ and $\gamma \in (0,1)$, the sign of the expression $h(c)$ is negative, which shows that the bricks-and-mortar retailer get lower profits under the coexistence of reselling and agency selling than that in the reselling mode.

Appendix A6. Proof of Proposition 6

The supply chain profit can be calculated as follows:

$\pi_{t}^{j} = \pi_{m}^{j} + \pi_{r}^{j} + \pi_{e}^{j}$. Based on the above function, we subtract the supply chain’s profits under the reselling mode from the supply chain profit’s profits under the combined reselling and agency selling, that is:

$$\pi_{t}^{RA} - \pi_{t}^{R} = \frac{{\left[ \begin{gathered} - c_{e}^{2} k^{2} \gamma^{5} - 16c_{e}^{2} k^{2} \gamma^{4} + 429c_{e}^{2} k^{2} \gamma^{3} - 2421c_{e}^{2} k^{2} \gamma^{2} + 4032c_{e}^{2} k^{2} r - 2c_{e}^{2} k\gamma^{2} + 72c_{e}^{2} k\gamma \hfill \\ - 648c_{e}^{2} k + 42c_{e} c_{r} k^{2} \gamma^{5} - 668c_{e} c_{r} k^{2} \gamma^{4} + 3306c_{e} c_{r} k^{2} \gamma^{3} - 5094c_{e} c_{r} k^{2} \gamma^{2} + 14c_{e} c_{r} k\gamma^{3} \hfill \\ - 202c_{e} c_{r} k\gamma^{2} + 1038c_{e} c_{r} k\gamma + 2c_{e} k^{2} \gamma^{6} - 10c_{e} k^{2} \gamma^{5} - 190c_{e} k^{2} \gamma^{4} + 1536c_{e} k^{2} \gamma^{3} - \hfill \\ 2970c_{e} k^{2} \gamma^{2} - 10c_{e} k\gamma^{3} + 58c_{e} k\gamma^{2} + 258c_{e} k\gamma - 25c_{r}^{2} k^{2} \gamma^{5} + 428c_{r}^{2} k^{2} \gamma^{4} - 2139c_{r}^{2} k^{2} \gamma^{3} \hfill \\ + 3267c_{r}^{2} k^{2} \gamma^{2} - 86c_{r}^{2} k\gamma^{3} + 736c_{r}^{2} k\gamma^{2} - 1782c_{r}^{2} kr - 23c_{r}^{2} \gamma + 243c_{r}^{2} - 42c_{r} k^{2} \gamma^{6} + \hfill \\ 718c_{r} k^{2} \gamma^{5} - 4162c_{r} k^{2} \gamma^{4} + 9372c_{r} k^{2} \gamma^{3} - 6534c_{r} k^{2} \gamma^{2} - 14c_{r} k\gamma^{4} + 374c_{r} k\gamma^{3} - \hfill \\ 2510c_{r} k\gamma^{2} + 3564c_{r} k\gamma + 46c_{r} \gamma - 486c_{r} - k^{2} \gamma^{7} + 26k^{2} \gamma^{6} - 264k^{2} \gamma^{5} + 1313k^{2} \gamma^{4} \hfill \\ - 3201k^{2} \gamma^{3} + 3267k^{2} \gamma^{2} + 12k\gamma^{4} - 216k\gamma^{3} + 1126k\gamma^{2} - 1782k\gamma - 23\gamma + 243 \hfill \\ \end{gathered} \right]}}{{4(\gamma + 66k\gamma - 36k\gamma^{2} + 4k\gamma^{3} - 18)^{2} }} - \frac{{\left[ \begin{gathered} - c_{e}^{2} k^{2} \gamma^{4} + 20c_{e}^{2} k^{2} \gamma^{3} - 112c_{e}^{2} k^{2} \gamma^{2} + 192c_{e}^{2} k^{2} \gamma - 32c_{e}^{2} k \hfill \\ + 2c_{e} c_{r} k^{2} \gamma^{5} - 32c_{e} c_{r} k^{2} \gamma^{4} + 160c_{e} c_{r} k^{2} \gamma^{3} - 256c_{e} c_{r} k^{2} \gamma^{2} \hfill \\ - 4c_{e} c_{r} k\gamma^{2} + 48c_{e} c_{r} k\gamma - 8c_{e} k^{2} \gamma^{4} + 64c_{e} k^{2} \gamma^{3} - 128c_{e} k^{2} \gamma^{2} \hfill \\ + 4c_{e} k\gamma^{2} + 16c_{e} k\gamma - c_{r}^{2} k^{2} \gamma^{5} + 20c_{r}^{2} k^{2} \gamma^{4} - 112c_{r}^{2} k^{2} \gamma^{3} + \hfill \\ 192c_{r}^{2} k^{2} \gamma^{2} - 4c_{r}^{2} k\gamma^{3} + 32c_{r}^{2} k\gamma^{2} - 96c_{r}^{2} k\gamma + 12c_{r}^{2} - 2c_{r} k^{2} \gamma^{6} \hfill \\ + 34c_{r} k^{2} \gamma^{5} - 200c_{r} k^{2} \gamma^{4} + 480c_{r} k^{2} \gamma^{3} - 384c_{r} k^{2} \gamma^{2} + 12c_{r} k\gamma^{3} \hfill \\ - 112c_{r} k\gamma^{2} + 192c_{r} k\gamma - 24c_{r} + k^{2} \gamma^{6} - 13k^{2} \gamma^{5} + 68k^{2} \gamma^{4} \hfill \\ - 176k^{2} \gamma^{3} + 192k^{2} \gamma^{2} - 8k\gamma^{3} + 48k\gamma^{2} - 96k\gamma + 12 \hfill \\ \end{gathered} \right]}}{{4(k\gamma^{3} - 8k\gamma^{2} + 16k\gamma - 4)^{2} }}$$

Let $l(c_{e} ) = \pi_{t}^{RA} - \pi_{t}^{R}$, we can find that the function $l(c_{e} )$ is a convex function of $c_{e}$, i.e., ${\raise0.7ex\hbox{${\partial^{2} l(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial^{2} l(c_{e} )} {\partial c_{e}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e}^{2} }$}} > 0$. Further, we obtain that ${\raise0.7ex\hbox{${\partial l(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial l(c_{e} )} {\partial c_{e} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e} }$}} < 0$,$l(\underline{{c_{e} }} ) > 0$ and $l(\overline{{c_{e} }} ) < 0$ for the threshold $\Delta {*}$, and $0 < \gamma < 1$. Thus it follows that there exists a $c_{e3} \in (\underline{{c_{e} }} ,\overline{{c_{e} }} )$ such that $l(c_{e3} ) = 0$.Hence,$c_{e} \in (\underline{{c_{e} }} ,c_{e3} )$, $l(c_{e} ) > 0$, i.e.,$\pi_{t}^{RA} > \pi_{t}^{R}$, and for $c_{e} \in (c_{e3} ,\overline{{c_{e} }} )$,$l(c_{e} ) < 0$,i.e.,$\pi_{t}^{RA} < \pi_{t}^{R}$.

Appendix A7. Proof of Proposition 7

Consumer surplus (CS), which consists of two components: consumers purchasing products from the bricks-and-mortar retailer and through an e-tailer, is calculated as follows:

$CS = \int_{{1 - q_{e} - q_{m} - q_{r} }}^{{1 - q_{e} - q_{m} }} {(\gamma u - p_{e} )} du + \int_{{1 - q_{r} }}^{1} {(u - p_{r} )} du$. Thus, we can obtain the difference of consumer surplus between the combined mode and reselling mode as follows:

$$CS^{RA} - CS^{R} = \frac{{\left[ \begin{gathered} 9c_{e}^{2} k^{2} \gamma^{5} - 176c_{e}^{2} k^{2} \gamma^{4} + 1167c_{e}^{2} k^{2} \gamma^{3} - 3015c_{e}^{2} k^{2} \gamma^{2} + 2304c_{e}^{2} k^{2} \gamma + 14c_{e} c_{r} k^{2} \gamma^{5} \hfill \\ - 124c_{e} c_{r} k^{2} \gamma^{4} + 210c_{e} c_{r} k^{2} \gamma^{3} + 342c_{e} c_{r} k^{2} \gamma^{2} - 28c_{e} c_{r} k\gamma^{3} + 270c_{e} c_{r} k\gamma^{2} - + 19\gamma \hfill \\ 582c_{e} c_{r} k\gamma - 18c_{e} k^{2} \gamma^{6} + 338c_{e} k^{2} \gamma^{5} - 2210c_{e} k^{2} \gamma^{4} + 5820c_{e} k^{2} \gamma^{3} - 4950c_{e} k^{2} \gamma^{2} \hfill \\ + 28c_{e} k\gamma^{3} - 270c_{e} k\gamma^{2} + 582c_{e} k\gamma - 39c_{r}^{2} k^{2} \gamma^{5} + 412c_{r}^{2} k^{2} \gamma^{4} - 1293c_{r}^{2} k^{2} \gamma^{3} + \hfill \\ 1089c_{r}^{2} k^{2} \gamma^{2} - 44c_{r}^{2} k\gamma^{3} + 378c_{r}^{2} k\gamma^{2} - 594c_{r}^{2} k\gamma + 19c_{r}^{2} \gamma + 81c_{r}^{2} - 14c_{r} k^{2} \gamma^{6} + \hfill \\ 202c_{r} k^{2} \gamma^{5} - 1034c_{r} k^{2} \gamma^{4} + 2244c_{r} k^{2} \gamma^{3} - 2178c_{r} k^{2} \gamma^{2} + 28c_{r} k\gamma^{4} - 182c_{r} k\gamma^{3} \hfill \\ - 174c_{r} k\gamma^{2} + 1188c_{r} k\gamma - 38c_{r} \gamma - 162c_{r} + 9k^{2} \gamma^{7} - 162k^{2} \gamma^{6} + 1004k^{2} \gamma^{5} - \hfill \\ 2393k^{2} \gamma^{4} + 1353k^{2} \gamma^{3} + 1089k^{2} \gamma^{2} - 28k\gamma^{4} + 226k\gamma^{3} - 204k\gamma^{2} - 594k\gamma + 81 \hfill \\ \end{gathered} \right]}}{{8(\gamma + 66k\gamma - 36k\gamma^{2} + 4k\gamma^{3} - 18)^{2} }} - \frac{{\left[ \begin{gathered} - 3c_{e}^{2} k^{2} \gamma^{4} + 28c_{e}^{2} k^{2} \gamma^{3} - 80c_{e}^{2} k^{2} \gamma^{2} + 64c_{e}^{2} k^{2} \gamma + 2c_{e} c_{r} k^{2} \gamma^{5} \hfill \\ - 16c_{e} c_{r} k^{2} \gamma^{4} + 32c_{e} c_{r} k^{2} \gamma^{3} + 4c_{e} c_{r} k\gamma^{2} - 16c_{e} c_{r} k\gamma + 4c_{e} k^{2} \gamma^{5} \hfill \\ - 40c_{e} k^{2} \gamma^{4} + 128c_{e} k^{2} \gamma^{3} - 128c_{e} k^{2} \gamma^{2} - 4c_{e} k\gamma^{2} + 16c_{e} k\gamma - \hfill \\ 3c_{r}^{2} k^{2} \gamma^{5} + 28c_{r}^{2} k^{2} \gamma^{4} - 80c_{r}^{2} k^{2} \gamma^{3} + 64c_{r}^{2} k^{2} \gamma^{2} - 4c_{r}^{2} k\gamma^{3} + 24c_{r}^{2} k\gamma^{2} \hfill \\ - 32c_{r}^{2} k\gamma + 4c_{r}^{2} - 2c_{r} k^{2} \gamma^{6} + 22c_{r} k^{2} \gamma^{5} - 88c_{r} k^{2} \gamma^{4} + 160c_{r} k^{2} \gamma^{3} \hfill \\ - 128c_{r} k^{2} \gamma^{2} + 4c_{r} k\gamma^{3} - 32c_{r} k\gamma^{2} + 64c_{r} k\gamma - 8c_{r} - k^{2} \gamma^{6} + 9k^{2} \gamma^{5} \hfill \\ - 20k^{2} \gamma^{4} - 16k^{2} \gamma^{3} + 64k^{2} \gamma^{2} + 8k\gamma^{2} - 32k\gamma + 4 \hfill \\ \end{gathered} \right]}}{{8(k\gamma^{3} - 8k\gamma^{2} + 16k\gamma - 4)^{2} }}$$

Let $z(c_{e} ) = CS^{RA} - CS^{R}$, to prove $CS^{RA} > CS^{R}$, we have to show that $z(c_{e} ) > 0$. The function $z(c_{e} )$ is convex, i.e., ${\raise0.7ex\hbox{${\partial^{2} z(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial^{2} z(c_{e} )} {\partial c_{e}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e}^{2} }$}} > 0$. Further, we obtain that ${\raise0.7ex\hbox{${\partial z(c_{e} )}$} \!\mathord{\left/ {\vphantom {{\partial z(c_{e} )} {\partial c_{e} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial c_{e} }$}} < 0$ and $z(\overline{c}_{e} ) > 0$ for the threshold $\Delta {*}$, and $0 < \gamma < 1$. Thus the function $z(c_{e} )$ is positive, that is to say, $CS^{RA} > CS^{R}$, Hence, the consumers under the coexistence of reselling and agency selling is always better than consumers under the reselling mode.