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A manufacturer distribution issue: how to manage an online and a traditional retailer

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Abstract

The Internet’s popularity and success have shaped new relationships in B2B market. The use of dual channels becomes a widespread practice and new challenges face channel members. We assess the benefit of two coordinating mechanisms namely the whole-channel price and the quantity discount when a manufacturer sells his product through a traditional and an online store and uses a single wholesale price for both retailers. Then, we extend the analysis to two-wholesale pricing scenario. Our model suggests that product compatibility to the web is a key factor impacting the decision to coordinate the channel or not and which coordination mechanism to use. We found also that the whole channel is always better-off when coordination is implemented though channel members have different positions with regards to such decision. Hence, a profit-sharing mechanism is required to satisfy all members. Finally, we analyze the effect of varying channel substitutability on channel members’ profitability.

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Notes

  1. Ingene and Parry (1995) determine optimal pricing strategies in the case of quantity discount by maximizing the whole-channel profit. However, Jeuland and Shugan (1983) determine optimal pricing strategies by maximizing the individual profit (i.e., the manufacturer’s profit).

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Correspondence to Nawel Amrouche.

Appendix: proofs of propositions

Appendix: proofs of propositions

1.1 Appendix 1

  • No-coordination under single-wholesale pricing scenario

We determine first the reaction function of each retailer \(p_{1}\left( W\right) \) and \(p_{2}\left( W\right) \). Then, we insert the reaction functions into the manufacturer’s profit function. We solve the manufacturer problem to determine the wholesale price W. The results are:

$$\begin{aligned} W= & {} \frac{2C+\left( g+1\right) \alpha _{2}}{4} \\ p_{1}= & {} \frac{4C+\alpha _{2}\left( 2-3\theta \right) +2C\theta +g\alpha _{2}\left( 10+\theta -4\theta ^{2}\right) }{4\left( 2-\theta \right) \left( \theta +2\right) }\\ p_{2}= & {} -\frac{1}{4}\frac{4C+2\alpha _{1}+10\alpha _{2}+2C\theta -3\theta \alpha _{1}+\theta \alpha _{2}-4\theta ^{2}\alpha _{2}}{\left( \theta -2\right) \left( \theta +2\right) }\\ Q_{1}= & {} \frac{-4C+6\alpha _{1}-2\alpha _{2}+2C\theta +\theta \alpha _{1}-3\theta \alpha _{2}+2C\theta ^{2}-3\theta ^{2}\alpha _{1}+\theta ^{2}\alpha _{2}}{4\left( \theta -1\right) \left( \theta +1\right) \left( \theta -2\right) \left( \theta +2\right) }\\ Q_{2}= & {} \frac{-4C-2\alpha _{1}+6\alpha _{2}+2C\theta -3\theta \alpha _{1}+\theta \alpha _{2}+2C\theta ^{2}+\theta ^{2}\alpha _{1}-3\theta ^{2}\alpha _{2}}{4\left( \theta -1\right) \left( \theta +1\right) \left( \theta -2\right) \left( \theta +2\right) } \\ \pi _{ OR}= & {} -\frac{\left( -4C+6\alpha _{1}-2\alpha _{2}+2C\theta +\theta \alpha _{1}-3\theta \alpha _{2}+2C\theta ^{2}-3\theta ^{2}\alpha _{1}+\theta ^{2}\alpha _{2}\right) ^{2}}{16\left( \theta -1\right) \left( \theta +1\right) \left( \theta -2\right) ^{2}\left( \theta +2\right) ^{2}}\\ \pi _{ TR}= & {} -\frac{\left( -4C-2\alpha _{1}+6\alpha _{2}+2C\theta -3\theta \alpha _{1}+\theta \alpha _{2}+2C\theta ^{2}+\theta ^{2}\alpha _{1}-3\theta ^{2}\alpha _{2}\right) ^{2}}{16\left( \theta -1\right) \left( \theta +1\right) \left( \theta -2\right) ^{2}\left( \theta +2\right) ^{2}}\\ \pi _{M}= & {} -\frac{\left( 2C-\alpha _{1}-\alpha _{2}\right) ^{2}}{ 8\left( \theta +1\right) \left( \theta -2\right) }\\ \pi _{ Channel}= & {} \frac{\left( \begin{array}{c} {-36C}^{2}{\theta }^{2}{+12C}^{2} {\theta }^{3}{+8C}^{2}{\theta }^{4} {-48C\alpha }_{1}{-48C\alpha }_{2}{+28\alpha }_{1}^{2}{+28\alpha }_{2}^{2}{-8C\theta }^{4}{\alpha }_{2} \\ {-21\theta }^{2}{\alpha }_{1}^{2}{-21\theta }^{2}{\alpha }_{2}^{2}{-5\theta }^{3} {\alpha }_{1}^{2}{-5\theta }^{3}{\alpha }_{2}^{2}{+6\theta }^{4}{\alpha }_{1}^{2} {+6\theta }^{4}{\alpha }_{2}^{2}{-8\alpha }_{1}{\alpha }_{2}{+32C\theta \alpha } _{1}{-32C}^{2}{\theta } \\ \,{+8\theta \alpha }_{1}^{2}{+8\theta \alpha } _{2}^{2}{+48C}^{2}{+6\theta }^{2}{\alpha }_{1}{\alpha }_{2}{+22\theta }^{3} {\alpha }_{1}{\alpha }_{2}{-4\theta }^{4}{\alpha }_{1}{\alpha }_{2} {-8C\theta }^{4}{\alpha }_{1} \\ \,{+32C\theta \alpha }_{2}{-48\theta \alpha }_{1} {\alpha }_{2}{+36C\theta }^{2}{\alpha }_{1}{+36C\theta }^{2}{\alpha }_{2} {-12C\theta }^{3}{\alpha }_{1}{-12C\theta }^{3}{\alpha }_{2} \end{array} \right) }{8\left( 1-\theta ^{2}\right) \left( \theta ^{2}-4\right) ^{2}} \end{aligned}$$
  • Whole-channel price under single-wholesale pricing scenario

We determine first the reaction function of each retailer \(p_{1}\left( W\right) \) and \(p_{2}\left( W\right) \). Then, we insert the reaction functions into the whole channel profit function. We solve the channel problem to determine the wholesale price W. The results are:

$$\begin{aligned} W= & {} \frac{4C-2C\theta +\theta \alpha _{1}+\theta \alpha _{2}}{4} \\ p_{1}= & {} \frac{4C+4\alpha _{1}+2C\theta +3\theta \alpha _{1}-\theta \alpha _{2}}{4\left( \theta +2\right) } \\ p_{2}= & {} \frac{4C+4\alpha _{2}+2C\theta -\theta \alpha _{1}+3\theta \alpha _{2}}{4\left( \theta +2\right) } \\ Q_{1}= & {} -\frac{-4C+4\alpha _{1}+2C\theta +\theta \alpha _{1}-3\theta \alpha _{2}+2C\theta ^{2}-\theta ^{2}\alpha _{1}-\theta ^{2}\alpha _{2}}{4\left( \theta -1\right) \left( \theta +1\right) \left( \theta +2\right) }\\ Q_{2}= & {} -\frac{-4C+4\alpha _{2}+2C\theta -3\theta \alpha _{1}+\theta \alpha _{2}+2C\theta ^{2}-\theta ^{2}\alpha _{1}-\theta ^{2}\alpha _{2}}{4\left( \theta -1\right) \left( \theta +1\right) \left( \theta +2\right) }\\ \pi _{ OR}= & {} -\frac{\left( -4C+4\alpha _{1}+2C\theta +\theta \alpha _{1}-3\theta \alpha _{2}+2C\theta ^{2}-\theta ^{2}\alpha _{1}-\theta ^{2}\alpha _{2}\right) ^{2}}{16\left( \theta -1\right) \left( \theta +1\right) \left( \theta +2\right) ^{2}}\\ \pi _{ TR}= & {} -\frac{\left( -4C+4\alpha _{2}+2C\theta -3\theta \alpha _{1}+\theta \alpha _{2}+2C\theta ^{2}-\theta ^{2}\alpha _{1}-\theta ^{2}\alpha _{2}\right) ^{2}}{16\left( \theta -1\right) \left( \theta +1\right) \left( \theta +2\right) ^{2}}\\ \pi _{M}= & {} \frac{\theta \left( 2C-\alpha _{1}-\alpha _{2}\right) ^{2}}{8\left( \theta +1\right) } \\ \pi _{ Channel}= & {} \frac{\left( \begin{array}{c} {12C}^{2}{\theta }^{2}{+4C}^{2} {\theta }^{3}{+16C\alpha }_{1}{+16C\alpha }_{2}{-8\alpha }_{1}^{2}{-8\alpha }_{2}^{2}{-\theta }^{2}{\alpha }_{1}^{2} \\ {-\theta }^{2}{\alpha }_{2}^{2}{+\theta }^{3}{\alpha }_{1}^{2}{+\theta }^{3} {\alpha }_{2}^{2}{-8\theta \alpha }_{1}^{2} {-8\theta \alpha }_{2}^{2}{-16C}^{2} {+14\theta }^{2}{\alpha }_{1}{\alpha }_{2}{+2\theta }^{3}{\alpha }_{1} {\alpha }_{2}\\ \,{+16\theta \alpha }_{1}{\alpha }_{2} {-12C\theta }^{2}{\alpha }_{1}{-12C\theta }^{2}{\alpha }_{2}{-4C\theta }^{3} {\alpha }_{1}{-4C\theta }^{3}{\alpha }_{2} \end{array} \right) }{8\left( \theta ^{2}-1\right) \left( \theta +2\right) ^{2}} \end{aligned}$$
  • Quantity-discount under single-wholesale pricing scenario

We determine first the reaction function of each retailer \(p_{1}\left( w,W\right) \) and \(p_{2}\left( w,W\right) \). Then, we insert the reaction functions into the whole channel profit function. We solve the channel problem to determine the wholesale price w and W. We find 2 choices of w:

If \(w=\frac{\theta -\theta ^{2}}{2}\) (considered as good solution):

$$\begin{aligned} W= & {} \frac{2C+\theta \alpha _{1}+\theta \alpha _{2}}{2\left( \theta +1\right) }\\ w= & {} \frac{\theta -\theta ^{2}}{2} \\ p_{1}= & {} \frac{C+\alpha _{1}}{2} \\ p_{2}= & {} \frac{C+\alpha _{2}}{2} \\ Q_{1}= & {} \frac{C\left( \theta -1\right) +\alpha _{1}-\theta \alpha _{2}}{ 2\left( 1-\theta \right) \left( \theta +1\right) } \\ Q_{2}= & {} \frac{C\left( \theta -1\right) +\alpha _{2}-\theta \alpha _{1}}{ 2\left( 1-\theta \right) \left( \theta +1\right) } \\ \pi _{ OR}= & {} \frac{\left( \theta +2\right) \left( -C+\alpha _{1}+C\theta -\theta \alpha _{2}\right) ^{2}}{8\left( 1-\theta \right) \left( \theta +1\right) ^{2}} \\ \pi _{ TR}= & {} \frac{\left( \theta +2\right) \left( -C+\alpha _{2}+C\theta -\theta \alpha _{1}\right) ^{2}}{8\left( 1-\theta \right) \left( \theta +1\right) ^{2}} \\ \pi _{M}= & {} \frac{\theta \left( \begin{array}{c} {2C}^{2}{\theta }^{2}{+6C\alpha } _{1}{+6C\alpha }_{2}{-\alpha }_{1}^{2} {-\alpha }_{2}^{2}{+\theta }^{2}{\alpha }_{1}^{2}{+\theta }^{2}{\alpha }_{2}^{2}{-4\alpha }_{1}{\alpha }_{2} \\ {+4C}^{2}{\theta +2\theta \alpha }_{1}^{2} {+2\theta \alpha }_{2}^{2}{-6C}^{2} {-4C\theta \alpha }_{1}{-4C\theta \alpha }_{2} {-2C\theta }^{2}{\alpha }_{1}{-2C\theta }^{2}{\alpha }_{2} \end{array} \right) }{8\left( \theta -1\right) \left( \theta +1\right) ^{2}}\\ \pi _{ Channel}= & {} \frac{2C\alpha _{1}+2C\alpha _{2}-\alpha _{1}^{2}-\alpha _{2}^{2}+2C^{2}\theta -2C^{2}-2C\theta \alpha _{1}-2C\theta \alpha _{2}+2\theta \alpha _{1}\alpha _{2}}{4\left( \theta -1\right) \left( \theta +1\right) } \end{aligned}$$

If \(w=\frac{\theta -\theta ^{2}+2}{2}\) (considered not a good solution because \(p_{1}=p_{2}=0\)):

$$\begin{aligned} W= & {} \frac{\alpha _{1}+\alpha _{2}}{2} \\ w= & {} \frac{\theta -\theta ^{2}+2}{2} \\ p_{1}= & {} 0 \\ p_{2}= & {} 0 \end{aligned}$$
  • Comparing profits functions

Comparing profits’expressions for all single-wholesale pricing scenarios gives the following:

 

NC-WC

NC-QD

WC-QD

\(\pi _{ OR}\)

?

?

?

\(\pi _{ TR}\)

?

?

?

\(\pi _{M}\)

\(\frac{\left( \theta -1\right) ^{2}\left[ 2C-\alpha _{2}\left( g+1\right) \right] ^{2}}{8\left( \theta +1\right) \left( 2-\theta \right) } >0 \)

?

?

\(\pi _{Cha}\)

\(-\frac{\left( \theta -1\right) ^{2}\left[ 2C-\alpha _{2}\left( g+1\right) \right] ^{2}}{8\left( \theta +1\right) \left( \theta -2\right) ^{2}}<0\)

?

\(\frac{\theta ^{2}\left[ \alpha _{2}\left( g-1\right) \right] ^{2}}{8\left( \theta -1\right) \left( \theta +2\right) ^{2}}<0\)

where (?) means that the expressions are too long and we resort to simulations to determine their signs. See Fig. 1 for more details about the signs of all expressions. The expressions are available from authors upon request.

1.2 Appendix 2

  • No-coordination under two-wholesale pricing scenario

We determine first the reaction function of each retailer \(p_{1}\left( W_{1},W_{2}\right) \) and \(p_{2}\left( W_{1},W_{2}\right) \). Then, we insert the reaction functions into the manufacturer’s profit function. We solve the manufacturer problem to determine the wholesale price \(W_{1}\) and \(W_{2}\). The results are:

$$\begin{aligned} W_{1}= & {} \frac{C+\alpha _{1}}{2} \\ W_{2}= & {} \frac{C+\alpha _{2}}{2} \\ p_{1}= & {} -\frac{2C+6\alpha _{1}+C\theta -\theta \alpha _{2}-2\theta ^{2}\alpha _{1}}{2\left( \theta -2\right) \left( \theta +2\right) } \\ p_{2}= & {} -\frac{2C+6\alpha _{2}+C\theta -\theta \alpha _{1}-2\theta ^{2}\alpha _{2}}{2\left( \theta -2\right) \left( \theta +2\right) } \\ Q_{1}= & {} \frac{-2C+2\alpha _{1}+C\theta -\theta \alpha _{2}+C\theta ^{2}-\theta ^{2}\alpha _{1}}{2\left( \theta -1\right) \left( \theta +1\right) \left( \theta -2\right) \left( \theta +2\right) } \\ Q_{2}= & {} \frac{-2C+2\alpha _{2}+C\theta -\theta \alpha _{1}+C\theta ^{2}-\theta ^{2}\alpha _{2}}{2\left( \theta -1\right) \left( \theta +1\right) \left( \theta -2\right) \left( \theta +2\right) } \\ \pi _{ OR}= & {} -\frac{\left( -2C+2\alpha _{1}+C\theta -\theta \alpha _{2}+C\theta ^{2}-\theta ^{2}\alpha _{1}\right) ^{2}}{4\left( \theta -1\right) \left( \theta +1\right) \left( \theta -2\right) ^{2}\left( \theta +2\right) ^{2}}\\ \pi _{ TR}= & {} -\frac{\left( -2C+2\alpha _{2}+C\theta -\theta \alpha _{1}+C\theta ^{2}-\theta ^{2}\alpha _{2}\right) ^{2}}{4\left( \theta -1\right) \left( \theta +1\right) \left( \theta -2\right) ^{2}\left( \theta +2\right) ^{2}}\\ \pi _{M}= & {} \frac{\left( \begin{array}{c} {2C}^{2}{\theta }^{2}{+4C\alpha } _{1}{+4C\alpha }_{2}{-2\alpha }_{1}^{2} {-2\alpha }_{2}^{2}{+\theta }^{2}{\alpha }_{1}^{2}{+\theta }^{2}{\alpha }_{2}^{2} {+2C}^{2}{\theta } \\ {-4C}^{2}{-2C\theta \alpha }_{1}{-2C\theta \alpha }_{2}{+2\theta \alpha }_{1}{\alpha }_{2}{-2C\theta }^{2}{\alpha }_{1} {-2C\theta }^{2}{\alpha }_{2} \end{array} \right) }{4\left( 1-\theta ^{2}\right) \left( \theta ^{2}-4\right) }\\ \pi _{ Channel}= & {} \frac{\left( \begin{array}{c} {-18C}^{2}{\theta }^{2}{+6C}^{2} {\theta }^{3}{+4C}^{2}{\theta }^{4} {-24C\alpha }_{1}{-24C\alpha }_{2}{+12\alpha }_{1}^{2}{+12\alpha }_{2}^{2}{-9\theta }^{2}{\alpha }_{1}^{2}{-9\theta }^{2} {\alpha }_{2}^{2} \\ \,{+2\theta }^{4}{\alpha }_{1}^{2}{+2\theta }^{4}{\alpha }_{2}^{2}{-16C}^{2} {\theta +24C}^{2}{+6\theta }^{3}{\alpha }_{1}{\alpha }_{2}{+16C\theta \alpha }_{1} {+16C\theta \alpha }_{2}{-16\theta \alpha }_{1} {\alpha }_{2} \\ \,{+18C\theta }^{2}{\alpha }_{1}{+18C\theta }^{2}{\alpha }_{2}{-6C\theta }^{3} {\alpha }_{1}{-6C\theta }^{3}{\alpha }_{2}{-4C\theta }^{4}{\alpha }_{1} {-4C\theta }^{4}{\alpha }_{2} \end{array} \right) }{4\left( 1-\theta ^{2}\right) \left( \theta ^{2}-4\right) ^{2}} \end{aligned}$$
  • Whole-channel price under two-wholesale pricing scenario

We determine first the reaction function of each retailer \(p_{1}\left( W_{1},W_{2}\right) \) and \(p_{2}\left( W_{1},W_{2}\right) \). Then, we insert the reaction functions into the whole channel profit function. We solve the channel problem to determine the wholesale price \(W_{1}\) and \(W_{2}\). The results are:

$$\begin{aligned} W_{1}= & {} \frac{2C-C\theta +\theta \alpha _{2}}{2} \\ W_{2}= & {} \frac{2C-C\theta +\theta \alpha _{1}}{2} \\ p_{1}= & {} \frac{C+\alpha _{1}}{2} \\ p_{2}= & {} \frac{C+\alpha _{2}}{2} \\ Q_{1}= & {} -\frac{-C+\alpha _{1}+C\theta -\theta \alpha _{2}}{2\left( \theta -1\right) \left( \theta +1\right) } \\ Q_{2}= & {} -\frac{-C+\alpha _{2}+C\theta -\theta \alpha _{1}}{2\left( \theta -1\right) \left( \theta +1\right) } \\ \pi _{ OR}= & {} -\frac{\left( -C+\alpha _{1}+C\theta -\theta \alpha _{2}\right) ^{2}}{4\left( \theta -1\right) \left( \theta +1\right) } \\ \pi _{ TR}= & {} -\frac{\left( -C+\alpha _{2}+C\theta -\theta \alpha _{1}\right) ^{2}}{4\left( \theta -1\right) \left( \theta +1\right) } \\ \pi _{M}= & {} \frac{\theta \left( 2C\alpha _{1}+2C\alpha _{2}-2\alpha _{1}\alpha _{2}+2C^{2}\theta +\theta \alpha _{1}^{2}+\theta \alpha _{2}^{2}-2C^{2}-2C\theta \alpha _{1}-2C\theta \alpha _{2}\right) }{4\left( \theta ^{2}-1\right) }\\ \pi _{ Channel}= & {} \frac{2C\alpha _{1}+2C\alpha _{2}-\alpha _{1}^{2}-\alpha _{2}^{2}+2C^{2}\theta -2C^{2}-2C\theta \alpha _{1}-2C\theta \alpha _{2}+2\theta \alpha _{1}\alpha _{2}}{4\left( \theta ^{2}-1\right) } \end{aligned}$$
  • Quantity-discount under two-wholesale pricing scenario

We determine first the reaction function of each retailer \(p_{1}\left( W_{1},W_{2},w_{1},w_{2}\right) \) and \(p_{2}\left( W_{1},W_{2},w_{1},w_{2}\right) \). Then, we substitute the reaction functions into the total channel profit function. We then solve the channel problem to determine the wholesale price \(W_{1}\), \(W_{2},w_{1}\) and \(w_{2}\). Though we can determine a system of equations to solve in the case of quantity discount under two-wholesale pricing scenario, the system is too complex to solve analytically.

1.3 Appendix 3

See Appendix Figs. 1, 2, 3, 4, 5 and 6; Table 7.

Fig. 1
figure 1figure 1figure 1

Single-wholesale price scenarios with \(\theta =0.5\). a NC–WC for online retailer’ profit, b NC–WC for traditional retailer’ profit, c NC–QD for online retailer’ profit, d NC–QD for traditional retailer’ profit, e WC–QD for online retailer’ profit, f WC–QD for traditional retailer’ profit, g NC–QD for manufacturer’ profit, h WC–QD for manufacturer’ profit, i NC–QD for whole channel’ profit

Fig. 2
figure 2

Two-wholesale price scenarios with \(\theta =0.5\). a NC–WC for online retailer’ profit, b NC–WC for traditional retailer’ profit, c NC–WC for manufacturer’ profit, d NC–WC for whole channel’ profit

Fig. 3
figure 3

Single-wholesale price scenarios with \(\theta =0.1\). a NC–WC for online retailer’ profit, b NC–WC for traditional retailer’ profit, c NC–QD for online retailer’ profit, d NC–QD for traditional retailer’ profit, e WC–QD for online retailer’ profit, f WC–QD for traditional retailer’ profit, g NC–QD for manufacturer’ profit, h WC–QD for manufacturer’ profit, i NC–QD for whole channel’ profit

Fig. 4
figure 4

Single-wholesale price scenarios with \(\theta =0.9\). a NC–WC for online retailer’ profit, b NC–WC for traditional retailer’ profit, c NC–QD for online retailer’ profit, d NC–QD for traditional retailer’ profit, e WC–QD for online retailer’ profit, f WC–QD for traditional retailer’ profit, g NC–QD for manufacturer’ profit, h WC–QD for manufacturer’ profit, i NC–QD for whole channel’ profit

Fig. 5
figure 5

Two-wholesale price scenarios with \(\theta =0.1\). a NC–WC for online retailer’ profit, b NC–WC for traditional retailer’ profit, c NC–WC for manufacturer’ profit, d NC–WC for whole channel’ profit

Fig. 6
figure 6figure 6

Two-wholesale price scenarios with \(\theta =0.9\). a NC–WC for online retailer’ profit, b NC–WC for traditional retailer’ profit, c NC–WC for manufacturer’ profit, d NC–WC for whole channel’ profit

Table 7 Managerial recommendations for channel members in each scenario

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Amrouche, N., Yan, R. A manufacturer distribution issue: how to manage an online and a traditional retailer. Ann Oper Res 244, 257–294 (2016). https://doi.org/10.1007/s10479-015-1982-6

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