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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Appendix: proofs of propositions
Appendix: proofs of propositions
1.1 Appendix 1
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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:
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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:
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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):
If \(w=\frac{\theta -\theta ^{2}+2}{2}\) (considered not a good solution because \(p_{1}=p_{2}=0\)):
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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
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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:
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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:
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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
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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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DOI: https://doi.org/10.1007/s10479-015-1982-6