Abstract
We consider a stylized distribution channel, where a manufacturer sells a single kind of good to a retailer. In classical setting, the profit of manufacturer is quadratic w.r.t. wholesales discount, while the profit of retailer is quadratic w.r.t. retail discount (pass-through). Thus, the wholesale prices and the retail prices are continuous. These results are elegant mathematically but not adequate economically. Therefore, we assume that wholesale discount and retail discounts are piece-wise constant. We show the strict concavity of retailer’s profits w.r.t. retail discount levels. As for the manufacturer’s profits w.r.t. wholesale discount levels, we show that strict concavity can be guaranteed only in the case when retail discount is constant and sufficiently large.
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Notes
- 1.
It seems realistic, cf. [12].
- 2.
Parameter \(\overline{\alpha }\in [A_1, A_2]\) takes into account the fact that the retailer has some expectations about the wholesale discount: the motivation is reduced if the retailer is dissatisfied with the wholesale discount, i.e., if \(\alpha (t) <\overline{\alpha };\) on the contrary, the motivation increases if \(\alpha (t)> \overline{\alpha }\).
- 3.
Note that \( x_{1}\left( \tau _{0}\right) =0\) while \(M_{1}\left( \tau _{0}\right) =\overline{M}\).
- 4.
Due to (3), these formulas are well defined.
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Acknowledgments
The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0018) and within the framework of Laboratory of Empirical Industrial Organization, Economic Department of Novosibirsk State University. The work was supported in part by the Russian Foundation for Basic Research, projects 18-010-00728 and 19-010-00910 and by the Russian Ministry of Science and Education under the 5–100 Excellence Programme.
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Bykadorov, I. (2020). Pricing in Dynamic Marketing: The Cases of Piece-Wise Constant Sale and Retail Discounts. In: Olenev, N., Evtushenko, Y., Khachay, M., Malkova, V. (eds) Optimization and Applications. OPTIMA 2020. Lecture Notes in Computer Science(), vol 12422. Springer, Cham. https://doi.org/10.1007/978-3-030-62867-3_3
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