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Annals of Operations Research

, Volume 254, Issue 1–2, pp 131–164 | Cite as

Multiproduct price optimization under the multilevel nested logit model

  • Hai JiangEmail author
  • Rui Chen
  • He Sun
Original Paper

Abstract

We study the multiproduct price optimization problem under the multilevel nested logit model, which includes the multinomial logit and the two-level nested logit models as special cases. When the price sensitivities are identical within each primary nest, that is, within each nest at level 1, we prove that the profit function is concave with respect to the market share variables. We proceed to show that the markup, defined as price minus cost, is constant across products within each primary nest, and that the adjusted markup, defined as price minus cost minus the reciprocal of the product between the scale parameter of the root nest and the price-sensitivity parameter of the primary nest, is constant across primary nests at optimality. This allows us to reduce the multidimensional pricing problem to an equivalent single-variable maximization problem involving a unimodal function. Based on these findings, we investigate the oligopolistic game and characterize the Nash equilibrium. We also develop a dimension reduction technique which can simplify price optimization problems with flexible price-sensitivity structures.

Keywords

Differentiated products Multiproduct pricing Multilevel nested logit Oligopolistic competition 

Notes

Acknowledgements

The authors are very grateful to the anonymous referees whose thoughtful comments and constructive suggestions have allowed them to improve the quality of this paper. This research is supported by the Natural Science Foundation of China [Grant 71622006] and the Center for Data-Centric Management in the Department of Industrial Engineering at Tsinghua University.

References

  1. Akcay, Y., Natarajan, H. P., & Xu, S. H. (2010). Joint dynamic pricing of multiple perishable products under consumer choice. Management Science, 56(8), 1345–1361.CrossRefGoogle Scholar
  2. Aksoy-Pierson, M., Allon, G., & Federgruen, A. (2013). Price competition under mixed multinomial logit demand functions. Management Science, 59(8), 1817–1835.CrossRefGoogle Scholar
  3. Anderson, S. P., & de Palma, A. (1992). Multiproduct firms: A nested logit approach. The Journal of Industrial Economics, 40, 261–276.CrossRefGoogle Scholar
  4. Aydin, G., & Porteus, E. L. (2008). Joint inventory and pricing decisions for an assortment. Operations Research, 56(5), 1247–1255.CrossRefGoogle Scholar
  5. Ben-Akiva, M. E., & Lerman, S. R. (1985). Discrete choice analysis: Theory and application to travel demand. Cambridge: MIT Press.Google Scholar
  6. Boyd, S., & Vandenberghe, L. (2009). Convex optimization. Cambridge: Cambridge University Press.Google Scholar
  7. Chen, K. D., & Hausman, W. H. (2000). Technical note: Mathematical properties of the optimal product line selection problem using choice-based conjoint analysis. Management Science, 46(2), 327–332.CrossRefGoogle Scholar
  8. Coldren, G. M., & Koppelman, F. S. (2005). Modeling the competition among air-travel itinerary shares: GEV model development. Transportation Research Part A, 39(4), 345–365.Google Scholar
  9. Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E. (1996). On the Lambert \(W\) function. Advances in Computational mathematics, 5(1), 329–359.CrossRefGoogle Scholar
  10. Dong, L., Kouvelis, P., & Tian, Z. (2009). Dynamic pricing and inventory control of substitute products. Manufacturing & Service Operations Management, 11(2), 317–339.CrossRefGoogle Scholar
  11. Erdem, T., Swait, J., & Louviere, J. (2002). The impact of brand credibility on consumer price sensitivity. International Journal of Research in Marketing, 19(1), 1–19.CrossRefGoogle Scholar
  12. Gabriel, S., & Painter, G. (2002). Intra-metropolitan mobility, residential location, and homeownership choice among minority and white households: Estimates of a nested multinomial logit model. Technical report, University of Southern California.Google Scholar
  13. Gallego, G., Huh, W. T., Kang, W., & Phillips, R. (2006). Price competition with the attraction demand model: Existence of unique equilibrium and its stability. Manufacturing & Service Operations Management, 8(4), 359–375.CrossRefGoogle Scholar
  14. Gallego, G., & Wang, R. X. (2014). Multiproduct price optimization and competition under the nested logit model with product-differentiated price sensitivities. Operations Research, 62(2), 450–461.CrossRefGoogle Scholar
  15. Ghose, A., & Han, S. P. (2014). Estimating demand for mobile applications in the new economy. Management Science, 60(6), 1470–1488.CrossRefGoogle Scholar
  16. Hanson, W., & Martin, K. (1996). Optimizing multinomial logit profit functions. Management Science, 42(7), 992–1003.CrossRefGoogle Scholar
  17. Hopp, W. J., & Xu, X. (2005). Product line selection and pricing with modularity in design. Manufacturing & Service Operations Management, 7(3), 172–187.CrossRefGoogle Scholar
  18. Hsiao, C.-Y., & Hansen, M. (2011). A passenger demand model for air transportation in a hub-and-spoke network. Transportation Research Part E, 47(6), 1112–1125.CrossRefGoogle Scholar
  19. Jiang, H. (2009). A nested-logit based approach to measuring air shopping screen quality and predicting market share. Journal of Revenue and Pricing Management, 8(2/3), 134–147.CrossRefGoogle Scholar
  20. Jiang, H., Qi, X., & Sun, H. (2014). Choice-based recommender systems: A unified approach to achieving relevancy and diversity. Operations Research, 62(5), 973–993.CrossRefGoogle Scholar
  21. Kök, A. G., & Xu, Y. (2011). Optimal and competitive assortments with endogenous pricing under hierarchical consumer choice models. Management Science, 57(9), 1546–1563.CrossRefGoogle Scholar
  22. Li, G., Rusmevichientong, P., & Topaloglu, H. (2015). The \(d\)-level nested logit model: Assortment and price optimization problems. Operations Research, 63(2), 325–342.CrossRefGoogle Scholar
  23. Li, H. M., & Huh, W. T. (2011). Pricing multiple products with the multinomial logit and nested logit models: Concavity and implications. Manufacturing and Service Operations Management, 13(4), 549–563.CrossRefGoogle Scholar
  24. Liaw, K.-L., & Frey, W. H. (2003). Location of adult children as an attraction for black and white elderly return and onward migrants in the United States: Application of a three-level nested logit model with census data. Mathematical Population Studies, 10(2), 75–98.CrossRefGoogle Scholar
  25. Liu, G. (2006). On nash equilibrium in prices in an oligopolistic market with demand characterized by a nested multinomial logit model and multiproduct firm as nest. Discussion papers.Google Scholar
  26. Morey, E. R., Rowe, R. D., & Watson, M. (1993). A repeated nested-logit model of Atlantic salmon fishing. American Journal of Agricultural Economics, 75(3), 578–592.CrossRefGoogle Scholar
  27. Song, J.-S., & Xue, Z. (2007). Demand management and inventory control for substitutable products. Working paper.Google Scholar
  28. Train, K. (2009). Discrete choice methods with simulation (2nd ed.). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  29. Vives, X. (2001). Oligopoly pricing: Old ideas and new tools. Cambridge: MIT Press.Google Scholar
  30. Wen, C.-H., & Koppelman, F. S. (2001). The generalized nested logit model. Transportation Research Part B, 35(7), 627–641.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Industrial EngineeringTsinghua UniversityBeijingChina

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