Journal of Optimization Theory and Applications

, Volume 127, Issue 3, pp 565–577

Dynamic Pricing via Dynamic Programming1

  • Y. Y. Fan
  • H. K. Bhargava
  • H. H. Natsuyama

DOI: 10.1007/s10957-005-7503-z

Cite this article as:
Fan, Y.Y., Bhargava, H.K. & Natsuyama, H.H. J Optim Theory Appl (2005) 127: 565. doi:10.1007/s10957-005-7503-z


This article specifies an efficient numerical scheme for computing optimal dynamic prices in a setting where the demand in a given period depends on the price in that period, cumulative sales up to the current period, and remaining market potential. The problem is studied in a deterministic and monopolistic context with a general form of the demand function. While traditional approaches produce closed-form equations that are difficult to solve due to the boundary conditions, we specify a computationally tractable numerical procedure by converting the problem to an initial-value problem based on a dynamic programming formulation. We find also that the optimal price dynamics preserves certain properties over the planning horizon: the unit revenue is linearly proportional to the demand elasticity of price; the unit revenue is constant over time when the demand elasticity is constant; and the sales rate is constant over time when the demand elasticity is linear in the price.


Dynamic pricing dynamic programming optimal control 

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Y. Y. Fan
    • 1
  • H. K. Bhargava
    • 2
  • H. H. Natsuyama
    • 3
  1. 1.Civil and Environmental Engineering DepartmentUniversity of CaliforniaDavis
  2. 2.Graduate School of ManagementUniversity of CaliforniaDavis
  3. 3.School of EngineeringCalifornia State UniversityFullerton