Journal of Optimization Theory and Applications

, Volume 127, Issue 3, pp 565-577

First online:

Dynamic Pricing via Dynamic Programming1

  • Y. Y. FanAffiliated withCivil and Environmental Engineering Department, University of California
  • , H. K. BhargavaAffiliated withGraduate School of Management, University of California
  • , H. H. NatsuyamaAffiliated withSchool of Engineering, California State University

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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