Available online at: https://doi.org/10.1007/s10957-005-7503-z
Fan, Yueyue, Hemant Bhargava, Harriet Natsuyama (2005) Dynamic Pricing via Dynamic Programming. Journal of Optimization Theory and Applications 127 (3), 565 - 577
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.
Key words: Dynamic pricing, dynamic programming, optimal control