Abstract
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.
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1We acknowledge professor robert e. kalaba for initiating this work and suggesting solution methods.
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Fan, Y.Y., Bhargava, H.K. & Natsuyama, H.H. Dynamic Pricing via Dynamic Programming1 . J Optim Theory Appl 127, 565–577 (2005). https://doi.org/10.1007/s10957-005-7503-z
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DOI: https://doi.org/10.1007/s10957-005-7503-z