Abstract
We consider a deterministic continuous time model of monopolistic firm, which chooses production and pricing strategies of a single good. Firm’s goal is to maximize the discounted profit over infinite time horizon. The no-backlogging assumption induces the state constraint on the inventory level. The revenue and production cost functions are assumed to be continuous but, in general, we do not impose the concavity/convexity property. Using the results from the theory of viscosity solutions and Young-Fenchel duality, we derive a representation for the value function, study its regularity properties, and give a complete description of optimal strategies for this non-convex optimal control problem. In agreement with the results of Chazal et al. (Nonlinear Anal. Theor. 54(8):1365–1395, 2003), it is optimal to liquidate initial inventory in finite time and then use an optimal static strategy. We give a condition, allowing to distinguish if this strategy can be represented by an ordinary or relaxed control. General theory is illustrated by the example of a non-convex production cost, proposed by Arvan and Moses (University of Illinois at Urbana-Champaign, Working paper No. 756, 1981, p.31).
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The research of the D. B. Rokhlin is supported by Southern Federal University, Project 213.01-07-2014/07.
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Rokhlin, D.B., Mironenko, G. Optimal production and pricing strategies in a dynamic model of monopolistic firm. Japan J. Indust. Appl. Math. 33, 557–582 (2016). https://doi.org/10.1007/s13160-016-0235-7
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DOI: https://doi.org/10.1007/s13160-016-0235-7