Optimal production and pricing strategies in a dynamic model of monopolistic firm

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).