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

TL;DR

This paper analyzes a continuous-time monopolistic firm model to determine optimal production and pricing strategies, revealing conditions for inventory liquidation and static control use, with insights into non-convex costs.

Contribution

It provides a novel representation of the value function and complete optimal strategies for a non-convex control problem using viscosity solutions and duality.

Findings

Optimal inventory liquidation occurs in finite time.

Static strategies are optimal after liquidation.

Production cycles are characterized by relaxed controls.

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 form 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. (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 static strategy can be represented by an ordinary or relaxed control. The latter is related to production cycles. General theory is illustrate by the example of a non-convex production cost, proposed by Arvan and Moses (1981).