The chaotic monopolist revisited with bounded rationality and delay dynamics

Abstract

Two types of boundedly rational monopolists are studied when the marginal revenue is not necessarily negative sloping. A limited monopolist (ℓ-monopolist) knows only the price and output values in two previous periods. A knowledgeable monopolist (k-monopolist) knows the analytic expression of the marginal profit function and forms sales expectations based on past market information. The k-monopolist adjusts its output levels according to the usual gradient process, while the ℓ-monopolist approximates the marginal profit with a two-point finite difference formula. Stability conditions and complex dynamics are studied in discrete and continuous time scales. In the discrete model, the two monopolists exhibit different dynamics after the stability loss. The ℓ-monopolist has a cascade of a Neimark-Sacker bifurcation. In contrast, the k-monopolist has two different routes, a period-doubling and a Neimark-Sacker bifurcation, depending on how to form sales expectations. In the continuous case, the discrete models are transformed into continuous models via Euler transformation. The stability switching curves are analytically constructed and the directions of stability switching are characterized by computing the stability index for each point of the curves. It is numerically demonstrated that two monopolists exhibit similar dynamics and the k-monopolist approximates the ℓ-monopolist under a special circumstance.