A multiscale time model with piecewise constant argument for a boundedly rational monopolist

• Published in: Journal of difference equations and applications 2016-10, Vol.22 (10), p.1480-1489
• Main Authors: Cavalli, F., Naimzada, A.
• Format: Article
• Language: English
• Subjects: bifurcation; Boundedly rational monopolist; Constants; Difference equations; Differential equations; Dynamic models; Learning; Mathematical models; multiscale; nonlinear difference equation; Nonlinear dynamics; Nonlinear equations; piecewise continuous differential equations; Reagents; stability; Time
• ISSN: 1023-6198; 1563-5120
• DOI: 10.1080/10236198.2016.1202940

We present a dynamic model for a boundedly rational monopolist who, in a partially known environment, follows a rule-of-thumb learning process. We assume that the production activity is continuously carried out and that the costly learning activity only occurs periodically at discrete time periods, so that the resulting dynamical model consists of a piecewise constant argument differential equation. Considering general demand, cost and agent's reactivity functions, we show that the behavior of the differential model is governed by a nonlinear discrete difference equation. Differently from the classical model with smooth argument, unstable, complex dynamics can arise. The main novelty consists in showing that the occurrence of such dynamics is caused by the presence of multiple (discrete and continuous) time scales and depends on size of the time interval between two consecutive learning processes, in addition to the agent's reactivity and the sensitivity of the marginal profit.