Nonlinear Delay Monopoly with Bounded Rationality

The purpose of this paper is to study dynamics of a monopolistic …rm in a continuous-time framework. The …rm is assumed to be boundedly rational and to experience time delays in obtaining and implementing information on output. A dynamic adjustment process is based on the gradient of the expected pro…t. The paper is divided into three parts: We examine delay e¤ects on dynamics caused by one-time delay and two-time delays in the …rst two parts. Global dynamics and analytical results on local dynamics are numerically con…rmed in the third part. Four main results are demonstrated. First, the stability switch from stability to instability occurs only once in the case of the one-time delay. Second, the alternation of stability and instability can continue if two time delays are involved. Third, the birth of Hopf bifurcation is analytically shown if stability is lost. Finally, in a bifurcation process, there are a period-doubling cascade to chaos and a period-halving cascade to the equilibrium point in the case of two time delays if the di¤erence between two the delays is large. Keywords: single and multiple …xed time delays, Hopf bifurcation, bounded rationality, stability switch, nonlinear monopoly, gradient dynamics JEL Classi…cation: C62, C63, D42 The authors highly appreciate …nancial supports form the Japan Society for the Promotion of Science (Grant-in-Aid for Scienti…c Research (C) 21530172), Chuo University (Joint Research Grant 0981) and the Japan Economic Research Foundation. The research leading to this paper started when the …rst author visited the Department of Systems and Industrial Engineering of the University of Arizona and …nished when the second author visited the Department of Economics, Chuo University. They appreciated hospitalities of those universities over their stay. The usual disclaimer applies. 1 2 AKIO MATSUMOTO AND FERENC SZIDAROVSZKY