A characterisation of duopoly dynamics with frictions in production adjustments
This article revisits the classical work of Puu (Chaos Soliton Fract 1(6):573–581, 1991) on duopoly dynamics by gathering two distinct aspects of the functioning of markets: production of goods requires time and is subject to some gestation lags, but trading takes place continuously. Dynamics are characterized by a two-dimensional system of delay differential equations. The main aim of this work is to show that regular and non-regular fluctuations may emerge endogenously because of the existence of heterogeneous interacting agents that choose production over time in a myopic way. Chaotic dynamics in the discrete-time model of Puu (Chaos Soliton Fract 1(6):573–581, 1991) appear to be close enough to the origin of axes (implying that quantities produced by both firms are close to zero). In contrast, in our continuous-time version of the model with discrete delays, the dynamic system is more suitable of generating complex dynamics far enough from the origin when marginal costs vary. This is because of the role played by time delays and inertia. From a mathematical point of view, we show the existence of Hopf bifurcations and detect how time delays and inertia affect the stability of the system by using the recent techniques of stability crossing curves introduced by Gu et al. (J Math Anal Appl 311(1):231–253, 2005) and generalized by Lin and Wang (Can Appl Math Quart 20(4):519–533, 2012). The article also provides some findings about global bifurcations and chaotic dynamics by combining analytical studies and simulation exercises.
KeywordsChaos Cournot duopoly Stability crossing curves Time delays
JEL ClassificationC61 C62 D43 L13
The authors gratefully acknowledge Tönu Puu, Ferenc Szidarovszky and participants at NED 2015 held at Chuo University, Tokyo (Japan), and AMASES 2015 held at University of Padova, Italy, for insightful comments and suggestions on an earlier draft. The authors are also indebted to two anonymous reviewers for valuable comments that have contributed to improve the work. The usual disclaimer applies.
Compliance with ethical standards
Conflict of interests
The authors declare that they have no conflict of interest.
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