Abstract
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.
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Notes
This approach is also used in other works with different assumptions about the degree of knowledge of economic agents (Bischi et al. 1998).
Notice that we are assuming that the model does not include problems about inventories.
The existence of delays and mismatch between choices and their achievements is recognized to be a central issue in management science (Harrison and Van Hoek 2008).
We note that we are considering the simplifying assumption that changing the quantities produced do not cause endogenous adjustment costs for firms. However, a sufficiently large value of σ i (roughtly speaking, this implies that \(\dot {x}(t)\) is close to zero) describes a situation in which there are strong frictions in production adjustments. In this regard, it will be interesting to study models that incorporate non-constant adjustment costs. See Bertola and Caballero (1990) for a general treatment on this issue.
The authors thank an anonymous reviewer for pointing this out.
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Acknowledgements
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.
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Gori, L., Guerrini, L. & Sodini, M. A characterisation of duopoly dynamics with frictions in production adjustments. J Evol Econ 27, 963–988 (2017). https://doi.org/10.1007/s00191-017-0515-7
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DOI: https://doi.org/10.1007/s00191-017-0515-7