A characterisation of duopoly dynamics with frictions in production adjustments

Abstract

This article revisits the classical work of Puu (1991) on duopoly dynamics by gathering\ud two distinct aspects of the functioning of markets: production of goods requires time and\ud is subject to some gestation lags, but trading takes place continuously. Dynamics are\ud characterised by a two-dimensional system of delay differential equations. The main aim\ud of this work is to show that regular and non-regular fluctuations may emerge endogenously\ud because of the existence of heterogeneous interacting agents that choose production period\ud by period in a myopic way. Chaotic dynamics in the discrete-time model of Puu (1991)\ud appear to be close enough to the origin of axes. In contrast, in our continuous-time version\ud of the model with discrete delays, the dynamic system is more suitable of generating\ud complex dynamics far enough from the origin when marginal costs vary. This is because\ud of the role played by time delays and inertia. From a mathematical point of view, we\ud show the existence of Hopf bifurcations and detect how time delays and inertia affect\ud the stability of the system by using the recent techniques of stability crossing curves\ud introduced by Gu et al. (2005) and generalized by Lin and Wang (2012). The article\ud also provides some findings about global bifurcations and chaotic dynamics by combining\ud analytical studies and simulation exercises

Delay Cournot duopoly models revisited

“…See Matsumoto and Szidarovszky 12 and Gori et al 13 that adopt the methods for analyzing delay dynamics of Cournot duopoly. Also, see Matsumoto et al 14 , Gori et al, 15,16 and some papers in Matsumoto et al 17 for further applications of the methods to another economic model.…”

A continuous-time heterogeneous duopoly model with delays

“…High values of σ i > 0, i = 1, 2, imply that the firm i is less reactive than the action of its rival and the system moves towards the equilibrium (see e.g. Matsumoto and Szidarovszky (2014), Gori et al (2015a), Gori et al (2017) for the role played by the parameter σ i ). As stressed by Gori et al (2016), one of the main advantages of introducing time delays concerns the possibility of taking into consideration the time needed by technology to bring production to completion.…”

“…As stressed by Gori et al (2016), one of the main advantages of introducing time delays concerns the possibility of taking into consideration the time needed by technology to bring production to completion. Following Matsumoto and Szidarovszky (2014) and Gori et al (2017), we consider heterogeneity of the firms focusing on the role of the frictions in the adjustment mechanism of quantities and in the heterogeneous time delays. The parameter α, which maintains the same role of the benchmark work, is now attenuated by σ 1 and σ 2 that represent two specific parameters related to the market circumstances.…”

Bifurcation Analysis of a Duopoly Game with R&D Spillover, Price Competition and Time Delays

“…Li and Ma [10] considered a small rational dual-channel game and simulate their model's complex dynamic behaviour in their research. Many researchers have explored the complex dynamical behaviours of this type of models from different aspects, such as differentiated goods [11][12][13][14][15], bounded rationality [16], heterogeneous firms [7,[17][18][19], delayed decisions [20][21][22][23][24][25] and other factors [26][27][28].…”