Hicks’ trade cycle revisited: cycles and bifurcations

• Published in: Mathematics and computers in simulation 2003-11, Vol.63 (6), p.505-527
• Main Authors: Gallegati, M., Gardini, L., Puu, T., Sushko, I.
• Format: Article
• Language: English
• Subjects: Applied sciences; Bifurcation diagram; Business and economics; Business cycle models; Economics; Ekonomi; Exact sciences and technology; Finite differences and functional equations; Mathematical analysis; Mathematics; Nationalekonomi; Operational research and scientific management; Operational research. Management science; Piecewise-linear maps; SAMHÄLLSVETENSKAP; Sciences and techniques of general use; SOCIAL SCIENCES; Tongues of periodicity
• ISSN: 0378-4754; 1872-7166; 1872-7166
• DOI: 10.1016/S0378-4754(03)00060-0

In the Trade Cycle, Hicks introduced the idea that endogenous fluctuations could be coupled with a growth process via nonlinear processes. To argue for this hypothesis, Hicks used a piecewise-linear model. This paper shows the need for a reinterpretation of Hicks’ contribution in the light of a more careful mathematical investigation. In particular, it will be shown that only one bound is needed to have non explosive outcome if the equilibrium point is an unstable focus. It will also be shown that when the fixed point is unstable the attracting set has a particular structure: It is a one-dimensional closed invariant curve, made up of a finite number of linear pieces, on which the dynamics are either periodic or quasi-periodic. The conditions under which the model produces periodic or quasi-periodic trajectories and the related bifurcations as a function of the main economic parameters are determined.