Mathematics and Computers in Simulation 63 (2003) 505–527 Hicks’ trade cycle revisited: cycles and bifurcations M. Gallegati a , L. Gardini b,∗ , T. Puu c , I. Sushko d a Department of Economics, Universita Politecnica delle Marche, Piazzale Martelli, 8, 60121 Ancona, Italy b Department of Economics, University of Urbino, via Saffi 42, 61029 Urbino (PU), Italy c Centre for Regional Science, University of Umeå, SE-901, 87 Umeå, Sweden d Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st, 01601 Kiev, Ukraine Received 13 March 2003; accepted 28 May 2003 Abstract 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. © 2003 Published by Elsevier B.V. on behalf of IMACS. Keywords: Business cycle models; Piecewise-linear maps; Tongues of periodicity; Bifurcation diagram 1. Introduction The accelerator-multiplier class of models in economics dates back to [1] (the origin of these models is described by [2]). Samuelson’s article, which is very short and formal, provides a linear second-order difference equation that produces either stable or diverging trajectories, but no self-sustained business cycles except at very particular values of the parameters, and in such cases it produces cycles of fixed period and amplitude. Something which is at odds with the empirical evidence. In [38], Hicks makes some important changes to the model: by adding a floor and a ceiling to a linear model, he formulates ∗ Corresponding author. Tel.: +39-0722-305510 (University secretary)/3055568 (University direct)/30550 (University)/580539 (R); fax: +39-0722-305550 (University). E-mail addresses: gallegati@deanovell.unian.it (M. Gallegati), gardini@econ.uniurb.it (L. Gardini), tonu.puu@econ.umu.se (T. Puu), sushko@imath.kiev.ua (I. Sushko). URL: http://www.econ.uniurb.it/gardini/gardiniweb.htm. 0378-4754/$30.00 © 2003 Published by Elsevier B.V. on behalf of IMACS. doi:10.1016/S0378-4754(03)00060-0