From bi-stability to chaotic oscillations in a macroeconomic model

In this paper a discrete-time economic model is considered where the savings are proportional to income and the investment demand depends on the difference between the current income and its exogenously assumed equilibrium level, through a nonlinear S-shaped increasing function. The model can be ultimately reduced to a two-dimensional discrete dynamical system in income and capital, whose time evolution is “driven” by a family of two-dimensional maps of triangular type. These particular two-dimensional maps have the peculiarity that one of their components (the one driving the income evolution in the model at study) appears to be uncoupled from the other, i.e., an independent one-dimensional map. The structure of such maps allows one to completely understand the forward dynamics, i.e., the asymptotic dynamic behavior, starting from the properties of the associated one-dimensional map (a bimodal one in our model). The equilibrium points of the map are determined, and the influence of the main parameters (such as the propensity to save and the firmsʹ speed of adjustment to the excess demand) on the local stability of the equilibria is studied. More important, the paper analyzes how changes in the parametersʹ values modify both the asymptotic dynamics of the system and the structure of the basins of the different and often coexisting attractors in the phase-plane. Finally, a particular “global” (homoclinic) bifurcation is illustrated, occurring for sufficiently high values of the firmsʹ adjustment parameter and causing the switching from a situation of bi-stability (coexistence of two stable equilibria, or attracting sets of different nature) to a regime characterized by wide chaotic oscillations of income and capital around their exogenously assumed equilibrium levels.