## Routes to Complexity in a Macroeconomic Model Described by a Noninvertible Triangular Map

##### Author

Dieci, Roberto

Bishi, Gian-Italo

Gardini, Laura

##### Abstract

In this paper a particular discrete-time macroeconomic model is considered, where the saving 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 functions. The model proposed can be seen as a particular case of a general class of business cycle models, known as Kaldortype models, which are characterized by the fact that the investment demand also depends on the capital stock (and this assumption is usually considered the main requirement for the occurrence of oscillating behavior of income and capital). The resulting model is described by a dynamical system in income and capital whose time evolution is given by the iteration a two-dimensional map of singular type: this means that one of the components of the map, namely the one driving the income evolution, is an independent one-dimensional map. Due to the particular triangular structure of the system, the asymptotic dynamic behavior and the bifurcations can be completely described starting from the properties of the associated one-dimensional map. the dynamic behaviors of the model are explored under different ranges of the main parameters, such as the firms speed of adjustment to the excess demand and the propensity to save. Although our exercise shows that the basic dynamic scenario is given by a situation of bi-stability, i.e. coexistence of two stable steady states, the existence of more complex dynamics is proved, for sufficiently high values of the adjustment parameter. The effects of the switching to the regime of noninvertibility of the map on the basins structure are considered, together with the bifurcations which modify the structure. Moreover, some bifurcations which change the qualitative properties of the attracting sets are analyzed, in particular a global (homoclinic) bifurcation which causes the transition from a situation of bi-stability to a regime characterized by wide chaotic oscillations of income and capital around their exogenously assumed equilibrium levels, i.e. a typical situation of irregular business cycles.