A Geometric Approach to Local Indeterminacy & Local Bifurcations in Three-Dimensional Dynamical Systems.

In discrete three-dimensional dynamical systems frequently encountered in dynamic general equilibrium models, local indeterminacy -sunspot equilibria - and local bifurcations -endogenous deterministic fluctuations - issues are quite difficult to handle. Pitfalls derive from the fact that in generic models a standard analysis of the spectrum of the 3 x 3 Jacobian matrix, though achievable, usually turns out to be of no use. It is the purpose of this contribution to develop geometrical tools which avoid cumbersome computations or numerical investigations. Rather tractable necessary and sufficient conditions are provided, that ensure the occurence of generic local bifurcations and also a useful "toolkit" for assessing the local determinacy / indeterminacy of intertemporal equilibria. The usefulness of this method is finally illustrated in a productive model of overlapping generations with elastic labour supply.