A Geometric Description of a Macroeconomic Model with a Center Manifold

This paper presents a unified framework of different algorithms to numerically compute high order expansions of invariant manifolds associated to a steady state of a dynamical system. The framework is inspired in the parameterization method of Cabré, Fontich and de la Llave [7], and the semianalytical algorithms proposed by Simó [13], and those of Gomis-Porqueras and Haro [9]. Within this methodology, one can compute high order approximations of stable, unstable and center manifolds. In this last case the use of high order approximations (not just linear) are crucial in understanding the dynamic properties of the model near the steady state. To illustrate the algorithms we consider a model economy introduced by Azariadis, Bullard and Smith [6]. Besides its intrinsic importance, this four dimensional macroeconomic model is an ideal testing ground because it delivers steady states with stable and unstable manifolds (of dimensions 1 or 2), and each of them has also a one dimensional center manifold. Moreover, the numerical computations lead to a further theoretical study of the dynamical system completing some of the results in the original paper.