A Multi-Sector Goodwin Model with n Kinds of Capital Goods

In a work concerning stability analysis, Burmeister, Dobell, and Kuga(1968) showed the properties of Solow type growth model with many capital goods. They suggested that the one-sector Solow model is stable not because it has only one capital good, but because it has particularly simple saving function. In the analysis, they proved the global stability of the multi-sector Solow model. The model is, however, unrealistic due to their neoclassical flamework in which full employment is permanetly kept. In this note, we allow for the existence of unemployment in their model. We substitute linear functions for Cobb-Douglas production functions. Also we introduce some elements of Goodwin's model (Goodwin(1965)), thus generalizing Sato's model in Sato(1985). The purpose of this paper is to analyze of the stability of a thus extended Goodwin model. We consider the relationship between the number of capital goods and the stability of the system. This paper is organized as follows. In section 2, we propose an extended Goodwin model and show the existence of a positive equilibrium. In section 3 concerning a neighbourhood of the equilibirium the property of three dimensional case is compared with that of two dimensional case. Section 4 is devoted to a couple of analyses by computer simulations. Some comments are given in the final section.