The Ramsey model with logistic population growth and benthamite felicity function revisited

Abstract

In this paper we extend the study done by Ferrara and Guerrini [12], where two different research lines within the Ramsey model were joined together: the one studying the role of a logistic population growth rate (Accinelli and Brida [2]), and the one analyzing the effects of a Benthamite formulation for the utility function. The results obtained in [12] for the special case of a constant intertemporal elasticity of substitution (CIES) utility function and a Cobb-Douglas production function are provided to be still true for a general utility function and a neoclassical production function. We have that the model is described by a three dimensional dynamical system, whose unique non-trivial steady state equilibrium is a saddle point with a two dimensional stable manifold. Consequently, the speed of convergence is determined by two stable roots, rather than only one as in the basic Ramsey model. In addition, in the special case of a CIES utility function and a Cobb-Douglas technology, an explicit solution for the model can be derived, when capital's share is equal to the reciprocal of the intertemporal elasticity of substitution.