The Ramsey model with a bounded population growth rate

This paper presents an extension of the canonical Ramsey growth model of optimal capital accumulation by departing from the standard assumption of constant population growth rate. More concretely, this rate is assumed to be variable over time, subject only to be between prescribed upper and lower limits. In this kind of setup, the model is represented by a two dimensional dynamical system, which happens to be non-autonomous. In contrast to the Ramsey model, the linearization method cannot be used. It will be necessary to use different techniques to derive the dynamic properties of the model's solution, as well as investigate its long-run growth and asymptotic stability. This will be done using the method proposed by Guerrini (2006), when studying the Solow model with a variable population growth rate. The paper also shows that closed-form analytic solutions can always be derived for the model, when capital's share is equal to the reciprocal of the intertemporal elasticity of substitution.