Closed-Form Solutions to an Economic Growth Logistic Model with Migration

Abstract

This paper presents a Solow Growth Model with the labor force ruled by the logistic equation added by a constant migration rate, I. We prove the global asymptotic stability of the capital and production per capita. Considering a Cobb-Douglas production function, we show this model to have a closed-form solution. This transient behavior is expressed in terms of the Beta and Appell F1 functions. We also show that if I > 0, it implies in a higher (minor) output of the economy, and in a minor (greater) capital and production per capita in the short term (middle and long term); furthermore, these ratios converge to the same steady-state value of the no-migration model; if I < 0 the behavior is the opposite; and if I = 0 we recover the solution obtained by Mingari Scarpello and Ritteli [12] for the pure logistic case. Finally, we replace the Cobb-Douglas by a Perfect Substitution CES Production Function, accompanied with sample problems.

Keywords: Solow Growth Model, Logistic Labor Force, Migration, Appell Function

JEL Classification: C61, C62, O41