A closed-form solution to the Ramsey model with logistic population growth

In this paper, we consider the Ramsey growth model with CIES utility function, Cobb-Douglas technology, and logistic-type population growth law. We show the model to have a unique non-trivial steady-state equilibrium (a saddle point) and prove the optimal path to be non-monotonic over time. Moreover, we derive a closed-form solution for the case where capital's share is equal to the reciprocal of the intertemporal elasticity of substitution.