On the Solutions to the Ramsey Model with Logistic Population Growth via the Partial Hamiltonian Approach

In two recent papers Naz et al. developed a new methodology for solving the dynamic system of first-order ordinary differential equations arising from optimal control problems, by using the partial Hamiltonian approach. They derived closed-form solutions for the Ramsey model and for the Lucas-Uzawa model. As it is well-known, on the two models there are some papers which have found closed-form solutions like those of Smith, Ragni et al., Viasu, Bucekkine and Ruiz-Tamarit, Hiraguchi and Chilarescu. We knew that with no restrictions on the parameter the solution of the model of Ramsey could not be found. Almost all papers published until now tried to describe the trajectories of the variables along the steady state (equilibrium). Of course, this is not enough to understand the evolution of an economy. The most important thing is to be able to describe the trajectories from the starting point to the steady state. We apply this new technique developed by Naz et al. in order to examine the existence of analytical or closed-form solutions for the Ramsey model with logistic population growth, a model recently studied by Guerrini.