Comparison of Closed-form Solutions for the Lucas-Uzawa model via the Partial Hamitonian Approach and the Classical Approach

TL;DR

This paper derives and compares closed-form solutions for the Lucas-Uzawa growth model using the partial Hamiltonian approach and the classical method, revealing a new solution and confirming convergence to the same long-term growth path.

Contribution

It introduces a novel closed-form solution for the Lucas-Uzawa model using the partial Hamiltonian approach and compares it with classical solutions, demonstrating their convergence.

Findings

A new closed-form solution was derived.

All solutions converge to the same long-run growth path.

The partial Hamiltonian approach effectively generates multiple solutions.

Abstract

In this paper we derive the closed-form solutions for the Lucas-Uzawa growth model with the aid of the partial Hamiltonian approach and then compare our results with those derived by the classical approach \cite{chil}. The partial Hamiltonian approach provides two first integrals \cite{naz2016} in the case where there are no parameter restrictions and these two first integrals are utilized to construct three sets of closed form solutions for all the variables in the model. First two first integrals are used to find two closed form solutions, one of which is new to the literature. We then use only one of the first integrals to derive a third solution that is the same as that found in the previous literature. We also show that all three solutions converge to the same long run balance growth path.