Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach

Abstract By use of our newly developed methodology (Naz et al., 2014 [1]), for solving the dynamical system of first-order ordinary differential equations (ODEs) arising from first-order conditions of optimal control problems, we derive closed-form solutions for the standard Lucas–Uzawa growth model. We begin by showing how our new methodology yields a series of first integrals for the dynamical system associated with this model and two cases arise. In the first case, two first integrals are obtained and we utilize these to derive closed-form solutions and show that our methodology yields the same results as in the previous literature. In the second case, our methodology yields three first integrals under certain restrictions on the parameters. We use these three integrals to obtain new solutions for all the variables which in turn yield new solutions for the growth rates of these variables. Our results are significant as our approach is applicable to an arbitrary system of ODEs which means that it can also be invoked for more complex models.