Analytical approaches for growth models in economics

Abstract

This study deals with the group-theoretical analysis of nonlinear optimal control problems called the optimal growth model with the environmental asset and capitalist decision model of endogenous growth, which are expressed in terms of the current and present value Hamiltonian functions. The first-order conditions for optimal control problems based  on Pontryagin’s maximum principle are considered. In addition, Lie point symmetries and their relation with the  Prelle-Singer, λ-symmetries, adjoint symmetries, Darboux polynomials, and Jacobi’s last multiplier approaches are  studied by analyzing the coupled nonlinear first-order ordinary differential equations corresponding to the first-order  conditions. For both current and present Hamiltonian cases, the first integrals and the corresponding analytical (closed- form) solutions and graphical representations for the optimal control problems are represented.