A Partial Hamiltonian Approach for Current Value Hamiltonian Systems

TL;DR

This paper introduces a partial Hamiltonian method to find solutions of current value Hamiltonian systems, especially in economic growth models, providing a systematic way to derive known and new results.

Contribution

The paper develops a novel partial Hamiltonian framework that simplifies solving current value Hamiltonian systems in economic models, offering an algorithmic approach applicable to various variables.

Findings

Successfully applied to Ramsey and AK models

Can reproduce existing results in literature

Finds new solutions in economic growth models

Abstract

We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.