Nonlocal Solutions to Dynamic Equilibrium Models: The Approximate Stable Manifolds Approach

TL;DR

This paper introduces a novel method based on dynamical systems theory to construct and analyze approximate solutions for general equilibrium models on nonlocal domains, ensuring convergence to true solutions.

Contribution

It develops a rigorous approach using the Contraction Mapping Theorem and Stable Manifold Theorem to approximate and prove convergence of solutions in nonlinear equilibrium models.

Findings

Method guarantees convergence to true solutions under certain conditions.

Provides explicit bounds on approximation errors.

Validates approach as a rigorous proof for the extended path algorithm.

Abstract

This study presents a method for constructing a sequence of approximate solutions of increasing accuracy to general equilibrium models on nonlocal domains. The method is based on a technique originated from dynamical systems theory. The approximate solutions are constructed employing the Contraction Mapping Theorem and the fact that solutions to general equilibrium models converge to a steady state. The approach allows deriving the a priori and a posteriori approximation errors of the solutions. Under certain nonlocal conditions we prove the convergence of the approximate solutions to the true solution and hence the Stable Manifold Theorem. We also show that the proposed approach can be treated as a rigorous proof of convergence for the extended path algorithm to the true solution in a class of nonlinear rational expectation models.