Orbits in a stochastic Goodwin-Lotka-Volterra model

TL;DR

This paper analyzes the cycling dynamics of a stochastic Goodwin-Lotka-Volterra model, characterizing orbits in the deterministic case and demonstrating stochastic cycles around equilibrium using Lyapunov methods.

Contribution

It provides a novel analysis of stochastic cycles in the Goodwin model, extending deterministic orbit characterization to stochastic settings with new conditions for system recurrence.

Findings

Deterministic model exhibits complex non-linear orbits.

Stochastic noise induces cycles around equilibrium.

Sufficient conditions for system recurrence are established.

Abstract

This paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the Lotka-Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic highly non-linear model. We then study the cycling behavior for a stochastic version, where a Brownian noise is introduced via an heterogeneous productivity factor. Sufficient conditions for existence of the system are provided. We prove that the system produces cycles around an equilibrium point in finite time for general volatility levels, using stochastic Lyapunov techniques for recurent domains. Numerical insights are provided.