Optimal Production Planning Under Macroeconomic Regime Switches

TL;DR

This paper presents a mathematical framework for optimal production planning in economies with stochastic regime switches, providing explicit solutions and analyzing their properties through theoretical and numerical methods.

Contribution

It introduces a novel multi-regime control framework with explicit solutions, extending classical models to account for macroeconomic regime switches.

Findings

Explicit quadratic solutions for HJB systems under regime switches

Analysis of time consistency and parameter sensitivity in two-regime models

Numerical validation of theoretical results and regime dynamics

Abstract

We develop a comprehensive mathematical and computational framework for optimal production planning in economies governed by stochastic regime switches driven by a continuous-time Markov chain. The value functions of the underlying stochastic control problem satisfy a weakly coupled system of quasilinear elliptic Hamilton--Jacobi--Bellman (HJB) equations. We establish a global multi-regime comparison principle and prove the existence and uniqueness of classical solutions under optimal growth conditions, generalizing the scalar framework to the multi-regime setting. Furthermore, we derive exact, radially symmetric quadratic solutions for both the scalar and the fully coupled HJB systems, providing explicit, dimension-free representations of the optimal value functions and production policies. These theoretical results are applied to a two-regime economy to analyze time consistency and the sensitivity of optimal responses to macroeconomic parameters. The article is supplemented by numerical implementations that validate the analytical findings and demonstrate the regime-switching dynamics.