Global stability of a piecewise linear macroeconomic model with a continuum of equilibrium states and sticky expectation

TL;DR

This paper analyzes the global stability of a piecewise linear macroeconomic model with a continuum of equilibria, showing conditions under which the system converges to these states or diverges due to noise or policy effects.

Contribution

It introduces a novel analysis of macroeconomic models with sticky expectations, demonstrating global stability and domain estimates using Lyapunov functions and operator formulas.

Findings

Continuum of equilibria is globally attracting in noise-free systems.

Bounded noise or policy destabilization affects the attraction domain.

Lyapunov functions and operator formulas are effective analytical tools.

Abstract

We consider piecewise linear discrete time macroeconomic models, which possess a continuum of equilibrium states. These systems are obtained by replacing rational inflation expectations with a boundedly rational, and genuinely sticky, response of agents to changes in the actual inflation rate in a standard Dynamic Stochastic General Equilibrium model. Both for a low-dimensional variant of the model, with one representative agent, and the multi-agent model, we show that, when exogenous noise is absent from the system, the continuum of equilibrium states is the global attractor. Further, when a uniformly bounded noise is present, or the equilibrium states are destabilized by an imperfect Central Bank policy (or both), we estimate the size of the domain that attracts all the trajectories. The proofs are based on introducing a family of Lyapunov functions and, for the multi-agent model, deriving a formula for the inverse of the Prandtl-Ishlinskii operator acting in the space of discrete time inputs and outputs.