The global stability of a class of history-dependent macroeconomic models

¹ Department of Mathematical Sciences George Mason University, VA, USA.
² Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic.
³ Faculty of Civil Engineering, Czech Technical University, Prague, Czech Republic.
⁴ Department of Mathematical Sciences, The University of Texas at Dallas, TX, USA.

^* Corresponding author: dmitry.rachinskiy@utdallas.edu

Received: 24 January 2019
Accepted: 30 November 2019

Abstract

We consider piecewise-linear, discrete-time, macroeconomic models that have a continuum of feasible equilibrium states. The non-trivial equilibrium set and resulting path-dependence are induced by stickiness in either expectations or the response of the Central Bank. For a low-dimensional variant of the model with one representative agent, and also for a multi-agent model, we show that when exogenous noise is absent from the system the continuum of equilibrium states is the global attractor and each solution trajectory converges exponentially to one of the equilibria. 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.