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.
Highlights
Notions of friction and stickiness are widely accepted to exist in organizations, economies, and financial systems
It is a fundamental economic question to determine what kinds of friction can arise from the behavior of economic agents and how Keywords and phrases: Piecewise linear discrete time system, play operator, Dynamic Stochastic General Equilibrium model, multi-agent model, global stability
A population may concurrently be subject to many human limitations, but for our purposes it is enough to assume that, when aggregated, individual actors behave according to the play operator we describe, and we remain relatively agnostic as to which mechanism may be driving the general features of the play operator
Summary
Notions of friction and stickiness are widely accepted to exist in organizations, economies, and financial systems. There are various empirical regularities found in micro-, macro-, and organizational economic data, such as path-dependence, permanence, hysteresis, boom blessings and recession curses that are hard to replicate or explain with unique-equilibrium models They can be accounted for quite naturally by the meta-stable states, with associated long timescale dynamics, that frictions at the micro-level introduce into such systems. The future is not fully predictable, agents’ expectations are assumed not to be systematically biased and to collectively use all relevant information in forming expectations of economic variables The latter includes the widely-used sticky models of Calvo [12] and the sticky-information of Mankiw and Reis [40]. We conclude with a summary of the main results and some suggestions for future work
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