Stationary Distributions in Monotone Markov Models: Theory and Applications

TL;DR

This paper provides a unified necessary and sufficient condition for the existence and stability of stationary distributions in monotone Markov models, applicable across various settings including noncompact spaces and nonlinear operators.

Contribution

It introduces a single characterization based on asymptotic contractivity and tightness, extending previous patchwork conditions to a broad class of models.

Findings

Characterization applies to both compact and noncompact state spaces.

Results demonstrated in wage dynamics, Bayesian learning, and income tail analysis.

Extends to nonlinear Markov operators depending on aggregate states.

Abstract

Many economic models feature monotone Markov dynamics on state spaces that may be noncompact. Establishing existence, uniqueness, and stability of stationary distributions in such settings has required a patchwork of sufficient conditions, each tailored to specific applications. We provide a single necessary and sufficient condition: a monotone Markov process has a globally stable stationary distribution if and only if it is asymptotically contractive and has a tight trajectory. This characterization covers both compact and noncompact state spaces, discrete and continuous time, and extends to nonlinear Markov operators that depend on aggregate state. We demonstrate the result through applications to wage dynamics, Bayesian learning with belief shocks, and income processes that generate Pareto tails.