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Stationary Distributions in Monotone Markov Models: Theory and Applications

Takashi Kamihigashi, John Stachurski

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.

Stationary Distributions in Monotone Markov Models: Theory and Applications

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.

Paper Structure

This paper contains 34 sections, 32 theorems, 105 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Stability of Asymptotic Contractions
  3. Background
  4. Semigroups
  5. Results
  6. Stability on Probability Spaces
  7. Background
  8. Semigroups over Distributions
  9. Probabilistic Markov Models
  10. Stochastic Kernels
  11. Transition Probability Functions
  12. Monotone Mixing
  13. Theory
  14. Application: wage dynamics
  15. Noncompact State Space, Discrete Time
  16. ...and 19 more sections

Key Result

Theorem 2.1

Let $(T_t)$ be an order-preserving semigroup on $\mathsf{X}$. If a:enva:env2 hold, then the following statements are equivalent: $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Diagonal property: if $x, y \in [a,b]$, then $d(x,y) \leq d(a,b)$
  • Figure 2: Simulation of the wage dynamics model in \ref{['ss:awd']}.
  • Figure 3: Simulation of the belief shock model in \ref{['ss:bshock']}.
  • Figure 4: Simulation of the pure jump income model in \ref{['ss:purejump']}.
  • Figure 5: Simulation of the income dynamics model in \ref{['ss:driftmodel']}.

Theorems & Definitions (67)

  • Example 2.1
  • Example 2.2
  • Theorem 2.1
  • Corollary 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Example 3.1
  • Proposition 3.1
  • Lemma 3.2
  • ...and 57 more