Tail Behavior of Stopped Lévy Processes With Markov Modulation

Abstract

This article concerns the tail probabilities of a light-tailed Markov-modulated Lévy process stopped at a state-dependent Poisson rate. The tails are shown to decay exponentially at rates given by the unique positive and negative roots of the spectral abscissa of a certain matrix-valued function. We illustrate the use of our results with an application to the stationary distribution of wealth in a simple economic model in which agents with constant absolute risk aversion are subject to random mortality and income fluctuation.

Productivity Dispersion, Between‐Firm Competition, and the Labor Share

“…In terms of new theoretical results, I also provide a sufficient condition for the existence of a stationary equilibrium (Assumption 2) and a characterization of the Pareto exponent of the firm‐size distribution (Appendix Proposition 9). I use theoretical results on power laws (i.e., Toda (2014); Beare, Seo, and Toda (2021); Beare and Toda (2022)) and the “Pareto extrapolation” solution method (developed in Gouin‐Bonenfant and Toda (2022)) to solve the firm size distribution. The emergence of fat‐tailed size distribution in my model echoes findings in Luttmer (2007) and Luttmer (2011), who show that homogeneous firm dynamics models naturally give rise to fat‐tailed distributions.…”

Determination of Pareto Exponents in Economic Models Driven by Markov Multiplicative Processes

“…Examples 3.5 and 3.6 in Beare, Seo, and Toda (2022) concern a continuous‐time reformulation of Example 1 in which the log‐size of agents evolves as a two‐state Brownian motion with drift. There the determination of the Pareto exponent boils down to examining the roots of a quartic polynomial, or of a quadratic polynomial in the absence of a diffusive component.…”

“…Continuous‐time versions of the results in this article have been established in a companion article, Beare, Seo, and Toda (2022). There we study the tail probabilities of a Markov‐modulated Lévy process stopped at a state‐dependent Poisson rate.…”

“…Gouin‐Bonenfant and Toda (2022) show how to improve grid‐based calculation of aggregate quantities in dynamic economic models by using equation (1) to extrapolate beyond the grid. Beare, Seo, and Toda (2022) use a continuous‐time version of equation (1) to study the tails of wealth in a Huggett economy inhabited by agents with constant absolute risk aversion.…”

Wealth Inequality in a Low Rate Environment

“…Recently, this literature has examined more realistic models with persistent growth rate heterogeneity (Luttmer (2011); Jones and Kim (2018); Gabaix, Lasry, Lions, and Moll (2016)). In this case, the Pareto exponent can be obtained as the principal eigenvalue of an operator related to the transition matrix between states (see de Saporta (2005); Beare, Seo, and Akira Toda (2022); Beare and Akira Toda (2022)). Relative to that literature, a theoretical contribution of our paper is to obtain a closed‐form expression for the derivative of the Pareto exponent with respect to a parameter (here, the interest rate).…”