Determination of Pareto exponents in economic models driven by Markov multiplicative processes

TL;DR

This paper introduces new analytical tools to determine the Pareto tail exponents in economic models where units undergo Markov-driven multiplicative shocks and resets, revealing how the spectral properties of associated matrices influence the tail behavior.

Contribution

It provides a novel method to compute Pareto exponents in Markov-modulated multiplicative processes, linking spectral radius solutions to the tail distribution shape.

Findings

Pareto tail exponents are characterized by spectral radius equations.

Eigenvectors associated with the exponent describe the type distribution in the tail.

The approach applies under a non-lattice condition on growth rates.

Abstract

This article contains new tools for studying the shape of the stationary distribution of sizes in a dynamic economic system in which units experience random multiplicative shocks and are occasionally reset. Each unit has a Markov-switching type which influences their growth rate and reset probability. We show that the size distribution has a Pareto upper tail, with exponent equal to the unique positive solution to an equation involving the spectral radius of a certain matrix-valued function. Under a non-lattice condition on growth rates, an eigenvector associated with the Pareto exponent provides the distribution of types in the upper tail of the size distribution.