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

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 nonlattice condition on growth rates, an eigenvector associated with the Pareto exponent provides the distribution of types in the upper tail of the size distribution.

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.…”

Pareto extrapolation: An analytical framework for studying tail inequality

“…Second, we focus on models without aggregate risk. At present, the mathematical theory of Pareto tails such as Beare and Toda (2022) only allow for idiosyncratic risk, and extending it to the case with aggregate uncertainty remains beyond the frontier. Finally, the Pareto exponent formula (3.9) is proved only for Markov multiplicative processes (where Gibrat's law holds exactly).…”

“…Thus, in the asymptotic problem, the law of motion for wealth necessarily satisfies Gibrat (1931)'s law of proportional growth. Assuming that agents enter/exit the economy at constant probability , Beare and Toda (2022) show that under mild conditions the stationary wealth distribution has a Pareto upper tail and characterize the Pareto exponent ζ , as follows. For , let be the moment generating function of the log growth rate conditional on transitioning from state s to , and be the matrix of conditional moment generating functions (3.7).…”

“…(In the case of the KS model, we simply have using (3.5) and (3.7). ) Then under mild conditions Beare and Toda (2022) show that the equation where denotes the Hadamard (entrywise) product of P and , has a unique positive solution , and that the stationary wealth distribution has a Pareto upper tail with exponent ζ . The following proposition gives a simple test for the solvability of (3.9).…”

“…Although the asymptotic linearity of policy functions with homothetic preferences is intuitive, a rigorous proof with an exact analytical characterization of asymptotic slopes was obtained only recently by Ma and Toda (2021, 2022) after the working paper version of this paper (Gouin‐Bonenfant and Toda (2018)) was circulated. To analytically characterize the Pareto exponent of the wealth distribution in a general Markovian environment, we apply the recent results from Beare and Toda (2022), who characterize the Pareto exponent as a solution to an eigenproblem. Toda (2019) pointed out the usefulness of the asymptotic problem for computing the Pareto exponent in general models that admit no closed‐form solutions.…”