# đź‘ŤPareto Extrapolation: An Analytical Framework for Studying Tail Inequality

Published in Quantitative Economics, 2023

During the review process of my 2019 JME paper, the handling associate editor (Alisdair McKay) asked me to numerically solve a Bewley-type heterogeneous-agent model with both idiosyncratic capital and labor income risks (that do not admit a closed-form solution) to illustrate the usefulness of the Beare & Toda (2022) Pareto exponent formula for analyzing such models that applied economists are interested. When I did so, I found that the solution accuracy is sensitive to the grid for wealth. The reason is that, when the wealth distribution has a Pareto upper tail, the true wealth distribution has a large probability mass in the upper tail, which is missed in the numerical solution due to the truncation error caused by the finiteness of the grid. So I thought about a method for correcting this truncation error by extrapolating the wealth distribution off the grid using the theoretical Pareto exponent.

I had a feeling this could be another Econometrica paper, so I didnâ€™t talk about these issues in the JME paper. My student Ă‰milien, who was already familiar with frontier research on power laws, showed interest and made many great suggestions, so I recruited him as a coauthor. During my visit to Australian National University in the summer of 2018 (it was winter in Australia), we worked on this project day and night taking advantage of the time zone difference. Initially I thought about this project as a numerical method, but it is more than that. It is a framework to analyze heterogeneous-agent models by allowing the user to characterize tail properties such as Pareto exponents, top wealth shares, upper tail entry and exit probabilities (e.g., â€śWhat is the probability an individual exits the Forbes 400 list?â€ť), upper tail type distributions (e.g., â€śWhat is the fraction of entrepreneurs in the top 1%?â€ť), and much more. If a researcher ever solves a heterogeneous-agent model with multiplicative shocks, there is no excuse for not using this method.

Unfortunately, the referees at ECMA and REStud were split, and some complained about the application, which was not the central point of the paper. We thus removed the original application (which Ă‰milien spent so much time on calibration etc.) and replaced it with a stylized model to just illustrate the solution accuracy, sent the paper to QE exercising our option to recycle ECMA reports, and the paper got accepted as is. I feel that improving the paper (by proving sharper theorems, say) often reduces the chance of publication.

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