Kindred Distributions: An Application to Wealth and Income Densities

Abstract

We describe a simple method to obtain kindred stationary densities of random Markov processes with respect to an Ito transformation function. As applied to income and wealth densities, they are akin to one another because they share the income growth and income volatility shape parameters. The procedure first assumes a linear infinitesimal drift and a quadratic infinitesimal variance, which together with the stationary Kolmogorov forward equation gives rise to a power law income distribution. To find a kindred distribution we assume that the rate of change of wealth is a function of savings from income and the returns from accumulated wealth. Applying the Ito transformation, we then obtain an explicit wealth-income density, which turns out to be another power law density with an augmented exponential term. It shares certain shape parameters with the income density. Another objective of the paper is to learn how volatility affects the specific wealth-income prediction in Piketty-Zucman (2014).