The sandbox does it better Dr. Piketty

Here we have some new economic methods in which the household distribution is carried, ratherthan the mean and assumed variance process.  So, throughout the model, the effect of a trade on income distributions can be had via a convolution of two distributions, resulting in a 2-d distribution, but that can be collapsed back to one dimensional with orthogonalization.  So you carry mean, variance and skew along for the ride.  It is a bit like Mankiw's paper on taxing height.

In a recent paper, we attempt to create inequality statistics for the US that overcome the limitations of existing data by creating distributional national accounts (Piketty et al. 2016). We combine tax, survey, and national accounts data to build a new series on the distribution of national income. National income is the broadest measure of income published in the national accounts and is conceptually close to gross domestic product, the broadest measure of economic growth.³ Our distributional national accounts enable us to provide decompositions of growth by income groups consistent with macroeconomic growth.

Anyway, my point is that the sandbox gives all agents the secure element which contains any significant spending,  encapsulated into a histogram, just like the good doctor wants.  Once we have that, then we have half a dozen forensic methods to figure out why some thumb prints are poor and some rich. S&L technology works in the bandbox by inferring the sizes of our consumption baskets, and tells us, soon enough, which baskets are too empty or too full.