# How is the "value added" of firms distributed

While it can be justified that the sales (or revenues) of firms in an industry follow a Pareto-distribution, I wonder how the value added (sales minus input costs minus taxes, excluding depreciation) can be summarized by a parametric distribution. Pareto seems not to work, since negative values are possible.

Is there an econometric theory that gives ground for a specific class of distributions? My dataset suggest something like Gumbel.

• Why not to draw an empirical distribution of EBITDA from Compustat or open sources? Jun 2 '16 at 16:27
• My data is incomplete and I like to use theoretical insights to fit a distribution that represents all data. Jun 2 '16 at 23:14

Indeed, it's somewhat difficult to model value added because in many models firms don't make profits all the time. But therein lies your answer I think. Model VA as $Z$ in a distribution $g$ with two dimensions $(Z,Y)$, where Y is the size of the firm, Pareto distributed, and then for any given $Y$, $Z$ is distributed as $Z \sim Y \cdot (1+ X)$ with $X$ a normal variable or something like that, so that $Z$ can be negative, but it is still on average proportional to $Y$. You can maybe then solve for $f_Z=\int g(Z,Y)dY$ analytically or numerically otherwise.