“Growth and the Difference-of-Log-Normals” [PDF]
The growth of natural, social, and economic phenomena including firms, cities, and pandemics is known to be heavy-tailed. Most growth is modest, but we often observe explosive growth rates. Neither a simple explanation nor a well-fitting distributional form for these heavy-tailed growth phenomena is known. Here I show that an obscure yet simple statistical distribution | the difference-of-log-Normals | describes these and other growth phenomena remarkably well. I discuss why it arises as a natural consequence of the Central Limit Theorem, propose a novel production function it implies for modeling such phenomena, and sketch dynamic models using this production function for firms, cities, and populations. I further use this production function to generalize the growth concept so as to measure growth from/to negative values as in, e.g., firm income (profit and loss).
“Why Are Firm Growth Distributions Heavy-tailed?” [PDF]
Firm growth and return distributions are heavy-tailed. Accounting for the interplay of sales and expenses is sufficient to explain this fact without relying on time-varying volatility or factors external to the firm. Embedding the implied production function into a standard q-theory model yields novel and specific predictions regarding the distributions of income, growth, and returns. The predictions are supported by the data. The model is the first to correctly replicate the distribution of firm income and is hence useful as a foundational model for future work. It proposes extended definitions of firm income scale, efficiency, and growth.
“Facts of US Firm Scale and Growth 1970-2019: An Illustrated Guide” [arXiv]
This work analyzes data on all public US firms in the 50 year period 1970-2019, and presents 18 stylized facts of their scale, income, growth, return, investment, and dynamism. Special attention is given to (i) identifying distributional forms; and (ii) scale effects — systematic difference between firms based on their scale of operations. Notable findings are that the Difference-of-Log-Normals (DLN) distribution has a central role in describing firm data, scale-dependent heteroskedasticity is rampant, and small firms are systematically different from large firms.