The Size Variance Relationship of Business Firm Growth Rates

TL;DR

This paper presents a proportional growth model explaining the power-law relationship between firm size and growth rate variance, and validates it with industry data showing consistency with empirical observations.

Contribution

The paper introduces a model of proportional growth that accounts for the size-variance relationship and demonstrates its validity using detailed industry-specific data.

Findings

The model predicts a crossover in the size-variance exponent from 0 to 0.5.

Empirical data shows the exponent is approximately 0.14 to 0.2.

Model aligns well with observed industry-specific growth rate variances.

Abstract

The relationship between the size and the variance of firm growth rates is known to follow an approximate power-law behavior $\sigma(S) \sim S^{-\beta(S)}$ where $S$ is the firm size and $\beta(S)\approx 0.2$ is an exponent weakly dependent on $S$. Here we show how a model of proportional growth which treats firms as classes composed of various number of units of variable size, can explain this size-variance dependence. In general, the model predicts that $\beta(S)$ must exhibit a crossover from $\beta(0)=0$ to $\beta(\infty)=1/2$. For a realistic set of parameters, $\beta(S)$ is approximately constant and can vary in the range from 0.14 to 0.2 depending on the average number of units in the firm. We test the model with a unique industry specific database in which firm sales are given in terms of the sum of the sales of all their products. We find that the model is consistent with the empirically observed size-variance relationship.