  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.
