Gibrat’s Law, firms’ growth (and decline) and the evolution of firms’ size distribution
Then it is easy to see, and we will report later in some detail, that – as time goes by – there exist quite general conditions under which xi,n, starting from an arbitrary xi,0, is distributed, in the limit, i.e. for n large, as a lognormal. This is a simple consequence of the central limit theorem (CLT) for random variables, holding under the assumptions about εi,n required for this theorem to hold. Since the assumptions supporting the CLT are quite general, it is clear that in economics – as in many other fields of science – statistical regularities about the distribution of xn tend to emerge in the form of a lognormal distribution of some attributes of a given phenomenon. For a long time virtually all of the contributions in the area have relied on this general result and on some extensions aimed at overcoming some counterfactual implications, where data did not exactly fit the limit distribution. Starting from this background the main questions addressed in the economics literature are basically of the following type:
Q1 Under which conditions (basic stylized facts) is the growth process in equation (1) a good approximation so that a lognormal (or other distributions within the class of power laws) limit distribution for xn should be the expected outcome to be observed?