A comment on the relationship between firms' size and growth rate

Since the seminal work of Pareto, many empirical analyses suggested that the distribution of firms size is characterized by an asymptotic power like behavior. At the same time, recent investigations show that the distribution of annual growth rates of business firms displays a remarkable double-exponential shape. A recent letter propose a possible connection between these two empirical regularities. By assuming a bivariate Marshall-Olkin power-like distribution for the size of firms in subsequent time steps, and performing a qualitative asymptotic analysis, it is suggested that the implied growth rates distribution takes a Laplace shape. By performing a complete analytical investigation, I show that this statement is not correct. The implied distribution does in general possess a non-continuous component and becomes degenerate when perfect correlation is assumed between size levels at different time steps. Essentially, the approach is faulty as it treats firm size levels as stationary stochastic variables and neglects the integrated nature of the growth process.Firm Growth, Size distribution, Power law, Laplace distribution.

On the relationship between firms' size and growth rate

Since the seminal work of Pareto, many empirical analyses suggestedthat the distribution of firms size is characterized by an asymptoticpower like behavior. At the same time, several investigations showthat the distribution of annual growth rates of firms displays aremarkable double-exponential shape. Recently it has been suggestedthat both these statistical properties can be explained by assuming abivariate Marshall-Olkin power-like distribution for the size of firmsin subsequent time steps. Through analytical investigation, I showthat the marginal distribution of growth rates implied by thisassumption does not possess, in general, a Laplace shape and becomesdegenerate when subsequent size levels are perfectly correlated. Assuch, the bivariate Marshall-Olkin distribution is unable to properlyaccount for the observed regularities. The original suggestion isfaulty as it treats firm size levels as stationary stochasticvariables and neglects their integrated nature.Firm Growth

On the Pareto Type III distribution

This short note analyzes the distributional properties of Pareto Type III random variables. We introduce a three parameters version of the orignal two parameters distribution proposed by Pareto and derive both the density and the characteristic function. The analytic expression of the inverse distribution function is also obtained, together with a simple series expansion of its moments of any order. Finally, we propose a simple statistical exercise designed to show the increased reliability of the Pareto Type III distribution in describing asymptotically dumped power-like behaviors.Pareto Distribution, Fat Tails, Power Law Distribution, Zipf Law

On the Irreconcilability of Pareto and Gibrat Laws

If business firms face a multiplicative growth process in which their growth rates are independent from their sizes, then these sizes cannot be distributed according to a stationary Pareto distribution. At the same time , the Laplace distribution of growth rates cannot be easily reconciled with a Pareto distribution of firm sizes. Recent contributions, using formal arguments, seems to contrast these statements. We prove that the proposed formal results are wrong.Firm Growth, Gibrat's Law, Power law distribution, Laplace distribution.

Inequality and the GB2 income distribution

The generalized entropy class of inequality indices is derived for Generalized Beta of the Second Kind (GB2) income distributions, thereby providing a full range of topsensitive and bottom-sensitive measures. An examination of British income inequality in 1994/95 and 2004/05 illustrates the analysis.inequality, generalized entropy indices, generalized Beta of the second kind distribution, GB2 distribution, Singh-Maddala distribution, Dagum distribution