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.

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.

Subbotools User's Manual

The SUBBOTOOLS package is intended as an help to the use of the Subbotin family of probability densities in a statistical analysis environment. The package contains various programs for the maximum likelihood estimation of an unknown distribution and for the generation of pseudo random variables. The last version of SUBBOTOOLS can be found at http://bottazzi.sssup.it/~giulio. Comments and bug reports are welcome. E-write to [email protected].

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 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 Laplace Distribution of Firms Growth Rates

A very robust stylized fact has recently emerged concerning the distribution of growth rates of manufacturing business firms. We briefly review past analysis and present new evidence on this topic from the Italian Manufacturing Industry. We then propose a very simple model that, under rather general assumptions, provides an excellent explanation for the observed regularities. The model is based on a very simple stochastic process describing the random partition of a number of ``business opportunities'' among a population of identical firms. A theoretical result is presented for the limiting case in which the number of firms and opportunities go to infinity. Moreover, using simulations, we show that even in a moderately small industry the agreement with asymptotic results is almost complete.-

Selection in asset markets: the good, the bad, and the unknown

In this paper, we use a series of simple examples to illustrate how wealth-driven selection works in a market for Arrow securities. Our analysis delivers both a good and a bad message. The good message is that, when traders invest constant fractions of their wealth in each asset and have equal consumption rates, markets are informationally effcient: the best informed agent is rewarded and asset prices eventually reflect this information. However, and this is the bad message, when asset demands are not constant fractions of wealth but dependent upon prices, markets might behave suboptimally. In this case, asymptotic prices depend on preferences and beliefs of the whole ecology of traders and do not, in general, reflect the best available information. We show that the key difference between the two cases lies in the local, i.e. price dependent, versus global nature of wealth-driven selection.Market Selection; Evolutionary Finance;Informational Efficiency; Asset Pricing; CRRA Preferences

Wage structure in Italian Manufacturing firms

This paper jointly considers some pieces of evidence regarding peculiarities of industries' structure which are often separately addressed. Italian industrial sectors are known to be characterized by a high proportion of small enterprises that suffers from 'constraints to growth'. We look at the interplay of variables accounting for size, productivity and labor cost, and assess the relevance of labor force structure in determining the structure of cost for wage. We start by exploring the firm size-wage effect in Italian manufacturing sectors and investigate the extent to which such a trend is offset by a positive and counterbalancing relation linking together productivity and size. We investigate how size contributes to the wage differential within a firm on the earnings of distinct categories of employees. The empirical findings we present reveal that labor force structure matters in determining the wage cost structure of firms.Cost of labor; Productivity; Wage-size effect

Explaining the Distribution of Firms Growth Rates

Empirical analyses on aggregated datasets have revealed a common exponential behavior in the shape of the probability density of the corporate growth rates. We present clearcut evidence on this topic using disaggregated data. We explain the observed regularities proposing a model in which the firms’ ability of taking up new business opportunities increases with the number of opportunities already exploited. A theoretical result is presented for the limiting case in which the number of firms and opportunities go to infinity. Moreover, using simulations, we show that even in a small industry the agreement with asymptotic results is almost complete.Firm Growth, Laplace Distribution, Urn Processes