Is Gibrat’s “Economic Inequality” lognormal?

Abstract

In the seminal book “Les Inégalités Économiques,” Gibrat (Les Inégalités Économiques, Librairie du Recueil Sirey, Paris, 2013) proposed the law of proportional effect and claimed that a variety of empirical size distributions—such as income, wealth, firm size, and city size—obey the lognormal distribution. Gibrat’s law went on to become a stylized result stimulating a voluminous subsequent research that has contributed to our understanding of stochastic growth processes and a statistical regularity of the size distribution. However, many of the motivating examples used by Gibrat in his original work were subject to various data issues, and Gibrat’s reasoning of lognormal fit was based solely on graphical analysis. In this paper, we revisit the original 24 data sets considered by Gibrat (Les Inégalités Économiques, Librairie du Recueil Sirey, Paris, 2013) and show that in the majority of cases, the Pareto-type distribution actually provides a better fit to the data than lognormal. We show that Gibrat’s erroneous conclusion is partly due to data binning, truncation, and failure to weight data points properly.

Appendix A: Summary of Gibrat (1931)

Part I, Chapters 1–3 Gibrat introduces some concepts in probability and statistics.

Part I, Chapter 4 In Sect. 2, Gibrat discusses the binomial distribution using the example of drawing r balls with replacement from an urn containing n balls, of which fraction p and $q=1-p$ are white and black, respectively. In Sect. 3, citing Laplace, he discusses that when n is large but p is fixed, the distribution converges to the normal distribution. In Sect. 5, he mentions that when $n\rightarrow \infty$ but $np=m$, we then obtain the Poisson distribution. He writes:

M. Jordan a donné des développements très intéressants théoriquement à partir de la formule de Poisson, ils ne paraissent pas avoir d’intérêt pratique.

(Mr. Jordan has given very interesting theoretical developments from Poisson’s formula, [but] it does not seem to have practical interest.)

From Sect. 6 on, Gibrat starts to apply the normal distribution to actual data. Interestingly, he does explain why he ignores the Poisson distribution on p. 52:

La formule de Poisson ne paraît avoir d’utilisation que pour les très petit nombres (spécialistes). Nous la laisserons donc de côté. Il ne rest que la formule de Laplace dont l’importance est très grande.

(Poisson’s formula does not seem to have a use except for very small numbers (specialists). Hence we will leave it aside. The only remaining formula that is of great importance is Laplace’s.)

In Sect. 10, Gibrat applies the reasoning of Kapteyn (1903) to explain why the binomial model can be applied to economics (in particular, wage distribution across cities in a certain profession).¹⁴ Assume that initially all cities have the same wage, S. In month 1, it becomes $S'+s$ in half of the cities and $S'-s$ in the other half, where $S'$ is the average wage in month 1. In month 2, wage grows by $a+s$ in half of the cities and $a-s$ in the other half, but independently from the previous month. Hence in month 2, the wage is $S'+a+2s$ in 1/4 of cities, $S'+a$ in 1/2, and $S'+a-2s$ in 1/4.

He then mentions that we get a binomial distribution after n months, and hence “la loi de Laplace” (the law of Laplace, i.e., normal distribution) if n is large, assuming that the wage increase/decrease is independent from the past. At the bottom of p. 59, Gibrat mentions that it is easy to relax the simplifying assumptions: as long as “l’effet des causes” ($=\text {shocks}$) are

• 1. “elles sont nombreuses” (they are in large numbers),

• 2. “leur effet est indépendant de celui des autres” (its effect is independent from those of others), and

• 3. “l’effet de chaque cause est petit vis-à-vis de la somme des effets” (effect of each cause is small compared to the sum of the effects),

we get the normal distribution. In the footnote he cites Lindeberg (1922).

Part I, Chapter 5 Gibrat mentions that the normal distribution is symmetric, while many size distributions in economics are asymmetric. He cites again Kapteyn (1903) to motivate a function of size for which the binomial model applies. He thus arrives at “l’effet proportionnel” (proportional effect): instead of additive shocks to the wage above, what if they are multiplicative? He derives what is now called the lognormal distribution and spends many pages on how to estimate its parameters and then applies it to actual data (including the famous French firm size distribution, data set 1 in Table 1).

Part I, Chapter 6 Gibrat elaborates on “la loi de l’effet proportionnel” (the law of proportional effect). He mentions that a necessary and sufficient conditions for obtaining “formula A” (which is basically the lognormal distribution) are:

• 1. “Les causes de fluctuation du personnel sont nombreuses” (large number of shocks),

• 2. “Leur effet relatif sur le nombre d’ouvriers (ou leur effet absolut sur le logarithme), ne dépend pas de ce nombre d’ouvriers.” (the relative effect on the number of workers (or the absolute effect on the logarithm) does not depend on the number of workers),

• 3. “L’effet de chaque cause de fluctuation est petit vis-à-vis de l’effet de toutes” (shocks are small relative to the total).

Part I, Chapter 7 Gibrat discusses some measures of inequality, including the Gini coefficient.

Parts II–IV The remainder of the book is devoted to the statistical analysis of various data sets (see Table 1), where Gibrat shows the remarkable fit of the lognormal distribution. He repeatedly criticizes Pareto (1897); for example, on p. 111, footnote 2, he mentions that Pareto’s law is merely empirical and does not fit the data except at the upper tail, whereas the lognormal distribution is based on theory and fits the entire distribution.