Peru ca. 1800 1.79

Source: Pareto, 1897a, Tome II, p. 312.

if the social constitution were to radically change; were, for example, collectivism to

replace private property” (p. 376). He also admitted that “during the course of the

19th century there are cases when the curve (of income) has slightly changed form,

the type of curve remaining the same, but the constants changing.” [See, e.g.,

Bresciani Turroni (1905) for empirical evidence using German data from the

nineteenth century.]

However, Pareto still maintained that “statistics tells us that the curve varies very

little in time and space: different peoples, and at different times, give very similar

curves. There is therefore a notable stability in the ¬gure of this curve.”

The ¬rst fact discovered by Ammon (1895, 1898) and Pareto at the end of the

nineteenth century was that “the distribution of income is highly skewed.” It was a

somewhat uneasy discovery since several decades earlier the leading statistician

Quetelet and the father of biometrics Galton emphasized that many human

characteristics including mental abilities were normally distributed.

Numerous attempts have been made in the last 100 years to explain this paradox.

Firstly it was soon discovered that the original Pareto function describes only a

portion of the reported income distribution. It was originally recognized by Pareto

but apparently this point was later underemphasized.

9

1.3 HISTORY OF ECONOMIC SIZE DISTRIBUTIONS

Pareto™s work has been developed by a number of Italian economists and

statisticians. Statisticians concentrated on the meaning and signi¬cance of the

parameter a and suggested alternative indices. Most notable is the work of Gini

(1909a,b) who introduced a measure of inequality commonly denoted as d. [See also

Gini™s (1936) Cowles Commission paper: On the Measurement of Concentration

with Special Reference to Income and Wealth.] This quantity describes to which

power one must raise the fraction of total income composed of incomes above a

given level to obtain the fraction of all income earners composed of high-income

earners.

If we let x1 , x2 , . . . , xn indicate incomes of progressively increasing amounts and

r the number of income earners, out of the totality of n income earners, with

incomes of xnÀrþ1 and up, the distribution of incomes satis¬es the following simple

equation:

xnÀrþ1 þ xnÀrþ2 þ Á Á Á þ xn d r

¼: (1:5)

x1 þ x2 þ Á Á Á þ x n n

If the incomes are equally distributed, then d ¼ 1. Also, d varies with changes

in the selected limit (xnÀrþ1 ) chosen and increases as the concentration of

incomes increases. Nevertheless, despite its variation with the selected limit,

in applications to the incomes in many countries, the d index does not vary

substantially.

Analytically, for a Pareto type I distribution (1.2)

a

d¼ , (1:6)

aÀ1

however, repeated testing on empirical income data shows that calculated d often

appreciably differs from the theoretical values derived (for a known a) from this

equation.

As early as 1905 Benini in his paper “I diagramma a scala logarithmica,” and

1906 in his Principii de Statistica Metodologica, noted that many economic

phenomena such as savings accounts and the division of bequests when graphed on a

double logarithmic scale generate a parabolic curve

log Nx ¼ log A À a log x þ b( log x)2 , (1:7)

which provides a good ¬t to the distributions of legacies in Italy (1901 “1902),

France (1902), and England (1901 “ 1902). This equation, however, contains two

constants that may render comparisons between countries somewhat dubious. Benini

thus ¬nally proposes the “quadratic relation”

log Nx ¼ log A þ b( log x)2 : (1:8)

10 INTRODUCTION

Mortara (1917) concurred with Benini™s conclusions that the graph with the

coordinates (log x, log Nx ) is more likely to be an upward convex curve and

suggested an equation of the type

log Nx ¼ a0 þ a1 log x þ a2 ( log x)2 þ a3 ( log x)3 þ Á Á Á Á

In his study of the income distribution in Saxony in 1908, he included the ¬rst four

terms, whereas in a much later publication (1949) he used only the ¬rst three terms

for the distribution of the total revenue in Brazil in the years 1945“ 1946. Bresciani

Turroni (1914) used the same function in his investigation of the distribution of

wealth in Prussia in 1905.

Observing the fragmentary form of the part of the curve representing lower

incomes (which presumably must slope sharply upward), Vinci (1921, pp. 230 “ 231)

suggests that the complete income curve should be a Pearson™s type V distribution

with density

f (x) ¼ CeÀb=x xÀpÀ1 , x . 0, (1:9)

or more generally,

f (x) ¼ CeÀb=(xÀx0 ) (x À x0 )ÀpÀ1 , x . x0 , (1:10)

where b, p . 0, x0 denotes as above the minimum income, and C is the normalizing

constant.

Cantelli (1921, 1929) provided a probabilistic derivation of “Pareto™s second

approximation” (1.3), and similarly D™Addario (1934, 1939) carried out a detailed

investigation of this distribution that (together with the initial ¬rst approximation)

has the following property: The average income f(x) of earners above a certain level

x is an increasing linear function of the variable x. However, this is not a

characterization of the Pareto distribution(s). D™Addario proposed an ingenious

average excess value method that involves indirect determination of the graph of the

function f (x) by means of f(x) utilizing the formula

&° 1 '

af0 (x) f(z)

dz :

f (x) ¼ exp

x À f(x) z À f(z)

x

This approach requires selecting the average f(x) and its parameters based on the

empirical data. The method was later re¬ned by D™Addario (1969) and rechecked

by Guerrieri (1969 “ 1970) for the lognormal and Pearson™s distributions of type III

and V.

For a complete income curve, Amoroso (1924 “1925) provided the density

function

1=s

f (x) ¼ CeÀb(xÀx0 ) (x À x0 )( pÀs)=s , x . x0 , (1:11)

11

1.3 HISTORY OF ECONOMIC SIZE DISTRIBUTIONS

x0 being the minimum income, C, b, p . 0, and s a nonzero constant such that

p þ s . 0 and ¬t it to Prussian data. This distribution is well known in the English

language statistical literature as the generalized gamma distribution introduced by

Stacy in 1962 in the Annals of Mathematical Statistics”which is an indication of

lack of coordination between the European Continental and Anglo-American

statistical literature as late as the sixties of the twentieth century. The cases s ¼ 1 and

s ¼ À1 correspond to Pearson™s type III and type V distributions, respectively.

Rhodes (1944), in a neglected work, succeeded in showing that the Pareto

distribution can be derived from comparatively simple hypotheses. These involve

constancy of the coef¬cient of variation and constancy of the type of distribution of

income of those in the same “talent” group, and require that, on average, the

consequent income increases with the possession of more talents.

D™Addario”like many other investigators of income distributions”was

concerned with the multitude of disconnected forms proposed by various

researchers. He attempted to obtain a general, relatively simply structured formula

that would incorporate numerous special forms. In his seminal contribution La

Trasformate Euleriane, he showed how transforming variables in several expressions

for the density of the income distribution lead to the general equation

1 Àw(x)