f (x) ¼ e (1:12)

G( p)

[here G( p) is the gamma function]. Given a density g(z), transforming the variable

x ¼ u(z) and obtaining its inverse z ¼ w(x), we calculate the density of the

transformed variable, f (x), say, by the formula

f (x) ¼ g[w(x)]jw0 (x)j:

Here, if we use D™Addario™s terminology, g(z) is the generating function, z ¼ w(x)

the transforming function, and f (x) the transformed function. If the generating

function is the gamma distribution

1 Àz pÀ1

g(z) ¼ e z , z ! 0,

G( p)

then the Eulerian transform is given by (1.12). This approach was earlier suggested

by Edgeworth (1898), Kapteyn (1903) in his Skew Frequency Curves in Biology and

Statistics, and van Uven (1917) in his Logarithmic Frequency Distributions, but

D™Addario applied it skillfully to income distributions. More details are provided in

Section 2.4.

In 1931 Gibrat, a French engineer and economist, developed a widely used

lognormal model for the size distributions of income and of ¬rms based on

Kapteyn™s (1903) idea of the proportional effect (by adding increments of income

to an initial income distribution in proportion to the level already achieved).

Champernowne (1952, 1953) re¬ned Gibrat™s approach and developed formulas

12 INTRODUCTION

that often ¬t better than Gibrat™s lognormal distribution. However, when applied to

U.S. income data of 1947 that incorporate low-income recipients, his results are not

totally satisfactory. Even his four-parameter model gives unacceptable, gross errors.

Somewhat earlier Kalecki (1945) modi¬ed Gibrat™s approach by assuming that the

increments of the income are proportional to the excess in ability of given members

of the distribution over the lowest (or median) member. (A thoughtful observation by

Tinbergen, made as early as 1956, prompts to distinguish between two underlying

causes for income distribution. One is dealing here simultaneously with the

distribution of abilities to earn income as well as with a distribution of preferences

for income.)

A somewhat neglected (in the English literature) contribution is the so-called

van der Wijk™s law (1939). Here it is assumed that the average income above a limit

P

x, xi .x xi =Nx , is proportional to the selected income level x, leading to the “law”

P

xi .x xi

¼ gx, (1:13)

Nx

where g is a constant of proportionality. For instance, if g ¼ 2, then the average

income of people with at least $20,000 must be in the vicinity of $40,000 and so on.

Bresciani Turroni proposed a similar relationship in 1910, but it was not widely

noticed in the subsequent literature.

Van der Wijk in his rather obscure volume Inkomens- en Vermogensverdeling

(1939) also provided an interpretation of Gibrat™s equation by involving the concept

of psychic income. This was in accordance with the original discovery of the

lognormal distribution inspired by the Weber “ Fechner law in psychology (Fechner,

1860), quite unrelated to income distributions.

Pareto™s contribution stimulated further research in the speci¬cation of new

models to ¬t the whole range of income. One of the earliest may be traced to the

French statistician Lucien March who as early as 1898 proposed using the gamma

distribution and ¬tted it to the distribution of wages in France, Germany, and the

United States. March claimed that the suggestion of employing the gamma

distribution was due to the work of German social anthropologist Otto Ammon

(1842 “ 1916) in his book Die Gesellschaftsordnung und ihre naturlichen ¨

Grundlagen (1896 [second edition]), but we were unable to ¬nd this reference in

any one of the three editions of Ammon™s text. Some 75 years later Salem and Mount

(1974) ¬t the gamma distribution to U.S. income data (presumably unaware of

March™s priority).

Champernowne (1952) speci¬ed versions of the log-logistic distribution with two,

three, and four parameters. Fisk (1961a,b) studied the two-parameter version in detail.

Mandelbrot (1960, p. 79) observed that

over a certain range of values of income, its distribution is not markedly in¬‚uenced either

by the socio-economic structure of the community under study, or by the de¬nition chosen

for “income.” That is, these two elements may at most in¬‚uence the values taken by certain

parameters of an apparently universal distribution law.

13

1.3 HISTORY OF ECONOMIC SIZE DISTRIBUTIONS

and proposed nonnormal stable distributions as appropriate models for the size

distribution of incomes.

Metcalf (1969) used a three-parameter lognormal distribution. Thurow (1970)

and McDonald and Ransom (1979a) dealt with the beta type I distribution.

Dagum in 1977 devised two categories of properties for a p.d.f. to be speci¬ed as

a model of income or wealth distribution: The ¬rst category includes essential

properties, the second category important (but not necessary) properties. The

essential properties are

Model foundations

.

Convergence to the Pareto law

.

Existence of only a small number of ¬nite moments

.

Economic signi¬cance of the parameters

.

Model ¬‚exibility to ¬t both unimodal and zeromodal distributions

.

(It seems to us that property 3 is implied by property 2.) Among the important

properties are

Good ¬t of the whole range of income

.

Good ¬t of distributions with null and negative incomes

.

Good ¬t of the whole income range of distributions starting from an unknown

.

positive origin

Derivation of an explicit mathematical form of the Lorenz curve from the

.

speci¬ed model of income distributions and conversely

Dagum attributed special importance to the concept of income elasticity

x dF(x) d log F(x)

Z(x, F) ¼ ¼

F(x) dx d log x

of a distribution function as a criterion for an income distribution.

He noted that the observed income elasticity of a c.d.f. behaves as a nonlinear and

decreasing function of F. To represent this characteristic of the income elasticity,

Dagum speci¬ed (in the simplest case) the differential equation

Z(x, F) ¼ ap{1 À [F(x)]1=p }, x ! 0,

subject to p . 0 and ap . 0, which leads to the Dagum type I distribution

h x Àa iÀp

x . 0,

F(x) ¼ 1 þ ,

b

where a, b, p . 0.

14 INTRODUCTION

It was noted by Dagum (1980c, 1983) [see also Dagum (1990a, 1996)] that it is

appropriate to classify the income distributions based on three generating systems:

Pearson system

.

D™Addario™s system

.

Generalized logistic (or Burr logistic) system

.

Only Champernowne™s model does not belong to any of the three systems.