k¼À1

Champernowne further assumed (his “basic assumption”) that transitions are

possible only in the range between Àn and 1. If the process continues for a long

time, the income distribution reaches an equilibrium in which the action of the

transition matrix leaves the distribution unchanged. The equilibrium state is thus

described by

X

1

xjÀk pk , j . 0:

xj ¼ (3:20)

k¼Àn

The solution of this difference equation is obtained upon setting xj ¼ z j , yielding

the characteristic equation

X

1

z1Àk pk À z ¼ 0:

g(z) :¼ (3:21)

k¼Àn

This equation has two positive real roots, one of which is clearly unity; in order to

ensure that the other will be between zero and 1, Champernowne introduced the

stability assumption

X

1

0

kpk . 0:

g (1) ¼ À (3:22)

k¼Àn

Since g(0) ¼ p1 . 0, and from the stability condition g 0 (1) . 0, the other root must

satisfy

0 , b , 1,

67

3.2 HISTORY AND GENESIS

yielding the required equilibrium distribution of the form

xj ¼ b j :

The total number of incomes is therefore 1=(1 À b) and for any other given number

of incomes N the equilibrium distribution becomes

xj ¼ N (1 À b)b j :

The number of incomes greater than or equal to xj :¼ x0 c j is therefore

˜

Nxj ¼ Nb j

˜

or

log Nxj ¼ log N þ j log b:

˜

Setting a ¼ À log b= log c and g ¼ log N þ a log x0, we ¬nally obtain

˜

log Nxj ¼ g À a log xj : (3:23)

˜

˜

This means that the logarithm of the number of incomes exceeding xj is a linear

˜

function of log xj , thus giving the Pareto distribution in its original form.

An essential feature of the model is the stability condition (3.22), which means

that the expectation of possible transitions is always a reduction in income, from

whatever amount income one starts with. Steindl (1965) argued that the economic

justi¬cation of this assumption is implicit in another feature of Champernowne™s

model: He considered a constant number of incomes and accounted for deaths by

assuming that for any income earner who drops out, there is an heir to his or her

income. But this means that on changing from an old to a young income earner,

there will usually be a considerable drop in income, especially in the case of high

incomes. In fact, the proper economic justi¬cation for the stability assumption is that

the growing dispersion of incomes of a given set of people is counteracted by the

limited span of their lives and the predominantly low and relatively uniform income

of new entrants.

Champernowne discussed several generalizations of his basic model. If

transitions are possible in an extended range between Àn and m, m . 1, only a

distribution asymptotic to a Pareto distribution can be derived. This is still possible if

people are allowed to fall into groups (by age or occupation) and movements from

one group to another are allowed. However, a Markov process would not yield a

stationary distribution unless the transition matrix is constant. It is hard to imagine a

society whose institutional framework is so static. Also, a crucial assumption is that

the probabilities of advancing or declining are independent of the size of income.

68 PARETO DISTRIBUTIONS

Mandelbrot (1961) constructed a Markov model that approximates a Pareto

distribution similarly to Champernowne but does not require a law of proportionate

effect (random shocks additive in logarithms). He emphasized weak Pareto laws

whose frequency distributions are asymptotic to the Pareto. Total income is a sum of

many i.i.d. components (e.g., income in different occupations, incomes from

different sources), at least one of which is nonnegligible in size. If the overall income

also follows this probability law, we have “stable laws,” that is, either normal

distributions or a family of Pareto-type laws. To get a normal distribution, one

requires that the largest component is negligible in size. Mandelbrot argued that in

common economic applications the largest component is not negligible; hence, the

sum as well as the limit of properly normalized partial sums can be expected to

follow a nonnormal stable law (with a Pareto exponent a , 2).

Wold and Whittle (1957) offered a further model (in continuous time) that

generates the Pareto distribution, in their case as the distribution of wealth. They

assumed that stocks of wealth grow at a compound interest rate during the lifetime of

a wealth holder and then divide equally among his or her heirs. Death occurs

randomly with the known mortality rate per unit time. Applying the model to wealth

above a certain minimum, Wold and Whittle arrived at the Pareto distribution and

expressed a as a function of (1) the number of heirs and (2) the ratio of the growth

rate of wealth to the mortality rate. [Some 25 years later Walter (1981) derived all

solutions of the Wold “ Whittle differential equation, showing that not all of them are

of the desired Paretian form.]

It is remarkable that the Pareto coef¬cient is here determined as the ratio of

certain growth rates”namely, the ratio of the growth of wealth to the mortality rate

of wealth owners”that apparently represent the dissipative and stabilizing

tendencies in the process. As noted by Steindl (1965, p. 44), this con¬rms the

intuition of Zipf (1949), who considered the Pareto coef¬cient the expression of an

equilibrium between counteracting forces.

3.2.4 Lydall™s Model of Hierarchical Earnings

Lydall (1959) assumed that the people working in an organization or ¬rm are

arranged hierarchically and that their salaries re¬‚ect the organization of the ¬rm.

Suppose that the levels li , i ¼ 1, 2, . . . , k, are numbered from the lowest upward

and let xi be the salary at li and yi the number of employees at that level. Two

assumptions are made:

yi

where n . 1 is constant for every i,

¼ n, (3:24)

yiþ1

and

xiþ1

where p , 1 is constant for all i:

¼ p, (3:25)

nxi