5
distribution is apparently relatively stable although so
many elements relevant to it are changing day by day; the
explanation is that the stability lies in the distribution
of wealth, education, training etc which change only
slowly.
In the present paper we shall confine ourselves to the
consideration of wealth and thus consider only the income
of the wealthy. 4
The dependence of income on wealth.
In the following we shall initially consider the income
of the wealthy as flowing from wealth. They have,of
course,not only unearned but also earned income, and the
two are not easy to distinguish even apart from lack of
suitable data. But as a first step we may pretend that all
their income is interest or profit.
Instead of a matrix of income transitions we have now to
consider a matrix wealth-income,which shows for each
amount of wealth the corresponding probability of
different incomes. The basis of the analysis is thus the
conditional distribution of income,given the wealth.
Economically speaking this is the probability of a certain
rate of return to wealth, or profit rate. From this, if we
know it,we can derive the distribution of income,provided
we know the distribution of wealth. But the distribution
of wealth is known; It follows the Pareto law over a
fairly wide range and its pattern can also be explained
theoretically (see the preceding paper in this volume ).
Denoting wealth by W ,let us write for the density of the
wealth distribution
p*(W) = c W a-1 dW
or,putting w = In W
p(w) = c e -aw dw for w > 0
p(w) = 0 for w < 0. (7)
If Y denotes income and y = In Y ,the conditional density
function of income can b^ represented in the form f*(y-
w),the density of a certain return on wealth. Even without
knowing this function we might manage to derive the
distribution of income from that of wealth provided we can
make certain assumptions about independence.We shall
provisionally assume that the distribution of the rate of
return is independent of the amount of wealth. The method
will be to "mix" the conditional distribution of income
given the wealth (the distribution of return) with the
density function of wealth. For the purposes of this
calculation we shall replace the density f*(y-w) by the
mirror function f(w-y) which is also independent of