Full text: The Personal Distribution of Income

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 
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

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