# Full text: The Personal Distribution of Income

```9
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 be 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
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