8
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 has also been explained
theoretically / 13/.
Denoting wealth by W , let us write for the density
of the wealth distributinn
p* = c W"^~ 1 dW
or putting
w = ln I
p(w) = c e
Ms
for
3
iV
O
P (w) = 0
for
w •< 0
(4)
If Y denotes income and y= ln 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.
In terms of random variables, if ^ j Cu and
denote income, wealth and the rate of return, we hafae
If the random variables wealth and the rate of return are independent,
their sum can be represented by a convolution of the corresponding
density functions, and we shall in this way obtain the
distribution of income.
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 wealth. The two functions are symmetric and have
the same value ( in fact, the only difference is in the dimension :
While the former refers to a rate of return per year, the reciprocal
value refers to the mumber of years income contained in the wealth ).
The calculation of the density of income q(y)
proceeds then by mixing the function f(w-y) with the density
of wealth: