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 distribution
p* = c W-*” 11 dW
or putting
w =
In W
p(w) =
c e
- c4tt
for
w >
0
(4)
ak^yxi
p(w) =
0
for
W <
0
If
Y denotes income
and
y= In
Y the
conditional
dens it;/
function
of income can be
represented :
Ln the form
f*(y-w), the density of a certain return on wealth. Sven 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 oof return is independent of the amount of wealth.
In terms of random variables, if y J Cu and ^
denote income, wealth and the rate of return, we have
If the random variables wealth and the rate of return are independen
their sum can be represented by a convolution of the corresponding
density functions, and 7/e 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 sjmmetric 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 number 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: