6
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 Tf , let us write for the density
of the wealth distribution
p* = c TT^~ 11 dW
or putting w * ln W
p(w) = c e ' du?
for
*
IV
o
p(w) = 0
for
w < 0
(4)
If Y denotes income and y= ln I the conditional
density function of income can be represented in 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.
T ,7e shall provisionally assume that the distribution of
the rate -of return is independent of the amount of wealth.
In terms of random variables, if (yf , h-C and J\j
</ J
denote income, wealth and the rate of return, we have
U - '4: ;/ “
/
u -
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 ve 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 :
TTnile 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: