8
Denoting wealth by W, let us write for the density of the
wealth distribution
p(W) = cW
- c<-1
dW
for w 0
(41
p X (w) = 0
for w < 0
w = In W
If we know something (though not everything) about the joint
distribution of income and wealth, we might use this in order
to derive from the wealth distribution the income distribution.
Under certain restrictions this is indeed possible. We shall
use the conditional density function of income, given the wealth,
and shall mix (randomize) this with the wealth density. The
conditional density function of income can be represented
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in the form T (y-w), the density of a certain "rate of return"
on wealth. We assume tentatively that this rate of return, for
given wealth is stochastically independent of the wealth.
This assumption is necessary because we are going to represent
the income density as a convolution of the rate of return and
the wealth densities; The random variable income y is represented
as the sum of the rate of return and the wealth