We can then represent the density of income g (y) by means of
randomisation as follows:
g (y) =
f t*) *
where is
(w-y)e
0
the
dw =
>
transform of
(w)
The above mixture is a Laplace transform of (w), shifted to the
right by y.
The Laplace transform requires that (w) is defined as equal to
zero for w o . If the density function is shifted to the
right, the densties for w v will therefore be zero. We have
thus to assume that w y (in other words, that there are no cases
of wealth smaller than income, which means the rate of return must
be less than 100%).
Equation (2) shows that the Pareto form of the wealth distribution
is reproduced in the income distribution, provided the independence
condition is fulfilled, and y w.
We have now to face the fact that the rate of return on wealth
will in reality not be independent of wealth. The cross-classifi
cations of wealth and income ^wealth ownersffor Holland, Sweden)
show that income is a linear function of wealth, the regression
coefficient being smaller than unity. We can easilv take account
of that by defining a conditional rate of return density or rather
its mirror function as (kw-v), where k is the regression coefficient
of y on w. Assuming that the variance and the higher moments of