# Full text: The Personal Distribution of Income

```9
For the purposes of the following calculation, it is necessary
to use the mirror function of f (y-w), that is f (w-y), which
will be as much independent of wealth as the former.
In terms of random variables we have then
- y/zA
We can then represent the density of income g (y) by means of
randomisation as follows:
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jVt, or > y 3" 0
!<n., <ur < -p
g (y) = //(w-y) e dw = C [*) € ^
g (y) = o
where ^«)is the Laplace transform of J (w)
(V
The above mixture is a Laplace transform of
right by y.
shifted to the
The Laplace transform requires that r (w) is defined as equal to
zero for w <C o . If the density function fis shifted to the
i
right, the dens'ties for w y 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%).
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