Denoting wealth by w, let us write for the density of the
wealth distribution
p(w) = cw - ^dw
or p*(W) = ce'^^dW
p*(w) = 0
W = In w
If the distribution of the rate of return is given in the
form of a density function f (Y,W) dY (where Y = In y,
y denoting income) we obtain the income distribution by
randomisation as follows:
q (Y) = c dY f f (Y,V) e"°° W dW (2)
Jo
The minimum wealth (above which the distribution conforms
to the Pareto law) is taken here as a unit, so that we can
integrate from 0 tooo , The income density is thus the
Laplace transform of the conditional density of income.
Y - W is the rate of return on the wealth.
If this rate of return is independent of the wealth then
the above relation (2) becomes a convolution.
For convenience we shall use instead of f(Y - ¥) the
symmetric density function f* (W - Y) which is of the same
magnitude. We obtain, then, as a special case of (2)
for W> 0 (4)
for W < 0