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