Q 
Denoting wealth by w, let us write for the density of the 
wealth distribution 
- <* ^ 
p(w) 
cw 
dw 
or 
p*(¥) * ce“ U dW 
P* 00 
0 
W * In w 
for W> 0 
for W < 0 
(D 
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: 
J/V^) 9 ft) - = dY / f (Y,V) e-“"d¥ (2) 
The minimum x^ealth. (above which the distribution conforms 
% to the Pareto lax*) is taken here as a unit, so that x^e can 
integrate from 0 to QO . The income density is thus the 
Laplace transform of the conditional density of income. 
Y - ¥ is the rate of return on the wealth. 
k 
If this rate of return is independent of the wealth then i^ 
the above relation (2) becomes a convolution. 
iCh>* 
tfiM 
For convenience we shall use instead of f (Y - W) the t<9^ 
symmetric density function f* (W - Y) which is of the same 
rU "