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 "