10
given the wealth (the distribution of return) with the
density function of wealth. For the purposes of this
calculation we shall replace the density f*(y-w) by the
mirror function f(w-y) which is also independent of
wealth. The two functions are symmetric and have the same
value.In fact, the only difference is in the
dimension:while f* refers to the rate of return per year,
f refers to the number of yearly incomes contained in the
wealth (the reciprocal value of the return).
We calculate then the density of income q(y) by mixing the
function f(w-y) with the density of wealth:
q(Y) =
oo
f(w-y) e-aw dw = c 0(a) e-ay
for w > y > 0
q(y) = 0 for w < y.(8)
where 0(a) is the Laplace transform of f(w).
The above mixture is a Laplace transform of f(w) shifted
to the right by y.
The Laplace transform requires that the argument of the
function f be non-negative. We have therefore to assume
that w > y (we shall show later how this restriction can
be relaxed).
Equation (8) shows that the Pareto pattern of the wealth
distribution is reproduced in the income