9
S' (JO
*(y) - /f(w-y) e* * w dw
q(y) a 0
f V w > y >
(5)
r w < y *
where
is the
Laplace transform of f(w).
The ab >ve mixture is a Laplace transform of f(w) shifted to the
right by y.
The Laplace transform requires that the argument £ the
function f be non-faegative. We have therefore to assu- e that
w -^Ly ( we shall further below haw this restriction can be relaxed ).
Equation (5) shows that the Pareto form of the ealth
distribution is reproduced in the income distribution, provided
the independence condition is fulfilled and y <£ w.
We have now to face the fact that the rate of r turn n wealth
will in reality not be independent of wealth, ^he cross-class ifications
of wealth and income f wealth owners for Holland and Sweden
ah w that mean income is a linear function f wealth, the regression
c efficient being smaller than unity.
P r the define of the raad >f return with increasing
wealth xxxiixx variola reas ns ar ® r ^f||• 1
The earned income will be/leas imp; riant the greater
the wealth. In particular the income fr a ( non-c rp rate) business
?/tll be higher in relation to wealth in the lower wealth classes.
Further, capital gains are nt counted as income, but they affect
wealth, and they will be nnre important f or 1 ayge wealth, because
the proportion of shares held increases with wealth. The internal
accumulationof firms will not find expressi>n in the income,
but quite probably in the wealth of the share holders. Also
appreciation of real estate may affect the lar e wealth pr p rtionately
more.
Now the rate of return is independent of wealth if
its conditional distribution is the same whatever the size of
wealth. It would seem that we might perhaps restore the
condition of independence sipply by turning the system of coordinates
in the appropriate way, so that we would reduce the resent to the
former case. If we can mak the c variance of w and w-y zero
then the coefficient of regression of y on w should be one,
as in the former case}