9 
for w y~t O 
or y , 
(5) 
where 
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-hegative. We have therefore to assume that 
y ( we shall further below haw this restriction can be relaxed ). 
Equation (5) shows that the Pareto form of the wealth 
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 return on wealth 
will in reality not be independent of wealth. The cross-classifications 
of wealth and income of wealth owners for Holland and Sweden 
show that mean income is a linear function of wealth, the regression 
coefficient being smaller than unity. 
For the decline of the ratd of return with increasing 
wealth xsrtixx variois reasons are^r^g^oHgjble; 
The earned income will be/less important the greater 
the wealth. In particular the income from ( non-corporate) business 
will be higher in relation to wealth in the lower wealth classes. 
Further, capital gains are not counted as income, but they affict 
wealth, and they will be more important for lagge wealth, because 
the proportion of shares held increases with wealth. The internal 
accumulationof firms will not find expression in the income, 
but quite probably in the wealth of the share holders. Also 
appreciation of real estate may affect the large wealth proportionately 
more. 
Now the rate of return is independent of we a lth 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 present to the 
former case. If we man make the covariance of w and w-y zero 
then the coefficient of regression of y on w should be one, 
as in the former case: