6 
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: 
'oo 
q(y)= f(w-y) e -aw dw = c 0(a) e~ a Y for w > y > 0 
J ° 
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 
distribution,provided the independence condition is 
fulfilled and y < w. 
We have now to face the fact that the rate of return on 
wealth is in reality not 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,and the regression coefficient is 
smaller than one. 
Various reasons are responsible for the decline of the 
rate of return with increasing wealth.Most important, the 
income of wealth owners contains more or less considerable 
amounts of earned income which are loosely connected with 
the ownership of wealth but which can hardly be separated 
even conceptually quite apart from the lack of data. The 
earned income will be less important the greater the 
wealth,simply because one can get rarely as much income 
from work as from large wealth.In particular, income from 
non-incorporated business is to a considerable extent 
earned income,and this type of business is less frequently 
present the greater the wealth. A number of other factors 
also contribute to explain the regression coefficient. The 
retained income of corporations will not find expression 
in the income of shareholders but it will, at least in 
many cases, affect the shareholders wealth via the market 
value. Also speculative capital gains from appreciation of 
shares or of"real estate will affect the wealth but not 
the income,and it will presumably be more important in the 
higher wealth classes.