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.