9
f(*-y) e'^* dw - c ß^J e“ for w > jr? C
q(j) - Ö or w< y ,
where V/^' ) 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. Tie have therefore to assume that
tt( 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.
Tie have now to face the fact that the rate of" return on wealth
will in reality not be independent of wealth, -he 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 rata of return with increasing
wealth iiriiix variois reasons are responsible:
relatively
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 affect
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
mors. •
How 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 night perhaps restore the
condition of independence simply by turning the system of coordinates
in the appropriate way, so that we would reduce the present to the
former case. If we nan 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: