10
Equation (f) 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-classifi
cations of wealth and income of wealth owners (for Holland, Sweden)
show that mean Income is a linear function of wealth, the regression
coefficient being smaller than unity. We can easily take account
of that by defining a conditional rate of return density or rather
its mirror function as j' (kw-y), where k is the regression coefficient
of y on w. Assuming that the variance and the higher moments of
kw-y) are independent of w we can proceed as before:
~oCCJ~
9 (y) -
— po (s. f 1
f (kw-y)e dw « Crr
It may be noted that the condition kw y is more restrictive
than the former condition w^ y.
The result is now that the Pareto shape of the wealth distribution
is reproduced in the income distribution, but with a larger Pareto