10
Cor (w, w-y ) = Tar (w) - Cov ( w,y) = 0;
Cov U,y ) ± t .
7ar (w)
If the regression line of income on wealth is
7 = AT* + y 0
and^d! if the variance and higher moments of the conditional
income distribution are independent of wealth then we should
use instead of f(w-y) the function f (y£w+ yc - 7 )
and this distribution will be independent of wealth.
We can then proceed as before:
q( 7 ) = [Ktt + 7<r- 7 ) e'** dw =
for ,^w> 7 - 7 C
a(7) = 0
for u< w < 7 -7 C •
The result is now that the Pareto shape of the wealth
distribution is reproduced in. the income distribution, but with a
larger Pareto coefficient ( since k'O )♦ This is exactly what
has to be explained ( income distributions are in fact more
"equal" than the wealth distributions, empirically, in the sense
described ). The particular shape of the rate of return distribution
has no influence onthe tail of the income distribution, as long as
it fulfills the independence conditions mentioned.
Concerning the restriction ;< w__> 7 - j c
it should be remarked that we are free to shift the coordinate system
to any y c we choose so as to make the above condition valid,
with no consequence except that the conclusion about the Pareto tail
will be confined to incomes in excess of y c .
It would seem that in practice, in view of the value of
, j 0 must often be more or less high, so that the Parto pattern
will be confined to a rather narrow range of the income distribution
while in the caseof wealth it usually extends to the whole of the
assessed wealth data. -This, it is true, partly results from the
fact that the wealth data are more truncated than the income data,
in view of the underlying tax laws.