7
Now the rate of return will be independent of wealth if
its conditional distribution is the same whatever the size
of wealth.lt would seem that we might perhaps restore the
condition of independence simply by turning the system of
coordinates in the appropriate way,so that we could reduce
the present case to the former one.If we can 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:
Cov (w,w-y) = Var (w) - Cov (w,y) = 0;
Cov__(w i _y)__ =
Var (w)
If the regression line of income on wealth is
y=ftw + y Q ( ft < 1 )
and if also 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 ( k> w +
y Q - y ) because this distribution will be independent of
wealth.
Although we do not really know anything about these higher
moments we shall nevertheless try to use the above
function and proceed in the same way as before by a
mixture of the conditional distribution:
pOO
q(y) = I f(fcw + y Q - y) e“ aw dw =
Jo*
= 9 l/ft
</>(“/
) exp{-a/ (y-y n )}
for Ittw > y-y
q(y) = o for kw < y-y Q . (9)
The result is now that the Pareto coefficient of the
wealth distribution is reproduced in the income
distribution, but with a larger Pareto coefficient (since
K < 1 ). This is exaxtly what has to be explained (income
distributions are in fact more "equal" than the wealth
distributions,empirically,in so far as they show a larger
Pareto coefficient).The particular shape of the rate of
return distribution has no influence on the tail of the
income distribution as long as it fulfills the
independence conditions mentioned.
Concerning the restriction w > y-y 0 it should be
remarked that we are free to shift the coordinate system
to any y 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©*