Full text: The Personal Distribution of Income

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©*
	        

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