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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©*