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6 wealth. The two functions are symmetric and have the same value.In fact, the only difference is in the dimension:while f* refers to the rate of return per year, f refers to the number of yearly incomes contained in the wealth (the reciprocal value of the return). We calculate then the density of income q(y) by mixing the function f(w-y) with the density of wealth: 'oo q(y)= f(w-y) e -aw dw = c 0(a) e~ a Y for w > y > 0 J ° q(y) = 0 for w < y. (8) where 0(a) is the Laplace transform of f(w). The above mixture is a Laplace transform of f(w) shifted to the right by y. The Laplace transform requires that the argument of the function f be non-negative. We have therefore to assume that w > y (we shall show later how this restriction can be relaxed). Equation (8) shows that the Pareto pattern 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 is in reality not independent of wealth. The cross classifications of wealth and income of wealth owners for Holland and Sweden show that mean income is a linear function of wealth,and the regression coefficient is smaller than one. Various reasons are responsible for the decline of the rate of return with increasing wealth.Most important, the income of wealth owners contains more or less considerable amounts of earned income which are loosely connected with the ownership of wealth but which can hardly be separated even conceptually quite apart from the lack of data. The earned income will be less important the greater the wealth,simply because one can get rarely as much income from work as from large wealth.In particular, income from non-incorporated business is to a considerable extent earned income,and this type of business is less frequently present the greater the wealth. A number of other factors also contribute to explain the regression coefficient. The retained income of corporations will not find expression in the income of shareholders but it will, at least in many cases, affect the shareholders wealth via the market value. Also speculative capital gains from appreciation of shares or of"real estate will affect the wealth but not the income,and it will presumably be more important in the higher wealth classes.