Full text: The Personal Distribution of Income

10 
given the wealth (the distribution of return) with the 
density function of wealth. For the purposes of this 
calculation we shall replace the density f*(y-w) by the 
mirror function f(w-y) which is also independent of 
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: 
q(Y) = 
oo 
f(w-y) e-aw dw = c 0(a) e-ay 
for w > y > 0 
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
	        

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