Full text: The Personal Distribution of Income

provided we know the distribution of wealth. But the distribution 
of wealth is known: It follows the Pareto law - over a fairly 
wide range - and its pattern has also been explained 
theoretically / 13/- 
Denoting wealth by W , let us write for the density 
of the wealth distribution 
p* = c W-*” 11 dW 
or putting 
w = 
In W 
p(w) = 
c e 
- c4tt 
for 
w > 
0 
(4) 
ak^yxi 
p(w) = 
0 
for 
W < 
0 
If 
Y denotes income 
and 
y= In 
Y the 
conditional 
dens it;/ 
function 
of income can be 
represented : 
Ln the form 
f*(y-w), the density of a certain return on wealth. Sven without 
knowing this function we might manage to derive the distribution 
of income from that of wealth provided we can make certain 
assumptions about independence. 
We shall provisionally assume that the distribution of 
the rate oof return is independent of the amount of wealth. 
In terms of random variables, if y J Cu and ^ 
denote income, wealth and the rate of return, we have 
If the random variables wealth and the rate of return are independen 
their sum can be represented by a convolution of the corresponding 
density functions, and 7/e shall in this way obtain the 
distribution of income. 
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 sjmmetric and have 
the same value ( in fact, the only difference is in the dimension : 
While the former refers to a rate of return per year, the reciprocal 
value refers to the number of years income contained in the wealth ) 
The calculation of the density of income q(y) 
proceeds then by mixing the function f(w-y) with the density 
of wealth:
	        

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