Volltext: Konvolut The Personal Distribution of Income 1

6 
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 Tf , let us write for the density 
of the wealth distribution 
p* = c TT^~ 11 dW 
or putting w * ln W 
p(w) = c e ' du? 
for 
* 
IV 
o 
p(w) = 0 
for 
w < 0 
(4) 
If Y denotes income and y= ln I the conditional 
density function of income can be represented in 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. 
T ,7e shall provisionally assume that the distribution of 
the rate -of return is independent of the amount of wealth. 
In terms of random variables, if (yf , h-C and J\j 
</ J 
denote income, wealth and the rate of return, we have 
U - '4: ;/ “ 
/ 
u - 
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 ve 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 symmetric and have 
the same value ( in fact, the only difference is in the dimension : 
TTnile 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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