Full text: The Personal Distribution of Income

8 
provided we know the di tribution of wealth. But the distribution 
of wealth :s kn wn! It follows the Pareto law - over a fairly 
wide range - and its pattern has also been explained 
theoretically / 13/- 
Denotin- wealth by W , let us write for the density 
of the wealth distributinn 
p* * c W** 1 dW 
or putting w ® In W 
p(w) - 
c e 
•o( W^W 
for 
w 
0 
(4) 
0 
for 
w 
< 0 
If 
Y den tes 
inco; e and 
y* 
In Y 
the conditional 
density functi n f incune can be represented in the f ria 
f*(y- w ) t the density of a certain return on wealth. liven ithout 
knowing this functi n we might manage to derive the distribution 
of income from that of wealth provide! we can make certain 
assuapti ns about independence. 
e sha 1 provisionally assume that the istribution of 
the rate of return is independent of the mount of wealth. 
In terms of random variables, if / CtT and 
denote inco e, wealth and the rate of return, we have 
If the random variables wealth and the rate of return are independent, 
their sum can be represented by a convolution of the corresponding 
density functions, and we 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 mirr r functi n f(w-y) which is ala' 
independent of wealth. The two functi ns are symmetric and have 
the same value ( in fact, t e nly difference is in the dimension ; 
While the farmer refers t- a rate f return er year the recipr cal 
value refers to the number f years inc me c ntained in the wealth ). 
The calculation of the density of income q(y) 
proceeds then by mixing the functi n f(w-y) with the density 
of wealth*
	        

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