Full text: Konvolut Wealth and Income Distribution 2

- 
<b 
1J 
Denoting wealth by \t/, let us write for the density of the 
wealth distribution 
-of -1 
^ ( u ) 
ft*) 
or 
cw 
x 
cJ' 
for Jff 0 
for JH < 0 
If we know something (though not everything) about the joint 
or 
distribution of income and wealth, We might use this in order 
to derive from the wealth distribution the income distribution. 
Under certain restriction^this is indeed possible. We shall 
use the conditional r function of income, given the wealth, 
and shall mix (randomize) this with the wealth density. The 
conditional density function of income can be represented 
a/ 
in 
/ * ou 
(y-w), the density of yt& certain "rate of return" 
on wealth. We assume tentatively that this rate of return, for 
given wealth is stochasticallv/of the wealth. 
This assumntion is necesary because we are going to represent 
the income density as a convolution of the rate of return and 
d&Wjititt (04 
the wealthy The random variable income y is represented j&ti the 
sum of the rate of return and the wealth 
( Qtzp-' 
v -= R—h W 
(R &•y - W) 
For the purposes of the following calculation, it is necessary 
or 
to use the mirror function of f , that is f (w-v) , which 
will be as much independent of wealth as the former. 
In terms of random variables we have then 
)• \ \££J\ A#-' v ?£'V' 
^y~~=- W -- A (where -A ^-W—y)
	        

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