Full text: The Personal Distribution of Income

There are only two alternative transition for a person in 
this system: Either a rise of income from one year to the 
next (note that the income classes are on the log scale) 
which has probability p; or death of the person, i.e. 
transition to the zero class, which has probability q 
(p + q = 1). In addition, there are entries from the zero 
class/lreplenish the stock of income receiver^ 
1 ^ 
The essence of this model is described by Feller ' in the 
following terms; The state Ek represents the age of the 
system. When the system reaches age K, it either continues 
to age or it rejuvenates and starts afresh from age zero.’ 
The successive passages through the zero state represent a 
recurrent event. The probability that the recurrence time 
equals K. is p q. 
We are interested in the question; How many years have 
passed, i.e. how many income steps have been mounted, since 
the last rejuvenation? This is the "spent waiting time" 
of the renewal process. Choosing an arbitrary starting point 
we can say that in the year n the system will be in state En- 
if and only if the last rejuvenation occured in year n-K. 
Letting n-K increase we obtain in the limit the steady state 
probability of the 'fepent waiting time" 2 \ It is proportionate 
Vol. I. XV,3, p. 382. 
^ Chapter V.

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