Full text: The Personal Distribution of Income

2 
the next higher stage in the next year which has 
probability p; or the death of the person,i.e. transition 
to the zero class which has probability q (p+q =1). 
In addition there are new entries from the zero class to 
replenish the stock of persons. The entries are supposed 
to compensate the exits of persons so that the stock 
remains constant. 
In this form the model is exactly the same as a renewal 
process described by Feller in the following terms (Feller 
1968 Vol I XV. 3 ,p. 382) :The state E k 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 k-1 q. 
We are interested in the question:How many years have 
passed, or rather,, how many steps in the hierarchy have been 
mounted,since the last rejuvenation? This is the "spent 
waiting time" of a renewal process. Choosing an arbitrary 
starting point we can say that in the year n the system 
will be in state E k if and only if the last re-juvenation 
occurred in year n-k. Letting n-k increase we obtain in 
the limit the steady state probability of the "spent 
waiting time" (Feller 1968 Chapter V ). It is 
proportionate to the tail of the recurrence time 
distribution i.e. to p . 
More directly the vector of steady state probabilities u k 
can be derived from the following two conditions: 
u k = p u k-1 . 
u o = <3 u o + <5 u i + qu 2 + (1) 
The first condition ensures the invariance of the steady 
state;the second ensures that entries to and exits from 
the population balance. 
It follows that 
u k = P u o * 
u Q = 1 - p. p < 1. (2) 
The result is,of course, identical with the distribution 
of the spent waiting time derived above. 
So far we have described the process without mentioning 
income,and have identified the states with a kind of age 
(seniority). We have now to define the income in relation 
to the class intervals of the matrix. Note that income is 
to be measured logarithmically. The lower limit of class 1 
is to be taken as the minimum income. We may choose the 
income units in such a way as to make the minimum income 
equal to unity, i.e.on the logarithmic scale it will be 
zero. The income y k at the lower limit of successive 
income classes k is defined by
	        

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