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