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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