# Full text: The Personal Distribution of Income

```3
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^ ■ L 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 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
• • • • lc
distribution i.e. to p .
More directly the vector of steady state probabilities u^
can be derived from the following two conditions:
u k = P u k-l*
Uq = qu Q + qu x + 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 ku o*
```

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