The Personal Distribution of Income

Steindl, Josef

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Document type:: Works

Collection:: Josef Steindl Collection

Title:: The Personal Distribution of Income

Author:: Steindl, Josef

Scope:: Typoskript, 24 Seiten, mit handschriftlichen Ergänzungen bei mathematischen Formeln

Year of publication:: 1990

Source material date:: [vermutlich um 1990]

Language:: English

Shelfmark:: S/M.52.11

DOI:: https://doi.org/10.48671/nls.js.AC14446376

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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