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, 14 Seiten, mit handschriftlichen Anmerkungen

Year of publication:: 1990

Source material date:: [vermutlich um 1990]

Language:: English

Topic:: Stochastic processes and size distribution

Shelfmark:: S/M.52.10

Rights of use:: In Copyright

Access:: Free access

DOI:: https://doi.org/10.48671/nls.js.AC14446375

3 y k = e kh or In y k = kh (3) where h is the size of the class interval on the logarithmic scale. 1 To find the tail of the steady state distribution P(y k ) we sum (2) from k to °o and obtain p .Thus In P(y k l = In p (1/h) In y k and, putting -h 1 ln p = a , we have In P(y k ) = - a In y k . (4) Evidently the crucial feature of the model is the geometric distribution of the recurrence time. This relates here to the rank in an economic hierarchy linked to income.As promotion is assumed to be automatic the "age" k of the system ,or "spent life time" is geometrically distributed.Since the income is also an exponential function of k the Pareto law results from an elimination of k from the two exponential functions. It is natural to object here that Champernowne's model (Champernowne 1953)has been drastically simplified in the above argument. In his model there are more alternatives:People can either rise one step in the ladder or stay in the same state or recess to a lower state,although the possibilities of movement are limited to a certain range.The steady state solution which Champernowne obtains for his model is, however,essentially the same as the simplified case treated above. Champernowne assumes that the probability of transitions from state k-v to state k is independent of k and depends only on v.That means that the transitions depend on the size of the jump but not on where it starts from or where it ends. On this basis the following equation for the steady state is established: X k Pv x k-v < 5 > In terms of generating function the equation becomes S p., z 1-v - z = 0. (6) —n The solution of the equation is X k = b K (b<l). The steady state distribution of the population according to rank is (l-b)b k and the tail of the distribution is b . The steady state solution in Champernowne's case is thus equivalent to the simplified case treated further above, if p is replaced by b.

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