The Personal Distribution of Income

Steindl, Josef

Bibliographic data

Description

Document type:: Works

Collection:: Josef Steindl Collection

Title:: The Personal Distribution of Income

Author:: Steindl, Josef

Scope:: Typoskript, 14 Seiten, mit handschriftlichen Anmerkungen

Source material date:: [vermutlich um 1990]

Publication date:: 1990

Language:: English

Related work:: Steindl, Josef: The Personal Distribution of Income. In: Steindl, Josef: Economic Papers 1941-88. London: Macmillan, 1990, S. 356-371

Topic:: Stochastic processes and size distribution

Shelfmark:: S/M.52.10

Access:: Free access

Full text

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.

Description

Works

Image

Image fragment

Citation links

Citation recommendation

Goobi viewer