Full text: The Personal Distribution of Income

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
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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