1
THE PERSONAL DISTRIBUTION OF INCOME
When D.G.Champernowne showed how you can explain by means
of a stochastic process why the pattern of the Pareto
distribution is found with such great regularity in
various fields he very naturally chose as an example the
distribution of income,because that is the classical
case.It seems to me that the approach is more easily
applied to firm,towns,or wealth. The case of income is the
hardest,so that the great pioneering paper (Champernowne
1953) while fully demonstrating a new method has not
entirely disposed of the special problem to which it was
directed.
Champernowne's Model.
I shall give a simplified version of Champernowne's model
which will throw a new light on its relation to other
models of the Pareto law.
With Champernowne the income of a person is the state of
the system,and its evolution is described by a Markov
chain.The stochastic matrix of probabilities of income
transitions from one year to the next,in desperate
simplification, looks as follows:
income in year t+i
income in year t
0 1 2 3 4 5 6
q P
q p
q p
q p
q p
q p
We now re-interpret this matrix so that the states of the
system are not the incomes but the stages in a hierarchy
or the seniority,a kind of'age",which,however,as we shall
see later,is linked to the income.Let us assume only two
alternative possibilities of transition for a person in
this system:Either a rise from one stage in one year to
0
1
2
3
4
5