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