Full text: The Personal Distribution of Income

To find the tail of the steady state distribution P(y^.) we 
sum (2) from X to oo and obtain p K , Thus 
In P(y K ) = In p . J In 
and, putting -h In p = oC we have 
in P(y^) = -cc In y& (3) 
Evidently the crucial feature of the model is the geometric 
distribution of the recurrence time. This relates here to 
the life-time of the persons as income receivers; since 
promotion id automatic, the age of the system is measured 
in income classes K, The age, or spent life-time, is 
geometrically (approximately exponentially) distributed. 
Since the income is also an exponential function of K , 
the Pareto law results from an elimination of time K from 
1 1 
the two exponential functions, ' 
This is exactly the same pattern of explanation as was used 
IS - 1-1,1- 2-y 2 3 
in other fields by Simon £ and myself ,/i-3,-1|>, 14J and 
which is ^directly descended from Yule ^ r ’35*'_7, who used it 
to explain the frequency of species in genera. According to 
11 \ 
'Although Champernowne s" model is more complicated than the 
above, the essential features remain the same (only p in 
the solution is replaced by b, the solution of a difference 

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