Pareto Distribution
Josef
Steindl
AL00661966
Pareto Distribution
Using certain data on personal income V.Pareto (1897 )
plotted income on the abscissa and the number of people
who received more than that on the ordinate of logarithmic
paper and found a roughly linear relation. This
Pareto distribution or Pareto law may be written as
—* / . .
x = a y or log x = a — log y (1)
where oi ( the negative slope of the straight line )
is called the Pareto coefficient. The density of the
distribution is
dx = a c< y dy
irv
The Pareto coefficient is occasionally used as a measure
of inequality: The larger 0( the less unequal is the
distribution. According to Champernowne 0( is useful
as a measure of inequality for the high income range
whereas for medium and low incomes other measures are
preferable (Champernowne 1952 ).
(£ takes only positive values. If o( ^ 2 the distribution
has no variance; if oC ^ 1 it has no mean either.
In practice the Pareto law applies only to the tail of
the empirical distributions i.e. to incomes above a certain
size. Thus the law (1) is valid assymptotically as y —^ oo
The range in which the empirical distributions conform
i
to thejlaw is different in different cases. It seems to
be larger for wealth than for income ( perhaps because
we have only data for large wealth ) and even larger for
tuwns. In the case of firm sizes only the very large firms
are covered by the law.
2
In the case of the distribution of towns by size of
population the rank-size relation has been used ( Zipf 1949 )
which is the same as the Pareto distribution except that
it uses rank as a measure of the tail ( instead of the
number of twfons above a certain size ) so that the higher
the rank (beginning with rank one for the largest town )
the smaller the size of the town. Zipf believed ( incorrectly)
that the coefficient is always about one so that
the product of rank and size is constant. But Pareto,
of course, was even more "out" wwith his belief that
the Pareto coefficient for income cx always equals unity.
In highly industrialised countries to-day it is above 2
and sometimes above 3.
The main interest of the Pareto distribution lies not in
its rather limited use as a measure of inequality but
inthe explanations it has provoked, naturally so since
regular patterns are felt to be a challenge to the mind.
There are two types of approach to the problem, that of
Champernowne, Yule and Simon which explains the characteristic
pattern as the steady state of a stochastic process
which has been evolving in time, so that the pattern reflects
something which has been going on in the past. In contrast
to that Mandelbrot has been looking for a "synchronic"
explanation which does not depend on a process in time.
He is mainly concerned with the reproductive quality of
the Pareto distribution: If a large number of independent
random variables are identically distributed according to
Paretos law then the sum of these random variables will
also be distributed according to this law. ^
3
Thus it could be expected that the income of the various
counties in England would be Pareto distributed because
it results in each case from the addition of individual
incomes which are Pareto distributed.
Champernowne's pioneering work (1953) in essence goes back
to his fellowship dissertation of 1936, published 1973.
He builds on a tradition which explains the normal distribution
as the result of the addition of random unit steps
(left or right ) on the line over a long time ( random walk;
for the terms and concepts relating to random processes
refer to Feller Vol I ).If the random walk takes place on
the logarithmic scale the distribution of the sum of steps
will tend to log normality. This does not give, however,
a stable distribution, because the dispersion will go on
increasing all the time. Champernowne chooses the technique
of the Markov chain: Each yearjs income depends only on the
previous year's income/plus a random increment proportionate
/
to last year's income; the probability of various increments
remains constant from one year to the other. This feature
is called the law of proportionate effect. Thus the required
data will be embodied in a matrix which contains the
probabilities of transition from one income in one year to
another income in the following year. The number of
income receivers remains stable in Champernowne's model
because‘each exit is assumed to be automatically compensated
by a new entry. To guaratee that the system reaches a
steady state it is assumed that on the average the change of
income is downwards; this is necessary to compensate the
tendency of the system to diffusion which is characteristic
of the unrestrained random walk. The assumption reflects
4
the low income of new entrants. In fact the role of new
entry is crucial not only in this model but in other
applications as well ( size of firms, towns, wealth ).
H.Simon (1955) studied the number of times a particular
word (vocable) occurs in a text. The number of vocables
which occur with a given frequency decreases with that
frequency in a Pareto like fashion. Simon’s treatment is
based on the work of Yule ( 1924 ) who dealt with a
biological problem : The frequency of genera with different
number of species which is distributed according to Pareto.
He explained this pattern by means of a pure birth process
deriving from this the Yule distribution with density
f(n) = 0( f""'(1 + o4 ) n-1-°i' as n “.
The model of evolution assumes that mutations occur
randomly with a frequency g per time unit, creating new
genera, and with a frequency s per time unit creating
new species, where g ^ s . Since each species has the
same chance of creating a new species we have here a
proportionate growth, in analogy to the law of proportionate
effect. The steady state is produced by the emergence of
new geqera. The Pareto coefficient equals the ratio of
the frequencies with which the two kind?of mutations appear,
that is g/s. Simon whose merit it is to have drawn
attention to this brilliant work has suggested application
to incomes ( not very convincingly ) and has himself applied
it to firm sizes (1967). A very direct application relates
to the size of “the "number of
Ct ( / lt> ~u ( £f L S 'J *
5
towns grows at the rate of and the number of inhabitants
of the town grows at the rate of ^ then after a
sufficiently long time there will be a steady state
distribution with Pareto coefficient yp •
Mandelbrot (1960,1961) deals with the problem from the
point of view of a mathematician and therefore on a
very general level. He starts from the concept of stable
a
laws (compare Feller Vol II ch.VI ). If thre sum of
independent identically distributed random variables
is distributed in the same way afjits components, except
for a scale factor and possibly of a location factor,
then this distribution is stable. The best known example
is the normal distribution. It has been shown by P.Levy
that there is a class of distributions with infinite
variance which are stable and which converge to the law
of Pareto when the variable in question ( say income )
tends to infinity. The Pareto law in this context is
confined to the range 1 2. Mandelbrot surmises,
owing to the reproductive quality, in the above sense,
of the Pareto law, that its importance empirically must
be very great. He also considers that this must have
implications for some statistical meljtods which depend
on the assumption of normalcy.
As to income Mandelbrot suggests that it can be regarded
as composed of a number of independent elements which
are identically distributed. We can easily imagine
f
decomposition into a few parts such as earned income,
property income and transfer income. Mandelbrot requires^,
however, in order to assure convergence, a large number of
components, and these, as he admits, have hardly any
6
counterparts in reality ( 1961, p.525 ). The explanation
is analogous to the well known explanation of the stature
of adult men as a random variable composed of a great
number of independent small random variables ; this
ex^Lains the normal distribution of height. The precise
identity of these small random variables is, here again,
not specified and rather speculative. This may perhaps
explain why this "synchronous" approach has not, so far,
found much resonance among economists.
The interest of the alternative approach ( Champernowne or
Yule ) of explaining the law as a steady state of
a stochatystic process is that it establishes a relation
between the stratification found in a cross section
and the past history which has produced it,and which is
mapped in the cross section. This is analogous to the
stratifications in geology and the rings in the trunk of
a tree. Irregularities or shifts in the empirical
distributions can according to this view be explained
\
by major disturbances of the process in certain points of
time in the past.
Concretely, the Pareto distribution has been shown^
in the case of a birth and death process model, to depend
on growth; in an economy which has always been stationary
it would not exist ( Steindl 1965 ). The Pareto coefficient
in such models is usually a ratio of growth rates; thus
in the case of firm size it is a ratio of the growth rate
of the number of firms to the growth rate of the firms
themselves ( Steindl 1965). The importance of new entry
as a factor making for less inequality has also been
shown, inter alia in the case of wealth ( Steindl 1972).
7
The stochastic models have often been criticised for
their lack of economic content. Perhaps it has been
overlooked that they only represent the first steps
in a new exceedingly difficult terrain. It may be
thought that the work of Champernowne, Yule, Simon,
Wold-Whittle contains the seed of future studies which
will reveal their full potentiality only when they
are extended to distributions in several dimensions.
Josef Steindl
Bibliography
Champernowne,D.G.(1952) The Graduation of Income Distribution.
Econometrica Vol.20, October
" (1953) A Model of Income Distribution, in
The Distribution of Income between Persons
N-qw Y-o-rk-—Cambridge, Cambridge University Press 1973.
Feller,W. ( 1950,1966 ) An Introduction to Probability Theory
and its Applications, Vol I (repr.1968)Vol II(rep.1971)
New York, John Wiley &Sons.
Ijiri,Y.and Simon H.A. (1964)Business Firm Growth and Size.
American Economic Review 54- PP 77-89
Mandelbrot,B.(I960) The Pareto-Levy Law and the Distribution
of Income. International Economic Review Vol 1 No 2
pp 79-106 May
" (1961) Stable Paretian Random Functions and
the Multiplicative Variation of Income.
Econometrica 29 no 4 October pp517-543
Paretp,V. (1 897) Coi/us d'Economie Politique Lausanne-Paris.
Rouge & Co
Simon,H.A.(1955) On a Class of Skew Distribution Functions
repr.in Models of Man; Social and Rational.New York. John$jli
Steindl,J.(1965) Random Processes and the Growth of Firms.
A Study of the Pareto Law. London^ Griffin
" (1972) The Distribution of Wealth after a Model
of Wold and Whittle, Review of Economic Studies
Vol 39 no 3 July pp263-279.
Wol^c'jH.O.A.and Whittle,P. (1957) A Model Explaining the Pareto
Distribution of Wealth.Econometrica Vol 25 October
9
Yule,G.U.(1924) A Mathematical Theory of Evolution Based
on the Conclusions of Dr.J.C.Willis. Philosophical
Transactions of the Royal Society of London B 213.
Zipf,G.K. (1949) Human Behaviour and the Principle of Least
Effort Reading,Mass. Addison-Wesley.
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