The Personal Distribution of Income
Josef
Steindl
AL00656538
Josef Steindl
August-September 1972
The Personal Distribution of Income
When D.G. Champernowne showed how you can explain the
Pareto law hy a stochastic approach he very naturally
chose as an example the distribution of income, because
that is the classical case. It appears now that the approach
is more easily applied to firms, towns or wealth.-)
The case of income is the hardest, so that the great
•f
r
pioneering paper £*& while fully demonstrating a powerful
new method, has not entirely disposed of the individual
problem which it was dooignod - to -o-olve♦
Champernowne's Model
I shall give a simplified version of Champernowne's model
7
r
which will throw a new light on its relation to other
models of the Pareto law.
The income of a person is the state of the system, and
its evolution is described by a Harkov chain. The stochastic
matrix of income transitions from one year to the next, in
desperate simplification, looks as follows;
0 1 2 5 4 5 6 .
0 v q p
1 q P
2 q P
5 q P
q p
'5 q p
• • • • ♦ •
U
CO
CD
R
•H
<D
9
O
O
fi
H
2
There are only two alternative transition for a person in
this system: Either a rise of income from one year to the
next (note that the income classes are on the log scale)
which has probability p; or death of the person, i.e.
transition to the zero class, which has probability q
(p + q = 1). In addition, there are entries from the zero
to
class/lreplenish the stock of income receiver^
1 ^
The essence of this model is described by Feller ' in the
following terms; The state Ek represents the age of the
system. When the system reaches age K, it either continues
to age or it rejuvenates and starts afresh from age zero.’
The successive passages through the zero state represent a
recurrent event. The probability that the recurrence time
equals K. is p q.
We are interested in the question; How many years have
passed, i.e. how many income steps have been mounted, since
the last rejuvenation? This is the "spent waiting time"
of the renewal process. Choosing an arbitrary starting point
we can say that in the year n the system will be in state En-
if and only if the last rejuvenation occured in year n-K.
Letting n-K increase we obtain in the limit the steady state
probability of the 'fepent waiting time"2\ It is proportionate
Vol. I. XV,3, p. 382.
^ Chapter V.
3
to the tail of the recurrence time distribution, i.e. to p\
More directly, the vector of steady state probabilities
can be derived from the following two conditions:
.,t ■
ufC = p UK_1
Uo = qUo + qu1 + qu2 + •**
The first condition ensures the invariance of the steady
state; the seco.nd ensures that entries to and exits from
the income population balance.
It follows that
% - p\ ci )
uQ = 1- p p< 1
The result,is, of course, identical with the distribution
of the spent waiting time,derived above.
-Sa-
lt will be noted that the process can be described up to
this point without any reference to income. We may regard the
states of the system represented by the elements of the matrix
as "grqdes" or "age" - possibly carreer age or age in earning
life, although in conformity with Champernowne we referred to
them as income classes.
We have now to define the income In relation to the class
intervals of the matrix; The lower limit of class 1 is
taken to be the minimum income. We may choose the income
units such that the minimum income is unity, i.e. on the
logarithmic scale it is zero. The income y^ at the lower
limit of successive income classes K is defined by
Jk -
e*h
or In yK = Kh
c*-)
where h is the size of the class interval on the log scale
D
TT
The dilfLculties arising from the discrete representation of
a continuous income variable in the matrix do not concern
us here. See £,oJ p. 62. 7Z*, /
'/
'
^ /
b * C i- a7 *1 c- by
■'%
To find the tail of the steady state distribution P(y^.) we
sum (2) from X to oo and obtain pK, Thus
In P(yK) = 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
<2ji-
which is ^directly descended from Yule ^r’35*'_7, who used it
2.
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
equation).
Footnotes to p.4
1)
Although Champernowne*s theory is more complicated than the
simple model which takes its place in the above reasoning,
we can easily extend the conclusions: With Champernowne, the
promotion is stochastic, with probabilities'of non-promotion
and demotion. In this more general case p in the above solution
has to be replaced by b which is the root of the characteristic
equation of the matrix*?^ Oc -h & -
, The Pareto coefficient in the simple case is - ^ £ , the ratio
of the parameters of the two exponential distributions; in the
more general case of Champernowne the Pareto coefficient is
X n Id
~ —— . b could be regarded as the parameter of an age dis-
r
tribution, if the classes (states of the system) are regarded
as age classes j l /u feu T<r ^ ic-cct ^ r u
(,J cC o/s •yir'S’i fe
2) Champernowne apparently did not know Yule's paper: It was
H.Simon's merit to have brought it to the attention of
economists; unfortunately he reproduced it in a form which
obscured its essence, which is the interplay of the two ex-
ponential distributions, i.e. of two stochastic processes.
5
this approach, size distribution is a transformed age
distribution, and the pattern of the Pareto law appears
so often simply because of the empirical importance of
exponential growth which makes both the age distribution
and the tranformation function exponential. Owing to the
conceptual density of Champernowne's model the two
elements of life- cycle and promotion are merged into
one.
(which resides
There is, however, a difference^in the interpretation not
in the form) between Champernowne's model and the others;
Since physical persons sooner or later die, the age in
his model is limited, while in the others (relating to
firms or wealth) there is always the probability of
virtually infinite life which accounts for a very
peculiar character of the steady states concerned £ iJ •
Further developments
¥e may consider the following stages in the treatment of
the income distribution:
I. Champernowne's model. Q
II. Rutherford's model £ 1$ J• He treated persons'life-
times explicitly.
Ill* The above models are open to criticism on two grounds;
First, income is not very suitable as a state variable
for a Markov process. It does not embody the "influence
6
globale" (Polya) of the past, Second, the model is con-
fined to a life-cycle from /the*entrance to exit. But the
relevant stochastic process goes far beyond that. In
fact, when somebody starts in life, his chances of re-
ceiving certain incomes are already settled to a large
extent by the condition of his parents: By their wealth,
status, connections, reputation and the education or training
they have been able to give him. In other words, the exits
and the entries in the life-time model are linked by in-
heritance etc., and the process of diffusion continues on
a higher stage. J
to (JMA »0H
Both arguments point/\the same conseqtre5Sdet ¥e must relate
the chances of getting certain.incomes to the amount of
wealth, material and immaterial, and its distribution.
In this way we can link the income to a suitable state
variable (wealth) which is evolving in a long run process
through the generations.
Also, we shall then be able to ana?er the question why
income distribution is relatively stable, although so many
elements relevant to it are changing day by day; The answer
is that the stability lies in the distribution of wealth,
education, training etc., which change only slowly.^
L 12 J p. 142
seq,
l) Champernowne was, of course, aware of these facts, as his thesis
of 1337 C Q ) Show#. His formalised mod®! of 1933 i@, howsVar,
not well suited to reflect all the economic factors so well
stated in general terms in 1937-
IV. In a further stage we should cease to take the wealth
distribution as given, and instead treat wealth and
income as Joint variables in a process evolving over
the generations. Propensity to save and rate of return
would be the double link between the two random
variables.
We shall not further refer to this last stage in the
following paper, but shall try to fill some of the
empty space of stage III.
Property income
We shall distinguish property income and earned income
and deal with the case of property income first, because
it is simpler than the general case.
Instead of the matrix of income transitions used by
Champernowne we have to imagine an analogous matrix
Wealth-Income which shows for each amount of wealth
the probability of different incomes.
The basis of the analysis is thus the conditional dis-
tribution of income, given the wealth. Economically
speaking this is the probability of a certain rate of
return to wealth or profit rate. Prom this we can derive
the distribution of income, provided we know the distribu-
tion of wealth. But the distribution of wealth is known:
It follows the Pareto law (over a fairly wide range) and
ftfl pattsex*n hea also Poen oacplaiaea 'jf •
1 L
8
Denoting wealth by W, let us write for the density of the
wealth distribution
p(W) = cW
- c<-1
dW
for w 0
(41
pX(w) = 0
for w < 0
w = In W
If we know something (though not everything) about the joint
distribution of income and wealth, we might use this in order
to derive from the wealth distribution the income distribution.
Under certain restrictions this is indeed possible. We shall
use the conditional density function of income, given the wealth,
and shall mix (randomize) this with the wealth density. The
conditional density function of income can be represented
/ ¥
in the form T (y-w), the density of a certain "rate of return"
on wealth. We assume tentatively that this rate of return, for
given wealth is stochastically independent of the wealth.
This assumption is necessary because we are going to represent
the income density as a convolution of the rate of return and
the wealth densities; The random variable income y is represented
as the sum of the rate of return and the wealth
9
For the purposes of the following calculation, it is necessary
to use the mirror function of f (y-w), that is f (w-y), which
will be as much independent of wealth as the former.
In terms of random variables we have then
- y/zA
We can then represent the density of income g (y) by means of
randomisation as follows:
__ /s/ />
jVt, or > y 3" 0
!<n., <ur < -p
g (y) = //(w-y) e dw = C [*) € ^
g (y) = o
where ^«)is the Laplace transform of J (w)
(V
The above mixture is a Laplace transform of
right by y.
shifted to the
The Laplace transform requires that r(w) is defined as equal to
zero for w <C o . If the density function fis shifted to the
i
right, the dens'ties for w y will therefore be zero. We have
thus to assume that w ^ y (in other words, that there are no cases
of wealth smaller than income, which means the rate of return must
be less than 100%).
10
Equation (£) shows that the Pareto form of the wealth distribution
is reproduced in the income distribution/ provided the independence
condition is fulfilled, and y w.
We have now to face the fact that the rate of return on wealth
will in reality not be independent of wealth. The cross-classifi-
cations of wealth and income of wealth owners (for Holland, Sweden)
show that mean income is a linear function of wealth, the regression
coefficient being smaller than unity. We can easily take account
of that by defining a conditional rate of return density or rather
its mirror function as f{kw-y), where k is the regression coefficient
of y on w. Assuming that the variance and the higher moments of
^(kw-y) are independent of w we can proceed as before:
It may be noted that the condition kw ^ y is more restrictive
than the former condition w^>y.
The result is now that the Pareto shape of the wealth distribution
is reproduced in the income distribution, but with a larger Pareto
coefficient (since k <0 ) • This is exactly what had to be explained
(the income distributions are "more equal" than the wealth distribu-
tions, empirically). The particular shape of the rate of return distrl
bution has no influence on the result, as long as it fulfills the
independence conditions mentioned. Unfortunately, as we shall see,
this is not always the case.
The income of property owners
Some empirical data will illustrate the above theory. While this
theory deals with property income, the data below rather refer to
income of property owners, which in part is earned income. It is not
easy to separate the earned and unearned income!). Nor are the two
parts independent, so that a convolution of two separately derived
distributions would not be appropriate.
In the following, incomes of property owners will be
treated as a whole. The distribution of the rate of return
or conditional distribution of income therefore includes
earned income here. The regression of property owners' total
income on their wealth can be studied on the basis of Dutch
data,^ (fig.3). The regression is linear and the variance of
income is not much different in different wealth classes;
the correlation coefficient is 0,5, the regression coefficient
1) Attempts to separate them such as in /5/ are unconvincing.
In this study, compensation for risk is included in earned income,
but by its nature it is obviously related to capital.
12
is 0,626 + 0,004 (data for 1962/63). The regression coefficient
corresponds to our k. That k <! 1 can be explained in the first
place by the presumed fact that with increasing wealth earned
income is less and less important; in the second place perhaps
by the fact that income from shares which dominates for the
larger wealth does not contain the undistributed profits.
Since the Pareto coefficient for wealth was 1,38 in 1962/63,
we should expect it to be 2,20 for income on the basis of
the theory. In reality it was 2,08. A better correspondence
is hardly to be expected, since the independence condition holds
only very approximately.
A similar calculation with Swedish data /20/ gives apparently very
bad results, although the regression of income on wealth is linear.
To take an example: For married couples, both taxed, in 1971, the re
gression coefficient of income on wealth is 0,49, the Pareto coeffi-
cient for/income ought thus to be 3,4, but it is in reality 2,5.
The explanation is that the standard deviation of income increases
with wealth (from 0,3 to 0,4 in the highest wealth class). This
produced a thinning out of the tail of the income distribution, thus
leading to a smaller Pareto coefficient than would otherwise obtain.
The effect of increasing standard deviation is actually the same
as that of a steepening of the regression line of income on wealth.
/- C'V / /?s.? W L'- 1 y-<J-
I'i
19
II* Earned Income
In dealing with, unearned income, wealth, had been used
as a state variable. It is natural to extend this idea.
Income depends also on education, training, status,
family connections etc, which are in a sense inherited
like wealth, and result from a sort of "investment" of
income by preceding generations. On an abstract level
we can speak of "income potential" which embodies all
the forms of "investment" mentioned. There are certain
differences between the various forms of income poten-
tial: Education is passed on to the children as they
grow up, while the inheritance of wealth has usually
to wait for the death of the parent.
It is tempting to speculate on an equalization of the rate
of return between the different forms of investment; this
would make it easy to generalize results on the distri-
bution of income. We know, however, that equalization
of return is not likely. And we must beware of drawing
to close a parallel between material wealth and
educational investment. It seems that beyond a certain
range it is less and less easy to increase educational
investment further. Of course, there is no absolute
limit: Indeed there are always possibilities of further
investment (in learning on the Job), and the quality of
N
20
possible education is greatly differentiated, so that
there is more opportunity than might have seemed at
first to increase the investment in education. It is
clear, however, that the frontier is not as open as
in the case of wealth which is impersonal , so that
the individual capitalist is faced with a world of
opportunities for using additional capital: He has
merely to buy twice as many houses or factories while
the educational innstor may have to think hard before
he finds a reasonable way of doubling his investment.
And yet it appears that the tendency for diffusion of
incomes, i.e. of the pushing out of the top scale of
incomes to ever higher levels, is no less prominent
in the case of earned than of unearned income. The up-
shot is that we have to. take a closer look on the
opportunities offered by the market which for the reasons
just mentioned are more important in the case of earned
income than in the case of wealth.
We might also put it as follows; The range of the rate
of return to wealth is limited enough to be restricted
to values below 100 % or lower.In the case of earned in-
come the relation to an educational investment will
quite often include returns of several hundred percent-.-
Or, to put it still another way: It seems that the in-
equalities of earned income could not be adequately
21
explained by educational investment alone. Quite
evidently, there are other forces, too, which lead to
a diffusion of incomes; we have to think only of the
emergence of more and more elevated managers and
supermanagers, of the specialisation of skills and
aptitudes (doctors, scientists, engineers) and of
the rise of some sportsmen as well as actors, singers
and painters to the highest ranges of the income scale.
What are the forces at work here? On the one hand we
have the growing scale and complexity of organizations
in our society - industrial concerns, hospitals and
health services, schools and universities, government
agencies of all kinds - which involve the growth of
managerial hierarchies - pyramids of increasing height.
Next we have the differentiations of skills resulting
from the growing complexity of our techniques. Finally
we have the growth of the mass media (films, gramophone
records, broadcasting and television, magazines, mass-
circulation papers, sports and show business) which has
enormously increased the public which a sucessful star
can reach in one performance. This removes the constraints
against a diffusion of income (in a society lacking the
feudal maecenas) and produces larger and larger top
stars.
J6
22
* r • • •
,1 ' .
These seemingly incongruous phenomena have certain
elements in common. Organisation is a problem of in-
formation. It is necessary to have channels of in-
formation so as to coordinate the activities of the
employees. But since everybody's capacity for com-
munication is limited, it is advisable to limit the
number of channels. As H. Simon //^/7/ has argued, the
hierarchical system, in which every executive is re-
sponsible to one person in the next higher level,
and supervises or controls a certain number of people
(called the "span of control") on the next lower level,
serves to keep the number of channels down to manage-
able levels. In so far as the span of control is inelastic,
this implies, however, that with the growth of an organ-
ization (the growth of its base, that is) the number of
levels will increase; The pyramid will become higher,
and higher grades of managers will emerge.
The hierarchy of managers has a parallel in certain
hierarchies of skills. There is a hierarchy of teachers.
Those in the university teach the teachers of medium
schools and those again teachers in elementary schools.
A similar teacher-pupil relation, of a more durable
character, however, exists between the foreman and his
workers, between an engineer, and his technicians,
between the chief surgeon and his assistants in a hospi-
tal department, between a flight captain and his crew.
I?
23
Thus there are grades linked by a direct teacher - pupil
relation, and the teacher-pupil ratio corresponds to the
span of control in management. Indeed, the "teaching" in
a rather general sense plays a large role in production
and all economic activities; to a large extent the
function of the skilled person is giving (specialized)
information and advice, or being ready to give information
and advice, serving as a repository or "memory" for
certain special knowledge. This will naturally establish
hierarchies of skill, with different grades.
The diffusion of new knowledge from the researcher to
the routine production will tend to proceed over a
series of gradations of "skill". This scale, however,
interlocks in a curious way with others: The diffusion
of technical knowledge will have to go un the managerial
'
scale at some points!
There are also skills, however, which do not need diffusion
to be applied: The highly specialized surgeon applies his
knowledge directly to the consumer, his patient.
This brings us to another group of income earners: the
stars. They are in direct contact (in a sense) with their
audience, and owing to the mass media, their "range" is,
or can be, very wide. A hierarchy of actors etc. has
of course always existed owing to their different talent
or success with, the public, but the mass-media have
pushed out the limits of growth of this hierarchy. Thus
there is a certain analogy with the super-managers;
Just as they have increased their range of indirect
control of employees at the lowest grade, so the big
stars have increased their range of communication (i.e,
their audience), only in their case this communication
is direct. The super managers have arisen from the growth
of organization in the face of rigid span of control,
while the superstars, on the contrary, have arisen from
an enormous extension of the size of audiences. Accord-
ingly the "grade" in the case of the stars must be
measured by the size of audience, For the managers, on
the other hand, te grades are a sequence of random
numbers which represent the successive spans of control*
similarly for the teachers, technicians etc.
How can all these consideration help us in the explana-
tion of incomes? We see a hierarchy (or hierarchies) of
grades established in connection with the growth of
information systems. The grade can serve as state varia-
ble in a stochastic process resulting in a kind of
steady state hierarchy. From the grades we can derive
the incomes provided we can plausibly assume a stochastic
relation between the two, A complication still to be
mastered is the fact that the grades change with the age
of the person concerned , the time curve during the life
being called the carrfier. The carrier will, of course,
depend to some extent on the education, so that we have
a link with the element from which we started.
Can we assume that the hierarchy of grades evolves on
known patterns leading to skew distributions?
Such a pattern would result from a general consideration
of the growing complication of society. As a result of
scientific and technological developments the amount of
information which must be held in store ready for use
increases steadily. This leads to specialization; Here
or there a specialist splits off from a qualification
because the information cannot be managed any more, it
has to be divided. The specialist usually will represent
a higher grade than the qualification from which^splits
off. If specialists are generated as in a birth process,
each grade bringing forth new specialists one grade
higher in proportion to the parent population of each
grade then we should obtain the logarithmic growth
characteristic of the diffusion processes in economics.
In addition w£ require as a second assumption that/
growing pyramids (or hierarchies) of the type described
exist at/different ages/- in different stages/of develop-
ment -/one beside the/other at the same tiale; and further
that/this age is exponentially distributed.
20
An Attempted Generalisation
In dealing with income of property owners we have chosen as a
state variable wealth which evolves slowly in a stochastic process;
unearned income can be derived from it by means of another random
variable, the rate of return. Can we generalise this two-step expla-
nation to include also earned income?
In a somewhat formal way we could speak of the rank which an
individual occupies in one or several hierarchies. Examples
of such hierarchies are wealth, education, status, grade (level)
of an official or manager, rank of officers, ability, degree of
specialisation, grading by popularity of stars etc. Each of these
would represent a dimension in what might be called hierarchical
space. An individual would occupy a certain point in that space,
corresponding to its rank in the various hierarchies, and it
would have certain probabilities of transition within a certain
time to another point in that space. In other words, an indi-
viduals' hierarchical position in society (a vector) would be the
state variable of a stochastic process.
To each point in the hierarchical space corresponds a certain probabi
lity distribution of income; the basic rule is that the higher
rank means expectation of a higher income.
21
In the course of his life-carreer, the individual moves from one
position to another. It has been repeatedly described how the
hierarchical advance during the life-time leads to skewed income
distribution^/ /, and, in fact, may give rise to a Pareto distri-
bution; this is shown in the example treated at the very beginning
of this paper.
The position reached by an individual influences, however, also
the inital position and the progress of this heirs. The stochastic
process thus continues over the generations. This has been studied
by sociologists under the title of "social mobility" /4/.
In order to lend just a little more concreteness to our theory,
let us consider a special hierarchy, that of the managers. Their
income distribution has been studied by several authors /13, 15,
16 or 3/ and we shall refer to the very simple but illuminating
picture given by H.Simon /3/. He assumes that each manager can control
directly a certain number of subordinates and no more; this number
is called the span of control. If this span of control is the same
on all levels, then the employment at various levels of the hierarchy
from top to bottom will increase in geometric proportion.
He further assumes that each manager gets a salary which is determinec
as a certain proportion (larger than unity) of his subordinate's
22
salary. The salary thus decreases geometrically as we go from
top to bottom.
In terms of algebra:
n: span of control
b: manager's salary in proportion to that of his direct
subordinates
L: level of the hierarchy (counting the base as unity)
N CL): number of managers at level L
'K'
L-!
/-L
C(L): Salary at level L
No: number of managers at the base level (2 J - A
Salary at the base
^ N - - 4^ f** C '■ ^A)
'{r
N - A * C ■ *
The salaries under the assumptions given conform to Pareto's
distribution. Simon bases his model on empirical facts (Roberts'
regression of top manager's salary on the size of firm, regression
coefficient 0,37, which would correspond to the value of
and Davis' Pareto distribution of managers' salaries in General
Motors, Pareto coefficient 3).
The above demonstration is purely deterministic, but if
we regard the span of control as the reciprocal of a probability
23
of advance to the next higher level, we get the special version
of Champernowne's model described at the beginning of this
paper: There are again two geometric distributions, from which
the result is obtained by elimination of L, which stands here
in place of time.
This, in fact, leads to criticism of Simon's explanation. It is,
on the face of it, timeless, it does not show how the pattern
arises from a stochastic process in time.
One does not have to go very far however, in order to see the
dynamic implications of the matter. A certain span of control
implies that the managers of a given level have a limited chance
of advancing to the next level. To the span of control corres-
ponds a certain transition probability. It might be argued that
the transition probabilities only reflect the given structure
of the organisation. This, however, has itself arisen as a result
of an evolution (including trial and error) and it is changing
continously albeit slowly. Thus the chances of advancement in the
^ -th
individual's life carreer determine the structure: If —
n
of the occupants of a certain level expect to move m levels in
a life-time then there must be n times as many occupants on the
lower level than on the higher (compare for these topics Bartholo
new /4/.
To be precise we have also to take account of movements into and
out of management, ftoin other oooupatlsm (£o* example, polities) •
24
The transitim probabilities will also reflect long run develop-
ments: Growth, organisational changes and innovations etc.
After the explicit introduction of time (age, and also "historical"
time) the model could also be made more realistic by making the
span of control as well as the income relation b into random variable.
The pattern of the explanation could then, X think, be extended
from the managers to other groups of income earners.
So far we have only refered to the separate groups (like managers
etc) each of which is represented by a dimension in the hierarchy
space. The relation between these dimensions remains open, and
therefore also the question how these separate distributions
combine into a total income distribution which still shows the
familiar Pareto pattern. Prima facie the relation between the various
hierarchy dimensions is undetermined; our society does not definitly
rank business managers, doctors, officers etc. The only common
denominator is income. There is, however, some sort of vague
hierarchy of the hierarchies themselves, indicated by the mean
income and by the inequality as measured by the Pareto coefficient.
On both counts wealth is at the top of the hierarchies; the stars,
the managers and some professions follow in a rather uncertain order.
By and large, however, you will find the groups with lower mean
income also have higher Pareto coefficient and are larger groups.
1 should insist on the greatly irrational (or "traditional")
25
character of the income relation between these groups and yet think
that there is this vague ordering which would explain how a regular
pattern of the total income distribution comes about at all. In
particular, it seems essential that the tail of the distribution
is mostly dominated by income from wealth, which assures that
the total income distribution conforms to the Pareto pattern.
References
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