Konvolut Wealth and Income Distribution 1
Josef
Steindl
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coefficient (since k^Cl). 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 distri
bution has no influence on the result, as long as it fulfills the
independence conditions mentioned. Unfortunately, as we shall see7
this—is not always the case.
The income of property owners
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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 incomel). Wor 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 incecoa therefore includes
earned income here. The regression of property owners' total
income on their wealth can be studied on the basis of Dutch
dataj$ (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 kO 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.
19
II, Earned Income
In dealing with unearned income, wealth had been used
as a state variable. It is natural to extend this idea.
Incone 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 pn on 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 les3 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 dob), and the quality of
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 capitals He has
merely to buy twice as many houses or factories while >
the educational iraestor may have to think hard before
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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 over 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 rote
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 incoma could not be adequcMy
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©2t iod by educational investment alone. Quite •
evidently, there are other forces, too, which load to
s diffusion of incomes: we have to think only of the
emergence of more and laore elevated managers and
supermansgers, 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 organisations
in our society  industrial concerns, hospitals and
health services, schools and universities, government
agencies of oil 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, magasines, mass
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against a diffusion of income (in a society lacking the Ah*At fit
feudal maecenas) and produces larger and larger top •
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22
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 / / 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 teacherpupil relation, of a more durable
character, however, exists between the foreman and his
workers, between on engineer, and his technicians,
between the chief surgeon and his assistants in a hospi
tal department, between a flight captain and his crew.
23
Thus there are grades linked by a direct teacher  pupil
relation, and the teacherpupil 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 (specialised)
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, wita 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,
inter: oc in a curious way with others: The diffusion
of technical knowledge will have to go vn the managerial
scale at some points!
There are also skills, however, which do not need diffusion
to be applied: The highly specialised 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
24
or success with the public, but the aassnedia have
pushed out the limits of growth of this hierarchy. Thus
there is a certain analogy with the supermanagers:
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 organisation in the face of rigid span of control,
while the superstars, on the contrary, have arisen from
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an enormous extension of the size of audiences. Accord
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ingly the "grade" in the case of the stars must be
measured by the size of audience. For the managers, on
the other hand, tie 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
25
of the person concerned, the tine curve during the life
"being called the carrier. 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 he held in store ready for use
increases steadily. This leads to specialisation: Here
or there a specialist splits off from a qualification
because the information cannot he managed any more, it
has to be divided. The specialist usually will represent
a higher grade than the qualification from which/,split*
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 we require as a second assumption that
growing pyrasUads (or hierarchies) of the type described
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exist at different ages  in different stages of develop
ment  one beside the other at the same time; and further,
that this age is exponentially distributed.
The difficulty which remains is that the grades in
different systems  management, skill hierarchies of
various fields*/star hierarchies  seem to he incommen
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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 twostep expla
nation to include also earned income?
In a somewhat formal way we could 3peak 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 lifecarreer, the individual moves from one
position to another. It has been repeatedly described how the
hierarchical advance during the lifetime 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 mobility1* /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 determined
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(L): number of managers at level L
N(L ) ~ No 'Vo
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C(L): Salary at level L
No: number of managers at the base level
A: Salary at the base
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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 Chainpernowne' 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
.jth
individual's life carreer determine the structures If —
n
of the occupants of a certain level expect to move m levels in
a lifetime 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 from other occupatism (for example, politics).
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 variables.
The pattern of the explanation could then, I 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. Prlma 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.
I 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
/I/ Atkinson, A.B. (ed) The Personal Distribution of Incomes,
London 1976
/2/ Atkinson, A.B. Unequal SharesIncomes in Great Britain.
Allan Lane, 1972
/3/ Atkinson, A.B. (ed) Wealth, Income and Inequality. Penguin 1973
/4/ Bartholomew, D.J., Stochastic Models for Social Processes,
New York 1967.
/5/ Central Bureau voor de Statistiek, Statistische en econametri
sche OnderzoekIngen, No 3, 1965s Inkomens ongelij'kheid,
/6/ Central Bureau voor de Statistiek, Inkomensverdeling 1962
en vermogensverdeling 1963
/!/ Champernowne, A.G., M Model of Income Distribution. E.J. 1953
/8/ Champernowne, D.G., The Distribution of Income Between
Persons, Cambridge 1973
/9/ Cox, R.D. Renewal Theory. London 1962
/10/ Cramer, J.S. Empirical Econometrics, Amsterdam 1969
/II/ Davis, H.T. The Theory of Econometrics. Bloomington , 1941 *
/12/ Feller, W., An Introduction to Probability Theory and
its Applications. Vol. I. Third ed.
Vol.II, Second ed.
/13/ Mayer T., The Distribution of Ability and Earnings.
R.Econ. and Statistics 1960.
/14/ Mincer, J., "Investment in Human Capital and Personal
Distribution of Income" J.Pol.Econ.Vol.66, 1958
.
 II 
/15/ Roberts, D.R., Executive Compensation. Free Press Glencoe,
Illinois 1959.
/16/ Simon, H.X., The Compensation of Executives. Sociometry 1957.
/17/ Simon, H.A., The New Science of Management Decision, 1960
/18/ Simon, H.A., "On a Class of Skew Distribution Functions"
Biometrica 1955.
/19/ Rutherford, R.S.G., "Income Distribution: A New Model",
Econometrics 1955.
/20/ Statistika Centralbyran (Stockholm), Inkomst och F8rm6genhet.
/21/ Steindl, J., Random Processes and the Growth of Firms.
London 1965.
/22/ Steindl, J., The Distribution of Wealth after a Model
of Wold and Whittle. R.E.St. July 1972
/23/ Steindl, J., "Size distributions". International Encyclopedia
of the Social Sciences. New York 1967.
/24/ Yule, G.U., "A Mathematical Theory of Evolution Based on
the Conclusions of Dr.J.C.Willis"
Philosophical Transactions of the Royal Society of London
B213 (1924)
/25/ Woodward, J., Management and Technology, London 1958
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contexts as well. Thus the distribution of wealth
results from a convolution of the accumulation of
previous generations with the accumulation of the
living generation /15/* Denote the accumulation of
previous generationshy Wj and the total accumulation
by W , both measured on the log scale. The accumulation
of the living is VW( » r where r is the rate of
accumulation and T the "spent life" of the living wealth
owners, reckoned from the time of their inheritance. If
we may regard ^ (VW, ), the distribution of the spe^fc
life time (telescoped by r) as independent of the inher
ited wealth tf) we can write for the density of the total
wealth distribution q(¥):
where (j) (<A/) is the Laplace transform of
/ (X) the duaay WtfMto replaced by ±
and where W> W, > 0.
The wealth will conform to the Pareto law with coefficient
00, and we do not need to assume scything about the
distribution of the spentlife time, except that it is in
dependent of inherited wealth.
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Q
Denoting wealth by w, let us write for the density of the
wealth distribution
 <* ^
p(w)
cw
dw
or
p*(¥) * ce“ UdW
P* 00
0
W * In w
for W> 0
for W < 0
(D
If the distribution of the rate of return is given in the
form of a density function f (Y,W) dY (where Y * In y,
y denoting income) we obtain the income distribution by
randomisation as follows:
J/V^) 9 ft)  = dY / f (Y,V) e“"d¥ (2)
The minimum x^ealth. (above which the distribution conforms
% to the Pareto lax*) is taken here as a unit, so that x^e can
integrate from 0 to QO . The income density is thus the
Laplace transform of the conditional density of income.
Y  ¥ is the rate of return on the wealth.
k
If this rate of return is independent of the wealth then i^
the above relation (2) becomes a convolution.
iCh>*
tfiM
For convenience we shall use instead of f (Y  W) the t<9^
symmetric density function f* (W  Y) which is of the same
______________ rU "
9
q (T) = cdY f* (¥  Y) e"^WdW =
r
oo
= c (f(oC) e *'Y dY
(3)
(W> Y > 0)
where Cj? (oo) is the Laplace transform of f*(x).
The first result is thus: If the rate of return is inde
pendent of wealth, the Pareto law will be reproduced in
the distribution of income, and the coefficient will be
the same as for wealth. The particular form of the return
rate distribution is of no account, except for a scale
factor.
This result can be somewhat generalized. If the rate of
return is notSiependent of wealth  and indeed it will
hardly be in reality  then there will lit u correlation
between income and wealth. If this dependence of income
on wealth is linear in the logs and if the linear re
gression is also homoscedas.tic we can virtually reestablish
the previous case of equation (3) by stretching or con
tracting the scale of W (see fig. 1). That is to say, we
10
can write for the density of the rate of return f (Y  k¥)
and for its symmetric function f* (kW  Y). In this way we
manage to express the argument of the function f* (which
actually represents the reciprocal profit rote, a—dwaea '
s±onle3«=gg&fe«r^ in terms of W and Y again, and yet keep
it independent of ¥, provided the regression is homo
scedastic. k is a constant which equals fce regression co
efficient of Y on (see fig. 1).
rate of
If thc/ireturn decreases with wealth, we have to take
k<1, if it increases with wealth, we take k^1. In fig. 1
the first case is assumed.
Proceeding os "before, the symmetric function f* (k¥  Y)
will now he randomised by means of the wealth function(
which means taking the Laplace transform of the former:
q (X)
¥<* ,
e d¥
c d V f * (k¥  Y)
e(<*/]$ dY for k¥>Y> 0
(4)
q (X)
0
for k¥<Y> 0
This is now the second result: If there is a loglinear
dependence of income on wealth which is homoscedastic, with
formal grounds we 'might ari'
on coefficient, put since for\ecori
clearly the independent vari
comgtor weaUrhiias'iTO'be used.'V
J
a regression coefficient k, then again the Pareto lav.1 of
the wealth distribution will be reproduced in incone,
but this tine the Pareto coefficient will be modified to
<*/k.
If k is below unity  which we may anticipate, is in
reality the likely case  then the Pareto coefficient for
income will be larger than for wealth.
It is time now to turn to the restrictive assumptions which]
so far have been stated only in algebraic terms:
\Oy in the case of independence, kV^ Y in the case of
linear dependence.
,C,
This means that the rate of return must not be 100 j# or
t
larger in the first case; in the second case, if k<1,
■nijn [. i'j. ..7.Mi43A%Tftiri ;
wealth is defined as equal to zero for ¥<0; in con
sequence, the left tail of the function f* (W) (correspond
ing to negative values of W, thus to rates of return of
\
100 % and more) must also be defined as equal to zero (see
fig. 2). The function f* (W) relates to the case where
Hvs t'ji) *
1
12
the condition is even more restrictive: The rate of return
If «• i
must not be larger than w .loo p.c. whithin the range of
wealth sizes in which the empirical data lie.
The restriction is unavoidable because the Laplace transform
in defined only for positive values of the argument of f•
For negative values the devinity f is by definition zero.
If the argument is shifted to the right by Y the transform will
be defined only for densities of a rate of return below loo p.c.
Similarly, for an argument of kWY the transform will be defined
1c 1
only for rates of return below w .loo p.c.
In reality, rates of return in excess of the limit given may
exist. In this case we can, however, always ensure that the
above condition is fulfilled and the transformation (4) remains
valid provided we make the unit of welath, WQ (Wo = 0) sufficient
ly large.
Indeed the condition
i.<wP“1
w
,<wk (5)
m
will be more easily fulfilled if w and y are both measured in
a large unit, because th£n their values will be both lower in
the same proportion, and that will automatically make it easier
to fulfill the condition (5), if k<l.
The choice of a large unit, however, will mean that the conclusions
with regard to the distribution of income which are implied in (4)
 13
will be restricted to values of income not smaller than Wo
the unit of income and wealth. In other words, the income
/K>
distribution will conform to the Pareto only for
values of income above Wo; this may serve to explain why the
income distributions are usually much less Paretoconform than
the wealth distributions, starting to become straight lines only
near the top of the income distribution while the wealth dis
tribution is straight for almost the whole range of taxable
wealth
12
income is unity (thus Y = 0), and the reciprocal rate of
return is therefore given by ¥.
jilt should be noted that
ur
a doublesided Laplace transform/sould not help us, because
it could never converge at the empirical values of ©c /.
The density of the rate of return will thus be truncated.
In the case of dependence the truncation will ^ ,
if k <1, because the function f* (W) will be stretchy
\
by a factor 1/k.
I N.
There is, however,a deeper reason for the restriction:
The validity of the Pareto law cannot be assumed for low
values of ¥ (in fact,for negative values, if we put the
unit of wealth at a level which limits the range of
\ N\ A
linearity of the distribution)./0wing to the irregular
ity of the wealth distributing for low values of wealth
the possibility of very high ra\es of return might disturb
the regularity of the pattern of Income distribution. ¥e
musb, therefore, set p limit to the permitted rate of
return.
reality the return rates which have been truncated in
*r exercise may, however, exist. The point is, then, that
to the extent that they exist  and that will be the more
likely the smaller k is  the income distribution will be
less regular than the wealth distribution. In practice
;his will mean that the range dominated by the Pareto law
dll be narrower for income than for wealth.
,
In reality, profits in excess of 1oo % may exist. In
the caoe where k <1 we can,however, always ensure
that the conditions are net and (4) remains applicable
As before, we rotate the diagonal round a poii
it is parallel to the regression line (see fig.
and along that line the distribution is truncated. If
46*ft
we choose the point appropriately, we can reduce the
number of incomes truncated. In algebraic terms, we
define W* » Vi^W©t and we tfrite (4) in the following
form:
y\/o
W t
V
q (I)  cdX
j* (kWM4/0Y) if*®*' d¥ '
p) exp F f (Y4$(o)J ,
(5)
kV'> &Wfl\
hU/ ^>(y fwWe]
The resulting income distribution differs only by a
constant factor from (4). Since the transformation (5),
however, is only applicable to wealth not smaller than
V0, it can be intuitively seen that the income distrib
ution will start only at the level Y « V0 to cbnfftrm to
the Pareto Law. In other words, the Pareto law will
always be projected from wealth on to income, but\de
peading on the shape of the regression line, the ih^come
distribution will confirm to linearity only from a mbre
14
IcygQrefincome * In the lower part of the
income distribution, the distribution of the profit rate
trill play a role.
It could be justly said, of course, that the regression
itself calls for an explanation, fliis can only be given on
the plane of "stage four" (see above p. ..) by a stochastic
process in several variables.
ffhe income of  
JSS2ES2i*
"V”» C^S f
S"Wl
Some empirical data trill 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
1}
the earned and unearned income. J Nor are the tiro 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 "profit rate",
^Attempts to separate them such as in /1 / are un
convincing. In this study, compensation for risk is
included in earned income but by it3 nature it is
obviously related to capital.
15
or conditional distribution of income, therefore includes
MMft income here, The regression of property owners’ total
income on their wealth can be studied on the basis of
Dutch data '■ (fig. 5). The regression is linear, and homo
scedastic; the correlation coefficient is 0,5* the re
gression coofficient is 0.G2Si0.004 (data for 1962/63).
Tho regression coefficient corresponds to our k, Shot
<1 can bo explained in the first place by the presumed
fact that with increasing wealth earned income is less
and loss important; in the second pi co perhaps by the
fact that income from shares which dominates for the
larger wealth does notjk ant a in 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 2.03. A better correspondence
is hardly to be expected, since the wealth distribution at
that time has been distorted by the stock exchange boom
(see /l /).2)
A similar calculation with Swedish data gives very un
satisfactory results, although the regression line is
linear and homoscedastic. This may be explained by the
guosr: that the classification of income (which stops at
^ D*J
,.n .a.*
at *
1>See / 1/
2)
Since the holding of shares increases strongly with the
^0 ^(T ^ , z't
V\
16
11 w—yyy —
Footnote J from page 15 continued:
wealth the increase in the stock prices has made the
distribution of wealth nore unequal (and at the same
time decreased k which in 1950/59 was still 0,3). For
very large wealth  in excess of a million guilders ~ the
increase in share  holdings does not play such a large
role any more, and the Pareto coefficient in the range of
millionaires has not been 3trongly affected therefore
■/ 2j/\ we may guess that also It in that range 1 Nft higher
than our value of 0.63. Ibis will marginally also affect
the income distribution studied, and it might explain
why the estimated Pareto coefficient is too high.
17
at 100.000), in view of the higher Swedisch incone level,
conceals the relevant part of the Pareto distribution
which presumably would show a much higher Pareto co
efficient than the income below 100.000 crowns does.
For 1953/59 (see / / /) data ar*. liable on the various
types of income ofproperty owners in Holland:1) income
of unincorporated business and professions 2) unearned
income, and 3) other income (which is chiefly Income
from employment). Udie simple regressions of these three
incomes on the total income of proi>erty owners hove been
calculated (fig. 4). Tho regression coefficients ore
respectively b^ ~ 0.30, b2 = 1.34 and b^ = 1.01.
Phis shows that with increasing wealth the share of
income derived from property is increasing.
flic Pareto coefficient for wealth in 1953/59 is 1.57,
for income of property owners it is 2.03. If we estimate
it on the basis of the theory, the coefficient for in
come would be 1.94 (since k in that year is about 0,31).
On the basic of the above information we can derive
theoretical estimates of Pareto coefficients for the above
mentioned three income types separately from the Pareto
coefficient for total income of property owners, which in
turn can be derived from the coefficient for the wealth
distribution. We can write for the density of, for
—
—
1 r ■ ii
example, the unearned income
'OO
« 00  a % I ^ Mj) rnm tj
dJ
* c exn
fc r2 ] d
(6)
i>
According to this transformation, inserting for c?C 1#57
and for k 0.31, we obtain a Pareto coefficient of 1.45
for unearned income (1.55 if we start from the empirical
value 2.00 of the total income distribution)• By analogous
calculation we obtain Pareto coefficients of ^
2.42 (2.GO) for business income and^
1.92 (2.OS) for "other income" l'»
1}
/5?hv Pareto coefficient for "horn incomon", i.e.,
incomes of those who exclusively live on one kind
of income , is in the case of unearned income about
1.4, in the case of business income about 2,4 (see
/ i / P. 167).
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COPYRIGHT (
MADE IN GERMi
Ijffi Nr. 3691/2 ! 5 Beide Achsen logar. geteilt, 1 bis 1000 u. 1 bis 100, Einheit 90 mm
Sa
lt will be noted that the process can be described up to
this point without any reference to income. We may regard the
3tates of the system represented by the elements of the matrix
as "grades" or "age"  possibly carreer age or age in earning
life, although in conformity with Champernowne we referred to
them as income classes.
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 Shampernowne, the
promotion i3 stochastic, with probabilities of nonpromotion
and demotion. In this more general case p in the above solution
has to be replaced by b which is the roo€ of the characteristic
equation of the matrix.
The Pareto coefficient in the simple case is  E. , the ratio
of the parameters of the two exponential distributions; in the
more general case of Champernowne the Pareto coefficient is
« „ u
 — . b could be regarded as the parameter of an age dis
tribution, if the classes (states of the system) are regarded
as age classes.
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.
Footnote 1 to p.6
1)
Champernowne was, of course, aware of these
of 1937 ( ) shows. His formalised model
not well suited to reflect all the economic
stated in general terms in 1937*
facts, as his thesis
of 1953 is, however,
factors so well
12
the condition is even more restrictive: The rate of return
k1
must not be larger than w .loo p.c. whithin the range of
wealth sizes in which the empirical data lie.
The restriction is unavoidable because the Laplace transform
in defined only for positive values of the argument of •
For negative values the devinity f is by definition zero.
If the argument is shifted to the right by Y the transform will
be defined only for d ities of a rate of return below loo p.c.
Similarly, for an argument of kWY the transform will be defined
lr1
only for rates of return below w .loo p.c.
In reality, rates of return in excess of the limit given may
exist. In thi3 case we can, however, always ensure that the
above condition is fulfilled and the transformation (4) remains
valid provided we make the unit of welath, WQ (Wo * 0) sufficient
ly large.
Indeed the condition
x<«p'1
w
y<wk (5>
will be more easily fultilled if w and y are both measured in
a large unit, because than their values will be both lower in
the 3ame proportion, and that will automatically make it easier
to fulfill the condition (5), if k<l.
The choice of a large unit, however, will mean that the conclusions
with regard to the distribution of income which are implied in (4)
 13
will be restricted to values of income not smaller than Wo
the unit of income and wealth. In other words, the income
distribution will conform to the Pareto problem only for
values of income above Wo; this may 3erve to explain why the
income distributions are usually much less Paretoconform than
the wealth distributions, starting to become straight lines only
near the top of the income distribution while the wealth dis
tribution is straight for almost the whole range of taxable
wealth
3>
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