Konvolut Wealth and Income Distribution 2
Josef
Steindl
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•» 3 •»
Denoting wealth by W, let us write for the density of the
wealth distribution
p(W) » cw dlf
or p*(w) m ce" ^Wdw for w > 0 f ^
p*(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 (randomise) this with the wealth density. The
conditional density function of income can be represented
in the form j^yw), 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 is represented
Aj L.
as the sum of the rate of return and the wealth </f '
s (JZ +
= //
/
 9 
For the purposes of the following calculation. It Is necessary
to use the mirror function of f (yw), that Is f (wy), which
will be as much Independent of wealth as the former.
In terms of random variables we have then
UA
We can then represent the density of income g (y) by means of
randomisation as follows:
—uy
r ur
g (y) * //(wy) e dw  C
~~q0 i
''V
g (y)  o j'r
where is the Laplace transform of j (w)
td" A y A o
Cr' <1
1
The above mixture is a Laplace transform of /(w), shifted to the
right by y.
The Laplace transform requires that f (w) is defined as equal to
sero for w 4 o . If the density function / is shifted to the
right, the densties 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 (f) 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 crossclassifi
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 j' (kwy), where k is the regression coefficient
of y on w. Assuming that the variance and the higher moments of
kwy) are independent of w we can proceed as before:
~oCCJ~
9 (y) 
— po (s.f 1
f (kwy)e dw « Crr
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
11
coefficient (since k^1). 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 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 income1). 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 incessi 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 examples 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 tall 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.
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 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 probabii
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 lnltal 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,
15 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
•alary. The salary thus decreases geometrically as we go from
top to bottom.
In terras of algebra;
n: span of control
bs manager’s salary in proportion to that of his direct
subordinates
L; level of the hierarchy (counting the base as unity)
/ L
N(L)t number of managers at level L
C(L); Salary at level L
Nos number of managers at the base level
As Salary at the base
£* A)
A H ~ Jh.
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 site 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 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 taka account of movements into and
out of management from other occupatism (for example, politics)•
24 
The transitint 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 deflnitly
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 ShareaIncomea in Great Britain.
Allan Lane, 1972
/3/ Atkinson, A.B. (ed) Wealth, Income and Inequality. Penguin 1973
/A/ Bartholomew, D.J., Stochastic Models for Social Processes,
New York 1967.
/5/ Central Bureau voor do Statistiek, Statistische en econometri
sche Onderzoekingen, No 3, 1965: Inkomens ongelij'kheid,
/6/ Central Bureau voor de Statistiek, Inkomensverdeling 1962
en venaogensverdeling 1963
/7/ Champernowne, II.G., H 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.K., The Compensation of Executive®. Sociometry 1957.
/17/ Simon, H.A., The New Science of Management Decision, 1960
/18/ Simon, H.A., "On a Class of Skew Distribution Function#"
Biometrics 1955.
/19/ Rutherford, R.6.G., '‘Income Distributions A New Model",
Econometric® 1955.
/20/ Statistika Centralbyxan (Stockholm), Inkomst och FdrraSgenhet.
/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
I !
 8 
Denoting wealth by W, let us write for the density of the
wealth distribution
or
p (W)  cW ~o(“1 dW
p*(w)  ce~
p (w)  0
for w > o
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 (randomise) this with the wealth density. The
conditional density function of income can be represented
in the form J (yw), 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 densitiess The random variable income ^ is represented
as the sum of the rate of return and the wealth JY/
, 02 + (CZy M)
 9 
For the purposes of the following calculation, It is necessary
to use the mirror function of f *(yw), that is f (w~y), which
will he as much Independent of wealth as the former.
In terms of random variables we have then
^ \A> •  YY  LJ J
We can then represent the density of income g (y) by means of
randomisation as followst
where C is the Laplace transform of
/
(w)
The above mixture is a Laplace transform of
right by y.
shifted to the
The Laplace transform requires that j (w) is defined as equal to
aero for w o . If the density function f lm shifted to the
right, the denstles for y will therefore be sero. 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 
r
Equation (JO shown 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 crossclassifi
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 e conditional rate of return density or rather
its mirror function as j^kwy), where k is the regression coefficient
of y on w. Assuming that the variance and the higher moments of
(kwy) are independent of w we can proceed as before x
^ OC7
g (y) * t (kwy)e dw 
c iz cf(£ )**}*{
A Cj ^ h ~
O
* c
(6
;
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
)
11 
coefficient (since k 41). 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 Idle
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. Zt is not
easy to separate the earned and unearned income 1). 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
rsj
data'll) (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.
Xn 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 fey 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.
t
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 tall 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.
i&Ui l < (z j I Ku Puytffai C'S^/j^t
 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 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 probabil
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 stoves 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 lnltal 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 determined
as a certain proportion (larger than unity) of his subordinated
22
salary. The salary thus decreases geometrically as we 90 frost
top to bottom.
In terms of algebra;
m span of control
hr manager's salary In proportion to that of his direct
subordinates
L: level of the hierarchy (counting the base as unity)
W(L)x number of managers at level L
C(h)t Salary at level L
Ros number of managers at the base level
A; Salary v“"*
■/£. C ^ A)
H * h **■ 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. Zt is,
on the face of it, timeless, it does not show how the pattern
arises from a stochastic process in tins.
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 oarreer determine the structuret It
it
of the occupants of a certain level expect to move m levels in
s 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 occupation: (for example, politics).
 24 
The tranaitira 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. Prime facie the relation between the various
hierarchy dimensions is undetermined? our society does not deflnitly
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 incase 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 incoma distribution conforms to the Pareto pattern.
References
/I/ Atkinson, A.B. (ad) The Personal Distribution of Incenses,
London 1976
/2/ Atkinson, A.B. Unequal SharesIncomes in Great Britain.
Allan Lane, 1972
/3/ Atkinson, A.B. (ad) Health, Income and Inequality. Penguin 1973
/4/ Bartholomew, D.J., Stochastic Models for Social Processes,
New York 1967.
/5/ Central Bureau voor de Statistiek, Statlstlsche en econometri
sche Onderzoekingen, No 3, 1965: Inkomens ongellj'kheid,
/€/ Central Bureau voor de Statistiek, Xnkomensverdeling 1962
en veratogensverdeling 1963
/7/ Champernowne, B.G., II Model of Income Distribution. 8.J. 1953
/3/ 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, M., An Introduction to Probability Theory end
its Applications. Vol. I. Third ed.
Vol.XX, Second ed.
/I3/ 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.Boon.Vol.66, 1958
/15/ Roberts, D.R., Executive Compensation. Free Press Glencoe,
Illinois 1959.
/16/ Simon, H.K., The Compensation of Executives. Socioraetry 1957.
/17/ Simon, H.A., The New Science of Management Decision, 1960
/18/ Simon, H.A., "On a Class of Skew Distribution Functioni*
Bionetrica 1955.
/19/ Rutherford, R.G.G., ”Income Distributions A New Model",
Econometrics 1955.
/20/ Statistika Central by ran (Stockholm), I n koras t och FOnadgenhet.
/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 1953
Denoting wealth by w, let us write for the density of the
wealth distribution
p(w) = cw^dw
or p*(W) = ce'^^dW
p*(w) = 0
W = In w
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:
q (Y) = c dY f f (Y,V) e"°°WdW (2)
Jo
The minimum wealth (above which the distribution conforms
to the Pareto law) is taken here as a unit, so that we can
integrate from 0 tooo , The income density is thus the
Laplace transform of the conditional density of income.
Y  W is the rate of return on the wealth.
If this rate of return is independent of the wealth then
the above relation (2) becomes a convolution.
For convenience we shall use instead of f(Y  ¥) the
symmetric density function f* (W  Y) which is of the same
magnitude. We obtain, then, as a special case of (2)
for W> 0 (4)
for W < 0
9
q (Y)  cdY
OO
f* (¥  Y) e_<7CWdW *»
= c(f(oC) eoCY dY
(w> HO)
(3)
where Cf> (oo) is the Laplace transform of i*(x).
The first result is thus: If the rate of return is inde
pendent of wealth, the Pareto law will he 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 not ^dependent of wealth  and indeed it will
hardly be in reality  then there will be a 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 homoscedastic 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
/
 k W ^
to p. 9 (footnote)
—
' The relation (3) may have applications in different
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 /13/. Denote the accumulation of
previous generations by W, and the total accumulation
by W , both measured on the log scale. The accumulation
of the living is WW, » riT, where r is the rate of
accumulation and 3* the "spent life" of the living wealth
owners, reckoned from the time of their inheritance. If
we may regard ^ (WW,), the distribution of the spent
life time (telescoped by r) as independent of the inher
ited wealth W . we can write for the density of the total
'  r
wealth distribution q(W):
q(W) = (l W f (WW
The wealth will conform to the Pareto law with coefficient
c>C, and we do not need to assume anything about the
distribution of the spentlife time, except that it is in
dependent of inherited wealth.
i 7\ — c£ >
0 ______/>, o
10
can write for the density of the rate of return f (Y  kW)
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 rate, /4^41mon— A
s^BjBuhejSLQ; number) in terms of W and Y again, and yet keep
it independent of W, provided the regression is homo
scedastic. k is a constant which equals to regression co
efficient of Y on (see fig. 1)
rate of
If theAreturn 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 as before, the symmetric function f* (kW  Y)
will now be randomised by means of the wealth function,
which means taking the Laplace transform of the former:
q (Y) = c oty I f * (kW  Y) e"Wo6 dW «
«  dp (£) e"^^Y dY for kW > Y > 0 (4)
q (Y)  o
for kW<Y> 0
This is now the second result: If there is a loglinear
dependence of income on wealth which is homoscedastic, with
ion
11
a regression coefficient k, then again the Pareto law of
the wealth distribution will be reproduced in income,
but this time the Pareto coefficient will be modified to
iX//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:
W>Y in the case of independence, kW>Y in the case of
linear dependence^
c
This means that the rate of return must not be 100 % or
larger in the first case; in the second case, if k<1,
Thrill,
loo e/o ,
The res
tUcrf
is necessary, becaus^the Pare to. denaity
Ltfyvyru t/Li&M&jfiyUn, zx./uvu
ealth is defined as equal zero for^W<0; in con
equence, the left tail of the functiop^f* (W) (correspond
ing to negative values of W, thus to rates of return/of
100 %^ehd more) must also be defined as equal to^ero (see
The fqnOtion f* jxT relates to the case where
A/
0 / y
income is unity/(thus Y = 0), and the reciprocal rate of
’eturn is therefore given by W. Ult shoulcj/be noted that
a doublepdLded Laplace transform £ould i^dt help us, because
it codld never converge at the empirical values of ©c /•
Ttie density of the rate of return^will thus be truncated.
In~the case of dependence the ./truncationwill be worse,
if \k<1, because the function f* (W) will be stretched
by a factor 1/k.
There is, however,a deeper reason for the restriction:
The validity of the Pareto law cannot be assumed for low
\
values of W (in fact,for negative values, if we put the
unit of wealth at a level which limits the range of
linearity of the distribution). Owing to the irregular
ity of the wealth distribution for low values of wealth
the possibility of very high rates of return might disturb
the regularity of the pattern of incojne distribution. We
mudb, therefore, set a limit to the perkltted rate of
return.
In reality the return rates which have been truncated in
our exercise may, however, exist. The point is, th^n, that
to the extent that they exist  and that will be thexmore
likely the smaller k is  the income distribution will \£e
less Regular than the wealth distribution. In practice
this will mean that the range dominated by the Pareto law
will be narrower for income than for wealth.
13
In reality, profits in excess of 1oo % may exist. In
the case where k <1 we can,however, always ensure/
that the conditions are met and (4) remains applicable:
As before, we rotate the diagonal round a point until
/
it is parallel to the regression line (see fig
and along that line the distribution is truncated. If
we choose the point appropriately, we can reduce the
number of incomes truncated. In algebraic terms, we
define => WW0, and we write (4) ±js the following
form:
GO
q 00
cdY
j* (kW' + *W0 Y)/e"^ dW ' =
 I <f C) exp I (Y^Wb)
(5)
kW’ > (TWb)
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
WQ, it can be intuitively seen that the income distrib
ution will start only at the level Y =» WQ to confirm to
the Pareto Law. In other words, the Pareto law will
always be projected from wealth on to income, but de
peB
ding
on the shape of the regression line, the income
distribution will confirm to linearity only from a more
/
12
the condition is even more restrictive: The rate of return
must not be larger than wk '1'.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^w^.
For negative values the derq^iity f is by definition zero.
If the argument is shifted to the right by Y the transform will
be defined only for d&wities of a rate of return below loo p.c.
Similarly, for an argument of kWY the transform will be defined
k "I
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 welfath, WQ (Wo = 0) sufficient
ly large.
Indeed the condition
y <w
w
ki
k
or
y \w
(5)
will be more easily fulfilled if w and y are both measured in
a large unit, because then their values will be both lower in
the same proportion, and that will automatically make it easier
to fulfill the condition (5), if k <1.
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 confirm to the Pareto problem 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.
14
&rdrgsshi^j^d^v^lQf 4noome. In the lower part of the
income distribution, the distribution of the profit rate
will play a role.
It could be justly said, of course, that the regression
itself calls for an explanation. This can only be given on
the plane of "stage four" (see above p. ..) by a stochastic
process in several variables.
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
1}
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 "profit rate",
11
'Attempts to separate them such as in / 1 / are un
convincing. In this study, compensation for risk is
included in earned income but by its nature it is
obviously related to capital.
15
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 homo
scedastic; the correlation coefficient is 0,5, the re
gression coefficient is 0.626^0.004 (data for 1962/63).
The regression coefficient corresponds to our k. That
k O 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 wealth distribution at
that time has been distorted by the stock exchange boom
(see / /).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
guess that the classification of income (which stops at
1^See / 1 /
2)
Since the holding of shares increases strongly with the
16

Footnote J from page 15 continued:
wealth the increase in the stock prices has made the
distribution of wealth more unequal (and at the same
time decreased k which in 1958/59 was still 0,8). 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 strongly affected therefore
/ /; we may guess that also k in that range is much higher
than our value of 0.63. This 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 income 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 1958/59 (see / /) data are available on the various
types of income ofjproperty owners in Holland:1) income
of unincorporated business and professions 2) unearned
income, and 3) other income (which is chiefly income
from employment). The simple regressions of these three
incomes on the total income of property owners have been
calculated (fig. 4). The regression coefficients are
respectively b^ = 0.80, l>2  1.34 and b^ = 1.01.
This shows that with increasing wealth the share of
income derived from property is increasing.
The Pareto coefficient for wealth in 1958/59 is 1.57»
for income of property owners it is 2.08. 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 0,81).
On the basis 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
"18
example, the unearned income Y2:
q (Y2) = d Y2 f f2(k2 YY2) exp [~rY] dY =
Jo u J
 0 e*P £  FT£ Y2 J d Y2 (6)
k2 y>y2
According to this transformation, inserting for 1.57
and for k 0.81, we obtain a Pareto coefficient of 1.45
for unearned income (1.55 if we start from the empirical
value 2.08 of the total income distribution). By analogous
calculation we obtain Pareto coefficients of
2.42 (2.60) for business income and
1.92 (2.06) for "other income"^\ ^
1)
'The Pareto coefficient for "kern incomen", 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
/ / p. 167).
26
The difficulty which remains is that the grades in
different systems  management, skill hierarchies of
various fields, star hierarchies  seem to be incommen
surable.
u
<b
1J
Denoting wealth by \t/, let us write for the density of the
wealth distribution
of 1
^ (u)
ft*)
or
cw
x
cJ'
for Jff 0
for JH < 0
If we know something (though not everything) about the joint
or
distribution of income and wealth, We might use this in order
to derive from the wealth distribution the income distribution.
Under certain restriction^this is indeed possible. We shall
use the conditional r function of income, given the wealth,
and shall mix (randomize) this with the wealth density. The
conditional density function of income can be represented
a/
in
/* ou
(yw), the density of yt& certain "rate of return"
on wealth. We assume tentatively that this rate of return, for
given wealth is stochasticallv/of the wealth.
This assumntion is necesary because we are going to represent
the income density as a convolution of the rate of return and
d&Wjititt (04
the wealthy The random variable income y is represented j&ti the
sum of the rate of return and the wealth
( Qtzp'
v = R—h W
(R &•y  W)
For the purposes of the following calculation, it is necessary
or
to use the mirror function of f , that is f (wv) , which
will be as much independent of wealth as the former.
In terms of random variables we have then
)• \ \££J\ A#'v?£'V'

^y~~= W  A (where A ^W—y)
We can then represent the density of income g (y) by means of
randomisation as follows:
g (y) =
f t*) *
where is
(wy)e
0
the
dw =
>
transform of
(w)
The above mixture is a Laplace transform of (w), shifted to the
right by y.
The Laplace transform requires that (w) is defined as equal to
zero for w o . If the density function is shifted to the
right, the densties for w v 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%).
Equation (2) 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 crossclassifi
cations of wealth and income ^wealth ownersffor Holland, Sweden)
show that income is a linear function of wealth, the regression
coefficient being smaller than unity. We can easilv take account
of that by defining a conditional rate of return density or rather
its mirror function as (kwv), where k is the regression coefficient
of y on w. Assuming that the variance and the higher moments of
'VW
(kwy) are of w we can proceed as before:
g (y) = f (kwy)e dw =
y
A
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 1). This is exactly what had to be explained
(the income distributions are "more equal" than the wealth distribu
tion^ 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 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 income1). Nor are the two
parts independent, so that a convolution of two separately derived
distributions would not be appropriate.
1) Attempts to separate them such as in are unconvincing.
In this study, compensation for risk is included in earned income
but by its nature it is obviously related to capital.
;
(kwv) are dependent of w we can proceed as before:
g (y) = f (kwy)e dw =
It may be noted that the condition kw y is more restrictive
than the former condition w y.
The result is
0. 6 2c.ec1! /i t QiZ/toJ lA*
Si/^irn’ W7 'Cs'e*c &
y
In the following, incomes of property owners will be
treated as a whole. The distribution of the 'Ipxotit rate*^
or conditional distribution of inc^ome^ therefore includes
earned income here. The regression of property owners' total
income on their wealth can be studied on the basis of Dutch
datal) (fig.3). The regression is linear^ aaad the correlation
coefficient is 0,5, fv\e 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
/V 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 Paret© 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 independance condition holds
only very approximately.
A simular calculation with Swedish data /20/ gives ^^pporently very
bad results, although the regression of income on wepLjath is linear.
To take an example: For married couples, both ^&X#5(in 1971, the re
gression coefficient of income aed 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
of the&tf
produced a thi.
of the income distribution, thus
leading to a smaller Pareto coefficient than would otherwise obtain.
&
The effect of increasing standard dviation is actually the same
as that of a steepening of the regression line of income on wealth.
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;
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 speak of the rank which an
individual occupies in one or several hierarchises. Examples
of such hierarchies are wealth, education, status, grade (level)
OU *
of an official or manager, rank of officers, ^obility, degreSe of
specialisation, grading by popularity of stars etc. Each of these
would represent a dimension in what might be called hierarchical
ey
spe'ce. An individual would occupy a certain point in that space,
01/
corresponding to its r0nk in the various hierarchies, and it
would h&ar€; certain probabilities of transition with ip 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.
a
To each point in the hierchical space corresponds a certain probability
is
distribution of income; the basic rule that the higher rank means
A
expectation of a higher income.
measur'd
I
In the course of his litfecarreer, the individual moves from one
position to another. It has been repeatedly described how the
hierarchical advance during the lifetime leads to skewed income
disitribution / /, and, in fact, may give rise to a Pareto distri
ct'
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 gener^^lSs*. This has been studied
by sociologists under the title of 'Social mobility" /4/.
A
In order 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.SimJPn /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
MS
from top to bottom will increase in geometric proportion.
He further assumes that each manager gets a which is determined
As
proportion (larger than unity) of his bueerxnates so(lary.
£u/soJuU'>iatt s
as a certain
vv
iri//V'
0
hit
V
The salary thus decreases geometrically as go from
top to bottom.
In terms of algebra:
n: span of control
b: managers4' salary in proportion to that of his
^Subordinates
•T
L: level of the hierarchy (counting the base as unity)
N(L): number of managers at level L
C(L): Salary at level L
N©: number of managers at the base level
A: Salary at the base
The salaries under the assumptions given conform to Pereter h'otAjhZo’ S
distribution. Simon bases his model on empirical facts (Roberts'
SoJtxOvu,
regression of top manager's* on the size of firm, regression
J
coefficient 0, 3^, which would correspond to the value of
and Davis' Pareto distribution of managers' salaries in General
Motors, Pareto coefficient 3).
i
The above demonstration is ourelv deterministic, but if jwe"
we regard the span of control as the reciprocal of a probabi^ \
of advance to the next higher level, the special version of
Champerno^wne'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
..... . 1 V. JC . . L . . . . . 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
Qs /
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 slowly. Thus the chances of advancement in the
f i
individual’s life carreer determine the structure: If^th
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).
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 reation 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
i/M
space. The relation between these dimensions remaffl#k open, and
therefore also the question how these separate distributions
conbine into a total income distribution which still shows the
familiar Pareto pattern. Prima ... 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
CCA
income and by the inequality &£ 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")
se
character of the income relation between the^ groups and yet think
that thejt is this vague ordering which would explain how a regular
pattern of the total income distribution comes about at all. In
Os
particular, it seems essential that the toil of the distribution
is mostly dominated by income from wealth, which assures that /
the total income distribution .
References
/1/ Atkinson, A.B. (ed) The Personal Distribution of Incomes,
“ 1976 9
/2/ Atkinson, A.B. Unequal SharesIncomes in Allon Land, 1972
/3/ Atkinson, A.B. (ed) Wealth, Income and Inequality. g~gyrr»> 1973
PtyufrUAfru
/4/ Bartholomew, D.J., Stochastic Models for Social Processes,
New York 1967.
&
/5/ Central Bureau voor de Statisti^k, Statistische en econometri
sche ^nderzoekingen,^^nkomens ongelij^kheid, No 3, 19655
/6/ Central Bureau voor de Statisti^k, Inkomensverdeling 1962
en vermogensverdeling 1963
D.S.
/7/ Champernowne, A Model of Income Distribution. E.J. 1953
/8/ Champernowne, D.fi. The Distribution of Income Between ...*»'£• A
/9/ Cox, R.D. Renewal Theory. London 1962
/10/ Cramer, J.S. Empirical Econometrics, Amsterdam 1969
/11/ Davis, H.T. The Theory of Econometrics. Bloomington , 1941.
/12/ Feller, W., An Introduction to Probability Theory and
A
its Applications. Vol. I. Third ed.
Vol.II, Second ed.
{^13/ Mayer T., The Distribution of Ability and Earnings.
R.Econ.^ Statistics 1960.
/14/ Mincer, J., "Investment in Human Capital and Personal
Distribution of Income" J.Pol.Econ.Vol.66, 1958
/15/ Roberts, D.R., Executive Compensation. Free Press Glencoe^
Illinois 1959.
o A *
/16/ Simfn, H.^., The Compensation of Executives. Sociometry 1957.
r
v>
>
y
/17/ Simon, H.A., The New Science of Management Decision, 1960
/18/ Simon, H.A., "On a Class of Skew Distribution Function^
S
Biometrica 1955.
/19/ Rutherford, R.S.G., "Income Distribution: A New Model",
Econometrica 1955.
/TV
/20/ Statistika Centralbyrap (Stockholm), Inkomst och Formogenhet.
/21/ Steindl, J., Random Processes and the Growth of Firms.
London 1965.
/22/ Steindl, J., The Distribution of Wealth after a Model ^
u >
/23/ Steindl, J., "Size distributions
/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)
Woodward, J., Management and Technology, London 1958
* ■ ‘ .
, r 504
* r
R.F.St, W2
u
If
^Cl5>cf^^7
, A)a^xJ^'L ^7
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rfa Ce^V^lZu*^ v^v^U a^tleQfrUfy^^ <&&.■,'
1 nJ / 0 1962. 0 V
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J)‘ £>
<#•
t /V ',
A H<?*uejiS /x>6mvwt^. B7, /frs
—— . :j5E1 ~ •■—■—~—
&M J ) roewwve/C . I— CTWiPl^va P)6 2—
^y ^ ^ ^ ^ *■ C^'t <Z£*J~^ i>Vv, <Ahsi Co , /j'Mto T^Unft^'C^n. Ej(? ^
%4h
T~ ’V!lc JteJU* fff *&JJ_
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(P*^ts>C i C7 Cbpv^xbi C<&^* rr^n , $~t X . ~&U~ixL exi I
!/ &£. IT t ^ccx^d e^X
c T/Tz IXfTpJ^tr:1 *{ 'Biases $4&yex^ }B
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WO,
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%k ^7 «?U c^T^i e / f^d^* (i&oi~n / ]/^£* &£j rf'ic?
fUSojA } b , & ‘ EK€<X*fi +* Erx^j^ ^£*1^ * '^2^ E'ie^^e^ot^c.,
[—!S1 *K c?H f f//A I Be Jth fty ^J £fiC@ui> bto > $ &c' < o H£FTsZy • tyE7
I#JS / OlOfU t] A t OOc K)CW cr< \yj ft'LteCt f.^c Xi. c «u ^ ai! < /’jrVii^ t7b 0 ,
^Sjtfj f/'K+^j ri.A • >;!^ ^ (&e^ +f ^ ^‘^'n, B^cO^o 7 &*£*22&J'*i.
yen Q.uXKtAjrut, R.S6 <,7ww Ib^iUt,^ : A Vc~, hi~*X. 'k^jU^.V*
1i
J .jSl^iwat.^ pijc^^ \A~* i^ur^u >£ fvX~,.X^^a^ l\
^'TBc ChTCux^ usLm^ exf Uc^ejy* ce f /c^ t^LooCc^XL
r 7~~—— ' 4 ...
' «
//
t# _______
<£/
U) o L £> A ro D lf~(,
* /A~r
/■ rxxarr
XC
r^:~
V^yusU \ /c lAji^x^ J> ^ E'^&txU'z^ /3eW 4^
wu c^ca.o,'^ ^ b^, ~j, o, isj:c&'s ?
P ie«^irv<7 i^7 <^4>^ of 7 " ' ■ ^
Jh/fsjy bJt\ RD^J~, 7^777^7Z77c.^ LC'U^^ 1 s f y,
Denoting wealth tor w, let uo write for the density of the
wealth distribution
(Vt  M'*  V
or
p’O)
p*(w)
Oo^HlVt
0
w » In V
for W> 0
for vf< 0
the
fOI
7 done
rondos!
distribution of tho roto of return is given in the '
t^cC ——— d —— " ••■
of o density function f XXty) <21 (whoro r *l»y,
incone) wo obtain the
K» aa foliar, /v^
diatx*ibt#lan .
X
O*
t av*/) o’
• OG
(2)
above which the distribution co^foroa
The njninua wealth
to the Yoroto law) is 'token babe m a unit* so that we can
integrate froa 0 tooo • ubp incon© density in thus the
loploco tr'onoforo of the conditional danoity of incooo,
■Z) \P<X
^ * i(r
A
Y  y is the rate of rdtum on the wealth*
If this rote of return lo independent of the wealth then
the above relation (2) bocoxaoo © convolution*
For convoaionoo wo shall use inotead of t(Y\U) the
density function f* (u  X) which
r U@ obtain* then * as o special cooc of (2)s
Vj OP^lZZy
£ (dt,o ^ *V
/f
w
£K:
/W !
i
t^ue j 0 C vd~ C^^n.'Lr’ifi't /i~7f\ /•?^ /^t' ^ <0fy*+ <* a2~t^t~iC Cct—St
/tiAc, / ttJK /^i ? "tt. (J?~l~ d^Lx_ ^O jllsV'*'*~
/4C<r <sC ~> ~Z^ tr^x ^yU^'^AjZfx,
/V^ /ZJ^du&sd^ /t/L^7C CiTt^^ £~> 0~X~> •&>£ C/^(^t^i<~3>
c4*~Th !T^ cc*e^d^4 ^
((2<x~c£ / ?&*£&' <£<£>#&+£ er ^ c<s>y7%.
/^o* c&edZ&td i^Ce't^ i tcj < ~J^L^ ' dt>^ £&<*^>rt^
y^t^UI^T <T
J
,/^Le
1
\Jls Cg/L/^VZsl ^
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£*<&£ r
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/I * ** *• i / /? / / <5*
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<^2J> JXtt^i ~ _e?—
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ch:
c7> ftf'ict^o &jAtfusclv^ii
 * l4,
(_ iM^<. _^t=
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0
*3
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fa* L
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1?
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J C "&<>~&j? uae# ~^
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0$" fi y ^ in cnptd^, . /A~o ^ Lsz^e. fa% c*&
Ay Osie <SK AAt '*— ^iT**—r ^ su?t*. L\ n'T^eGs
? A ^
fCc A^ut ^ /^f^M.ae^L
f^_ji siZ C.xp>~— s? sitpC^^U* tf^stF'TLi Tf^"<=£**(
t^Ccp^e^^—cL^~* &c. c^r*—^4 h '7J, /^7 ^f^ /
Cfi'Visi^L tj
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/&t&/ ftLe
„Sz£'^yL'0>~tr?tu c^o^C pL.r ,t^ 0~2^c_^<£ <oCt Ci, 
C£L P&JxeXa c^e^f
c^' <''~ > /^O l^y &A, tg'CP&y &c<ixjL
/P'1~ i ) O/CL/l
*>1 '
CQJ\j£, ASL*~*s9~l*e~ (5^j? IsL UiJ^
> f ^ / Z~tfLc, C4f^c £cZPl,
sfaLc yi^£<<J2^c^ <&L~^ “ZA^v b
//V^s\}~ C^—9 £4" j L&^P
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* ^Oi ~> itcn, ‘^av 2y *7\ 5
irQ^c
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€<t z TTin •<*+' 'ip'
ft? * '~t£*C'%~
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<2J*JjjZue^L c^'r^ ,
14
looo
ovoi of incono. In the lowor part of the
>ution, 0O diotribution of the profit rate
polo. / /
/
/
ooid/bf oouroQt that the rogrocoion
callPkf"for on cseplanotioa. Shis con only be glvm on
jdm plane of "stage four” (000 above p. »,) Ijy o otachaotlc
procooa in aovopdl varlableo. / ■
2Iio Inoopo of property ownoro.
Gone onplrical data will illustrate the above theory#
While this theory’ deals with property income, the data
bclou rather refer to incooo of property owners, which
in part io earned incono. It la not easy to ooparato
the earned and unearned income. Uor are the two parts
indopondont, 00 that a convolution of two separately
derived diatributiona would not bo appropriate.
In the following* income of property ownoro will bo
trootod as a wholo# Tho distribution of tho “profit rato°,
^Attorpto to ooparato then ouch 00 in /1 / aro un
convincing. In this otudy, conponoation for rial: io
included in earned incono but by ito nature it ic
obviously related to capital.
■ (JU< Ce c<r~^e ^ ^ ^
t&'/feyb&c*. / c^£t£ £ £< cCtPS^ej ‘
13
or conditional distribution of ineocc v therefor© includes
earned Incoae boro* 2h© regression of property owners1 total
incoae on their wealth con be otudiod on the basis of
Dutch deto1 ^ (fig* 5). The regression is linear, and
oofeiaatifr; the correlation coefficient is Off, the re
gression coofficiont is 0#620£0#004 (data for 1902/63)#
2ho regression coofficiont corresponds to our k# flirt
k (1 con bo explains the first place be? til© premia*
fact that with increasing wealth earned income is loos
and lose important* in the second pi co perhopo by the
fact that incooo froa shores which dooinotos for tho
large wealth does no^ontciu the uniiatribuiof profits#
Since tho Pareto coefficient for wealth was 1*33 in 1962/63,
vo should expect it to bo 2*20 for incooo on the basic of
too theory# In reality 1 o ess 2#0cW A setter coircspoadonco
is hardly to bo ejected, ofneo the wooltlirdistribution ot
disto.'.'teO. by'“tlier'sioei. ei.chongcr boon"—
<£?£/<) <2< '*r6^ C2^t^Ce iBTicC t~7 'iitfl L
^fiCdo LTvUf '***?
^alC'v.latiojy wibh^wodish/dota &IF&C very un
>y/peou£ts# oltht^gh ,*ho rogprfboftm lino is
boopwOdosbic # by frft©
coao4which stop9etr
.lag
•e4 ohorgs ■■■■iscx>ocsee strongly with the
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