Konvolut The Personal Distribution of Income 1
Josef
Steindl
AC14446015
Josef Steindl
August-September 1972
The Personal Distribution of Income
’//lien D.G. Champernowne snowed now you can explain the
Pareto law by a stochastic approach he very naturally
chose as an example the distribution of income, because
that is the classical case. It appears now that the approach
is more easily applied to firms, towns or wealth.')
'-The case of income is'the hardest, so that the great
pioneering paper /p^/while fully demonstrating a powerful
new method, has not entirely disposed of the individual
problem which it was resigned to se^ve.
Ghamnernowne's Model
I shall give a simplified version of Champernowne's model
which will throw a new light on its relation to other
models of the Pareto law.
The income of a person is the state of the system, and
its evolution is described by a Markov chain. The stochastic
matrix of income transitions from one year to the next, in
desperate simplification, looks as follows:
income in year t + 1
0 1 2 3 4 r- 6
0 q p
1- q P
2 q P
3 q P
4 q P
$ q P
• « • • • •
<o
<D
PI
•H
<D
a
O
o
a
H
liiere are only two alternative transition for a person in
this system: Either a rise of income from one year to the
next (note that the income classes are on the log scale)
which has probability p; or death of the person, i.e,
transition to the zero class, which has probability q
(p + q = 1). In addition, there are entries from the zero
clasS/Areplenish the stock of income receiver!*
' i +?Vi a
y£ Uf Cfv C^
The essence of tfed-a model is described by Feller*J in the
following terms: The state Etc represents the age of the
system. When the system reaches age K, it either continues
to age or it rejuvenates and starts afresh from age zero.
The successive passages through the zero state represent a
recurrent event. The probability that the recurrence time
equals K is p q.
We are interested in the question: How many years have
passed, i.e. how many income steps have been mounted, since
the last rejuvenation? This is the "spent waiting time"
of the renewal process. Choosing an arbitrary starting point
we can say that in the year n the system will be in state EK
if and only if the last rejuvenation occured in year n-k.
Letting n-k increase we obtain in the limit the steady state
2)
probability of the hpent waiting time" '. It is proportionate
M*. ■ Toi. I. Z7.3, p. 382.
Chapter V.
■fco the tail of the recurrence tine distribution, i.e.
to p*.
More directly, the vector of steady state probabilities
can be derived from the following two conditions;
/r' A .
? * ■
Sff I
= PU*_1
= Nqu + qu1 + qu2 + ...
'ike
■The first condition ensures the invariance of the steady
state; the seco.nd ensures that entries to and emits from
the income population balance.
It follows that
u* = pKu0
u = 1 - p p < 1
o
The result.is, of course, identical with the distribution
of the spent waiting time.derived above.
-'S.
to the tail of the recurrence tine distribution,i.e. to p .
More directly, the vector of steady state probabilities u*,
can be derived from the following two conditions;
A/
= PUK_1
uQ = qq + qu1 + qu2 + ...
\
•The first condition ensures the invariance of the steady
state; the seco.nd ensures that entries to and emits fron
the income population balance.
It follows that
= pKuo jefo
u = 1 - p P < 1
o
The result.is, of course, identical with the distribution
of the spent waiting time.derived above.
-'tv*
gvv *>V»
It. will, be noted that the process can be described up to
this point without any reference to. income. We may regard the
states of the system represented by the elements of. the matrix
as ,rgrq.desor ,rags." -'possibly carreer, age on age in earning
■ .. *11 ■
life,, although in conformity with Champernowne we referred to
them as income classes. 1
We have now to define the income in relation to the class
intervals of the matrix: The lower limit of class 1 is
taken to he the minimum’income. We may choose the income
units such that the minimum income is unity, i.e. on the
logarithmic scale it is zero. The income at the lower
limit of successive income classes K is defined by
7k -
or In yK = Kh
where h is the size of the class interval on
the log scale
V
^) The difficulties arising from the discrete representation of
a continuous income variable in the matrix do not concern
us here. See £toJ p. 62. 72^ 1. Met
c 3 «-i
/ /
&
CLci <3)
‘A h
zy
.y
rts ' C
'■ '1
/y5
7
I
4
To find the tail of the steady state distribution P(y^) we
sum (/) from & to oo and obtain p . Thus
1 \ 1
3-n H7K) -^la p ) E 1» JB
«-I
and, putting -h In p = oC we have
In P(y^) = -cC In y*, (3)
Evidently the crucial feature of the model is the geometric
distribution of the recurrence time* This relates here to
the life-time of the persons as income receivers; since
promotion is automatic, the age of the system is measured
in income classes K. The age, or spent life-time, is
geometrically XaPPrcsiinai:e-7 exponentiallyf distributed.
I / I lit i \ l
Since the income is also an exponential function of K »
the Pareto law results from an elimination of time K from
5'
the two exponential functions.
This is exactly the same pattern of explanation as was used
m .. L* \ t ^ ^ -
/ 2. i, 1
in other fields
t)v
- _ —7 y - _
by Simon £ Jp and myself -43/ and
which is-directly■descended from Tule 1# J, who used it
(IQ2H> 2.
to explain the frequency of species in genera. According to
'Although Champernowne's^model is more complicated than the
above,v-the essential features remain the same (only, o in
the solution is replaced by b, the solution of a difference
equation).
I
Footnotes to-p.4 •
1) Although: Champernowne rs, theory is mors complicated than the
simple- model which takes its placs in the above reasoning,
we. can easily extend the conclusions:. With Shampernowne, the-
.." promotion is stochastic, with probabilities'of .non-promotion
and demotion» In. this more general-case p in, the above solution
■ T has to be replaced by b which is the root of ht he - characteristic.
_..f ~~ r -L . > .. rjftf ■
equation of the matrix aT
't'W W- X'hL h £> tc&i ,
/
-/ .
'"' •'The Pareto coefficient -in the simple case is - ^ °
the ratio
‘ of the- parameters of the'two-exponential distributionsin the
. . .... N
more- general case of' Champernowne the Pareto; coefficient is*
In. b
» h could be regarded as .-the-parameter of'an age dis-
. -T
as age classesj
tribution,. if the classes (states of the system) are regarded
, SlBsf&lr ■fcr 3f~ f Ci, VL ic fcW ti~t.
* £r &&<*•■. <«2*-
2-X Champernowne .apparently did nor know Yulers„ paper: It was
■ Hi.Simonis merit to have.brough“ it. to 'the- attention -of
i economists ;. unfortunately he reproduced! it: in a form which
obscured its essence* which is ehe interplay of the two ex-
ponential distributions >. i.e. of two stochastic processes-
/ a
ylA.
- ßJ) Lsts&ij' 1 Ij?* ,X-
.... ’ L .. ' '
} ^k ■ ,/y
c >uya~
t
(J? \p\, ^ tt*- l -
sy* £*TV^
this approach, size distribution is a transformed age
cr'cc/^i^
distribution, and the pattern of the Pareto law appears-
so_often simply because of the empirical importance of
exponential growth which anises both the age distribution
and the tranformation function exponential. Owing to the
conceptual density of Champernowne's model the twc
elements of life- cycle and promotion are merged into
one • feiuJ*
(which resides
There is, however, a differsnce^in the interpretation not
in the form) between Champernowne's model and the others:
Since physical persons sooner or later die, the age in
his model is limited, while in the others (relating to
firms or wealth) there is always the probability of
virtually infinite life which accounts for a very
2.2.
peculiar character of the steady states concerned £'yg J.
Further developments
■ ■■■■— « ■ ■ ■■■ ■ ■ ■■ .. » /
\ x
Ve may consider the following stages in the treatment of
the income distribution:
I. Champernowne's model. *
'7
H. Rutherford's model -e treated persons'life-
times explicitly.
III. The above models are open to criticism on two grounds:
First, income is not very suitable as a state variable
for a Markov process. It does not embody the "influence
JJ
5a
Th/ose few units whidh survive for good continue to grow; on the
other end of. the scale new entrants/enlarge the total sample
of firms or wealthy’ dynasties. The continuance of the steady xx
state with unchanged distribution is accompanied by a steadily
growing size of the sample whidh produces ever larger firms,
wealth holdings etc. These large ’units always existed potentially
but could not be realised x± as long as the sample was too small;
with the growth of the economy such potential sizes become actual
and in consequence the largest units represent a greater and greater
shas*e of the total although the theoretical distribution is
unchanged. With the finite life and the stable population
of Champernowne1s model this peculiar form of growing inequality
m
would not arise.
Further developments
We may consider the following stages in the treatment of the
income distribution:
I. Champernowne1s Model.
II. ßutherfordä model /I9/. He treated person's life-times explicitly.
III. The above models are open to criticism]» on two grounds:
First, income is not very suitable as a state variable
for a Markov process. It does not embody the "influence
7-
globale" (Polya) of the past. Second, the model is con-
fined to a life-cycle from ri?ho<entrance to exit. But the
relevant stochastic process goes far beyond that. In
fact, when somebody starts in life, his chances of re-
ceiving certain incomes are already settled to a large
extent by the condition of his parents: By their wealth,
status, connections, reputation and' the education or training
they have been able to give him. In other words, the exits
and the entries in the life-time model are linked by in-
heritance etc., and the process of diffusion continues on
I >
a higher stage. ;
W
to i,iT i
Both arguments point/\the same consequence: We must relate
the chances of getting certain incomes to the amount of
wealth, material and immaterial, and its distribution.
In this way we can link the income to a suitable state
variable (wealth) which is evolving in a long run process
through the generations.
Also, we shall then be able to answer the question why
income distribution is relatively stable, although so many
elements relevant to it are changing day by day; The answer
is that the stability lies in the distribution of wealth,
education, training etc., which change only slowly
X)
.-^2-7 P* ^4-2 seq.
i) Champernowne was, of course, aware of these facts,- as his thesis ^
of 1957 ( $ ) shows. His formalised model of 1953 is, however,
not well suited to reflect all the economic factors so well
stated in general terms in 1937 -
7
U. In a further stage ire should cease to take the wealthy distribution
as given and instead treat wealth and income as joint variables
in a process evolving over ±±m* the generations. he should continue
to regardthe influence of slowly accumulating wealth and other stocks
on the current income as a dominant feature of the -process but,
trying to includethe preceding history of accumulation, we should
now also recognise that in the course of time the income acts on the
wealthy, not* only in so far as it arises from wealth as a return and
is (partly) re-invested, but also insofar as anew wealth is formed
from earned income in favorable circumstances and after_a lapsrf of
time required for accumulation.
That means that income and wealth are linked by two rinds
of relations:0ne is the rate of return which links the current income
of property owners to their wealth, the other is the relation
between (unearned and) earned income of the past to the accumulation
of new wealth, in otherwords, the propensity to save. *n so far
as the past income is in practice fairly strongly correlated with
present income, the present earned income appears to have an ---.feet
on the wealth which corresponds to the savings relation.
The influence of wealth on income via the rats of return
is the subject to be treated first over the next few pages.
The dependence of income on wealth
In the following we shall consider income as flowing from
wealth. This applies strictly to what we call unearned income.
The reader may think of that in the first place,but h e has to be
prepared for a more far reaching and wider interpretation facer on;
in fact, in practice-we cahhot easily separate unearned and earned
income, and the difficulty is perhaps greater than one that could be
overcome by adequate data ( especially in the case of unincorporated
business ).
Instead of the matrix of income transitions ased by
Champernowne we have to imagine an analogous matrix health-income
whichshows for each amount of swealth the orobability of different
I ‘
incomes.
^he basis of the analysis is thud the conditional distribution
of income, given the wealth. Economically speaking this is the
probability of a certain rate of return to wealth or profit rate.
From this, if we know it, we can derive tije distribution of income f
^ Oh * ( i , h ( u>>fr '
t \ ,A
6
provided we know the distribution of wealth. But the distribution
of wealth is known: It follows the Pareto law - over a fairly
wide range - and its pattern has also been explained
theoretically / 13/•
Denoting wealth by Tf , let us write for the density
of the wealth distribution
p* = c TT^~ 11 dW
or putting w * ln W
p(w) = c e ' du? for * IV o
p(w) = 0 for w < 0
(4)
If Y denotes income and y= ln I the conditional
density function of income can be represented in the form
f*(y-w), the density of a certain return on wealth. Sven without
knowing this function we might manage to derive the distribution
of income from that of wealth provided we can make certain
assumptions about independence.
T,7e shall provisionally assume that the distribution of
the rate -of return is independent of the amount of wealth.
In terms of random variables, if (yf , h-C and J\j
</ J
denote income, wealth and the rate of return, we have
U - '4:;/ “
/
u -
If the random variables wealth and the rate of return are independen
their sum can be represented by a convolution of the corresponding
density functions, and ve shall in this way obtain the
distribution of income.
For the purposes of this calculation we shall replace
the density f*(y-w) by the mirror function f(w-y) which is also
independent of wealth. The two functions are symmetric and have
the same value ( in fact, the only difference is in the dimension :
TTnile the former refers to a rate of return per year, the reciprocal
value refers to the number of years income contained in the wealth )
The calculation of the density of income q(y)
proceeds then by mixing the function f(w-y) with the density
of wealth:
9
f(*-y) e'^* dw - c ß^J e“ for w > jr? C
q(j) - Ö or w< y ,
where V/^' ) is the Laplace transform of f(w).
The above mixture is a Laplace transform of f ( w) shifted to the
right by y.
The Laplace transform requires that the argument of the
function f be non-hegative. Tie have therefore to assume that
tt( we shall further below haw this restriction can be relaxed ).
Equation (5) shows that the Pareto form of the wealth
distribution is reproduced in the income distribution, provided
the independence condition is fulfilled and y ^ w.
Tie have now to face the fact that the rate of" return on wealth
will in reality not be independent of wealth, -he cross-classifications
of wealth and income of wealth owners for Holland and Sweden
show that mean income is a linear function of wealth, the regression
coefficient being smaller than unity.
For the decline of the rata of return with increasing
wealth iiriiix variois reasons are responsible:
relatively
The earned income will be/less important the greater
the wealth. In particular the income from ( non-corporate) business
will be higher in relation to wealth in the lower wealth classes.
Further, capital gains are not counted as income, but they affect
wealth, and they will be more important for lagge wealth, because
the proportion of shares held increases with wealth. The internal
accumulationof firms will not find expression in the income,
but quite probably in the wealth of the share holders. Also
appreciation of real estate may affect the large wealth proportionately
mors. •
How the rate of return is independent of wealth if
its conditional distribution is the same whatever the size of
wealth. It would seem that we night perhaps restore the
condition of independence simply by turning the system of coordinates
in the appropriate way, so that we would reduce the present to the
former case. If we nan make the covariance of w and w-y zero
then the coefficient of regression of y on w should be one,
as in the former case:
10
Cot (w, w-y ) ■ Var (w) - Cot ( w,y) - 0;
Cot (w.y ) ä 1
Var (w)
If the regression line of income on wealth is
7 - K* + 7o
( )
and^i! if the Tariance and higher moments of the conditional
income distribution are independent of wealth then we should
use instead of f(w-y) the function f (>£w+ yc - y )
and this distribution will be independent of wealth.
7e can then proceed as before:
q(y)
‘a?
f( *Cw + jc- y ) e" *** dw =
for ^w,> y
7c
q(y) = 0
for kw < y -ye .
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’-O )• -his is exactly what
has to be explained ( income distributions are in fact more
"equal" than the wealth distributions, empirically, in the sense
described ). The particular shape of the rate of return distribution
has no influence onthe tail of the income distribution, as long as
it fulfills the independence conditions mentioned.
Concerning the restriction w^> y - yc
it should be remarked that we are free to shift the coordinate system
to any yc we choose so as to make the above condition Talid,
with no consequence except that the conclusion about the Pareto tail
will be_confined to incomes in excess of yc .
It would seem that in practice, in Tiew of the value of
, yö must often be more or less high, so that the Parto pattern
will be confined to a rather narrow range of the income distribution
while in the caseof wealth it usually extends to the whole of the
assessed wealth data. -This, it is true, partly results from the
fact that the wealth data are more truncated than the income data,
in Tiew of the underlying tax laws.
11
Income and Tealth: -muirical Patterns
The following remaiprs refer to the cross-classification::
of wealth and income published in -olland and Sweden. These data
show certain characteristics which are found a].3o in other
cross-section data concerned with sise distributions, especially
data from official publications like censuses etc.
The first feature is that the great bulk of the observations
is concentrated in the corner of the first ( the north-east ) quadrant.
In other wards these distributions are very skew, A great many of the
units are small in either dimension.
The second feature is that the wealth distribution is
heavily truncated ( in Sweden for example at 150.000 crowns)
while the income distribution is given down to rather low levels.
If the mean income in the various wealth classes is
calculated at linear regression of a very regular pattern is obtained.
( This "regression of the first kind" as we may call it differs
from the usual least squares regression in that it does not
assume a -priori a certain mathematical function for the regression?
see Harald Cramir, Mathematical Meoheds of Statistics,Princeton
if the regression of the means turns out to be linear as is the case
here, then it shoild be the same as the result of a linear least
squares regression; this may be not quite true only in so far as
we fail to use weights for the means corresponding to the various
frequencies in the different wealth classes ).
The same regression of the first kind in the other dimension -
wealth on income - gives a completely different picture: 'Lhe mean
wealth in the lower income classes does not indrease with income
at all;- for higher incomes it increases very strongly, so that
a strongly curved regression line results.
At least one, and probably the most important reason for
the curvi—linearity of this regression line lies in the truncation
of the wealth data. If we try to fill in the missing wealth data
in our imagination, according to plausible and common sense prior
knowledge, we find that the regression of wealth on income might
well be quite linear and rather steep; at least it would be very
much nearer to linearity than it is now. The inclusion of cases■
with wealth below the tax limit, which is probably the lower and
the more frequent the lower the income, would reduce the mean weal oh
in all income classes but it w'uld reduce it the more the lower one
12
income. In other words, the mean wealth in low income classes,
as measured by the dataphioh we have on the basis of tax assessments
very strongly overstates the real mean wealth, and this the more
the lower the income.
There is no proof, of course, that some curvilinearity
would not remain even if full wealth data were available. There
may even be theoretical reasons for that, as will be shown later.
One reason for linearity of the presumed full data should be
mentioned: It is rather queer that non-linearity affects onjy
one of the regression lines and the other is perfectly linear.
Only the truncation of wealth data gives a plausible explanation
of this contrast between the two regression lines.
Tmuirical data: Holland and 3weden.
The cross-classificaiion of wealth and income, available
for the Netherlands and Sweden, will now be discussed in/the light
of the theory contained in equation . "Yerification" aan
hardly be expected: The rate of return explanation can not nearly
explain the income of property owners fully, since a large part of
it is earned income. Apart from other statistical difficulties
it must be kent in mind that the -arbto coefficient is always
more or less arbitrary, since it depends on the range of income
or wealth classes included when you measure it. 3ven inconclusive
data, however, are better than speculating in the void.
An evidence on which I rely heavily is the linear and
fairly regular character of the regression of mean income on wealth
( fig 1). The regression coefficient is in most cases around 2/3.
but it may be as low as l/2 . Whether the higher moments of income
are independent of wealth is not easy to decide. Yhile the
variance, in the Swedish data, increases in the higher wealth classes
this can plausibly be explained by the increase in the range of
these classes ( the last but one wealth class has a range about
four times as great as the lower wealth blasses). The same
fact is relevant for the comparison of the conditional distribution
of income in the various wealth classes: They all have a ^areto tail,
the -^areto coefficient being markedly lower in. the last two or three
wealth classes * nan in the other s. This, again, may be
plausibly explained by the greater range of these high wealth
size classes.
I
^ C Otri c
ACC OCV
!M KG
I “
' I
i i
i i
I i
l
I !
I ►
A
AO OOO
A Qif.OCO
d
gz /(£ ?
HotLcW A%zf&
: T'9* 5
I I
I I
I 1
A c w oco
ilds.
Ve^Lt
r
i
13
Table
Sweden 1971
married
couples
single
persons
Holland 1962
Wealth distribution
Pareto coefficient
(whole range,
11 values) 1.78 1,73 1*53
Regression of
income on wealth
coefficient yL : 0.56 0,65 0.63
Calculated Pareto
coefficient for
income: 3,19 2.66
2.20
Actual Pareto
coefficiet for
income ( 5 values) 2.62 2.14 2.08
ditto, excluding
the open size
class of wealth 3*12 2.62
The above table gives the vilues for the coefficient
of regression of the mean income on wealth. The correlation is very
high ( r is ,997 ) for the mean income values; for the mean square
correlation of the grouped income data it is modest, r2= .25 in
Holland.
From the actual ^areto coefficient for wealth and the
coefficient we can calculate the Pareto coefficiet for income
in accordance with equation (S>). In Holland we obtain 2.20,
which compares with an actual Pareto coefficient cfor income of
wealth owners of 2.08. ( The coefficient for all incomes including
pepple with no tamable wealth, is not markedly different, which
is rather surprising).
Butch
While the/results are not worse than could be expected^
the Swedish data are less straightforward. A peculiarity is here
that married couples ( to some extent also children ) are jointly
assessed for wealth tax so that their joint wealth and income is given
in the data. This leads to a splitting up of the mass of wealth
u
holders into four groups ( couples where both hare wealth, couples when
one only has wealth, single men and single women ). ^he reduction in
sample size impairs the regularity of the data and I have therefore
aggregated the four into two groups : married couples and single
pwESons.
The calculated Pareto coefficients for income of wealth
owners are much higher than the actual ones ( ^able 1).
These calculated coefficients correspond more nearly to those of
all income receivers including the wealthless ones -the great
majority. They correspond also more nearly to these of the
conditional income distributions in all wealth classes except the
last three. In fact, if we exclude the open wealth class
frojj the income distribution ( which might perhaps be motivated
by the argument that it is not contained in the calculation of
either ) then we get ?areto coefficients entirely in line with
the calculated ones ( Table 1). The motivation is not entirely
convincing and the results are inconclusive.
Sinne the condi tional income distributions in the
wealth classes have been referred to several times, - give in the
following data for couples where both husband and wife have wealth.
conditiona 1 income distribution
health in 000 K [ft. mean Pareto coefficient
150-175 4.74 4.06
175-200 4.76 3.55
200-250 4.79 3.83
250-500 4.34 5.92
V>4 O O 1 £- O O 4.88 3.69
400-500 • 4.95 3.34,
500-750 5.01 5.29
750-1000 5-09 3.00
1000-2000 5.18 3.47
2000-5000 5.38 2.19
5000- (5.74) 1.17
Ail. 2.68
All without open
wealth class 5.16
15
The conditional distributions have all Pareto tails although the
fit is bad ( only 4 values can be used ). The Pareto coefficient
is between 3 and. 4 in all except the last two wealth classes,
where it is very low, and it is 2.63 for the whole income distribution.
It appears that the income distribution as a whole is -
as far as its tail is concerned - decisively influenced by the
last two wealth classes. This is due to the fact that most
of the top income receivers are in the last two wealth classes,
where the income distribution is very unequal simply owing
to the wide range of wealth in these two classes, as already
mentioned before. In this way the peculiar result arises that
the total income distribution is much more unequal than
almost all the conditional income distributions.
This in a way also answers the question which
might well be asked: Thy the pattern of income distribution
could not be derived from the conditional distributions
without reference to the wealth distribution.
Allometric growth of income and wealth
The discussion of the relations of income and 7,-wealth will
now be extended to take accoun^of influences in both directions.
The starting point will be the regression of income
on wealth which seems to be linear as far as the data go.
This might be regarded as a case of allometry, in analogy to
1)
a "law” well known to biologists :• Various parts of an organism
L I P yl
grow at different but constant rates and as a result the proportion
of their sizes on log scale /) remain constant in the growing body.
”*) Ludwig von ^ertalanffy, general System Theory. Penguin 1968 p.63-
Devendra Sahal has used the allometric law in combination with
the progress function in order to explain the Pareto distributio n
I in one dimension ); see A Formulation of the Pareto Distribution,
™, Science Centre, 1000 Berlin 33v mimeo ). Although the
use I am trying to make of the law is different, I oweAto
Devendra Sahal to have my attention drawn to it.
■
16
Thus in our case the income is y » }C w a where we suppose
that the distribution of income and wealth reflects in some way
a growth nrocess which has taken dace in the cast.
■x^.ys
In the ordinary formulation we should have
V ■ c, a
'» “
AC
Y = C2«
and after elimination of time
h'Wvv '1^ *'*• ’ ' £, C 3-
/W
7 - In c •
,/»
I 5
C*
I -tM6 ChX"*-
( w - In c, )
/V>V
0
■u.6^’ )l
* (7)
which corresponds to our regression with
<c » — xiy } ■
* yr,
How in economics random elements or shocks play a large
role and growth is influenced by them. We should therefore write,
instead .
the above:
W
SfC
e +
Y * c-» e +•
1
£(t)
'I (t)
where £ (t) and ^(t) are random variables. As a result we obtain
A r
In
( Y - yj (t) ) - In
c , =
0<s
j ln( W . £(t) ) - In c^- .
(3)
The random variables or ’’errors" ( not mere errors
of observation of course ) are in both variables. The common
I
tactics of regression is to ascribe all the "errors" to one
variable only, which yields two regression lines according to
. whether you choose the one or the other variable as the
repository of the errors. The equation (3) will not correspond
to either of the two regression lines. it may be guessed that it
will lie somewhere between them.
Let us now start from a different angle and consider
the two kinds of relations that exist between income and wealth.
One is the influence of wealth on income via the rate of return.
It mainly affects unearned income. The other is the influence of
past incomes on wealth via the propensity to save. Since
present and past incomes are fairly strongly correlated it
will appear as if current income acts on wealth via the
propensity to save.
At first sight it seems that the linear regression
of income on wealth represents simply the influence of wealth
via the rate of return. There is however an alternative interpretation.
17
Larger wealth presupposes larger income (if it had to be saved
out of it)j therefore the mean income has to be higher for
larger wealth. From this point of view it is not surprising
that the increase of income with wealth is less than proportionate,
because wealth i£ an integral of income taken over time;
if saving can be assumed to be a constant proportion of income.
The interpretation is only weakened to the extent to which it
could be argued that the creation of new wealth from earned
income will affect mostly the lower wealth classes ( only
earned income is really relevant in this context where income
is supposed to play an active role vis-a-vis wealth ).
The other regression - wealth on income - should
on the face of it represent the effect of the propensity to
save, the creation of new wealth from earned income
( continuing primary accumulation ). The curvilinear shape
might be compatible with this interpretation, since for the
lower incomes the saving will play no role and wealth will
only start emerging when income has reached a certain level,
and then it will rise steeply ( because it is an integral,
see above ), Here again, an alternative interpretation is
possible; If income is large, it probably has been derived
from large wealth, therefore to larger income will
on the average correspond larger wealth ( rate of return relation )
•* *4
Thus the two relations or theories or laws
behind the joint distribution of wealth and income seem
to be both relevant for each of the two regression lines,
although possibly not to the same extent ( each of the
regression lines may be be more strongly influenced by
one, relation than by the other ).
The preceding arguments implied that, to.' some exent,
each regression line is an inverse to the other regression line.
In order to make this clear let me choose an example from
another field-, where there is only one "law11 or at least we
can pretend there is only one. Take the size distribution of
manufacturing plants according to output and cost. The law
consists in the effects of scale on cost in relation to output.
The regression of cost ( or employment) on output will show
a coefficient less than one, decreasing cost to scale.
17e expect the other regression to be the inverse of the first
»
/
13
showing disproportionately increasing output with any increase
in input ( cost, or employment ). In fact, however, we often
find that it is not so and that both regression coefficients
are smaller than one, decreasing cost and decreasing returns
apparently coexisting.
How is this possible? -t can only occur with
wide dispersion round the regression line. The exceptionally
effieient plant will tend to be counted as small plant
in the input dimension whi?e the unusually inefficient ones
will be counted as large. In consequence there will be
a bias in favour of decreasing returns as measured in the
input dimension ( regression of output on cost or employment ).
The inversion af the regression corresponds
to the fact that the ratio;!, of the two standard deviations
is reciprocal in the two regression coefficients. If it
is 9/l0 in the regression of input on output, it is 1o/9
in the other regression. But, unless the correlation coeffieien
is sufficiently high, the regression coefficients
will both have values below unity.
The same mechanism must also be at work in' ••
the wealth-income distributions Those wich high return for
a given wealth will be classified unong large incomes, those
with low returns with the same wealth among small incomes,
which tends to counteract the natural tendency of wealth to
increase with income. This may have contributed to the
flatness of the wealth-income regression in the lower income
range, although the chief reason for that is no doubt the
truncation of the wealth distribution.
The preceding example ;f plant size, in which
only one underlying theoretical relation is • presumed to exist,
shows that while it is logical to expect in this case,
if one regression reflects the underlying relation, that the
other should as it were represent the inverse of it, yet in
reality this will not be true because the second regression
will be more or less distorted by the dispersion of values
round the first regression line.
If we have two underlying relations then each of
the regression lines will be influenced by both of them,
either directly or indirectly, because each will be to Some
19
extent an inversion of the other; in addition, each will be
influenced by the dispersion of the values round the other
regression line.
Thus . each of the regression lines will
represent a compromise between the two underlying relations
the weight of them being different in the one and the other
regression line. Ho regression line therefore will be
a true reflection of an underlying causal (or rather stochasti
relation. We shall have a better chance of understanding the
meaning of joint size distributions of this type if we regard
them as residues of a growth process. Set us therefore return
to the allometric law. As far as its relation to the joint
distribution wealth-income is concerned we have to make
two observations:
1) If the regression line income on wealth could
be regarded as an expression of the allometric law then,
as it will be remembered, the regression coefficient
is the ratio of the two Pareto,coefficients of the income
/
and the wealth distribution. '
2) Following up the idea that wealth can be
explained from saving over a certain time and saving can
be explained from income, taking saving propensity as given,
we can darive the distribution of wealth from that of income
in much the same way as the other way round:
We explain the saving distribution as a convnlution
of^the income distribution and of theidistribution of the
propensity to save ( savings ratio ):
q'(s) = q(y) * g ( Y 7 - s ), l (9)
and the wealth distribution as a convnlution of this and
the time the saving has accumulated ( which will be finite
in the case of earned income but not necessarily for unearned
income ):
q''(w)= q'(s)*h'(3-w) (10)
Prom this wealth distribution we should by means of the
original transformation (6) come back to the income
distribution
20
q»''(j) = qii(w) * f (^ w - y ) (11)
How can this series of transformations square
so as to produce a steady state of the joint distribution?
■*•0 explain this -are have to separate income at least theoretically
into earned and unearned income, -he distribution of earned
income has to be regarded as given from outside; its explanation
has to be sought separately; empirically at any rate it is
a Pareto distribution. It may be regarded as constant in the
simplest case. From it flows the continuing primary accumulation
which contributes to the accumulation of existing wealth holders
and also creates new wealth holders ( a kind of social mobility ).
This wealth then creates property income and secondary accumulation.
The process leads from a relatively moderate inquality of
earned income to more unequal primary wealth, and even core
unequal secondary wealth.
-*-n this way wa can imagine that total income
and total wealth will grow at constant rates. The corresponding
allometric proportion does not appear directly in the cross-section
where instead appear the two regression lines which are neither
of them a reliable estimate of the ratio of the two Pareto coefficients.
Can this be regarded ad the reason why the attempt
at empirical verification of equation (3^ on the basis of Swedish
and ^’hch data lead to unsatisfactory results?
References
/I/ Atkinson, A.B. (ed) The Personal Distribution of Incomes.
London 1976
/2/ Atkinson, A.B. Unequal Shares-Incomes 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 econometri-
sche Onderzoekingen, No 3, 1965: Inkomens ongelij'kheid,
/6/ Central Bureau voor de Statistiek, Inkomensverdeling 1962
en vermogensverdeling 1963
11/ Champernowne, D.G., A 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.A., 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",
Econometrica 1955.
/20/ Statistika Centralbyran (Stockholm), Inkomst och Förmögenhet.
/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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Postscript 1988.
A considerable part of the household saving takes the form of
contributions of employers to the pension funds and of premium
payments of employees to life insurance companies . From the point
of view of effective demand this should,in general, make no
difference.lt is in any case saving,invested in financial
assets,only the household does not have full and direct control
over it at any moment of time. From a statistical point of
view,however,a few complications arise.The U.S. National Income
and Product Accounts (briefly NIPA) as well as the Flow of Funds
(FF) of the Federal Reserve credit the assets of the pension funds
as a whole as well as the policy reserves of the life insurance
companies to the households. The implication is that the funds do
not make any saving,all accumulation is credited to the household.
In accordance with this approach the employers contributions to
pension funds and the life insurance premia are defined as labour
income (supplement to wages and salaries) and since they are not
deducted when the take home wage is calculated they are also
included in disposable income. A subsidiary feature of the
approach is that the benefits paid out by the pension funds and
life insurance companies are not credited to disposible income,but
instead the investment income (interest) of the funds and life
insurance companies is so credited.If the two are not equal the
balance goes to the households disposable income.
As already mentioned in this paper the two systems of data, the
NIPA and the Flow of Funds, give different estimates for the
household saving. The difference was formerly relatively
modest,but in the 1980s the estimates diverged very strongly.
The divergence is identical with the statistical discrepancy given
in the Flow of Funds which amounted to $40 to $90 billions per
von y?^KV im/
; :
4
/
renewal process birth to With
F,f-L
\^.Wl ,f
Josef SteincLl
August-September 1972
The Personal Distribution of Income
When D.G. Champernowne showed how you can explain the
Pareto law by a stochastic approach he very naturally
chose as an example the distribution of income, because
that is the classical case. It appears now that the approach
is more easily applied to firms, towns or wealth.^-
^The case of income is'the hardest, so that the great
T
r
pioneering paper while fully demomtrating a powerful
new method, has not entirely disposed of the individual
tPr , dZ'iecCztt'- * ,
problem which it was oe-s^gnod to se-ve.
^
, ,J ^ *
Champernowne's Model
I shall give a simplified version of Champernowne's model
r
which will throw a new light on its relation to other
models of the Pareto law.
The income of a person is the state of the system, and
its evolution is described by a Markov chain. The stochastic
matrix of income transitions from one year to the next, in
desperate simplification, looks as follows:
0 1 inco 2 me in 5 year t + 1 5 6
ü~ q p -
1• q P
2 q P
5 q P : • •
4 q - P
5 q P
• • : fr
• • - - _ . . . .
r ~ • . , .
• • .
u
a3
C
•H
(D
a
o
o
G
a
■lut.
T
/fyOt1
' /Ly^
There are only two alternative transiticrs for a person in
this system: Either a rise of income from one year to the
next (note that the income classes are on the log scale)
which has probability p; or death of the person, i.e.
transition to the zero class, which has probability q
(p + q = 1). In addition, there are entries from the zero
to «
class/^replenish the stock of income receiver?
j
The essence of this model is described by Peller ' in the
following terms: The state Ek, represents the age of the
system. When the system reaches age K, it either continues
to age or it rejuvenates and starts afresh from age zero.
j
The successive passages through the zero state represent a
recurrent event. The probability that the recurrence time
equals K is p q.
We are interested in the question: How many years have
passed, i.e. how many income steps have been mounted, since
the last rejuvenation? This is the ''spent waiting time"
of the renewal process. Choosing an arbitrary starting point
we can say that in the year n the system will be in state Ek
if and only if the last rejuvenation occured in year n-K.
Letting n-fc increase we obtain in the limit the steady state
2)
probability of the 'hpent waiting time" . It is proportionate
1)
2)
f\%7 vol. I. 17.3, p.
C9fJ
Chanter V.
0
3
je
to the tail of the recurrence tine distribution,i.e. to p .
More directly, the vector of steady state probabilities
can be derived from the following two conditions;
UK = ?UK_1
UQ = + qu1 + qu2 + ... jsf
\
■The first condition ensures the invariance of the steady
state; the seco.nd ensures that entries to and emits fron
the income population balance.
It follows that
= P*u0 /
uQ =1 - p p < 1
The result.is, of course, identical with the distribution
of the spent waiting time,derived above.
o
U o
* IC. cM
UA
to the tail of the recurrence time distribution,i.e. to p .
More directly, the vector of steady state probabilities
can be derived from the following two conditions;
aK " PUK-1
ao 1°1 ♦ 9°2 + "• *' % ' VC
M ^ \ \
(Ehe first condition ensures the invariance of the steady
state; the seco.nd ensures that entries to and exits from
the income population balance.
It follows that
Ofc = P^o ^
un = 1 - p p < 1
o s
The result.is, of course, identical with the distribution
of the spent waiting time.derived above.
-Sa-
lt. will, be noted that the process can be described up to
this point without any reference to. income- We may regard the
states of the system represented by the elements of*, the matrix
as "grades”■ or "age'* -'possibly carreer, age or age in earning
life,, although in conformity with Champernowne we referred to
them as income classes. "
We have now to define the income in relation to the class
intervals of the matrix: The lower limit of class 1 is
taken to he the minimum'income. ¥e may choose the income
units such that the minimum income is unity, i.e. on the
logarithmic scale it is zero. The income y^ at the lower
limit of successive income classes K is defined hy
Xh
7*r = e
or In yK = Kh
where h is the size of the class interval on the log scale
1)
The difficulties arising from the discrete representation of
a continuous income variable in the matrix do not concern
us here. See £ioJ p. 62.
s-i C > . 'U'/<L ^
' 7^4
1
Y
/
- ^ /li
■£L.
C-____' j-~- 1 £t' y
To find the tail of the steady state distribution P(y^) we
sum (£) from X, to oo and obtain p . Thus
(3)
1x1 P<7K) = _cC
Evidently the crucial feature of the model is the geometric
distribution of the recurrence time. This relates hers to
the life-time of the persons as income receivers; since
promotion is automatic, the age of the system is measured
in income classes K. The age, or spent life-time, is
geometrically (approximately exponentially) distributed.
Since the income is also an exponential function of K ,
the Pareto law results from an elimination of time K from
1)
the two exponential functions. 7
This is exactly the same rattern of explanation as was used
* - i/;it, 2 3 _
in other fields by Simon £ 1#/ and myself and
which is'directly. descended from Tule CJ i wil° used it
2.
to explain the frequency of species in genera. According to
'Although Champernowne'sxmodel is more complicated than t
above^the essential features remain the same (oniy^u in
the solution is replaced by t, the solution of a differs:
'equation).
Footnotes to p.4
1) Although Champernowne ' s theory is more complicated than the
simple- model which takes its place in the above reasoning,
we. can easily extend the conclusions:. With Champernowne, the-
promotion is stochastic, with probabilities ‘ of non-promotion
and demotion. In this more general-case p in the above solution
■ Thas to be replaced by jo which is the root of the characteristic
equation of the matrix^j" 0*. , ",
'In o
the ratio
The Pareto coefficient in the simple case is -
■ '' of the- parameters of the two-exponential distributions; in the
■'.■■ more general case of Champernowne' the Pareto, coefficient'is-
-r-— - h could be regarded as the-parameter of'an age dis-
tribution, if the classes (states of the system) are regarded
'’’'as age classes IV kU
2\ Champ ernowne, apparently did nor know Yulers paper: it was
; E.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 ehe interplay of the two ex-
ponential distributions,_ i.e. of two stochastic processes-
this approach, size distribution is a transformed age
distribution, and the pattern of the Pareto law appears-
so_often simply because of the empirical importance of
exponential growth which makes both the age distribution
and the tranformation function exponential. Owing to the
conceptual density of Champemowne' s model the two
elements of life- cycle and promotion are merged into
one.
(which resides
There is, however, a diff emnce/\ in the interpretation not
in the form) between Champemowne's model and the others;
Since physical persons sooner or later die, the age in
his model is limited, while in the others (relating to
firms or wealth) there is always the probability of
virtually infinite life which accounts for a very
2.2.
peculiar character of the steady states concerned C V& 7 •
Further developments
We may consider the following stages in the treatment of
the income distribution;
I. Chamnernowne's model.
II«' Hutherford's model C'&J* He treated persons'life-
times explicitly.
III. The above models are open to criticism on two grounds;
First, income is not very suitable as a state variable
for a Markov process. It does not embody the "influence
5 a
Th/ose few units whidh survive for good continue to grow; on the
other end of. the scale new entrants/enlarge the total sample
of firms or wealthy dynasties. The continuance of the steady jcc
state with unchanged distribution is accompanied by a steadily
growing size of the sample whidh produces ever larger firms,
wealth holdings etc. These large units always existed potentially
but could not be realised x± as long as the sample was too small;
with the growth of the economy such potential sizes become actual
and in consequence the largest units represent a greater and greater
sha^e of the total although the theoretical distribution is
unchanged. With the finite life and the stable population
of Champemowne' s model this peculiar form of growing inequality
*
would not arise.
Further developments
We may consider the following stages in the treatment of the
income distribution:
I. Champemowne 1 s Model.
II. fiutherfordd model /I9/. He treated person's life-times explicitly.
III. The above models are open to criticisms on two grounds:
First, income is not very suitable as a state variable
for a Markov process. It does not embody the "influence
globale" (Polya) of the past. Second, the model is con-
fined to a life-cycle from <the«entrance to exit. But the
relevant stochastic process goes far beyond that. In
fact, when somebody starts in life, his chances of re-
ceiving certain incomes are already settled to a large
extent by the condition of his parents: By their wealth,
status, connections, reputation and' the education or training
they have been able to give him. In other words, the exits
and the entries in the life-time model are linked by in-
heritance etc., and the process of diffusion continues on
I >
a higher stage. J
to
. Both arguments point/\the same consequence; Ve mus” relate
the chances of getting certain incomes to the amount of
wealth, material and immaterial, and its distribution.
In this way we can link the income to a suitable state
variable (wealth) which is evolving in a long run process
through the generations.
Also, we shall then be able to answer the question why
income distribution is relatively stable, although so many
elements relevant to it are changing day by day; The answer
is that the stability lies in the distribution of wealth,
education, training etc., which change only slowly. '
V) zC ~
- L P» 342 seq.
l) Champernowne was, of course, aware of these facts,, as his thesis ^
■. of 1957 ( $ ) shows. His formalised model of 1953 is, however,
not well suited to reflect all the economic factors so well
stated in general terms in 1937 -
7
TT. In a further stage we should cease to take the wealthy distribution
as given and instead treat wealth and income as joint variables
in a process evolving over ±±mm the generations. Tie should continue
to regardthe influence of sltrwly accumulating wealth and other stocks
on the current income as a dominant feature of the process but,
trying to includethe preceding history of accumulation, we should
now also recognise that in the course of time the income acts on the
wealthy not- only in so far as it arises from wealth as a return and
is (partly) re-invested, but also insofar as .-new wealth is formed
from earned income in favorable circumstances and after_a lapsT, of
time required for accumulation.
That means that income and wealth are linked by two sands
of relations:One is the rate of return which links the current income
of property owners to their wealth, the other is the relation
between (unearned and) earned income of the past to the accumulation
of new wealth, in otherwords, the propensity to save, -n so far
as the past income is in practice fairly strongly correlated with
present income, the present earned income appears to have an -cfect
on the wealth which corresponds to the savings relation.
The influence of wealth on income via ihe rats of return
is the subject to be treated first over the next few pages.
The dependence of income on wealth
In the following we shall consider income as flowing from
wealth. This applies strictly to what we call unearned income.
The reader may think of that in the first place,but h e has to be
prepared for a more far reaching and wider interpretation barer on;
in fact, in praiicewe cahhot easddy separate unearned and earned
income, and the difficulty is perhaps greater than one that could be
overcome by adequate data ( especially in the case of unincorporated
business ).
Instead of the matrix of income transitions ased by
Champernowne we have to imagine an analogous matrix 1ffealth--ncome
whichshows for each amount of swealth the orobability of different
, \ *
incomes. •
ihe basis of the analysis is thud the conditional distribution
of income, given the wealth. Economically speaking this is the
probability of a certain rate of return to wealth or profit rate.
From this, if we know it, we can derive tlje distribution of incomej
8
provided we ’«crow the distribution of wealth. But the distribution
of wealth is known: It follows the Pareto law - over a fairly
wide range - and its pattern has also been explained
theoretically / i3/-
Denoting wealth by of the wealth distribution p* - c W“*” 1 dtf Tf , let us write for the density
or putting w » ln W
p( w) = c e ~ cLc** for w -^> 0 (4)
p(w) => 0 for w <. 0
If Y denotes income and y= ln T the conditional
density function of income can be represented in the form
f*(y-w), the density of a certain return on wealth. Sven without
knowing this f’unction we might manage to derive the distribution
of income from that of wealth provided we can make certain
assumptions about independence,
' we shall provisionally assume that the distribution of
the rate ^of return is independent of the amount of wealth.
In terms of random variables, if ^ and 'Xj
denote income, wealth and the rate of return, we have
If the random variables wealth and the rate of return are independent,
their sum can be represented by a convolution of the corresponding
density functions, and we shall in this way obtain the
distribution of income.
For the purposes of this calculation we shall replace
the density f*(y-w) by the mirror function f(w-y) which is also
independent of wealth. The two functions are symmetric and have
the same value ( in fact, the only difference is in the dimension :
TTnile the former refers to a rate of return per year, the reciprocal
value refers to the number of years income contained in the wealth ).
The calculation of the density of income q(y)
proceeds then by mixing the function f(w-y) with the density,
of wealth:
9
q(y) - f(w-y) dw - 3 e“ for w £
q(7) -• 0 or j *
where V'/jv'1 is the Laplace transform of f(w).
/ "
The above mixture is a Laplace transform of f(w) shifted to the
right 07 7.
The Laplace transform requires that the argument of the
function f be non-hegative. He have therefore to assume that
w^ ( we shall further below haw this restriction can be relaxed ).
Equation (3) shows that the Pareto form of the wealth
distribution is reproduced in the income distribution, provided
the independence condition is fulfilled and y •‘C w.
He have now to face the fact that the rate of' return on wealth
will in reality not be independent of wealth, -he cross-classifications
of wealth and income of wealth owners for Holland and Sweden
show that mean income is a linear function of wealth, the regression
coefficient being smaller than unity.
For the decline of the ratd of return with increasing
wealth zzzüzs various reasons are responsible;
relatively
The earned income will be/less important the greater
the wealth- In particular the income from ( non-corporate) business
will be higher in relation to wealth in the lower wealth classes.
Further, capital gains are not counted as income, but they affect
wealth, and they will be more important for lagge wealth, because
the proportion of shares held increases with wealth. The internal
accumulationof firms will not find expression in the income,
but quite probably in the wealth of the share holders. Also
appreciation of real estate may affect the large wealth proportionately
more. *
Now the rate of return is independent of wealth if
its conditional distribution is the same whatever the size of
wealth. It would seem that we might perhaps restore the
condition of independence simply by turning the system of coordinates
in the appropriate way, so that we would reduce the present to thd
former case. If we nan make the covariance of w and w-y zero
then the ceefficient of regression of y on w should be one,
as in ~he former case;
10
Cot (w, w-y ) - Var (w) - Cov ( w,y) - 0;
Cot (w.y ) a 1 .
7ar (w)
If the regression line of income on wealth is
7 - K.* + 7o
and/i! if the Tariance and higher moments of the conditional
income distribution are independent of wealth then we should
use instead of f(w-y) the function f (ATw* yc- - 7 )
and this distribution will be independent of wealth.
’He can then proceed as before:
q(j)
f( Aiw + Jc- 7 ) e*Ä1f dw =•
= c 0?^jexp ~ ( 7 - 7o )
]
q(y) - 0
for ^W> 7 - 7C
for t<w < 7 -7C .
The result is now that the Pareto shape of the wealth
distribution is reproduced in. the income distribution, but with a
larger Pareto coefficient ( since k’<0 )• This is exactly what
has to be explained ( income distributions are in fact more
"equal" than the wealth distributions, empirically, in the sense
described ). The particular shape of the rate of return distribution
has no influence onthe tail of 4he income distribution, as long as
it fulfills the independence conditions mentioned.
Concerning the restriction w^> y - jc
it should be remarked that we are free to shift the coordinate system
to any yc we choose so as to make the aboTe condition Talid,
with no consequence except that the conclusion about the Pareto tail
will be confined to incomes in excess of yc .
It would seem that in practice, in Tiew of the Talue of
, j0 must often be more or less high, so that the Parto pattern
will be confined to a rather narrow range of the income distribution
while in the caseof wealth it usually extends to the whole of the
assessed wealth data. This, it is true, partly results from the
fact that the wealth data are more truncated than the income data,
in Tiew of the underlying tax laws.
^0/j- '^lyiur) y
Income sind health: -muirical Patterns
The following remaiprs refer to the cross-classifications
of wealth and income published in Holland and Sweden. These data
show certain characteristics which are found aJ.so in other
cross-section data concerned with size distributions, especially
data from official publications like censuses etc.
The first feature is that the great bulk of the observations
is concentrated in the corner of the first ( the north-east ) quadrant.
In other wards these distributions are very skew. A great many of the
units are small in either dimension.
The second feature is that the wealth distribution is
heavily truncated ( in Sweden for example at 150.000 crowns)
while the income distribution is given down to rather low levels.
If the mean income in the various wealth classes is
calculated a. linear regression of a very regular pattern is obtained.
( This "regression of the first kind" as we may call it differs
from the usual least squares regression in that it does not
assume a -priori a certain mathematical function for the regressionf
see Harald Cramir, Mathematical Metheds of Statistics,Princeton 19^6^3»—
if the regression of the means turns out to be linear as is the case
here, then it shoild be the same as the result of a linear least
squares regression; this may be not quite true only in so far as
we fail to use weights for the means corresponding to the various
frequencies in the different wealth classes ).
The same regression of the first kind in the other dimension -
wealth on income - gives a completely different picture: rhe mean
wealth in the lower income classes does not indrease with income
at all;- for higher incomes it increases very strongly, so that
a strongly curved regression line results.
At least one, and probably the most important reason for
the curvi—linearity of this regression line lies in the truncation
of the wealth data. If we try to fill in the missing wealth data
in our imagination, according to plausible and common sense prior
knowledge, we find that the regression'of wealth on income might
well be quite linear and rather steep; at least it would be very
much nearer to linearity than it is now. The inclusion of cases ■
with wealth below the tax limit, which is probably the lower and
the more freauent the lower the income, would reduce the mean wealth
* ^ . 7
mf - . ‘ .
in all income classes but it w'uld reduce it the more the lower the
12
income. In other words, the mean wealth in low income classes,
as measured by the data^rhioh we have on the basis of tax assessments
very strongly overstates the real mean wealth, and this the more
the lower the income.
There is no proof, of course, that some curvilinearity
would not remain even if full wealth data were available. There
may even be theoretical reasons for that, as will be shown later.
One reason for linearity of the presumed full data should be
mentioned: It is rather queer that non-linearity affects onjy
one of the regression lines and the other is perfectly linear.
Only the truncation of wealth data gives a plausible explanation
of this contrast between the *wp regression lines.
3mpirical data: Holland and Sweden.
The cross-classificaiion of wealth and income, available
for the Netherlands and Sweden, will now be discussed in/the light
of the theory contained in equation £5") • "Verification" aan
hardly be expected: The rate of return explanation can not nearly
explain the income of property owners fully, since a large part of
it is earned income. Apart from other statistical difficulties
it must be kent in mind that the ?ar*to coefficient is always
more or less arbitrary, since it depends on the range of income
or wealth classes included when you measure it. 3ven inconclusive
data, however, are better than speculating in the void.
An evidence on which I rely heavily is the linear and
fairly regular character of the regression of mean income on wealth
( fig 1). The regression coefficient is in most cases around 2/3.
but it hay be as low as l/2 . Whether the higher moments of income
are independent of wealth is not easy to decide. While the
variance, in the Swedish data, increases in the higher wealth classes
this can plausibly be explained by the increase .in the range of
these classes ( the last but one wealth class has a range aboht
four times as great as the lower wealth blasses). The same
fact is relevant for the comparison of the conditional distribution
of income in the various wealth classes: They all have a ^areto tail,
the ^arfcto coefficient being markedly lower in.the last two or three'
wealth classes chan in the other s. This, again, may be
plausibly explained by the greater range of these high wealth
size classes.
13
Table
Sweden 1911
married
couples
single
oersons
Holland 1Q62/3
health distribution
Pareto coefficient
(whole range,
11 values) 1.78
Regression of
income on wealth
coefficient yC : 0.56
Calculated Pareto
coefficient for
income: 5»18
Actual Pareto
coefficiet for
income ( 5 values) 2.62
1.73
0.65
'2.66
2.14
ditto, excluding
the open size
class of wealth 5*12 2.62
1.53
0.63
2.20
2.08
The above table gives the vilues for the coefficient
of regression of the mean income on wealth. The correlation is very
high ( r is ,997 ) for the mean income values; for the mean square
correlation of the grouped income data it is modest, r?~ .25 in
Holland.
From the actual ^areto coefficient for wealth and the
coefficient we can calculate the Pareto coefficiet for income
in accordance with equation (6>). In Holland we obtain 2.20,
which compares with an actual Pareto woefficient (for income of
wealth owners of 2.08. ( The coefficient for all incomes including
pepple with no taxable wealth, is not markedly different, which
is rather surnrising).
Butch
TThile the/results are not-worse than could be expected f
the Swedish data are less straightforward. A peculiarity is here ,
that married couoles { to some extent also children ) are jointly
assessed for wealth tax so that their joint wealth and income is given
in,the data. This leads to a splitting up of the mass of wealth
14
holders into four groups ( couples where both hare wealth, couples where
one only has wealth, single men and single women ). ^he reduction in
sample size impairs the regularity of the data and I hare therefore
aggregated the four into two groups : married couples and single
pwrsons.
The calculated Pareto coefficients for income of wealth
owners are much higher than the actual ones ( ^aole l).
These calculated coefficients correspond more nearly to those of
all income receivers including the wealthless ones -the great
majority. They correspond also more nearly to these of the
conditional income distributions in all wealth classes except the
last three. In fact, if we exclude the open wealth class
fro© the income distribution ( which might perhaps be motivated
by the argument that it is not contained in the calculation of f<C
either ) then we get ?areto coefficients entirely in line with
the calculated ones ( Table 1). The motivation is not entirely
convincing and the results are inconclusive.
Sinae the conditional income distributions in the
wealth classes have been referred to several times, - give in the
following data for couples where both husband and wife have wealth.
conditional income distribution
Health in 000 K Lf>- mean J Pareto coefficient
150-175 4.74 4.06
175-200 4.76 5.55
200-250 4.79 3.63
250-500 4.84 5.92
c 0 1 0 c N"N 4.88 5.69
400-500 • 4-95 5.34
500-750 5.01 3.29
750-1000 5-09 5.00
1000-2000 5.18 -5.47
2000-5000 5.38 2.19
5000- (5.74) 1.17
All. 2.68
All without open . .•
wealth class 5.16
15
The conditional distributions have all Pareto tails although the
fit is bad ( only 4 values can be used ). The Pareto coefficient
is between 3 and 4 in all except the last two wealth classes,
where it is very low, and it is 2.68 for the whole income distribution.
It appears that the income distribution as a whole is -
as far as its tail is concerned - decisively influenced by the
last two wealth classes. This is due to the fact that most
of the top income receivers are in the last two wealth classes,
9
where the income distribution is very unequal simply owing
to the wide range of wealth in these two classes, as already
mentioned before. In this way the peculiar result arises that
the total income distribution is much more unequal than
almost all the conditional income distributions.
This in a way also anarwers the question which
might well be asked: Thy the pattern of income distribution
could not be derived from the conditional distributions
without reference to the wealth distribution.
Allometrie growth of income and wealth
The discussion of the relations of income and »wealzh will
if
now be extended to take accounyof influences in both directions.
The starting point will be the regression of income
on wealth which seems to be linear as far as the data go.
This might be regarded as a case of allometry, in analogy to
a "law" well known to biologists"' ' : Various parts of an organism
grow at different but constant rates and as a result the proportions
of their sizes ( on log scale ) remain constant in the growing body.
1) Ludwig von ^ertalanffy, general System Theory. Penguin 1968 p.65-
Devendra Sahal has used the allometrie law in combination with
the progress function in order to explain the Pareto distributio n ,
( in one dimension ); see A Formulation of the Pareto Distribution.
IMM, Science ^entre, 1000 Berlin 33( mimeo ). Although the
use I am trying to make of the law is different, I oweAto
Devendra Sahal to have my attention drawn to it. '
16
Thus in our case the income is 7 - JC w jo where we suppose
that the distribution of income and wealth reflects in some way
a growth process which has taken place in the past.
W
c, e
In the ordinary formulation we should have
xt
AC
,0
■j
(?)
Y - c±e
and after elimination of time
<D / V
y - In c , =• — (, w - In c, )
^ C<,
which corresponds to our regression with kl ~ ~
How in economics random elements or shocks play a large
role and growth is influenced by them. We should therefore write,
instead . of the above:
at*
W - c , e +
c e
(It
E (t)
'] (t)
where £ (t) and K) (t) are random variables. As a result we obtain
0 r
In ( 7 ~ yj(t) ) - In c 2 m - j ln( ^ (t) ) - In c/j".
(a)
The random variables or "errors" ( not mere errors
of observation^of course ) are in both variables. The common
tactics of regression is to ascribe all the "errors" to one
variable only, which yields two regression lines according to
. whether you choose the one or the other variable as the
repository of the errors. The equation (3) will not correspond
to either of the two regression lines, ^t may be guessed that it
will lie somewhere between them.
Let us now start from a different angle and consider
the two kinds of relations that exist between income and wealth.
One is the influence of wealth on income via the rate of return.
It mainly affects unearned income. The other is the influence of
past incomes on wealth via the propensity to save. Since
present and past incomes are fairly strongly correlated it
will appear as if current income acts on wealth via the.
propensity to save. .
At first sight it seems that the linear regression
of income on wealth represents simply the influehce of wealth
via the rate of return. There is however an alternative interpretation.
17
Larger wealth presupposes larger income ( if it had to be saved
out of it)^ therefore the mean income has to be higher for
larger wealth. Prom this point of view it is not surprising
that the increase of income with wealth is less than proportionate
because wealth in an integral of income taken over time,
if saving can be assumed to be a constant proportion of income.
The interpretation is only weakened to the extent to which it
could be argued that the creation of new wealth from earned
income will affect mostly the lower wealth classes ( only
earned income is really relevant in this context where income
is supposed to play an active role vis-a-vis wealth ).
The other regression - wealth on income - should
on the face of it represent the effect of the propensity to
save, the creation of new wealth from earned income
( continuing primary accumulation ). The curvilinear shape
might be compatible with this interpretation, since for the
lower, incomes the saving will play no role and wealth will
only start emerging when income has reached a certain level,
and then it will rise steeply ( because it is an integral,
see above ). Here again, an alternative interpretation is
possible: If income is large, it- probably has been derived
from large wealth, therefore to larger income will
on the average correspond larger wealth ( rate of return relation
.4 *
Thus the two relations or theories or laws
behind the joint distribution of wealth and income seem
to be both relevant for each of the two regression lines,
although possibly not to the same extent ( each of the
regression lines may be be more strongly influenced by
one relation than by the other ).
The preceding arguments implied that, to.' some exent,
each regression line is an inverse to the other regression line.
In order to make this clear let me choose an example from
another field', where there is only one "law" or at least we
can pretend there is only one. Take the size distribution of
manufacturing plants according to output and cost. The law
consists in the effects of scale on cost in relation to output.
The regression of cost ( or employment) on output will show
a coefficient less than one, decreasing cost to scale.
We expect the other regression to be the inverse of the first
19
showing disproportionately Increasing output with any increase
in input ( cost, or employment ). In fact, however, we often
find that it is not so and that both regression coefficients
are smaller than one, decreasing cost and decreasing returns
apparently coexisting.
How is this possible? -t can only occur with
wide dispersion round the regression line. The exceptionally
effieient plant will tend to be counted as small plant
in the input dimension while the unusually inefficient ones
will be counted as large. In consequence there will be
a bias in favour of decreasing returns as measured in the
input dimension ( regression of output on cost or employment ).
The inversion äf the regression corresponds
to the fact that the ration of the two standard deviations
is reciprocal in the two regression coefficients. If it
is 9/l0 in the regression of input on output, it is 1o/9
in the other regression. But, unless the correlation coeffiaient
is sufficiently high, the regression coefficients
will both have values below unity.
The same mechanism rust also be at work in' '
the wealth-income distribution: These with high return for
a given wealth will be classified unong large incomes, those
with low returns with the same wealth among small incomes,
which tends to counteract the natural tendency of wealth to
increase with income. This may have contributed to the
flatness of the wealth-income regression in the lower income
range, although the chief reason for that is no doubt the
truncation of the wealth distribution.
The preceding example ;f plant size, in which
only one underlying theoretical relutnon ispresumed to exist,
shows that while it is logical to expect in this case,
if one regression reflects the underlying relation, that the
other should as it were represent the inverse of .it, yet in
reality this will not be true because the second regression
will be more or less distorted by the dispersion of values
round the first regression line.
If we have two underlying relations then each of
the regression lines will be’ influenced by both of them,
either directly or indirectly, because each will be to some
19
extent an inversion of the other; in addition, each will be
influenced by the dispersion of the values round the other
regression line.
Thus . each of the regression lines will
represent a compromise between the two underlying relations
the weight of them being different in the one and the other
regression line. No regression line therefore will be
a true reflection of an underlying causal (or rather stochastic )
relation. We shall have a better chance of understanding the
meaning of joint size distributions of this type if we regard
them as residues of a growth process. Set us therefore return
to the allometric law. As far as its relation to the joint
distribution wealth-income is concerned we have to make
two observations:
1) If the regression line income on wealth could
be regarded as an expression of the allometric law then,
as it will be remembered, the regression coefficient
kC
7 - 70
--5----
is the ratio of the two Pareto,coefficients of the income
i
and the wealth distribution. *
2) Following up the idea that wealth can be
explained from saving over a certain time and saving cam
be explained from income, taking saving propensity as given,
we can darive the distribution of wealth from that of income
in much the same way .as the other way round:
We explain'the saving distribution as a convnlution
of^the income distribution amd of the(distribution of the
» (
propensity to save ( savings ratio ):
q'(s) - q(y) * g ( iry -is ), \ (9)
amd the wealth distribution as a convulution of this and
the time the saving has accumulated (which will be finite
in the case of earned income but not necessarily for unearned
income ): : „ . -
q''(w) - q'(s)*h(s-w) (lO)
From this wealth distribution we should by means of the
original transformation (o) come back to the income
distribution
20
qt • »(7) - q' '(▼) * f (/v w - 7 )
(11)
How cam this series of transformations square
so as to produce a steady state of the joint distribution?
explain this we have to separate income at least theoretically
into earned and unearned income, -he distribution of earned
income has to be regarded as given from outside; its explanation
has to be sought separately; empirically at any rate it is
a Pareto distribution. It may be regarded as constant in the
simplest case. From it flows the continuing primary accumulation
which contributes to the accumulation of existing wealth holders
and also creates new wealth holders ( a kind of social mobility')*
This wealth then creates property income and secundary accumulation.
The process leads from a relatively moderate inquality of
earned income to more unequal primary wealth, and even more
unequal secundary wealth.
-*-n this way wa can imagine that total income
and total wealth will grow at constant rates. The corresponding
allometric proportion does not appear directly in the cross-section
where instead appear the two regression lines which are neither
of them a reliable estimate of the ratio of the two Pareto coefficients.
^----S)
this be regarded ad the reason why the attempt '
verification of equation (8} on the basis of Swedish
lead to unsatisfactory results?
References
/I/ Atkinson, A.B. (ed) The Personal Distribution of Incomes.
London 1976
/2/ Atkinson, A.B. Unequal Shares-Incomes 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 econometri-
sche Onderzoekingen, No 3, 1965: Lnkomens ongelij'kheid,
/6/ Central Bureau voor de Statistiek, Inkomensverdeling 1962
en vermogensverdeling 1963
/7/ Champernowne, D.G., A 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. X. 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
t
/1 5/ Roberts, D.R., Executive Compensation. Free Press Glencoe, Illinois 1959.
/I 6/ ; Simon, H.A., The Compensation of Executives. Sociometry 1957.
/I 7/ Simon, H.A., The New Science of Management Decision, 1960
718/ Simon, H.A., "On a Class of Skew Distribution Functions" Biometrica 1955.
719/ Rutherford, R.S.G., "Income Distribution: A New Model", Econometrica 1955.
7207 Statistika Centralbyran (Stockholm), Inkomst och Förmögenhet.
721 / Steindl, J., Random Processes and the Growth of Firms. London 1965.
722/ Steindl, J., The Distribution of Wealth after a Model of Wold and Whittle. R.E.St. July 1972
723/ Steindl, J., "Size distributions". International Encyclopedia of the Social Sciences. New York 1967.
7247 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)
7257 Woodward, J., Management and Technology, London 1958