The Personal Distribution of Income
Josef
Steindl
AC14446372
Josef Steindl
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
pioneering paper while fully demo»fbrating a powerful
new method, has not entirely disposed of the individual
Zcr tietecli&l *
problem, which it was designed -to solver.
Champernowne's Model/ <*
(tx&sfa * & qti '
I shall give a simplified version of Champernowne's model
1/jJ 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
There are only two alternative transition for a person in
this system: Either a rise of income from one year to the
next (note that the income classes are on the log scale)
which has probability p; or death of the person, i.e.
transition to the zero class, which has probability q
(p + q = 1). In addition, there are entries from the zero
to
class^replenish the stock of income receiveif.
The essence of this- model is described by Feller1^ in the
following terms: The state 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 a, is i '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 renex^al process. Choosing an arbitrary starting point
we can say that in the year n the system will be in state E/t
if and only if the last rejuvenation occured in year n-M,.
Letting n-K increase we obtain in the limit the steady state
probability of the'fepent waiting time"£^. It is proportionate
!)Z"7 7 Vol. I. 2?.3, p. 382.
2V^T7 Chapter V.
Jr
to the tail of the recurrence time distribution, : .e. to p .
More directly, the vector of steady state probabilities
can be derived from the following two conditions:
UK = puK-1
uQ * qu,_ + + <auo + • • • O)
The first condition ensures the invariance of the steady
state; the second ensures that entries to and exits from
the income population balance.
It follows that
u£ * p*SiQ (2)
uQ»1-p p<1
The result is, of course, identical with the distribution
of the spent waiting tine.derived above.
\—•
We have now to define the income in relation to the class
intervals of the matrix: The lower limit of class 1 is
taken to be the minimum income. We may choose the income
units such that the minimum income is unity, i.e. on the
logarithmic scale it is zero. The income y^ at the lower
limit of successive income classes k is defined by
fc.h
H * e
or In y^ • kh
where h is the size of the class interval on the log scale '.
The difficulties arising from the discrete representation of
a continuous income variable in the matrix do not concern
us here. See fgl/ p. 62.
- 3a -
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 grades' or 'age" - possibly carreer age or age in earning
lite, although in conformity with Champernowne we referred” to
,, * »« i
them as income classes.
4
To find the tail of the steady state distribution P(y^) we
sum (2) from ft to and obtain p . Thus
-1
In P(y~) = In p . ^ In
and, putting -h“ In p * o( we have
In P(y-:) * -oo 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 ft. The age, or spent life-time, is
geometrically (appronimately exponentially) distributed,
^ince the income is also an exponential function of ft ,
the Pareto law results from an elimination of time ft from
the two exponential functions. '
This is exactly the same pattern of explanation as wsg used
in other fields by Simon J and myself /^2, 13, 147 anci
t
which is^directly descended from Xule /~15_7, who used it
1
to explain the frequencj of species in genera. According to
mmmmmmm ^ *TVv y ‘
‘^Although Champernowne's model is more complicated than the
above:, the essential features remain the same (only p in
the solution is replaced by b, the solution of a difference
equation).
&u t Ay '
/y -0efa&ti/x. fa* S-fi- t d"
c4 ■
_
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 dhampernowne, the
promotion i3 stochastic, with probabilities of non-promotion
and demotion. In this more general case p in the above solution
has to be replaced by b which is the root of the characteristic
equation of the matrix ^ .
The Pareto coefficient in the simple case is - ~-R , the ratio
of the parameters of the two exponential distributions; in the
more general case of Champernowne the Pareto coefficient is
*| ^
" —j-— . b could be regarded as the parameter of ajaf age dis-
tribution, if the classes (states of the system) are regarded
as age classes
2) Champernowne apparently did not know Yule's paper: It was
H.Simon's merit to have brought it to the attention of
economists; unfortunately he reproduced it in a form which
obscured its essence, which is the interplay of the two ex-
ponential distributions, i.e. of two stochastic processes.
5
this approach* slM distribution la o temaforaod age
CTCCMA4
distribution* and the pattern ot the far©to law eppeere
Ut- i4c&h***h**' vUeI«i **■
«o of ten/ simply bwiaH of Hi* onpfrlcel iapaartmoQ of
©spcuMOtlol growth which laaioc x>th tbs age distribution
and Wm tronfCroatian funotlon iipc—atliil# Owing to the
conceptual density of itmiinrinana* n aodtl the two
©laments of life- cycle and promotion ero aergod into
one*
(which resides
' ,9 .:?* . „ ..: .: A ; : r. ns;
In the fora$ between Cbcsqperoouno'a oodel end Use others*
dinco physical persons sooner or Inter die* the ago in
Ms nodal la United* while in the others (reXatiHG to
firm or wealth) there is always the probability of
virtually infinite life which accounts for ® very
peculiar eliswnstar of the steady states ecooersad
:\irs:x,y CCTOO.
Me nay consider the following stage© is the treatment of
the iso-one diotrlbutioni
I • 05,-ii.gjamoune “ a nodal•
XX« Sutherford'e aedel /~*10 J* Be trestsd personafliie~
tines explicitly#
do....vi~ 7) ...'. •
First* income is not very suitable as © state variable
for a Ifer’zov process# It does not ©sbody the !t influence
5a
Th^ose fww units whidh survive for good oontinue to grow* on the
other end of the scale new entrantsenlarge the total sample
of firms or wealthy dynasties. The c mtinueace of the steady xx
state with unchanged distribution is accompanied by a steadily
growing size of the sample whidh produces over larger firms,
wealth holdings etc. These large units always existed potentially
but could not be realijrod sdt as long as the sample was too smallj
with the growth of the economy such potential sizes become actual
and in consequenoe the largest units represent a greater and greater
share of the total although the theoretical distribution is
unchanged. With the finite life and the stable population
of Charapem wne’s m^del this peculiar form of growing inequality
would not arise.
Further devel omenta
e may c nsider the f 11)wing stages in the treatment f the
income distributi n»
I. Champem wne’s Model.
II. Rutherf rdd model /I9/. He treated person’s life-times explicitly.
III. The above models are open to criticism^ on two gr undsi
First, income is not very suitable as a state variable
for a Markov process. It does not embody the "influence
6
globale” (Polya) of the past. Second, the model is con-
fined to a life-cycle from »—* entrance to exit. But the
relevant stochastic process goes far beyond that. In
fact, when somebody starts in life, his chances of re-
ceiving certain incomes are already settled to a large
extent by the condition of his parents: By their wealth,
status, connections, reputation and the education or training
they have been able to give him. In other words, the exits
and the entries in the life-time model are linked by in-
heritance etc., and the process of diffusion continues on
a higher stage.
tO > Tly
Both arguments point/lthe same conseqaence: ¥e must relate
the chances of getting certain incomes to the amount of
wealth, material and immaterial, and its distribution.
In this way we can link the income to a suitable state
variable (wealth) which is evolving in a long run process
through the generations.
Also, we shall then be able to answer the question ifhy
income distribution is relatively stable, although so many
elements relevant to it are changing day by day; She answer
is that the stability lies in the distribution of wealth,
*)
education, training etc., which change only slowly.
12 7 p. 142
seq«
---- -—............” .c |
) • ^*******~
***» *j w ^ \
tj ho (
4u,t Xd> W Si *t» f~ st+f-£»sL
C iiHjt<46A *
m
_
Footnote 1 to p.6
1) Champernowne was, of course, aware of these
of 1937 ( ) shows. His formalised model
not well suited to reflect all the economic
stated in general terms in 1937.
facts, as his thesis
of 1953 is, however,
factors so well
7
TV. In a further stage we should cease to take the wealths distribution
as given and instead treat wealth and inc me as joint variables
in a process evolving over ±±m* the generations. We should continue
to regardthe influence of slwwly accumulating wealth and other stocks
on the current income as a dominant feature of the pr cess but,
trying to ineludethe preceding history of accumulation, we should
n w also recognise that in the course of time the income acts on the
wealth? not only in so far as it arises from we 1th as a return and
is (partly) re-inve ted, but also insofar as new wealth is formed
from earned income in favorable circumstances and after a lapse of
time required for accumulation.
That means that income and wealth are linked by two kinds
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, In so far
as the past incone is in practice fairly strongly correlated with
resent income, the present earned income appears to have an effect
on the wealth which corresponds to the savings relation.
The influence of wealth m income via hhe rate of r turn
is the subject to be treated first over the next few pages
The dependence of income on wealth
I the following we shall consider inc me as flowing from
wen th. This applies sirictly to what we call unearned inc me.
The reader nay think of that in he first place but h e has to be
prepared fra m re far reaching and wider interpretation later on}
in fact, in pratice we cahnot asiiy separate unearned and earned
income and the difficulty is perhaps greater than one that could be
vercome by adequate data ( especially in the case of unincorporated
business ).
Instea of the matrix of income transitions used by
Champemowne we have to imagine an analogous matrix Wealth-Income
whiehshows for each amount of wealth the probability of differ >t
ncomes.
I’he basis of the analysis is thud the conditional distribution
of income, given the wealth. Economically speak ng t is is the
probata ity of a certain rate of return to wealth or profit rate.
Prom this, if we know it, we can derive t&e distribution of income ^
8
provided we know the di tribution of wealth. But the distribution
of wealth :s kn wn! It follows the Pareto law - over a fairly
wide range - and its pattern has also been explained
theoretically / 13/-
Denotin- wealth by W , let us write for the density
of the wealth distributinn
p* * c W** 1 dW
or putting w ® In W
p(w) - c e •o( W^W for w 0 (4)
0 for w < 0
If Y den tes inco; e and y* In Y the conditional
density functi n f incune can be represented in the f ria
f*(y-w)t the density of a certain return on wealth. liven ithout
knowing this functi n we might manage to derive the distribution
of income from that of wealth provide! we can make certain
assuapti ns about independence.
e sha 1 provisionally assume that the istribution of
the rate of return is independent of the mount of wealth.
In terms of random variables, if / CtT and
denote inco e, 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 mirr r functi n f(w-y) which is ala'
independent of wealth. The two functi ns are symmetric and have
the same value ( in fact, t e nly difference is in the dimension ;
While the farmer refers t- a rate f return er year the recipr cal
value refers to the number f years inc me c ntained in the wealth ).
The calculation of the density of income q(y)
proceeds then by mixing the functi n f(w-y) with the density
of wealth*
9
S' (JO
*(y) - /f(w-y) e* *w dw
q(y) a 0
f V w > y >
(5)
r w < y *
where
is the
Laplace transform of f(w).
The ab >ve mixture is a Laplace transform of f(w) shifted to the
right by y.
The Laplace transform requires that the argument £ the
function f be non-faegative. We have therefore to assu- e that
w -^Ly ( we shall further below haw this restriction can be relaxed ).
Equation (5) shows that the Pareto form of the ealth
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 r turn n wealth
will in reality not be independent of wealth, ^he cross-class ifications
of wealth and income f wealth owners for Holland and Sweden
ah w that mean income is a linear function f wealth, the regression
c efficient being smaller than unity.
P r the define of the raad >f return with increasing
wealth xxxiixx variola reas ns ar®r^f||•1
The earned income will be/leas imp; riant the greater
the wealth. In particular the income fr a ( non-c rp rate) business
?/tll be higher in relation to wealth in the lower wealth classes.
Further, capital gains are nt counted as income, but they affect
wealth, and they will be nnre important f or 1 ayge wealth, because
the proportion of shares held increases with wealth. The internal
accumulationof firms will not find expressi>n in the income,
but quite probably in the wealth of the share holders. Also
appreciation of real estate may affect the lar e wealth pr p rtionately
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 sipply by turning the system of coordinates
in the appropriate way, so that we would reduce the resent to the
former case. If we can mak the c variance of w and w-y zero
then the coefficient of regression of y on w should be one,
as in the former case}
10
Cov (w, w-y ) « Var (w) - Cov ( w,y) = Oj
If the regression line f income on wealth is
y “ <» + yc
and/^ if the variance and higher moments >f the conditional
income distributi n are independent of wealth then we should
use instead f f(w-y) the function f ( ^w+ yt - y )
and this distribution will be independent of wealth.
We can ten proceed as beforei
y c>
q(y) - 0
for K, w < y -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 •< < 1 ). 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 the income distribution, as long as
it fulfills the independence conditions mentioned.
Concerning the res trie tin a. w>y - Vo
it should be remarked that we are free to shift the coordinate system
t any yo we choose so as t> make the ab ve condition valid,
with no c nsequence except that the c nclusi n about the Pareto tail
will be c nfin#d to incomes in excess of ya .
It would seem that in practice, in view of the value f
fC , must ften be more r less high s> that the Part > pattern
will be c nflned to a rather narrow range if the income distribution
while in the case f wealth it usually extends to the whole of the
assessed wealth data. This, it is true partly results fr n the
fact that the wealth data are more truncated than the income data,
in view of the underlying tax laws.
12
income. In other words, the mean wealth in low income classes,
as measured by the datajsrhich we have on the basis of tax assessments
very strongly verstates the real mean wealth, and this the m re
the 1 wer the income
There is no pr of f c ' ree that a me curvilinearity
w uld n t remain even if full wealth data were available. Th re
may oven be the retical reas ns f r that, as will be h >m later.
One reas n for linea ity >f the presumed full data sh >uld be
menti-ned* It is rather queer that n n-linearity affects njy
one of the regression lines and the other is perfectly linear.
Only the truncati n f wealth data gives a plausible explanation
of this c ntrast between the two regression linea.
Empirical datai Holland and Sweden.
The cross •.classification of wealth ad income available
for the Netherlands and Sweden, will now be discussed it-the light
of the theory contained in equation . ‘'Verification" can
hardly be expected* The rate of return explanation can not nearly
explain the income of prope ty owners fully, since a large part of
it is earned income. Apart from other statistical difficulties
it must be kept in mind that the ? rtto coefficient is always
m re or less arbitrary, since it demands on the range of inc -r..e
or wealth classes included when you measure it. -oren inconclusive
data, h wevar, are better than speculating in the v id.
An evidence n which I rely heavily is the iinear and
fairly regul r character f the regression :.f ean inc > e n viealth
( fig 1). The regression c efficient ie in most cases armad 2/j,
but it hay be as low as l/2 . Whether the higher moments of inc- e
are independent of wealth is n>t easy t decide While the
variance, in t e Swedish data, increases in the high r wealth classes
this can plausibly be explained by the increase in the range of
these classes ( the last but >ne wealth class has a range ab ht
f ur times aa great as the lower wealth blasseo). The same
fact is relevant for the comparis>n of the conditional distribution
of inc me in the vari us wealth classes* They all have a 1 arete tail,
the art to coefficient being markedly lower in the lost two or three
wealth classes than in the other s. This, again, may be
plausibly explained by the gre ter range of these high wealth
size classes.
11
Income and health* Ggpjrical Fatteras
The foil wing renakrs refer to the cross-classifications
of wealth and income published in Holland and Sweden. These data
show certain characteristics which are found ajso in other
cross-section data concerned with size distributions, especially
data from official publications like censuses etc.
The first feature is that the groat bulk of the observations
is concentrated in the corner of the first ( the north-east ) quadrant.
In other words 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 150.0C crowns)
while the inc me distribution is given down to rather lw levels.
If the mean income in the vari>us wealth classes is
calculated aalinear regressim of a very regular pattern is btained.
( This "regression of the first kind" as we may call it differs
fr >m the usual least squares regressin in that it d ees not
assume a or!rl a certain mothem tiwal functifor the re .ression;
see Gerald CramAr, Mathematical etheds f Statistics,Pro neeton 1946jfo tjc>
m
if the regression of the means turns out to be inear as is the case
here, then it sh lid be the same as the result of a linear least
squares regress! n; this may be not quite true only in so far as
we fail to use weights for the means corresponding to the various
fre uencies in the different wealth classes ).
The same regression of the first kind in the other dimension «
wealth on income — gives a completely different picture* he 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 ourved 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 o r imagination, according to plausible and common ses.se prior
knowledge, we find that the regression of wealth on incot e might
well be quite linear and rather steep} at least it w >u d be very
much nearer to linearity than it is n w The inclusion of cases
with wealth below the tax li it, which is rrbably he lower and
the more frequent the 1 wer the inc me, w uld reduce the mean wealih
in all inc sae c asses but it w u d reduce it the more the 1 wer the
13
Table 1
Sweden 1971
3|S^!?ns
H-.1I n 1962/5
Wealth distribution
Pareto coefficient
(whole range,
11 values) 1.78 1.73 1»38
Regression of
income on wealth
coefficient /( * 0.56
Calculated Pareto
coefficient for
incomes 3*18
0.65
0.63
2.66
2.20
Actual Pareto
c efficiet £ r
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 vAlues 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 ^'aret coefficle t for wealth and the
coefficient A5 we can calculate the Pareto coefficiet for income
in accordance with equation ($). In Holland we obtain 2.20,
which compares with an actual Paret oefficieit ptoT income of
wealth owners of 2,08. ( The coefficient for all Incomes includi g
pepple with no taxable wealth, is n t markedly different, which
is rather surprising).
Sutch . ,
hile the/results are not wrse than c uld be expected
the Swedish data are less straightforward. A peculiarity is here
that married c uploo ( to s me extent als children ) are j intly
assessed f r wealth tax s that their joint wealth and inc me is given
in the data This leads t a splitting up f the maos f we .1th
14
h lders int fur gr -ups ( o u lea where b th have wealth- couples wh re
ne • nly has wealth, single men and single r .men ), •‘■'he reducti n in
sample 3ise Impairs the regularity f the data and I have therefore
aggregated the four int two groups i married c uples and single
pees ns.
The calculated ftoet> coefficients for income of wealth woe
wners are much higher than the actual ones ( ^able 1),
These calculated c efficients c >rrespond more nearly to those of
all income receivers including the wealthless ones -t e great
majority. They correspond also more nearly to those of the
conditional income distribtti >ns in all wealth classes except the
last three. In fact, if we exclude the open wealth class
fr g the income distribution ( which might perhaps be motivated
by the argument that it is not contained in the calculation of AT
either ) the we get Pareto coefficients entirely in line with
the calculated ones ( Table 1). The motivation is not enti el^
convincing and the results are inconclusive.
9inde the conditional income distributions in the
wealth classes have been referred to several times. * give in the
fallowing data for couples where b th husband and wife have wealth.
conditional inc me distributi n
Wealth in ~ K mean Pareto coefficient
150.175 474 4.06
175-2' - 4 76 3-55
2 ^-250 4-79 3 03
250-30 4 04 392
3 -4o 4.88 3.69
4 0.5,00 4-95 3-34
5 -75 5.01 3.29
75-1 r 5. 9 3.0
10 -2 0 5.10 3.47
2 0-5 0 5.38 2.19
5'00- (5.74) 1.17
All 2.68
All without open
wealth class 3.16
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 la 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 inc >me distribution as a whole is -
as far a3 its tail is concerned - decisively influenced by the
last two wealth classes. This is due t the fact that n st
of the t >p inc ne receivers are in the last iw wealth classes,
where the inc me diotributi n is very unequal simply owing
to the wide range f wealth in these two classes, as already
menti nod bef re In this way the oeculiar result arises that
the t tal inc me distributi n is much ra^re unequal than
almost all the conditional inc me distributions.
This in a way also answers the quest! n which
might well be aakeds ;hy the pattern of inc me diotributi n
c>uld not be derived fr m the conditional distributions
without reference to the wealth distribution.
Allometric growth of income and wealth
The discussion of the relatione of income and wealth will
now be extended t take accoun^of influences in both direc i&ns.
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 allnmetry, in anal gy to
a ''law” well known to biologists1 t Various parts of an organism
grow at different but constant rates and as a result the pr portion
of their sizes ( on log scale ) remain constant in the growing body.
1) Ludwig v -n -^ortalanffy, feneral System The >ry Penguin 19^8 p.63
Devendra Sahal has used the aTT~metric law in combination with
the progress funoti>n in order t- explain the Paret distributi) n
( in ne dime .si n ); see A F naulati n f the Paret' Distributi n.
US.!, Science ^ontre, 10 0 Berlin 33( mime > ). ■Ala-*- Although the
use I am trying to male f the law is different I owe t
Devendra Sah.nl 1o have my attention drawn to it.
16
Thus in ur case the inc me is y » K w ^ where we supp se
that the distribution >f income and wealth reflects in some way
a growth pr cess which has taken place in the past
In the ordinary fonulation we should have
and after elimination of time
y - In c* •» ( w • In c . )
<* . _ A
which corresponds to our regression with »v *
X
(7)
how in economics random elements or shocks play a large
role and growth is influenced by them. We should therefore write,
instead og of the above:
W => c . e
Y = c ^
£ (t)
(t)
where £ (t) and Yj (t) are rand m variables. As a result we obtain
a (
In ( Y - >j(t) ) - In c;, * ~ iln( w - £ (t) ) - In cj
(8)
The random variables or "err rs" ( not mere ei*r rs
of observati n( f curse ) are in both variables The common
tactics of re ressi n is to ascribe all the "e Tors" to one
variable only, which yields tw regression lines acc rding to
srf whether you choose the one or the other variable as the
rep sitory f the errors. The equation (8) will not correspond
to either >f 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 t® save. Since
present and past incomes are fairly strongly correlated it
will appear as if current income acts qin wealth via the
li ' \
propensity to save. I; \
At first sight it seems that the linear regression
of income on wealth represents simply the influence of wealth
via the rate of ret rn. There is h wever an alternative interpretation.
;
17
Larger wealth presupposes larger income ( if it had to he saved
out of it) } t erefore the mean income has t> be higher for
lar er wealth. Prom thid point of view it is not surprising
that the increase of income with wealth is less than proportionate,
because wealth in an integral f income taken ver time ^
if saving can be assumed t be a cinstant proportion of inc me.
The interpretation is only weakened to the extent to which it
could be argued that the creation of new wealth from earned
Inc me will affect mostly the lower wealth classes ( only
earned inc me i3 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 ereation 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 st eply ( 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 f return relation ).
n f[
Thus the #tw> relatione or theories >r laws
behind the j->int distribution of wealth and inc me seem
to be both relevant for each of the two regression lines
although possibly not to the same extent ( each f the
regressi n lines may be be more strongly influenced by
one relation than by the other ).
The preceding arguments implied that.to^ s me exeat,
each regression line is an inverse t the other regression line.
In order to make this clear let me choose an example fr m
another field/ where there is only one "law" or at least we
can pretend there is only one. 'fake 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
18
showing disproportionately increasing output with any increase
in input ( cost, or employment ). In fact, however, we often
find that it is not so amd that both regression coefficients
are smaller than one, decreasing cost and decreasing returns
apparently coexisting.
How is this possible? It can only occur with
wide dispersion round the regression line, The exceptionally
efficient plant will tend to be counted as small plant
in the input dimension while the unusually inefficient ones
?;ill be counted as large. In c nsequence there will be
a bias in fav ur of decreasing returns as measured in the
input dimension ( regression of output on cost or employment ),
The inversi n bf the regression corresponds
t the fact that the rati % of the two standard deviations
is recipr cal in the two regress! n c efficients If it
is 9/11 in the regression f input n >utput, it is to/9
in the other re ressi n But unless the correl ti n c efficient
is sufficiently high, the regression coefficients
will both have values be law unity.
The same mechanism must also be at work ir.tfc
the wealth-income distribution: Those with high return for
a given wealth will be classified among large incomes, those
with low returns with the same we 1th among small incomes,
which tends to counteract the natural tendency of wealth to
increase with income. ;'his may have contributed to the
flatness of the wealth-income regression in the lower income
rage, although the chief reason for that is no doubt the
truncation of the wealth distribution.
The preceding example of 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 t e sec nd regress! n
will be m re or less distorted by the dispersi n of values
round the first regression line.
If we have two underlying relati-ns then each f
the regress! n lines will be influenced by b th f them,
either directly r indirectly, because each will be t sme
extent an inversion of the other; in addition, each will be
influenced by the dispersion f the values r^und the other
recession line.
Thus ltfiT. each-.*# 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 ither
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. Cet 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 he remembered, the regression coefficient
is the rati ■ of the tw P&ret ja efficients if the inc me
and the wealth distributi n
2) Foil wing up the idea that wealth can be
explained fr m saving ver a certain time and saving can
be explained fr m inc me taking saving r pensity as given,
we can deirive the distributi n f wealth fr m that of income
in much the same way as the ther way r undi
V/e explain the saving distributi n as a c nvnluti n
ofjthe income distribution and >f thesistribution f the
propensity to save ( savings ratio ):
q«(s) - q(y) * g ( Vy - s ), (9)
and the wealth distribution as a convalution 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 )s
q«»(w) « ^ q'(s) * h ( s - w ) (10)
From this wealth distribution we should by means f the
original transformati n (6) c me back to the income
distribution
(11)
20
q»‘*(y) - q*»(w) * f (K w - y )
Row can this series of transf rmati ms square
an as to produce a steady state of the joint distribution?
20 explain this we have t separate income at least theoretically
into earned and unearned income, ^he distribution of earned
inc me has to be regarded as given from outside; its explanation
has to be sught separately; empirically at any rate it s
a Pareto distribution. It may be regarded as constant in the
simplest case. Prom it flows the continuing primary a -cumulation
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 inquall ty 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 vzealth will grow at constant rates. The corresponding
alloiuetric 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 Fardto coefficients.
Can this be regarded ad the reason why the attempt
at empirical verification of equation (8‘) on the basis of Swedish
and ^vfcsh data lead to unsatisfactory results?
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