Konvolut Wealth and Income Distribution 5
Josef
Steindl
AL00659471
of the person concerned, the time curve during the life
being called the carrfier. The carrier will, of course,
depend to some extent on the education, so that we have
a link with the element from which we started.
Can we assume that the hierarchy of grades evolves on
known patterns leading to skew distributions?
Such a pattern would result from a general consideration
of the growing complication of society. As a result of
scientific and technological developments the amount of
information which must be held in store ready for use
increases steadily. This leads to specialization; Here
or there a specialist splits off from a qualification
because the information cannot be managed any more, it
has to be divided. The specialist usually will represent
a higher grade than the qualification from which^ splits
off. If specialists are generated as in a birth process,
each grade bringing forth new specialists one grade
higher in proportion to the parent population of each
grade then we should obtain the logarithmic growth
characteristic of the diffusion processes in economics.
In addition wd require as a Second assumption that,
growing pyramids (or hierarchies) of the type described
exist at/different ages/ in different stage^/of develop
ment  one beside the/other at the same tijrfe; and further
20
An Attempted Generalisation
In dealing with income of property owners we have chosen as a
state variable wealth which evolves slowly in a stochastic process;
unearned income can be derived from it by means of another random
variable, the rate of return. Can we generalise this twostep expla
nation to include also earned income?
In a somewhat formal way we could speak of the rank which an
individual occupies in one or several hierarchies. Examples
of such hierarchies are wealth, education, status, grade (level)
of an official or manager, rank of officers, ability, degree of
specialisation, grading by popularity of stars etc. Each of these
would represent a dimension in what might be called hierarchical
space. An individual would occupy a certain point in that space,
corresponding to its rank in the various hierarchies, and it
would have certain probabilities of transition within a certain
time to another point in that space. In other words, an indi
viduals' hierarchical position in society (a vector) would be the
state variable of a stochastic process.
To each point in the hierarchical space corresponds a certain probabi
lity distribution of income; the basic rule is that the higher
rank means expectation of a higher income.
21
In the course of his lifecarreer, the individual moves from one
position to another. It has been repeatedly described how the
hierarchical advance during the lifetime leads to skewed income
distribution^/ /, and, in fact, may give rise to a Pareto distri
bution; this is shown in the example treated at the very beginning
of this paper.
The position reached by an individual influences, however, also
the inital position and the progress of this heirs. The stochastic
process thus continues over the generations. This has been studied
by sociologists under the title of "social mobility" /4/.
In order to lend just a little more concreteness to our theory,
let us consider a special hierarchy, that of the managers. Their
income distribution has been studied by several authors /13, 15,
16 or 3/ and we shall refer to the very simple but illuminating
picture given by H.Simon /3/. He assumes that each manager can control
directly a certain number of subordinates and no more; this number
is called the span of control. If this span of control is the same
on all levels, then the employment at various levels of the hierarchy
from top to bottom will increase in geometric proportion.
He further assumes that each manager gets a salary which is determined
as a certain proportion (larger than unity) of his subordinate's
22
salary. The salary thus decreases geometrically as we go from
top to bottom.
In terms of algebra:
n: span of control
b: manager's salary in proportion to that of his direct
subordinates
L: level of the hierarchy (counting the base as unity)
N(Ii): number of managers at level L
C CD: Salary at level L
No: number of managers at the base level C,  A
.'A'l/t 'T'V
N(L)
w ,~L
/Vi ^
Li
(V
PJ
A: Salary at the base
AAv N 
oi, ^
/■A, c A)
(y
N  A" C ,
The salaries under the assumptions given conform to Pareto's
distribution. Simon bases his model on empirical facts (Roberts'
regression of top manager's salary on the size of firm, regression
coefficient 0,37, which would correspond to the value of y&L
and Davis' Pareto distribution of managers' salaries in General
Motors, Pareto coefficient 3).
The above demonstration is purely deterministic, but if
we regard the span of control as the reciprocal of a probability
23
of advance to the next higher level, we get the special version
of Champernowne's model described at the beginning of this
paper: There are again two geometric distributions, from which
the result is obtained by elimination of L, which stands here
in place of time.
This, in fact, leads to criticism of Simon's explanation. It is,
on the face of it, timeless, it does not show how the pattern
arises from a stochastic process in time.
One does not have to go very far however, in order to see the
dynamic implications of the matter. A certain span of control
implies that the managers of a given level have a limited chance
of advancing to the next level. To the span of control corres
ponds a certain transition probability. It might be argued that
the transition probabilities only reflect the given structure
of the organisation. This, however, has itself arisen as a result
of an evolution (including trial and error) and it is changing
continously albeit slowly. Thus the chances of advancement in the
^ th
individual's life carreer determine the structure: If —
n
of the occupants of a certain level expect to move m levels in
a lifetime then there must be n times as many occupants on the
lower level than on the higher (compare for these topics Bartholo
new /4/.
To be precise we have also to take account of movements into and
out of management from other occupatism (for example, politics).
24
The transitim probabilities will also reflect long run develop
ments: Growth, organisational changes and innovations etc.
After the explicit introduction of time (age, and also "historical"
time) the model could also be made more realistic by making the
span of control as well as the income relation b into random variables
The pattern of the explanation could then, 1 think, be extended
from the managers to other groups of income earners.
So far we have only refered to the separate groups (like managers
etc) each of which is represented by a dimension in the hierarchy
space. The relation between these dimensions remains open, and
therefore also the question how these separate distributions
combine into a total income distribution which still shows the
familiar Pareto pattern. Prima facie the relation between the various
hierarchy dimensions is undetermined; our society does not definitly
rank business managers, doctors, officers etc. The only common
denominator is income. There is, however, some sort of vague
hierarchy of the hierarchies themselves, indicated by the mean
income and by the inequality as measured by the Pareto coefficient.
On both counts wealth is at the top of the hierarchies; the stars,
the managers and some professions follow in a rather uncertain order.
By and large, however, you will find the groups with lower mean
income also have higher Pareto coefficient and are larger groups.
I should insist on the greatly irrational (or "traditional")
25
character of the income relation between these groups and yet think
that there is this vague ordering which would explain how a regular
pattern of the total income distribution comes about at all. In
particular, it seems essential that the tail of the distribution
is mostly dominated by income from wealth, which assures that
the total income distribution conforms to the Pareto pattern.
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