Size Distributions in Economics
Josef
Steindl
AL00662361
International
Encyclopedia of
STATISTICS
Edited by
WILLIAM H. KRUSKAL and JUDITH M. TANUR
University of Chicago State University of New York
at Stony Brook
VOLUME 2
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994
SIZE DISTRIBUTIONS IN ECONOMICS
SIZE DISTRIBUTIONS IN ECONOMICS
The size distributions of certain economic and
socioeconomic variables—incomes, wealth, firms,
plants, cities, etc.—display remarkably regular pat-
terns. These patterns, or distribution laws, are usu-
ally skew, the most important being the Pareto law
(see Allais 1968) and the log-normal, or Gibrat,
law (below). Some disagreement about the pat-
terns actually observed still exists. The empirical
distributions often approximate the Gibrat law in
the middle ranges of the variables and the Pareto
law in the upper ranges. The study of size distribu-
tions is concerned with explaining why the ob-
served patterns exist and persist. The answer may
be found in the conception of the distribution laws
as the steady state equilibria of stochastic processes
that describe the underlying economic or demo-
graphic forces. A steady state equilibrium is a
macroscopic condition that results from the bal-
ance of a great number of random microscopic
movements proceeding in opposite directions. Thus,
in a stationary population a constant age struc-
ture is maintained by the annual occurrence of
approximately constant numbers of births and
deaths—the random events par excellence of hu-
man life.
The steady state explanation is evidently inspired
by the example of statistical mechanics in which
the macroscopic conditions are heat and pressure
and the microscopic random movements are per-
formed by the molecules. Characteristically, the
steady state is independent of initial conditions,
i.e., the initial size distribution. In economic appli-
cations this is important because it means that the
pattern determined by certain structural constants
tends to be re-established after a disturbance is
imposed on the process. This will only be the case,
however, if the process leading to the steady state
is really ergodic, that is, if the influence of initial
conditions on the state of the system becomes
negligible after a certain time; and it will be rele-
vant in practice only if this time interval is suffi-
ciently short.
The idea that the stable pattern of a distribution
might be explained by the interplay of a multitude
of small random events was first demonstrated in
the case of the normal distribution. The central
limit theorem shows that the addition of a great
number of small independent random variables
yields a variable that is normally distributed, when
properly centered and scaled. A stochastic process
that leads to a normal distribution is the random
walk on a straight line with, for example, a 50 per
cent probability each of a step in one direction and
a step in the opposite direction. It is only natural
that attempts to explain other distribution patterns
should have started from this idea. The first exten-
sion was to allow the random walk to proceed on
a logarithmic scale. The resulting distribution is
log-normal on the natural scale and is known as
the log-normal or Gibrat distribution. The basic as-
sumption, in economic terms, is that the chance
of a certain proportionate growth or shrinkage is
independent of the size already reached—the law
of proportionate effect. This law was proposed by
J. C. Kapteyn, by Francis Galton, and, later, by
Gibrat (1931).
O Let size (of towns, firms, incomes) at time t be
denoted by Y(t), and let e(t) represent a random
variable with a certain distribution. We have
Y(t) = (1 + e(t))Y(t — 1)
= Y(0)(l+e(l))---(l+e(t)),
where Y(0) is size at time 0, the initial period. For
small time intervals the logarithm of size can be
represented as the sum of independent random
variables and an initial size which will become
negligible as t grows:
log Y(t) = log Y(0) +6(1)+ 6(2) • • •+€(*).
If the random variables e are identically distributed
with mean m and variance cr2, the distribution of
log Y(t) will be approximately normal with mean
mt and variance a2t.
This random walk corresponds to the process of
diffusion in physics which is illustrated by the so-
called Brownian movement of particles of dust
put into a drop of liquid. Since it implies an ever
growing variance, the idea of Gibrat is not itself
enough to provide an explanation for a stable dis-
SIZE DISTRIBUTIONS IN ECONOMICS 99 5
tribution. There must be a stabilizing influence to
offset the tendency of the variance to increase;
indeed, a distinguishing feature of the various
theories presently to be reviewed lies in the kind
of stabilizer they introduce to offset the diffusion.
Two interesting cases may be noted here. One
possibility is to modify the law of proportionate
effect and assume that the chances of growth de-
cline as size increases. This approach has been
taken by Kalecki (1945), who assumes a negative
correlation between the size and the jump and
obtains a Gibrat law with constant variance. An-
other possibility is to combine the diffusion process
of the random walk with a steady inflow of new,
small units (firms, cities, incomes). Some units
may continue indefinitely to increase in size, but
their weight will be offset by that of a continuous
stream of many new, small entrants, so that both
the mean and the variance of the distribution will
remain constant. This approach, which leads to
the Pareto law, has been taken by Simon (1955).
Review of various models. Descriptions of var-
ious models will illustrate the methods employed.
Models differ with regard to the distribution law
explained, the field of application (towns, incomes,
etc.), and the type of stochastic process used.
Champernoivne’s model. Champernowne (1953 )
presents a model that explains the Pareto law for
the size distribution of incomes. The stochastic
process employed is the so-called Markov chain [see
Markov chains]. The model is based on a matrix
of probabilities of transition from one income class
to another in a certain interval of time, say a year.
The rows are the income classes of one year, the
columns the income classes of the next year. The
income classes are chosen in such a way that they
are equal on the logarithmic scale (for example,
incomes from 1 to 10, from 10 to 100, etc.). The
probability of a jump from one income class to
the next income class in the course of a year is
assumed to be independent of the income from
which the jump is made (the law of proportionate
effect). The number of income earners is constant.
The number of income earners in income class s
is then determined as follows. The number of in-
comes in class s at time £ + 1 is
f(s, t+l) = Sf(s- W„t)p(ti),
u--n
where s, it, and t take on integer values, p(u) is
the probability of a jump over u intervals (i.e.,
the transition probability), and the size of the jump
is constrained to the range +1, —n. In the steady
state equilibrium reached after a sufficiently long
time has passed, the action of the transition matrix
leaves the distribution unchanged. We then have
f(s) = Y,f(s-u)p(u), s>°.
u = -n
as This difference equation is solved by
putting f(s) = z\ The characteristic equation
g(z) = Z z1 Up(u) -z = o
w=-n
has two positive real roots, one of which is unity.
To assure that the other root will be between 0
and 1, Champernowne introduces the following
stability condition:
g'(i) = - S up(u) > o.
M = -n
The relevant solution is f(s) = b*, 0 < b < 1, which
gives the number of incomes in income class s. If
the lower bound of this class is the log of the
income Ya, then the probability of an income ex-
ceeding Ys is given by
log P(Yf) = s log b.
Since s is determined by
log Y, = sh + log Ymjn)
where h is the class interval and Ymin is the lower
boundary of the lowest income class, it follows that
log P(Y*) = y-arlogY,,
where the parameters y and a are determined by
b, h, and Ymin. This is the Pareto law with Pareto
coefficient a.
Champernowne’s stability condition implies that
the mathematical expectation of a change in in-
come is negative. This counteracts the diffusion.
How can the stability condition be justified on
economic grounds? It may be connected with the
fact that in this model every income earner who
drops out is replaced by a new income earner.
Since, in practice, young people have on the aver-
age lower and more uniform incomes than old
people, the replacement of old income earners by
young ones usually means a drop in income. Thus,
Champernowne’s stability condition, as far as its
economic basis is concerned, is very similar to the
entry of new, small units that act as a stabilizer
in Simon’s model.
Rutherford's model. Rutherford’s model (1955)
leads, in his opinion, to the Gibrat law for the size
distribution of incomes. Newly entering income
earners, assumed to be log-normally distributed at
the start, are subject to a random walk and thus
996 SIZE DISTRIBUTIONS IN ECONOMICS
to increasing variance during their lifetimes. The
process of birth and death of income earners, which
is explicitly introduced into the model, acts as the
stabilizer.
The distribution of total income is obtained by
summing the distributions for all age cohorts that
contribute survivors. Rutherford's method is to de-
rive the moments of the distribution by integration
over time of the moments for the entrance groups.
The distribution is built up "synthetically” from the
moments, as it were. In the absence of an analyti-
cal solution with a definite distribution law, some
disagreement remains about the result.
Simons model. In Simon's model (1955), which
leads to what he calls the Yule distribution, the
aggregate growth of firms, cities, or incomes is
given a priori. The stochastic process apportions
this given increment to various units according to
certain rules, which are weakened forms of the law
of proportionate effect and rules of new entry. As
a consequence of this procedure, there is no possi-
bility of shrinkage of individual units. The given
aggregate emphasizes the interdependence of for-
tunes of different firms (the gain of one is the loss
of another)—a point that is neglected in other
models, such as that of Steindl (1965). On the
other hand, the aggregate is, in reality, not given;
it is not independent of the action of the firms,
which may increase their total market by adver-
tising, product innovation, and so on.
The process of apportionment may be described
as follows. We may conveniently think of popula-
tions of cities, so that f(rz, N) is the frequency of
cities with n inhabitants in a total urban popula-
tion of N; to be realistic, we shall assume that a
city exceeds a certain minimum number of inhabi-
tants; n will measure the excess over this mini-
mum, and N will correspondingly be the sum of
these excess populations. An additional urban in-
habitant is allocated to a new city with a proba-
bility a and to an existing city, of any size class,
with a probability proportionate to the number of
(excess) inhabitants in that size class. Then,
f(n, JV -h 1) — f(n, N)
= [(« - 1 )f(n - 1, N) - nf(n, N)],
f(l,N+l)-f(l,N)
We assume that there is a steady state solution
in which the frequencies of all classes of cities
change in the same proportion, that is, in which
f(tt, N + 1) N i 1
“ f(n, NJ~ ' N ""
for all n.
Using this relation and defining a relative frequency
of cities as fCn ) ~ f(n, N)/(aN), we obtain from
the above equations
// \ /v 1 n (l “ oc)(n — l)
or, setting l/( 1 - «) = p, d f )
(n- 1 )(n — 2) • • • 2 • 1
f(n) =
(n + p)(n + p - 1) • • • (2 + p)
f( 1)
r(n)IYp + 2)
F (n + p + 1)
f( 1).
This expression is the Yule distribution. Using a
property of the T-function, it can be shown that the
Yule distribution asymptotically approaches the
Pareto law for large values of rz, that is, f(n)~*
n p Af( l)T(p + 2) as rz -> °o.
This model is applicable to cases in which size
is measured by a stock, for example, number of
employees of a firm. Simon provides an alternative
interpretation of it that applies to flows, such as
income and turnover of firms. For example, the
total flow of income is given, and each dollar is
apportioned to existing and new income earners
according to the rules given above.
Using simulation techniques, Ijiri and Simon
(1964) show that the pattern of the Yule distribu-
tion persists if serial correlation of the growth of
individual firms in different periods is assumed.
This finding is important because, in reality, growth
is often affected by “constitutional'' factors, such
as financial resources and research done in the
past.
The 7iiodel of Wold and Whittle. Wold and
Whittle (1957) present a model of the size distri-
bution of wealth in which stability is provided by
the turnover of generations, as in Rutherford’s
model. On the death of a wealth owner, his fortune
is divided among his heirs (in equal parts, as a
simplification). The diffusion effect is provided by
the growth of wealth of living proprietors, which
proceeds deterministically at compound interest.
The model is shown to lead to a Pareto distribution,
the Pareto coefficient depending on the number of
heirs to an estate and the ratio of the growth rate
of capital to the mortality rate of the wealth owners.
Steindis models. Steindl’s models (1965, chap-
ters 2, 3) are designed to explain the size distri-
bution of firms, but they can equally well be applied
to the size distribution of cities. The distribution
laws obtained are. for large sizes, identical with
the Pareto law. Like Rutherford’s model, Steindl’s
models rest on a combination of two stochastic
SIZE DISTRIBUTIONS IN ECONOMICS 997
processes. One is a birth-and-death process of the
population of cities or firms; the other is a sto-
chastic process of the growth of the city or firm
itself.
The way in which the interplay of these two
processes brings about the Pareto law can be ex-
plained in elementary terms. We start with the size
distribution of cities. The number of cities can be
explained by a birth process, if we assume that
cities do not die. Let us assume that new cities are
appearing at a constant rate, e, the birth rate of
cities. The number of cities increases exponentially,
and the age distribution of cities at a given moment
of time is
(1) R(t) =R(0)er«*, 6 > 0,
where t is age and R(t) is the number of cities
with age in excess of t; in other words, R(t) is the
rank of the town aged t 4- dt, and R( 0) is the total
number of towns existing at the moment of time
considered. The size of the city—its number of
inhabitants—increases, on the average, with age.
If the rate of births plus immigration, X, and of
deaths plus emigration, fi, are constant, we obtain
an exponential growth function for the size of the
city:
(2) n(t) = X > /i.
Eliminating t between eqs. (1) and (2), we get
(3) In R = — -r———In n 4- In R(0).
A — fJL
This is the Pareto law (ife/(X — /u,) > 1), and the
Pareto coefficient is seen to be the ratio of the
growth rate of the number of cities to the growth
rate of a city.
This demonstration, which on the face of it is
deterministic in character, can be supplemented
by a graphical illustration in which the stochastic
features are included. In Figure 1 the distribution
of cities according to age is plotted in the vertical
(In R, t) plane. The abscissa shows the age of the
city, and the ordinate shows the log of the rank
of the city. Each city is thus represented by a dot,
and the regression line fitted to these points repre-
sents relation (1). In the horizontal (t, In n) plane,
we show the exponential growth of cities with age,
as in relation (2). Again each city may be repre-
sented by a dot showing age and size. The scatter
diagram in the horizontal plane may be regarded
as a stochastic transformation of the time variable
into the size variable. If the size of each city has
been found on the scatter diagram, the cities can
be reordered according to size; we then obtain,
in the third (In R, Inn) plane, the transformed
In R
relation (3) between the number of cities (rank)
and the size of a city.
If firms are studied, we must take into account
the death of firms. We might assume that a firm
dies when it ceases to have customers. We can
imagine that the age distribution in plane 1 of
Figure 1 includes the dead firms; they are auto-
matically eliminated in the transformation to size,
being transferred to the size class below one. In
the exponential relation (1), e must now represent
the net rate of growth of the number of firms if
the birth of firms is assumed to be a constant ratio
of the population.
Figure 1 illustrates how the evolution in time
of the number of firms (cities) is mapped onto
the cross section of sizes. This process may be
compared to sedimentation in geology, where a
historical development is revealed in a cross sec-
tion of the layers. We can also see how irregulari-
ties in the evolution over time will affect the size
distribution. If an exceptional spurt of births of
nqw firms occurs at one point of time (after a war,
for example), the regression line in plane 1 will
be broken and its upper part shifted upward in a
parallel fashion. The same thing will happen to the
transformed distribution in plane 3.
The complete model for firms may be described
as follows. The size of a firm is measured by the
number of customers attached to it. This is gov-
erned by a birth-and-death process. Let us denote
by o(At) a magnitude that is small in comparison
with At. There is a chance X At 4- o(At) of a cus-
tomer’s being acquired and a chance /jl At 4- o(At)
of a customer’s being lost in a short period of time,
At; multiple births and deaths have a chance of
o(At). The probability that a firm has n or more
than n customers is given by
P(n) - [°°P(n, t)r(t) dt,
Jo
998 SIZE DISTRIBUTIONS IN ECONOMICS
where P(n, £) is the probability that a firm of age
£ has n or more than n customers. The term r(£)
is the density of the age distribution of firms, in-
cluding dead firms; for large £ it is the steady state
of a renewal process and is given by r(t) = ce~ft,
where e is the net rate of growth of the firm popu-
lation and c is a constant. The number of firms
with less than one customer, P(0, £) — P(l, £),
equals the dead firms. The value of P(n, £) is ob-
tained as the solution of a birth-and-death process
for the customers of a firm:
j(l — a)n 1
J(1 — a' )n
where a = e a a' = (jjl/\)e~a and X > /x.
This expression can be expanded in series and fh-
serted in the above integral; this yields, integrating
term by term,
w-i /^W-nVn-lN
P(n) = CYY A fe /
p + k + a)
00
=cr.cWn) b(w, <o+p),
v-o ' /
where <o = e/(X — /x) > 0 and B(n, a) + v) is the
Beta integral. Hence,
P(«) = CyY-£Y—4—:B(n, oj + v).
If /x/X < 1, we can neglect the terms with v above
a certain value. Thus, if n °o and v has a mod-
erate value, we can use the approximations
B(n, a) -f v) n~<a~,/V(o) + v),
B(n, r>) « n~vT{v);
therefore, as n -» °o,
P(n) -» C'n-".
The following features of the solution may be
remarked: Since the approximation depends on
the value of /x/X, which is the mortality of firms
of high age, the smaller the mortality of firms, the
greater will be the proportion of the distribution
that conforms to Pareto's law. The mean of the
distribution will be finite if o > 1. This is important
in connection with disequilibria, which can arise
through changes in X, /a, and e. It can be shown
that the Pareto solution applies to the growing firm
(X > /x, the above case) and, in a modified form,
to the shrinking firm (X</x); but it does not
obtain for the stationary firm (X - /x).
The above solution for the distribution according
to customers can be shown to be valid also for the
distribution according to sales, if firms grow mainly
by acquiring more customers and not by getting
bigger customers. This is often true in retail trade
but not in manufacturing. An alternative model
assumes the other extreme—that firms grow only
by getting bigger orders. This model is based on
the theory of collective risk. The capital of the firm,
a continuous variable, is subject to sudden jumps
at the instant when orders are executed and to a
continuing drain of costs, which is represented
deterministically by an exponential decline. The
steady state solution obtained from this process is,
for large values of capital, identical with the Pareto
law; for moderate values, the distribution has a
mode and represents, albeit with some complica-
tions, a modification of the “first law of Laplace,"
which was proposed by Frechet (1939) for income
distributions.
Size as a vector. It would be natural to measure
the size of a firm by a vector, including employ-
ment, output, capital, etc., and apply the steady
state concept to the joint distribution of several
variables. Regression and correlation coefficients
obtained in a cross section could then be regarded,
like the Pareto coefficient, as characteristics of the
steady state. It may be guessed that the growth of
the number of firms will have an influence on these
parameters as well.
Practically no work has been done in this direc-
tion, but it is the only way to clear up the meaning
of cross-section data and their relation to time
series data and to the theoretical parameters of
the underlying stochastic process. The situation in
economics is totally unlike that in physics, where
the processes are stationary and the ergodic law
establishes the identity of time and phase averages.
(Only the cosmogony of F. Hoyle, in which the
continuous creation of matter offsets the expansion
of the universe to establish a steady state of the
cosmos, offers a parallel to the growth processes
considered above.) The surprise expressed at one
time at the difference in estimates of income elas-
ticities from cross-section data and from time series
data appears naive in this light because we could
only expect them to be equal if the processes gen-
erating households, incomes, and consumption were
stationary.
But the population of households or the popu-
lation of firms is not stationary. A cross section of
firms shows the growth path of the firm through
its different stages of evolution; but the number
of firms of a given age depends on the past growth
of the total number of firms, and this may influ-
ence the regression coefficient. Moreover, the growth
path is not unique, because there are several proc-
esses superimposed upon one another (growth
SIZE DISTRIBUTIONS IN ECONOMICS 999
paths depending on age of firm, age of equipment,
age of the management, etc.). For example, the
short-run and long-run cost curves are inevitably
mixed up in a cross section of firms. [See Cross-
section analysis.]
How much “stability” and why. The starting
point of the theories here reviewed is the stability
of distributions, but stability must not be taken
literally. The distributions do change in time, but
the change is usually slow. The tail of the distribu-
tion of firms or, to a lesser extent, of wealth is
composed of very old units, and time must pass
before it can be affected by, for example, a change
in new entry rates or in growth rates of firms.
Thus, the reason for the quasi stability of distribu-
tions is that the stock of firms, etc., revolves only
slowly. Indirectly this also accounts for the quasi
stability of the distribution of incomes, because in-
come is largely determined by wealth or its equiva-
lent in the form of education. An even more
enduring influence on the income distribution is
the differentiation of skills and professions, which
evolves slowly, as a secular process.
The explanations advanced in this article do not
exclude the possibility that distribution patterns
may change abruptly—for example, as a conse-
quence of taxation, in the case of net incomes; or
as a consequence of a big merger movement, in
the case of firms.
Josef Steindl
[Directly related is the entry Rank-size relations.
See also Lebergott 1968.]
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Postscript
The diffusion process assumed in some of the
above models has been studied directly on the basis
of individualized data for German retail firms
(Steindl 1965) but more recently also for Austrian
manufacturing firms (Steindl 1972a). The variance
of the logarithm of sales is shown to increase with
time at a rate that is different in different industries.
The question arises whether this diffusion constant
has an economic meaning; it is tentatively sug-
gested that it might be regarded, in some sense, as
a measure of the “dynamics” of an industry (tech-
nological change in the widest sense, with resulting
competition).
Wold and Whittle’s model of wealth distribution
has been reformulated by Steindl (1972b) using
an age-dependent branching process. The Pareto
coefficient of wealth distribution is seen to depend
on the speed of accumulation over the generations
within a wealth dynasty, and on the rate at which
new wealth dynasties appear.
It is noted that a constant Pareto coefficient is
compatible with growing concentration of wealth
in a few hands, if the sample of wealth holders
grows in time, and wealth sizes, which before were
mere theoretical possibilities, become actualized.
Josef Steindl
ADDITIONAL BIBLIOGRAPHY
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Ijiri, Yuji; and Simon, Herbert A. 1967 A Model of
Business Firm Growth. Econometrica 35:348-355.
j Ijiri, Yuji; and Simon, Herbert A. 1971 Effects of
Mergers and Acquisitions on Business Firm Concen-
tration. Journal of Political Economy 74:314-322.