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of Economic Research
Volume Title: Annals of Economic and Social Measurement, Volume 6, number 1
Volume Author/Editor: NBER
Volume Publisher:
Volume URL: http://www.nber.org/books/aesm77-1
Publication Date: 1977
Chapter Title: The Measurement of Concentrated Industrial Structure and
the Size Distribution of Firms
Chapter Author: John C. Hause
Chapter URL: http://www.nber.org/chapters/c10503
Chapter pages in book: (p. 73 - 107)
.111.5015 of LcOltOflih otiil Social tiI,'aci,re,,,e',,t 6 / 1 / 977
THE MEASUREMENT OF CONCENTRATFI) INDUSTRIAl
STRUCTURE AND THF SI7F DISTRIBUTION OF FIRMS
B
JOE!N C. HAUSI
The Iler/inda/il II voncentratwlt ecu/es is t/,earetieall, appropriate /or studvin' price -cost
margins 0/ industries in (ouritot-A ash equilibrium. I/u.s paper a' in's that i/ce II lade prorides a lower hound /or reasonable CO?tc'entra(jun icue'aceeres. since the' ('-A eiu,Jihri,m, inorec
possible large-firm cu/lotion dependent on i/ic si:i' di etrihunon Strong rectriclions on
t/ieoretie-a/lt' acceptable eoncentraflwr lndis,c (Ire i/educed icro new inc/u-sc cat,c/r,c the ci'
rest,, chaos are proposed, and hacu- de/reis of current inca sores are identified See ei/jch and
Japanese (/0(0 tield currelatwn.c heisseen i/it' wide/v lice 4-/trot coneentratw,, ratio. (4. 050!
II ran gin frasit /1 to 79 Jor lit ,'h/i coiiee,itratec/ isulustriet. ibis dems,nc,rates ('4 is a poor
pro.vv for theoretical!, superior measure's itt i/ic e'n/uri-af slut/u ol hieftic cwuentrat ccl
inc/u.s tries. desjsiie' widespread belief i..' the contrary
I. INrontjciio
Two central assumptions of the competitive market model are (I) a Iare
number of (2) independeuth' ac/lag producers and (buyers). These assumptions assure that no discrepancy between price and marginal receipts is
perceived by individual producers in their attempt to maximize profits,
and leads to the equality of price and marginal cost. The markets of the
real world differ greatly in the number of firms and their relative size.
American manufacturing includes such industries as priniary aluminum
with seven firms and a four-firm concentration ratio of .96 and wood fur-
niture (not upholstered), with 2,927 firms and a concentration ratio of
.11.' One of the ongoing embarrassments ol' economic theory (and there
arc several) is the absence of a persuasive model that links the number of
firms and their relative size with the expected degree of competition in an
industry. Nature and students of industrial organization (and others)
abhor a theoretical vacuum, and a plethora of thoroughly ad hoc indices
have been proposed for measuring industrial concentration.
The "degree of competition in an industry" has rio widely accepted
*1 am indebted to George Sugler's (1968) stimulating article, The Measurement at
Ccncentration,' for my initial interest in this problem 1-lerheri Mohring's substantive and
stylistic comments have been very beneticial. The generous cooperation of Gunnar Du RiOi.
sc'ho painstakingly assembled the industrial data lor his ascii studies, and of Ears Wohlin.
director of the Industriens litredningsinstitut in Stockholm. made possible the empirical
results reported in Section IV of the paper. Edccard Fageriund elliciently carried alit the
main calculations. Reactions from seminar presentations of the paper at Te'sas A&M. Stanford. UCLA, and the (Jntvcrsit', of Minnesota maiertall improved the stied . Proprieta
responsibility for any errors resides with the author.
See U. S. Senate, Committee on the Judiciary, Subcommittee on inhitrust and
Monopoly Report, ('oncentralion Ratios in MaiiuJaclurine I,idiuctri: /963. Part I, Washington. D. C.. 1966
73
meaning in the eCOnOmICS literature. It is defined in this study as a measure
determined by persistent discrepancy between prlcc. p. and marginal costs,
MC, in an industry. Consider first a monopolized industry. The Lerner inMCl/p. has been widely (hut not unidex. defined hy the ratio (p
versally) accepted by economists as a useful quantitative characterization
of monopoly power. The index is sometimes defined as the ratio of
monopoly rent, (p - MC)q, to the iiiOnopOli.Sts total receipts, pq. The
equivalence of the two definitions is apparent since the monopolist's output, q, cancel out in the latter ratio. For a single-price monopolist, it is
well-known that profit maximization by the monopolist leads to a value of
- I !q for the Lerner index, where s is the elasticity of demand for the
product.
The second definition of the Lcrner index is readily extended to in-
dustries containing n firms by the defInition L = (7- (p - MC1)q,j/
pqj, i.e., by the ratio of the sum of the monopoly rents to all firms
in the industry to total industry receipts.2 This delinition assumes a
homogeneous (single-product) industry, and a siiigle price, p, at which
the product is sold. In this equation, q and MC, are the output and marginal cost, respectively, of the i°' firm.
An index of industrial concentration is function of the size distribution of firms in an industry: the purpose of the index is to explain (in con-
junction with other relevant variables characterizing an industry) an important dimension of' industry performancc. A concentration index
should be considered "theoretically reasonable'' only if a plausible
theoretical link has been established between the functional form of the
index and the way in which it determines the industry performance
characteristic of interest. This study considers only the degree of competition in an industry, as measured by the L index. The reader should
bear this in mind in interpreting statements about "theoretically reasonable measures of concentration" in the sequel. in principle, a theoretically
reasonable concentration index may depend on the specific measure of
industry performance under study, although it seems plausible that such
indices of concentration are highly correlated with each other.
Section II presents notation and some general considerations useful
for the analysis of size distributions and industrial concentration. Section
Ill argues that the theory of Cournot-Nash (C-N) equilibria provides
strong restrictions on the class of theoretically acceptable measures of in2Equivalently, 1. can be defined as the weighted average ut the individual firm ratio of
price less marginal Cost to price, where the seiglits are the firm shares of iidustr output.
But the definition suggested in the text for I. is related inure transparently to the empirical
price-cost margin literature.
1-or a concentration index to nizike any sense in this context, the si/c distribution must
carrespond to some sort of indusir equilibrium. If firm shares arc essentiaIl random and
demonstrate extreme and rapid Iluctuations in magnitude, it is difficult to see wh a conceritratiun index sould hc related to any phenomenon 1)1 intereSt to economists.
74
dustrial concentration. The argument establishes that the Herfindahl index (H)(sum of the squared shares of firms in an industry) is related in an
extremely simple way to the Lerner index at the C-N equilibrium, and
provides a useful lower bound on reasonable industrial concentration
measures, since it ignores cooperation between the firms, based on their
perceived interdependence. Several new concentration indices are proposed, based on criteria for reasonable measures that follow from this
line of argument. Section lv presents cross-sectional correlations between
several industrial concentration measures, along with intertemporal correlations of the indices for samples of Swedish and Japanese manufacturing
industries. These calculations contradict the widespread belief that concentration indices are so highly correlated that they are likely to be indistinguishable in empirical studies. Previous work suggests that industrial
concentration is most likely to be associated with departures from competition primarily at fairly high levels of concentration. Yet it is primarily at high levels of concentration that the measures difter most from
each other. The results show that the correlation between the widely used
four-firm concentration ratio (C4) and the HerfIndahi index (I-I) is only
.614 when the sample is restricted to Swedish industries with I-I greater
than .l6. Thus the study of the effects of industrial concentration on significant dimensions of industry performance may be substantially influenced by the specific concentration measure used, when attention is
focussed on "highly concentrated" industries. This important conclusion
is discussed in more detail in the following paragraph. The empirical
results also suggest that 1-1 is likely to be a substantially better measure of
concentration than C4 for determining competitiveness of an industry.
Finally, Appendix A discusses the main concentration measures that have
been introduced into the literature, including C4 and the entropy index,
and the serious way in which they violate the theoretical criteria is demonstrated.5 Appendix B describes briefly the concentration data and presents
the various indices for 45 Swedish manufacturing industries.
Because of the important implications of this paper for empirical
studies of industrial concentration, it is desirable to he more explicit about
the current professional folklore. There seems to he a belief among some
economists that further attempts to refine concentration measures are
4The appropriateness of the Herlindahi index for restricting the industry sample ',ill
become apparent in Section lii. Here one simpl) notes that all industries sith a C4 of .8
or greater are included by this criterion. All industries in which the largest firm has a share
greater than .4, where the largest two firms have a total share greater than .566. or here
the largest three firms have a total share greater than .693 are also included by it,
5This study is concerned primarily with the meaSurement of economically relevant industrial concentration from the firm size distribution. See G. J. Stigler (1968, pages 29-38).
for a broader critique of the defects of current measures including the arbitrary time period
or one year generally used for measuring the firm size distribution. Foi other reviev,s ol
concentration measures, see G. Rosenhluth (1955). G. J. Stigler (1955). C. Marfels (1971),
P. Hart (1971). See T. R. Saving(19701 for in earlier theoretical discussion.
75
9
,.rcise, in the pelorative sense. ihis belief is
likely to he an acad
based on two sirts of evidence. irst a number of studies have reported
s'er\ high ctrrelations betssecn alternative nic:tirs. Rosenhiuth ( 95S)
calculated Spearman rank correlation cocl1cient ranging troni 979
.981, between the three-him concentration ratio. 11, atiti the number of
firms required to produce 80 percent of industry output, fi.r 96 Canadian
manufacturing industries. Scherer (1970) found a simple correlation coefficient of .936 between ('4 and Ii br a sample of 91 four-digit United
States census industries, using data assembled by Ralph Nelson. Bailey
and Boyle (1971) tabulated simple correlation coellictents ranging from
.96 to .98 between ('4 and estimated I-I indices for all 417 census industries
in 1963. Second, studies relating industrial concentration (usuaIl
measured by ('4) and measured profits or price behavior have generall
reported a rather weak empirical association.7 Given these lIndmgs of
very high correlations between concentration indices, but much lower
(simple and partial) correlations between the indices and price-cost differentials or profits, it might seem unlikel that any empirical tests will
establish much justilication for choosing between concentration measures.
1-lence, one might conclude, one should rely on the widely available ('
measures without purist apologies for their theoretical demerits in carrying out empirical studies.
There are several reasons for rejecting this conclusion. Some empirical evidence suggests that (' has little association with profits until
(' is fairly high, say .50 or .80.8 But it is precisely in this range of high
concentration levels that substantial dilTererices in the relative size of the
6Scherer (970. page 52) states...Foriunatel) the chances of making a griesous anaktic
error in the choice of a market structure measure are slender, for the principal concentra
non indicators all dispia similar patterns. For most mien nd ustr corn parison purposes.
thcn, ii is senseless to spend sleepless nights worrying about choosmg the right concentrinon nicasu re.' I). l3aik' and S. l(o Ic (I 97 I . page 705) ss rite. ''Approxinmatels the same
results are obtained for both the licrfindahl Index and the simple concentration ratio
Although researchers ss ill seldom kno the actual firm distribution in each indusir. the
results indicate that this information would generally he irrelevant
7G. J. Stigler ( 1965, pages 33 34) comments. ''A considerable lustors could he wrtien
on the search for high correlations between these concentration ratios and (purported
measures ot competitive performance). The main finding has hcn disappomntineru seldom
have good relationships been found hetseen the concentration ratio and these potenual
indexes of monopols . 1., Weiss (1971, page 369) alter rcvmes !ng sonlc 32 studies relating
to concentration and pro his, concludes. ''The tv pica I result of concentrationprofits
studies. espeema lv those based on firms, has hccn a sign (ica ft hut fairls weak posmtm'. c re-
lationship'
8See (i. J. Stigler (1964, page 59), 'In general the data suggest Out there is no re
lationship between profitzlbmlit) and concentration if II is less than 0.25)) or the shart' of the
four largest firruis us less than about 50 percent." See I.. Weiss (1971. pages 371 372) for a
brief reviess on whether there is a "critical level of concentration'' that leads to signilicanhls
stronger association of profits ind concentration levels
I
.
Telser
1972. page 3221 timid'
that four-firm concentration leeds exceeding S have larger slopes ss hen tr ing to determine
he effect of con cent rail on on profits.
76
largest firms are most likely to affect the degree of industry competitiveness. Disclosure rules govern most official statistics on industry concentration, and crude estimates of alternative concentration indices (other than
(4, ('s, and ('20) are likely to exaggerate the correlation between various
indices. Aniong the correlations based on Canadtan and U.S. data cited
above, only the one reported by Scherer is based on the actual site disdistribution of the firms. And if one recomputes the correlation, restricting the sample alternatively to industries for which (' exceeds .50 and
.70, the ('4 - H correlation coefficients fall to .866 and .705, which are, of
course, substantially smaller than Scherer's correlations. Section IV presents for the first time the correlation matrices of all the concentration
measures that have been widely discussed in the literature, based on the
actual size distributions of firms, and allows the reader to assess for hintself the modest correlations between sonic of the nicasues, especially in
samples of highly concentrated industries. These findings confirm that il
C is used as a proxy for a theoretically more satisfactory concentration
index in statistical work, the proxy is an extremely imperiect one.
II
11. CoNclNrRATIoN M FAStJRIS ANt) 1III SIZE
DIsTRIItuTI0N OF FIRls
ii
h
C
This section develops some notation and general considerations useful for thinking about size distributions and industrial concentration. Assume there are ,, firms in a well-delined industr. let s, denote the share
firm in industry sales (or some other measure of relative size),
of the
where the firms have been ordered by size so that s
definition. s, = I. An index of industrial concentration is a function C
of $s, the set of shares that define the firm size distribution. Highly concentrated industries take on a higher value for the index, and are expected to be closer to the monopoly end of the spectrum from monopoly
to competition than industries with low values for the index, taking due
account of other industry characteristics that are expected to play a major
role in affecting the "degree of competition in the industry.
If an n-firm industry is made up of equal-size firms, then s = I/n for
all firms. In this special case, the concentration index is denoted C (n).
It is assumed that C(it) is a nionotonically decreasing function of n. reflecting the belief that the structural competitiveness of an industry should
increase as the number of firms increases from one (pure monopoly) to
very large ii. The expected behavior of (' (n) is discussed more precisel's
in Section III.
In general, the relative sizes of firms in an industry differ substantially. The important concept of the equivalent number o/ eqiW!-5i7.e
firms, n, is defined in the following way. Let the concentration index as-
........... By
IC
.1
Cs,
íaite
C
ten
ted
lit
t al
ing
tits
re-
re.
the
'lilt
nds
ine
77
I
sume the value K0 for a specific size distribution s}, i.e., C(1s,) =
K0Y If C(n) is a strictly monotonically decreasing function of n, and is
(not just the positive integers), ( has
defined for all real numbers n
(Kg,), i.e., it, is the value (generally not an
an inverse, C - and ,,,. C
integer) such that C(ne) = K0. Two distinct concentration indices (' and
(* are said to he dispersion-isomorphic if the n, corresponding to any
specifIc size distribution is the same for C and C*.
Most measures of industrial concentration considered in this paper
give rise to a strictly decreasing monotonic function C - (ii) for all real
not just the integers. Since it usually denotes the integer
numbers n
number of firms that belong to an industry in this paper, the argument ,z
is used in the function (' (it,) whenever the behavior of the function ('
is discussed, unrestricted to integer values. Thus ( (ii,) is identical to the
original concentration function ('()sl). where n corresponds to the size
distribution fs as defined by the concentration measure.
}:or example, consider n, for the r-fIrm concentration ratio C',, the
fraction of industry sales accounted for by the largest r firms in the industry. This index can be written C', = (rX,)/(nX), where , is the average sales of the largest r firms, and , is the corresponding mean for all
firms in the industry. If the industry were actually made up of equal-size
firms, then Y, X; so C', = nfl and n = r/C,. Thus for firms of unequal size, n = n/C,. (This paragraph assumes ii
r.)
This discussion indicates that a concentration index possesses two
distinct, important characteristics. The first is the cardinal behavior of the
function C (n), i.e., how rapidly the index decreases with increases in the
number of equal-size firms.'0 The second is how the index maps arbitrary
size distributions into n,. If a proposed measure seems to behave reasonably with respect to this second property, but not the first, a modified concentration index could he obtained by using the original index to calculate
I
I
,
n,, and one could then take a monotonically decreasing function of n,
that provides more satisfactory behavior for the first characteristic. The
9h)r most size distributions, K0 will not correspond preciseI to the value of the index for some integer number of equal-size tirnis, ic., there is no positive integer ,i such that
C
tn) = K,,. Nevertheless, there will generally he a positive integer in such that
((m)
K0 > C (u, + I). One then says thai the ii, corresponding to the concentration
index C and the turn sii.e distribution [s lies hetsseeii in and (in ± i
Some students of industrial organization have ssaived the cardinal significance of the
concentration inde,x in empirical research h introducing one or more dumms variables
corresponding to intervals of the concentration index as independent variables to explain
prolits or other performance variab!es. For example. Ham
l9l div,dd his industries inio
to classes, depending on whether the eight-firm concentration ratio C is larger or smaller
than 7. Dichotomizing industry b conceniratjon levels in this was is prohab! responsible
for several empirical attempts to determine whether there is a "critical conceniration' level
required to make conecntraiinri a significant factor in determining industrs profit rates.
There is little theoretical basis for expecting a strong dicontinuiti in the effect .i coiicentrdtitin measure has on any performance characteristic and it is desirable to construct concentration measures in which the cardinal properties ale iiken seriousl,
78
modified index is thus the composite function of the siit distribution corresponding to these two operations. The idea of computing ii, for comparing alternative concentration measures is useful, because it clarifies
whether the major difrefelices between them lie iii ihe values they give for
n or in the way the equal-size function C (n) behaves for the alternative
indices.
Most measures of industry concentration that have been proposed
can be expressed as a weighted sum of the shares of fIrms jit the industry,
where the weights are functions of the share and/or the firm rank,
1).
s1
In this situation, C(s) =
1). For example, Ip(s, 1) = I for
i < r; w(s, i) = 0 for i > r for the i-firm concentration ratio, C,. For H,
p(s,, I) s. If is only a function of s, and is slrictly monotonic in SI,
ne is easily obtained. In this case, one immediately obtains ,i, = l/'(K0),
' is the inverse of the weight function
where
, and K1, is the value
of the index for the size distribution in question.''
Ill. CRITERIA FOR TUEORFTICAIIX REASONABLE INDUSTRIAL.
CONCENTRATION MEASURFS AND TIlE COURNOT-NASH EQUIlIBRIUM
This section discusses first the Cournot-Nash (abbreviated C-N)
equilibrium and shows how its use in the Lerner index of monopoly power
leads quite naturally to the Flerfindahi index as the measure of industrial
concentration. Then it is argued that the C-N equilibrium provides a
reasonable upper bound on the expected output of an oligopolistic industry, given the industry demand function and the firm cost functions.
This conclusion suggests that the Herfindahi index provides a lower bound
for "theoretically reasonable" industrial concentration indices that range
from zero for pure competition to one for pure monopoly. From this result, one can deduce several properties that a theoretically acceptable
measure of industrial concentration should possess. None of the concentration measures discussed in the literature satisfy these conditions, and
two new families of concentration measures are proposed that do possess
the desired characteristics.
The introduction pointed out that the competitive model assumes a
large number of independently acting firms. Cournot's model for
homogeneous oligopoly considers the equilibrium that would be obtained
if the large numbers assumption is rejected, but a certain kind of independent (noncooperative) behavior is imposed that assumes each firm
chooses its output as if other firms will hold their output constant. (This
restriction is relaxed later in the discussion.) It is assumed that no entry
takes place. There arc n firms in the industry that are assumed to have
strictly increasing marginal cost functions, which need not be identical.
Proof:
For cqual-sitc tir,n.
(s,)
(K0).
79
,,[ I/n (I in)! = A1. and so n, =
I
Industry demand is represented by the strictly decreasing function p =
q, (q is >up
wflcre ji is price and totiif industr) output is q =
of the j11 firm). Ilie firms attempt noncooperativefy to inaxnutc their owi
- e (q) where e (q, ) is
profits. 1 he profit function of the ith firm is
the cost function of the firm. The eqwlihrium condition for the 11h firm is
ohtaiiied by maximutirlg its profits with respect to q,
(I)
f) + q, (iI// dq) I I
+L
Iq / I'q, j = NI
In this equation MC, is the i' firm's marginal Cost. In the Cournot
model, the 11h believes all other firms will hold their output constant when
it changes its own output: hence the terms iii the sum mation on the LIIS
of cqt!itiOi1 (I) are tero. With this assumptioil. cqLiatton (I) can he rwritten in the fuii thai form
p(l + s,/i) = MC
(2)
where j is the elasticity of demand for the industry and s
= q/q, the
1th firm's share of industry output at equilibrium If the firms have identical marginal cost functions, .v, = I/n f'o r all Ii rins, i.e.. the firms have equal
shares in equilibrium and the same level of marginal cost. Note that if the
firms have difierent shares in the C-N equilibrium, the equilibrium levels
of marginal costs arc highest for the firms with the smallest shares. This
situation is consistent with empirical evidence that indicates the largest
firms in an industr tend to have higher rates of return and thus provides
a theoretical rationale for some tindings by Demsett (1974).
Now consider the relationship between this equilibrium and the
generafited Lerner index of monopoly power L. defIned in the introduction The 1. index assumed the following value at the C-N equilibrium:
(p
MC)]/
(s,/i)s, = (-
pq, = -
where II is the Ilerlindahi index, defined by the sum of' the squared shares
of the firms in the industr , and r is the elasticity of' demand. Thus the L
index for the C-N equilibrium is the product of ts o factors: The h-IcrfIndah I index (v hich depends only on the firm site distribution) and the
absolute value of the reciprocal of industry demand elasticity)2 This result shows that the I-I index is theoretically apprnpriate as an industrial
concentration index for deterni in ing the degree of competition in an industrv ii' the industry equilibrium corresponds to the Cournot model. This
result also indicates the appropriate functional form for explaining price2Since
riting ihe ininal
ran of this papei'.
J
hate dtstoered ihat a nuniher ol
relationship ol II arid I. uridi'r
R
Spariri i')('r. \. \
econoIlIist have rwceniI heoure aare ni the irrirniate
OUI not cqurirhritrrri nhrdiirons Se I Radcr li72.
tIeIle. Jr IY7'o. )jiish arid \Villrii I)7,i
I
.
.
cost differentials as a function of ifldustry Concentration and demand
elasticity.
We now argue that the C-N equilibrium yields an upper bound on
equilibrium industry output (assuming one exists) and a lower bound on
the Lerner index for an oligopolistic industry. Since the H index is appropriate for measuring industrial concentration For the C-N equilibrium,
this argument implies that the H index provides a lower bound for
theoretical/v reasonable measu res of in'Justrial concentration that are
scaled to have a maximum value of one for purc monopoly (one fIrm) and
a minimum of zero For perfect competition.
In this discussion Iwo interpretations of the Cournot model are distinguished: (a) it represents a single-decision, one-period game, and (b) it
represents an iterated game. Consider first interpretation (a). Given the
strategies available to the firms, a noncooperative Nash equilibrium exists
if it is not prolitable for any fIrm to change its strategy, given the eqttilibriunl strategies of the other firms. In the one-period model, the strategies
correspond to the selection of output by each firm. Nash (1951) has provided an important game-theoretic argument which implies that Cournot's
solution is a noncooperative Nash equilibrium to the one-period Cournot
model.'3 The completely cooperative solution, which would maximize
joint profits to the firms in the industry, requires the marginal costs of all
producing firms to he equal to marginal receipts to the industry at the
equilibrium level, and would entail a reduction of industry output from
the C-N equilibrium. To the extent that a cooperative solution is feasible,
one expects the firms to achieve an outcome no worse than the noncooperative solution, for which the H index is theoretically appropriate.'t
See Telser (1972. nages 132 1351 for a discussion suggesting thai the (ournot model
should he interpreted as a single-decision model.
o remarks arc in order. 1- irst, if the firms Ii ave di Itereri t marginal cost In nut uris.
the C-N equilibrium will generally correspond to lirms nith different shares of iiidustr
output, and hence different equilibrium niarginal cOsts as indicated in equation (2). Suppose the lirills cooperated b reallocatine product ion among tliemsel'. es such that total
output remained constant, hut thai each tirnr has the same niargmnai cost under the alternative arrangement. This form of cooperation would increase Joint profits. hence one can
imagine a less cooperative arrangement in which joint pvotits are greater than at the (-S
equilibrium. lirnts have the same marginal cost, btit industry output is greau'r than in the
original C-N equilibrium. Although this result contradicts the text assertion that iiicomplete cooperation should reduce output beloss the C-N equilibrium, such an outcome
seems extreme/v implausible. The degree of prof t-sharing that V. otild be required to induce
the firms to produce at equal marginal costs would surcl tiiake it feasible for them to obtain a superior outcome by reducing total output.
Second. if the terms iq1/itq, iii equation (I) are negative, the summation of these
terms in equation (H would niakc the k'fthimnd side 01 the equation a larger Posmite number.
and heitce requires a higher output h I he Ii mi to rccst,iblmsh eqtma lit the term 'l sq,
is called the "conjecttiral variation'' (see Fellner (1949. pages 71 77)) in irflhj 's output per
unit change in firm i's output anticipated b) lirni 1. arid ss hich firm i takes into account ii heti
irving to choose its optimal output. Thus a negatise conjectural saria!mon iiq,'iq
81
S
J
We turn now to the interpretatiOil of the Cournot model as an
H as a lower bound for a
iterated game and show again the relevance of
index. The iterated Version of the
theoretically reasonable concentration
denounced by Fe!lner (1949), Stigler (1968
mode! has been thoroughly
This line ol criticism maintains
page 36) and other prominent economiStS.
the appropriate theoretical
that joint maximization of industry profit is
concerned with identifygoal for firms in an oligopolistic industry, and is
this objective from being
those
factors
that
prevent
ing and elaborating on
achieved. Since joint profit maximization requires cooperation, this view
considers oligopoly a cooperative game. This criticism is persuasive. Unfortunately, there is no COflSCflSUS on the appropriate model for charoligopolistic equilibrium,b Hence one again conacterizing cooperative
implications
of noncooperative equilibrium for the
siders the theoretical
unspecified cooperative model.
Noncooperative iterative models are still undergoing considerable
theoretical elaboration, and it is premature to assert what consensus will
be reached on them. The traditional dynamic mechanism that has been
postulated to attain the C-N equilibrium point is an extremely myopic
process, in which each firm chooses its output for period (i + 1) so as to
maximize its profits, under the assumption that the rest of the firms maintain the output level of period 1. This is equivalent to saying that the firms
all attach a probability of one that the current output of all other firms
will be the same as the preceding period's output. Of course, if all firms
adopt the same strategy and do not start out at the equilibrium point, they
all discover in the next period that their forecast has been falsified. It is
hard to imagine that even noncooperative louts would fail to learn that the
constant output assumption is incorrect and inappropriate as the game is
iterated. However, some recent theoretical work considers far more palatable adjustment processes. Robert Deschamps (1975) has applied the
game theory algorithm "fictitious play" (F.P.), to homogeneous oliogopoly, and shows that under suitable restrictions on the industry demand
would imply that tirm i expects firm jto contract its output if firm i expands its output.
It is readily shown that with perfect cooperation, the sum 1 the conjectural variations In
Ih
Iirms share in this situaequation (I) is positive, and equals (I - .c)/s. where s, is the
tion. With any degree of cooperation, the conjectural variations should he posiiise. There
is no satisfactory theoretical basis for deducing negative conjectural variations for each
firm in a single-period noncooperative model, nor is there any justification for assumint
them. Hence the Cournot assumption of zero conjectural variation provides the bounds on
industry output and profits claimed iii the lest. Conjectural variations are discussed further
when considering "dominant firm" models below.
5There is even debate about xshether useful theoretical models can he produced thai
generate such equilibria. We do not examine this issue in the present stud>, and assunse instead the existence of cooperative equilibria to which noncooperative equilibria provide onesided bounds.
82
'
function and the firm cost functions, the firm output choices converge
over time to the one-period C-N equilibrium point)6 The F.P. process
starts out with arbitrary output decisions by the two firms in period I. In
period (1 4- 1) each firm considers the sequence of outputs of the other
firm from period I through period 1, and assumes that these empirical outputs define the probability distribution of output that the other firm will
choose in period (t + 1). Thus the optimizing strategy of firm I in period
(t + l)(to maximize expected profits) is to choose an output q1(t + I) that
maximizes:
(l/t)
r(q1(t + l),q2(i))
where q2(i) is the output actually produced by firm 2 in period i, and
ir1(q1, q2) denotes profits of firm I as a function of the output of the two
firms. Similarly, permuting the subscripts for firms I and 2 gives the decision rule for firm 2's output in period (t + 1). This model converges
(with appropriate restrictions) to the C-N equilibrium point by a simple
dynamic learning process that is far more plausible than the traditional
dynamic Cournot model)7
1 ant mdebted to David Schmeidler for catlinu niv atinniir,n In fl hmn' ci,,,i,
t1Deschamps points out that itis not necessary to assume that flrrns take the unadjusted
frequency distribution of the other firm's outputs as the probability distribution of the other
firm's output in period (
-+-
I). He gives an example of another set of weights that give
heavier weight to recent outputs of the other firm which also lead to convergence at the C.N
equilibrium point. Lester Telser (1972. pages 149-164) reports similar results for an n-firm
oligopoly model, where each firm bases its expectations of industry output from the other
firms on a weighted average (with fixed weights) of the observed outputs of firms from preceding periods. This work is primarily concerned with restrictions on the weights that would
lead the process to converge to the C-N equilibrium point A major limitation of this
analysis is the assuntptton that the ii firms have identical constant marginal costs. Further
work is required to determine necessary and sufficient conditions on the demand function
and the marginal cost functions of an n-firm oligopoly for the existence ol a unique C-N
equilibrium point in the one-period model, and on expectation-generating mechanisms that
converge to this equilibrium.
Another argument by Luce and Raifra (1957. pages 97-100) and applied by Telser
(1912, page 141) to the noncooperative duopoly problem, concludes that iterating a nonzero
sum game a finite number of times can lead to the choice of the same C-N equilibrium point
each period. This counterintuitive result comes about essentially by showing that the cisoice
in the last period will he the same as in a one-period model, and reasoning backwards,
period by period, to the first period. This austere model contains no convergence mechanism
--each finn must know the industry demand and the other lirm's cost function in order to
select the equilibrium point on the first move It scents desirable to incorporate some learning mechanism into these models, since this would be an essential component of models
with stochastic variations in demand and firm Costs.
83
/
The discussion So far has assumed a basic symmetry in the behavior
of firms. There is a class of models that rejects tli is assumption, and postulates that sonic firms act according to the Cournot hypothesis and take as
given the output of other firms iii choosing their output, while it feast one
firm takes this behavior into account in choosing its optimum output.'5
This class includes "dominant firm'' models, and the relevance of such
models for this study is now considered. '' (i iven the industry demafl(l
function, a dominant firm realizes that other firms adapt their output jp1
the current period as if the rest of the industry holds constant its output
level froni the preceding period. In formal terms, the iioIi-dotiijntnt firms
have a reactioii lunclioii, which gives the output that would rnaximjj
their profit, given the industry demand function, the level of output of all
other firms in the preceding period, and each firm's marginal cost function. The dominant firm then maximizes its profits, subject to its marginal
costs and the reaction functions of the other firms. In general, the crucial
feature of such models is that firms do not simultaneously assume that all
firms but themselves choose output according to a reaction function, since
110 equilibrium usually exists under that assumption.2°
A major objection by Fellner (1949, pages 66 69, I l 119) atid others
to dominant firm models in the context of static demand conditions lies
in the arbitrary asymmetry between the dominant and subordinate firms.
It is often assumed that the largest firm is the dominant firm, but this is
an ad hoc specification that lacks theoretical support. If the largest firm is
dominant, it usually (but not necessarily) turns out that in comparison
with the C-N equilibrium, the dominant firm is larger and has higher
profits, the subordinate firms are smaller and have lower profits, and
industry output is greater (which implies industry price is lower) and
the atlocatioial loss from monopolistic restriction is lower. lut why
should the smaller firms acquiesce to the largest firm if it lowers their
profits? The claim that the dominant firm is "more powerful" does not
I assume that one could jut io the fictitious pla
mccli art stir instead of the traditional CoUrnot assumption if one interprets the donti I na nit him model asan iterated izanie.
For hrevit . discussion in the test uses the traditional statement oi the (ournioi h poihesis.
191 am indebted to Herhcrt Mohring and Michael Darhs for espressing dissatisfaction
at the neglect of dominant firm consideration5 ii an earlier S ersion ot this paper.
20lloSSeVei. Ills possible to construct models i tb a coilsislenit hierarch of dominance.
possessing well-defined eq iii libria. Suppose i he nd List r can hc partitioned into us sets ci
lirms (svhich mas consist of simile tirnis) ssith the foliossing properts: each firm iii set i makes
the Cou rnoi assumption a bout a ns other liroi belonging to set i ii ad to all firms helonfing
to sets
- 2...... I.
I
It chooses its output to noasinulie its proits. giseis the reacilon
functions of urns iii sets j + I
ii,. and its ossni niargirial cost function Finn Kdlaisd
(1976) alludes to this estension of dominant firm models in a dnamic setting.
2tThe reader familiar with the duopok theory developed h Stackelherg and discussed
b Fellner (949) is assare of the uncomfortable ariets of results that can he produced iu
I
84
.1
t
bear scrutiny. Even if true (whatever it means), the relevant question is
whether the present value of the dominant firm would he greater by 1j.
vesting to teach other firms it intcn(Is to ITtilintiliti Output at the dominant
firm equilibrium level or by accepting the C-N cquifibiiuiit.
Despite these serious doubts about the relevance of the dominant
firm equilibrium, sonle economists find one \'erSi011 of the doniiriant fIrm
model more persuasive when there is a competitive fringe of quite small
firms that might passively adjust to whatever output level the large fIrms
choose. Suppose the firms in the competitive fringe are so small that they
regard price and marginal revenue as equivalent. How should this assumption be incorporated into the expected ratio of monopoly profits to
industry revenue? If all other firms in the industry correctly perceive the
supply curve of the competitive fringe, they can obtain the derived demarid net of the amount supplied by the competitive fringe by the function
q(p) q*(p), where q(p) is the industry demand function, qjp)
q(p)
is the quantity supplied by the competitive fringe, arid q*(p) is the net demand for the rest of the industry. Suppose the rest of the firms in the in-
the dominant firm model. As an iflustrtion. consider a duopol model ss ith firms a and 6
possessing unit elastic marginal Cost luncuons MC(q) = q/.8 anil MCj,(q5) = q/.2.
and assume the industry demand function q = t,q + q) lip. The iolloin table shos
equilibrium outputs and profits for each tirm. and equilibrium industry price under three
alternativeequilibria: C-N. firm a dominant, and firm 6 dominant.
Equilibrium Values
Solution
Cournot
Firma Dominant
Firm 6 Dominant
a
.4216
.4595
.4154
.2108
.2069
J908
.5556
.5575
.5774
'
h
.2222
.2035
.2237
.4444
.4256
.4696
.11(1
.0964
.1327
1.581
1.5(X)
1.650
is defined h monopoly rent of firm I: (p
Relative to the C-N equilibrium. if finn a is dominant, a's output and profit is in.
creased, and h's output and profit is decreased, total output uicreases. and industr price
falls. This result conforms to the claim in the text Rut 1 6 ix dominant, then compared
with the C-N equilibrium, both a and 6 have losser output and higher profit; indeed, in this
case the subordinate firm a has higher profits than it attains if it is the domrnant firm. .\
condition that assures the commonly expected result that C\iifl5iOfl b the dominant firm in-
d
duces,in oniput contraction h the subordinate lirm is the tollowing: The subordinate
firm's marginal receipts curve falls when the dominant tirni's oiitut inreaxes. R.G.1).
Allen (1942, pages 345 347) expresses the condition as dp/dq 1- qh(d p/c!q ) < 0. where a
is the dominant firm and 6 the subordinate fIrm. An alternative condition that may be
q) (dp/dq) . 0.
flare convenient is (dMR/dq)qj, (q
85
I
attain the C-N equilibrium, taking into account the supply by the
competitive fringe. One readily determines that the L index for the in-
dustry, given these assuuptiuns, is:
Li dusry
- [l/(i - (I -
k1))]H
where is the industry demand elasticity, E is the competitive fringe supq*/q, the fraction of equilibrium industry output from
ply elasticity, k1
the non-fringe firms, and 1-I is the l-lertindahl index for the industry!
Thus taking into account the competitive fringe modifies the relevant
elasticity that appears in the I index, but does not alter the use of the I-I
index as the relevant industrial concentration measure.23 To simplify exposition in the rest of the paper, the competitive fringe model will not be
considered again. However, the reader should keep in mind the adjust-
ment that should be made to the industry demand elasticity when the
fringe model is relevant.
The chief conclusion from this analysis is that the Herfindahl index
provides a lower hound for theoretically reasonable measures of industrial
concentration intended to explain the L index of an industry. It is a lower
bound because it does not allow for explicit or tacit cooperation by the
22Proof: First one determines the demand elasticity collectively facing the firms not in the
competitive fringe. Let q(p) = q(pl + q5(p) denote industry output as the sum of competitive fringe supply and output by the nonfringe firms. Then the demand elasticity collcctivelv facing the nonfringc part of the industry ij = (p/q*)d(q - q,- )dp = (I/ki)
- (I - k )). where k = q'/q. Now let t = qj/q'. the output of a nonfringe firm as a
(
fraction of output by all nonfringe firms. The equilibrium condition for the i firm is the
analogue of equation (2): p(l + sp/ii*) = MC,. From equation (4) monopoly rent as a fraction of total receipts of the nonfringe firms is (- l/i*)H&, where H =
Hence monopoly rent as a fraction of total industry receipts is (-k /ii*)Il*. Let .c
q,/q the output
of the
nonfringe firm as a fraction of indusrt output. Since s7 = s:/ki .
where II is the ordinary Herfindahl index for the industry. tThis last step assumes the individual firms in the competitive fringe are so small that they make a negligible contribution to II.) Suh:titutin the e\pressioiis for 11* and
into the lerner indes ( - I !it IF)'
icids the espression gisen in the icxt. It may he that the none model is useful for
-
analyzing industries exposed to significant foreign competition in their home market The H
function could either be based on the domestic share of domestic producers and foreign
supply ssould be incorporated in the demand curve perceived b the domestic producers
as a group. Or the share of foreign producers could he included in H. and foreign producers could be treated as a competitive fringe, us in the text.
230f course, the equilibrium and alloeational properties of the equilibrium do depend
on whether the industry equilibrium with a competitive fringe is (2-N, or whether the competitive fringe reacts passively to output decisions by nonfringe firms (and the nonfringe
lirms realize it). For example, consider an industry s ith demand function q
I/p (hence
= - I). and suppose there is one large firm ss ith marginal cost function MC' = (2q)uIt
and a competitive frinne with aggrezize margin;il cost function MC = (2q)'Jt The
equilibrium share of the large firm, industry price, Lerner index, and Flerfindahl index are
shown for the different marginal cost elasticities
in the folloss ing table For the C-N
equilibrium and the dominant firm equilibrium models.
86
additiofla restrictions or' reasonable concentration indices follow from
this conclusion. An index should approach the H index in the limit as the
number of firms increases indefinitely, and the share of the largest firm
tends to zero. Indeed, an even stronger restriction on reasonable indices
seems appropriate. Suppose that the largest finn's share, s1, is held constant, and the rest of the industry output is supplied by an extremely large
number of firms, all very small. In the limit, the H index is s, the squared
share of the largest firm.24 It seems reasonable that departures from the
C-N equilibrium (in the direction of greater monopoly) should occur only
if there are two or more "large" firms in the industry. It has been widely
argued that the costs of collusion rise with the number of (equal-size)
firms. Since many of the little firnis would have to collude with the large
firm for there to be any feasibility of higher profits for the colluders, it appears that the H index is again the correct limit for a satisfactory concentration index for one "large" firm, and an indefinitely large number of
small ones. A corollary to this argument is that the index should noi approach zero as the number of firms in the industry increases indefinitely,
if there are one or more "large" firms in the industry. This condition also
implies that the ne of a theoretically reasonable concentration index will
5*
(large firm share)
Model
p
(industry prices)
CournotNash
5
.22 19
1.0765
I
.4227
.4007
.3904
.3805
1.0622
1.0508
1.0425
1.0210
.1132
.0730
.0539
.0353
-
3
5
re
N
.1786
.1606
.1524
.1448
3
.1459
.1009
.0759
.0492
2
5
.1459
.1009
.0759
.0492
1.1118
1.1092
-
1
H Index
.3820
.3177
.2755
I
2
Dominant
Firm
L Index
1.097 I
This table indicates the significant fall in the Lerner index as the supply elasticity of the
competitive fringe increases for both the C-N and the dominant firm equilibria. The share
of the large firm declines much more slowly with increases in the marginal cost elasticity in
the dominant firm model than in the Cournot modeL It is worth noting that even though the
), the opposite
U index is larger for the dominant firm model than the Cournot case (given
importance
of
taking
into
account both
is true for the Lerner index. This result shows the
the H index and the relevant demand elasticity for predicting the L index.
Whethvr the industry equilibrium with a competitive fringe should be expected to
conform to the Cournot model or the dominant firm model is moot. If one beheves the
latter is relevant, one can adjust the demand elasticity in the Lerner ifldeA. as indicated
models
in the text, without modifying Fl. There is a technical difficulty in relating the two
treated as
subdivide
a
group
of
firms
until
they
are
to handle sequences of industries which
explored in
a passive competitive fringe by the rest of the industry. This problem is not
this study.
87
he finite il the share of the largest firm does not go to zero, regardkss of
the it urn her ol firms iii the industry.
The industrial concentration fund ott ic defined as the citninlative dis-
) tribution function of industry output, with firms ordered From the largest
s, for I j < a, wheft
to the smallest in the industry, i.e., Fa(j) =
,j is the number of firms in the industry, and the a subscript denotes a
specific industr. It is convenient to define industr a as strict/i' Wc.s conceiziratell than industry h if F(j) < i',,(j) for all]. arid F(j) J'() for at
least one 25 In words, the concentration eure of a must lie strictly heloss
.
the concentration curve of h at one point or more, but can tiever lie above
the eoncentration curve of h. It follows irnmediatel from this delinition
that an it-firm industry with equal-size firms is strictly less concentrated
than an\ othcr size distribution with a firms. It also lollows from the
definition that if two (or more) firms in an industry merge, and no change
takes place in the individual shares of the remaining firms, the sue distribution of the pre-merger industry is strictly less concentrated than the
post-merger distribution It is readily verified that if one distribution is
strictly less concentrated than another. it necessarily has a smaller value
for its Ilerlindahl index. There is nothing in the Cournot model, nor in
feasibility of collusion considerations, to suggest that any reasonably defIned measure of industrial concentration should not take a smaller value
for strictly less concentrated size distributions, and this property should he
verilIed when examining any newly proposed concentration measures.
The H index considered as a function of r is l/n, a convex function
that declines with n, but at a declining rate. In the absence of a plausible
theory of the relationship between collusion and the size distribution that
implies a concentration index that violates this condition, it also scents a
reasonable feature to expect front a satisfactory index.21
The most unsatisfactory aspect of this entire development is the lack
25The definition of "strictly less concentrated'' corresponds to the concept of
"stochastic dominance" that has been wideR used by economists in the analysis of risk.
Jack Meyer's coninlents Oil an earlier draft have greatly simp!ilied the discussion in this
paragraph. See Hanoeh and Levi (l9fl9) and Rothschild and Stiglitz t1970) for estensive
discussion of stochastic dominance and several equivalent delinitions.
260f course, if the marginal cost functions of the merging firms are unaltered, the share
of the merged tirms would he smaller in the ness equilibrium than the sum of the prenierger
shares !n the intliat equilibrium. This occurs because the marginal revenue perceived b the
merged firm at the premergcr level of output is tower than the marginal revenue for either
firni iii the prenierged equilibrium.
27F. M. Scherer (1970, page 1S3) has suggested, "As a serv crude general rule, ii evcnl
matched firms supply homogeneous products in a well-defined market. they arc likely to begin ignoring their inlluenee on price when their itumber esceeds ten or tsselvc. Ii is more
difficult to generalize when the size distrihutioii is highly skewed." This claim might he
interpreted as implying that C (n) may have a nonconvcx region Over an interval of
small vjius Ofne. since it suggests an abrupt decliiie in ( t'i) sshcn the firms begin to
ignore their interdependence. To ms knots ledge, no theoretical argunien t or empirical esdence is available that stould support the lirsi sentence in this claim.
88
t
of a convincing cooperatIve model that puts signmlicant restrictions on the
departures that illay be expected from the C-N equilibrium and ci
making these departures a function ci the firm size distribution
ft.
Stigler's important article (1964) on the theor of olí'pi indicatec an
intereSi!ng approach for attacking this problem, although the ellect of inequality of firm size on the feasibility of collusion has hardly been touched
upon.
In summary, the criteria for theoretically reasonable industrial concentrationl measures that have been deduced from eonsidenttion ol the
C-N equilibrium, the corresponding Lerner index implied by it, arid a i'ew
extremely natural assumptions about the i'easihility of collusion and the
size distribution of firms are as follows.
If the two largest firms in an industry site distribution each have
shares greater than a strictly positive constant, the concentration
index should take on a value greater than I-I. ibis simply reflects
tacit or explicit collusion, which leads to a larger value ci' the index
than indicated by the C-N equilibrium.
If the largest firm is greater than a strictly positive constant, and
one considers a sequence of industries where the rest of industry
output is produced by an increasing number of firms such that the
share of the second largest firm in the industry approaches zero,
the concentration measure should converge to s, the squared
share of the largest firm.
As the number of firms in the industry increases, and the share ol
the largest tirm approaches zero, the concentration measure
should converge to the Herfindahl index. For the special case of
equal-size firms, this implies the index approaches the function
I/n as ,i becomes large.
If the cumulative size distribution of one industry is sirietli' less
concentrated than another, the former industry should have a
lower value for its concentration index.28 As corollaries, the index
should attain its minimnuni. given n, when firms are eqtial-sized
and the merger of two or more firms should increase the value of
the index if the share of all other firms remains the same.
The index should be a decreasing, convex function of n, the
equivalent number of equal-size firms implied by the index.29
Finally, the following normalization convention is adopted:
The index should equal unity ii ,, = I, and should approach zero
as 'me increases without hound. This convention follows naturally
28The definition of "stiictly less concentrated'' is given earlier in the test. Fssentiail
it require.s the cumulative site distribution of the industry to lie strictly below the other
industry size distribution at least at one point, and never to lie above the other distribution.
The decrease in the index as a function ofn follows immediately from the thcoretcal
arguments: the convexity condition does not, hut seems highi) plausible.
89
if one considers that a theoretically satisfactory index should essentially be an I-I-in'dex, adjusted to allow for C011LiSk)fl This normalization facilitates comparison of alternative measures its onh
theoretical significance is that it suggests that Ct)flcefltratj(.)fl
indices should have a bounded range.
These criteria do not define a unique index, which is hirdly Surprising, since an adequate theory of collusion has not yet bce devised
But they are much more restrictive than others that have been proposed
in the literature, and they follow immediately from the analysis of the C-N
equilibrium, modified to allow for collusion.30
As far as I know, the measures of industrial concentration that have
appeaied in the literature all fail to satisfy one or more of these criteria
The appendix indicates the ways iii hich the niost wideh discussed
measures violate them. This section concludes with the introduction and
brief discussion of two new measures that are theoretically acceptable in
terms of the analysis. Both depend on a single parameter that allows for
the efTècts of collusion; both measures converge to the H index as the
parameter increases without bound. These measures were devised primarily by considering simple functions of s1 that satisfy the first four
criteria, then restricting them to satisfy the fifth.
1. The multiplicatively-modilled Cournot measure of industrial concentration (with parameter a) is defined by the equation
(4)
Hm(a; Js})
sJi
(a
.15)
In this formula, H is the ordinary Herlindahl index for the size distribution. The expression raised to the a power always lies between zero and
2(1/3)15 (<I), and so the exponent on s always exceeds one,3' lithe firms
are equal-size, the index takes on the value (I /n)(l /ny'' -
which
converges to the H index, 1/n for large n. It is straightforward to verify
that this index satisfies conditions I -6 in the text, except for the convexity
30M. Hall and N. Tidenian (1967) proposed the following propert!es for an indes. It
should be a (I) scalar function of (2) all the firm shares that (3) should decrease if part of
a share is transferred from a larger to a smaller firm, (4) should decrease by the factor I/A
if there is a k-fold increase in the number of fIrms while holding constant the ,elaiise sue
distribution, and (5) should decrease with n for distributions of equal-size firms. Property 4
is rejected by our theory, which modifies the Cournot theors to allos for collusion. The rest
of the Hail-Tidenian criteria follow Iron) our analysis. C. Marfels (1972) repoils an attempted axiomatic approach by Johrik (1970), which proposes (I) symmetry of the indes
with respect to firm shares, (2) continuity with respect to share changes, (3) an increase if
part of a share is transferred from a smaller to a larger him. (4) if K is the index and
- s') = (s' - u) IK(sc) - K(s)]
K(i') - K(ii)J. and (5) the index should range over the
O - I intersal. The arguments of K are vectors of firm shares. This condition is apparentl
intended to imply convexits of the index although the formulation is unclear. These conditions are all implied by the criteria developed in thc main text, and are much less restrictive in defining permissible indices than the arguments in thss paper
The upper bound 0f2(f/3)i.S is obtained in a ts; u-firm industrs
90
TAULE I
TIlE IN()LX Hm((: fs1f) FOR EQUAL-SIZE FIRMS
Cr
.25
.15
I
I
I
2
.830
4
.701
603
5
.522
10
.3H
20
.168
.032
.0023
ioo
1000
.5
I
2
I
.638
.449
.338
.267
.124
.058
.0105
.0010
.755
.591
.475
.395
.203
.097
.016
.0012
I
I
I
.545
.362
.267
.211
.102
.050
.010
.505
335
.200
.100
.050
.010
.500
333
.250
.200
.100
.050
.010
.001
.001
.001
.251
'I-lerlindahl
II
1
e
r
-
was found necessary to assure con5. The restriction
vexity. Table I compares this index for several values of a, and with the H
index (a = c ) for distributions with equal-size firms, It is clear from the
table that the tendency toward competition as n increases is much slower
for low values of a than the I-I index implies.
2. The additively-adjusted Cournot measure of industrial concentration (with parameter ) is defined by the equation
condition in
H0($; s1) =
(5)
ud
is
h
fy
ity
+
[s(H - .$)]
(> I)
if the firms are
Again, 11 denotes the ordinary l-Ierfindahl index for
,J2$ (I - n)]. The restriction
equal-sizes the measure becomes [n
> I is required to obtain the required convergence behavior for large ii
to satisfy condition 332 It can be verified that with this restriction, the
index H0(3; ts1}) satisfies conditions 1-6. Table Ii compares 110 for alternative values of and the H index ( = x) for equal-size firm distributions.
It
of
I/k
cst
at-
and
te
Further theoretical and empirical work is required to determine
where I-I,,, or some other index is clearly better than 1-1, as vell as the value
for the parameter a that seems most appropriate.33 In principle, it seems
plausible that the parameter a should be industry-specifIc since it reflects
the feasibility of collusion which surely depends on industry characteristics
besides the size distribution. If it is desired to create an index that uses the
same parameter for all industries, the parameter value should presumably
32IfJ = I, the measure converges to 2/n, instead of I/n. and converges
for<l.
as
l/n
33Although the values of the H0 index diljer substantially from the I-I index for small
tilL
,
the empirical results in Section IV indicate that IIa(k) and II have very high correlation
diller little from
even in this case. Thus the statistical results usin Ha sso'jld presumabI
those obtained with Fl.
91
I
l\I3I.I
liii
a
1.01
INF)FX
1.10
I
2
3
4
5
tO
20
100
1,0(X)
.745
.550
.432
.355
.186
.095
.019
.0019
.703
.505
.388
.313
.156
.076
.014
.0013
II
II,,(tJ; $s,1 ) 10k IQ(iAl.-Si/.I
1.25
1.50
2
I
I
I
.649
.337
.268
.128
.060
.011
.001
555
.394
.291
.229
.109
.052
.010
.001
Fii.is
3
I
.531
.504
.349
.259
.205
.fl5
500
.
.250
.200
.050
.010
.250
.200
.100
.050
.010
.001
.001
O(JI
.101
i
.050
ow
* } ertindahl
reflect in sonic sense the average level of interdependence that is present
in the individual industries. Appendix A demonstrates the serious theo.-
retical deliciencies of the four-firm concentration index, entropy, and
several other indices that have been introduced into the literature.
A very recent article b' Cowling and Waterson (1976) suggests an
interesting was' of analyzing the relationship of price-cost margins and
concentration which doesn't require an ci priori choice between the
Cournot model and the dominant firm class of models. Cowling and
Waterson propose the use of time series and crosssectional data in this
approach. They attempt to generalize the Cournot model by assuming
that firms in an industry have stable, but possibly nonzero, conjectural
variations about output responses of other firms to their own output
changes. The equilibrium condition for the individual firms is
p[l + .v(l + A)/iy] = MC,.
here A, =
,aq1/q, the conjectural variations expected b' the ith firm
in response to its change in output. It is readily verified that the Lerner
index corresponding to the industry equilibrium is given by
L = H(l .f X)/(-71),
where X =
A1.s-/1-I, a weighted average of the expected conjectural
variations, where the weights are the squared shares of' the firms.
If the aepo.v.c (mu' coeffic;ents ol variation of the iiìdustrv demand
elasticity, n, and of the adjustment factor for conjectural variation,
I + A). arc both substantially smaller than the
across time coefficient of
variation of' the H index, then the ratio of the Lerner indices for an industry at two points iii time is approximateI equal to the ratio of the II
indices for the same two points in time. Cowling and Waterson thus sug92
gest taking the ratio of the price-cost margin
in an industry (which is
equivalent to the ratio of the Lerner indices) at two points
in time as a
ay of getting rid of the dillIcult-to-ineasure parameters ij aiid A. They icport some regressions of the log of the price-cost iliargili ratio on the log
of the H index ratio and several other variables, and corn pare the results
with those obtained with the log of the (' ratio replacing the U index
ratio. As the theory suggests, the H index ratio tppearS to give a better
(although low) statistical fit.
Although Cowling-Vateison do not refer to any models that make a
specific, nonzero assumption for the expected conjectural variations, X,
the dominant firm models obviously belong to this class. If there is a significant difference in a priori beliefs about the relative plausibility of the
Cournot-Nash equilibrium and a dominant firm equilibrium, the CowlingWaterson paper suggests that sonic empirical results of interest may still
be obtained by using the ratio of price-cost margins at tWo points in time
as the dependent variable and the corresponding H ratio as an independent variable. If the assumption about the relative magnitude of the coei1ici.nts of variation of H, i. and (I + X) holds, this procedure enables us
to get rid of the tatter two parameters.
IV. CROSS-SECTION ANI) INTERTENIPOIOAI. CORRELATIONS OF
INDUSTRIAL CONCENTRATION INDICES ---Sosii ESIPIRICAI. RESULTS
This section presents and discusses correlations between various con-
centration indices based on employment for 45 Swedish manufacturing
industries in 1968, as well as the correlation of the individual indices between 1954 and 1968. The industries contain the metal, metal-working,
and plastic sectors, and include 4247 firms with about 405,000 employees
in 1968 (about 48°,ç of manufacturing employment). The indices for the
individual industries are given iii Appendix B.
The indices that have been discussed in the literature, their definitions, and their values for equal-size fIrms are as follows:
I. Four-firm concentration ratio C4 =
C;(n) = 4/nifn
s. C(n) =
I
if n
4,
4:
Herfindahl index H =
H(n) =
Entropy index34 F = fl'1s,i, F (n) = I/n:
34Eniropy, as delined in phssics and inforniatior iheor is --ln(I-.). and is obsiously
is oinot formed properls für a concentration index even if the sii.'n is chariued so that
dered in the correct direction. The entropy index as a measure of industry coiicenuaiion is
mentioned by Stigler (1965) and Nlarfels (1971). See M. 0. Fiiikelstein and Ed M. Friedherg
1
(1967) for an early discussion of using entropy as a concentration index. Although the
entropy index has serious theoretical deficiencies in terms of our criteria, this specific transformation of entropy at least has the merit of yielding a measure of concentration that is
consistent with the Cournot model for industries ssith equal-sue firms.
93
.869
.396
.275
.300
.664
.618
.536
.450
.419
.509
Mean
351)
.324
.400
(.15)
Hm(.25)
H ,,( .5
Ii,,,) 75J
13.
H
32 Industries, II
11,11.11
II
(1(1
R
H
.773
.306
.209
.229
.558
.514
.430
(4
Part B.
A. 22 1ndustrie,
Hal 1.1)
H(75)
H,(.50)
Hp.,(.25)
H,,,(.I5)
Cft'I
H
E
R
('4
Part A.
.09
.16
.740
.92k
.865
.832
.885
(33
.9'3
.791
.201
.194
203
.212
.213
.210
.222
.795
.773
.9/4
.855
.944
.877
.944
.933
.962
.990
.997
.982
.980
.887
.825
.724
.672
.931
.832
.943
.899
.945
.986
.996
.810
H
.6 4
.811
TABLE Ill
.905
.928
.947
.949
.924
.881
.974
.866
.930
.947
.854
.925
.947
.826
.964
.920
E
ANE)
1968
.891
.878
.848
.887
.897
898
.9/0
.877
.159
.857
.893
.865
.896
.897
R
.970
.963
.987
.991
.906
.990
.886
.986
.995
.983
967
.989
C1CI
.995
.973
.956
.980
.9/8
.961
.905
.992
.958
.933
H,,,(.15)
954 CORRFI.ATI0Ns
.979
.993
.991
.9/0
.986
.970
.986
.891
Hm(.25)
.998
.997
.862
.997
.996
H,,,(.5)
(orrelat ions
.992
.992
H(.75)
DnvArIoNs, AN!) CoRRELATIO'.s OF AI.TFRNATIVF CONCENTRAT!ON INDICES
.710
.756
.898
(4
.177
.207
.187
.206
.134
.152
.168
.182
.187
.180
. 190
.191
.117
Deviation
Standard
1968 MFANS. SrANDAR
597
.475
H(i.1)
b
'C
.265
.245
.305
.939
.968
.992
.998
.985
.941
.954
.907
.90/
.93!
.980
.873
.893
.926
.952
.956
.898
.932
.856
.885
.908
.919
.916
.902
.950
.994
.989
.969
.956
.98!
.98!
.956
.97!
.958
.994
Source: Data from Gunnar Du Rietz (1975.
In each part. i1aIiczcd diagonal entries are corrC!ations between the rncasure in 954 and !96
C. 45 Industries
H(I.!)
Hrn(,15)
H,,,(.25)
Hm(,5)
Hm(,75)
dcl
R
.905
.774
.787
.963
.966
.935
.883
.857
.331
.831
.179
.193
.239
.246
.239
.224
.217
.240
Li
.211
.232
.155
.170
.452
.404
.963
.253
.651
I-I
Part C'.
C4
.995
.948
.99!
.98!
.996
.931
.998
.919
.996
937
S
Rosenhluth index
R = [(2
is,) -- IJ', R(n)
=
I/n
"ComprehenSive industrial concentration index" Cçi =
(3,,2 - l)/n.
(2 -- .s,), CICI (ii)
c
s ) and lla(/if Is f de.
indices
11,,(o
The two new concentration
fitted in Section 111 are also included in the correlation matric's for
IS, .25, .50, and .75, and for = 1.1.
Table ill shows the correlations between these ten indices. The off.
diagonal entries give the intercorrelations for 1968. while the italici,ed
diagonal entries give the intertempOral correlation for each index h.
tween 1954 and 1968. The first matrix is restricted to the subsample of 22
.16; the second matrix is restricted to the 32 industries
industries with H
.09; and the third matrix is for the entire set of 45 industries.
with H
The mean and standard deviation of each index arc given in colunins adjacent to the corresponding correlation matrix.
Consider lust the sample of 22 highly concentrated industries with
.16. If an industr is in C-N euuilibrium, this restriction includes oni)
H
those industries for which (lie ratio of monopoly rent to total receipts is
16 percent or more of the ratio that would exist under pure monopoly
(assuming the demand function has constant elasticity). Four features of
this table are particularly worth noting. First, the correlation between C4
and H is .6 14. a quite modest value when compared with the professional
folklore mentioned in the introductory section. The fact that ('. accounts
for only 38 percent of the variance of H contradicts the assumption that
('4 is a good proxy for II for the stud', of highly concentrated industries.
Hm(i 5) k the theoretically acceptable index having the highest correlation
with C, .887. Hence C4 accounts for about 79 percent of the variance in
l-1,,.l 5), still considerably less than the level of association between concentration measures widely assumed. It should he remembered that c =
.15 is nearly the extreme permissible value for the H,,, measure. Second,
H0(l. I). the additively adjusted Courriot measure, has a correlation of
.98 with H. Since 13 = 1.1 is near the extreme permissible value for '1a to
be a theoretically acceptable measure, this empirical result suggests that H
is a statistically adequate proxy for the whole family of l-1,(13) concentra.986.
.5. H and Hm(a) have a correlation
tion indices. Third, if a
The lowest correlation between H and H,,(a) in the table is .899, and
occurs for the extreme value a = .15. This result suggests that only the
.50 for l-1,,.(a) has much chance of producing
statistical results that can he distinguished from those that would be obparameter range .15
35The Rosenbiuth index is mentioned in Marleis (197la) who credits G. Rosenhiuth
(196!) with its introduction into the litcrature. Hall and Tideman (19671 ,ndependenti rediscovered it and proposed the measure in a psihhcation niore readily c:csihk no AnlcnLJfl
economists.
36The C!Cl index was originall proposed by J. Horvaih (1970).
96
tamed by using thc HerlIndahl mdcx as an explanatory variahle7 Ftirth
the interteniporal correlations between 1954 and I 96t br the alternative
concentration indices range ff0111 .905 for H,,( 1 5) to .810 br II. ( has
the second highest interteniporal correlation. .89) These results conlirni
the widespread belief that concentration indices measure an aspect of the
firm size distribution that tends to change slowly over significant periods
of time. However, it may he that the C4 index exaggerates the inter-
temporal stability, since the fani liv of lI,,,() indices have lower intertemporal correlations than ('4 for
.25.
The correlation matrices for the less restricted sample of 32 industries
.09, and for the entire sample of 45 industries, show similar
with H
patterns. Most (not all) of the correlations become larger, as one sotild
expect. Even SO, it is interesting to note that ( accounts for only 63 percent of the variance in II in the 33-industry saniple, and 69 percent in the
full sample, providing further evidence of the substantial statistical differences in these indices.8
Some empirical studies have been more interested in the relationship
between the change in industrial concentration and the change in some
other industry characteristic than in level of concentration and the industry characteristic. Table IV shows the simple correlations between the
changes of the various concentration indices between 1954 and 1968. The
bottom line shows the correlation between the various concentration
index changes and the industry growth rates in employment between 1954
and 1968. defined by p = [log(l968 employment) - iog(1954 employment))/14. The correlations between the various index changes are substantially smaller than the index level correlations reported in Table III.
For example, the correlation between \ (4 and \ H is .417, thus only l7,
of the variance of changes in the H index is accounted for by changes in
the C4 index. The correlation between the H index change and average industry growth rate is small (- .22), but twice the size of the C' change and
growth rate. It is not surprising that the change correlations are lower
than the level correlations, but these results provide further evidence on
the substantial dillerences iii the statistical characteristics of niany of the
concentration indices.
31This assertion assumes that the concentration index is not highly correlated with
other independent variables in regression analyses intended to estimate the partial effect
of concentration on the dependent performance variable.
There are three reasons wh the correlation between C4 and H IS SO much smaller for
the Swedish unrestricted sample than for the unrestricted calculation reported by Scherer
(1970) for Anvrican manufacturing. First. Census disclosure rules prevented reporting of
14 (and thus its use) for industries where II is very large. Second. the absolute sue of the
American market is much larger than the Swedish market, and concentration indices br
most industries are significantly higher in the Swedish market and have less disperston.
Third, the sample of Swedish industries tended to include those industries with aboveaverage levels of concentration.
97
p
.111
906
.929
.796
.804
.884
.968
.940
.220
- .270
.9i
.722
.809
.892
.671
.982
TABI.h IV
.830
- 286
- .223
.901
.703
.760
.838
.914
.848
973
.962
CICl
.978
- .295
.301
970
LJ
918
.940
985
-NI.I5)
l
INI)l,srRrAF CrNrLN1kA-I-J0 INI)I( IS
o wnii INDI;STRY GLWWI Li RAIL,' p. 954 1968
ENTLKF SAMI'1. (45 INL)USTkII.)
(.:F-IAN;I
*Jndustry growth rate. p. log (196K employment) - log( 954 ernployment)1l4
Source: I)ata from Gunnar Du Riet.' (l975)
H0(I.l)
CI(:I
E
.417
.303
.375
.779
.794
.698
.552
.612
wini EMIL OTUIR
01
959
242
H,( 5)
281
H,(l.i)
TABI.I V
COIW.II.ATION, NIFASS, ANt) S!Ar-a)Aiw t)IvIArIoNs
ta
('orrd:iiton
Level of
11 Index
(' ANt) II
"1)1 \
litcilty CoNclNrPcTFO NIANIJI-A(rilsiNc; lNni:Si iUIS l" JAPAN'
Number of
Industries
kan
between
-
-
Si,nlri
)1i
---.-
II and ('4
.36
31
H
.25
87
H.16
.127
.449
165
.701
Ii
.09
250
.882
.4g4
.366
.287
.230
975
.932
.1100
.1129
.0485
.0705
.1247
.1533
(3
.774
1014
* 1972 data from Japancse Fair Trade Commission (1975).
The Japanese Fair Trade Conimission has recently published detailed
data on the H index and Concentration ratios for the largest 3. 4, 5. , and
10 fIrriis for some 350 manultcturing industries. These data make it pos-
sible to compute the simple correlation coefficient of Ii and (. for hiuhlv
concentrated industries, using II as a cut-oO criterion for deliniiig diil'erent
levels of "highly concentrated'' industries. Table V shov s the means, standard deviations, and correlation between 11 and ( br four levels of high
concentration. It is worth noting that ('4 accounts for less than 2 percent
of the variance in H for very highly concentrated industries (1-1
.36).
while ('4 still accounts for slightly less than hall' the variance in 11 with a
much lower cut-oil' criterion (H
.16). These calculations clearly reveal
that C is a very poor statistical proxy for H in samples of highly concentrated industries.
Both the Swedish and Japanese data strongly support the conclusion
that there is little justilication for the bland, almost monolithic reliance on
the widely available (4 concentration measure for the empirical stud of
concentration and industry Performance. particularl in the analysis of
highly concentrated industries.
Concentration indices, ol' course. are onl part of the story of the con-
ditions that lead to competitive behavior. The simple Cournot-Nasli
model explored iii this paner attaches high importance to the industr
elasticity of demand, and much more remains to he done iii establishing
the empirical significance of entr. But it does seem clear that economists
should be able to do better than the casual theoriiing about concentration
and the equally casual choice of a concentration index For empirical work
that dominates the literature.
AI'1'lNulx
1\
iheoretical I.)e/eeis of ('urreni Meavares of Inilu.v(rwl (onee,uralion
This appendix considers live measures ot industrial concentration
that have been discussed in the literature and estimated in Section IV:
99
8
yes
no
no
n t)
yes
shares)
if two or
more "large"
exceeds H
(Index
1?
no
(severely)
no
no
(severely)
yes
no
share)
If only
One "large"
(Index
equals H
(Strict
Ih
no
ye.s
yes
yes
tO 4/,)
(con verges
ri)
equal-size
firms)
to I/n for
Converges
4a?
f'orn 2: 12/3:
J/3l rninimjl.cs)
Usually
(anomalous case
yes
yes
yes
sometimes
industries
should have
smaller
index values)
less conCcntrated
(Stricil,
< n,,
2)
U
vc
yes
yes
yes
yes
without bound)
increases
and approaches
zero aSn
(index is
one ifn =
6
US fl''ol ni-fold
rio
(concave for
yes
yes
yes
yes
of n)
function
decreasing
convex
(index a
S
lv,! will, .V slricliy floflgr< md is
ClsflcerflcrI with ihc ?tflj
no
no
no
yes
no
to zero)*
shares1 goes
to U index
as maximum
(Special
case-index
CONCENTRATION MEASURES
FOR THEORETICAlLY
ADEQUATE MEASURES
Case-index
Converges
relevant hcrc assumes an lnhttai dtsrr,bu,,on
th,l,.,,,, '"''""''.
lb.. .,oi. eidmm,.i Ml,,.
*The convergence
(IC'
Industrial (on.
cenratjon Index"
5. "Comprh
Index R
4. Rosen0
Index E
3. Entropy
Index H
2. Herlindahj
Concentration
Ratio C4
I. 4-firm
In dcx
Criterion
Does Index
Satisfy
IN TERMS OF CRITERIA
INDUSTRIA
TARLE VI
PERFORMANCE OF ALTERNATIVE
3
E
C4, H, F, R, and CICI. It determines the extent to which they satisfy the
criteria developed in the text for theoretically acceptable concentration
indices. Table VI assesses these indices in terms of the six theoretical
criteria developed in the text.
Table VI shows that none of the measures satisfy all three of the
especially important theoretical criteria 1, 2, and 3a. The implications of
these failures for each of the measures are discussed below.
The concentration ratio C4 is by far the most widely used index of industrial concentration, and dominates cross-sectional studies of American
manufacturing, since it has been computed and made available by the
Census Bureau for a large number of industries and product classes. since
1947. A major theoretical defect is that changes in the relative size of the
largest four firms have no effect on the index, so long as their total share
remains constant. It seems implausible that it makes no difference for industry behavior whether the top four firms each have a 20 percent share,
or whether one firm has a 78.5 percent share and the iiext three firms each
have only . percent. Its value does exceed the H index if there arc two or
more large firms (criterion I), but does not behave reasonably for many
different configurations of relative size of individual firms, including
possible mergers of moderate size.
The Herfindahi index satisfies all of the criteria except the first, which
it obviously must fail. This generally favorable perforniance is hardly surprising, since the text argued that the H index arises naturally out of the
Cournot model. Its major theoretical defect is that it makes no allowance
for tacit or overt collusion if several firms have large shares, a situation
which makes a C-N equilibrium rather implausible. Whether such collusion would be expected to obtain high returns depends, of course, on other
industry characteristics not captured in the size distribution data, such as
the demand elasticity, the ease of entry, and the feasibility of high rates of
expansion by other firms in the industry. And in a study of firm or industry profit rates these other factors should be taken into account, as
well as the concentration index.
The entropy index, F, fails all three of the important criteria I. 2. and
3a. It is always less than the H index, except for equality if the industry is
composed of equal-size firms. Thus it does not in general yield au index as
large as that produced by the Cournot model which ignores the additional
effect of collusion. An equally serious defect is the overwhelming influence
that a large number of very small firms can have on the index. It is easily
demonstrated that if one firm has a 99.99 percent share of industry sales,
if the remaining sales are from an indefinitely large number of small firms,
the equivalent number of equal-size firms tends to infinity, i.e.. industry
structure becomes perfectly competitive.39 For the entropy index to be a
39Proof: L.et the largest firm have share ,c1
k/rn. where k
(I - s1). Then
101
let the remainin
rn firms ha;c equil share
0
reasonable measure of industrial concentration lr studying the degree of
competition in a sample of industries, one would have to discard completely the (Tournot theory, modified to allow for collusion. Perhaps one
couki argue that ii an industry includes an extremeI large number of
small firms, this indicates that entry is very easy, and So even if industry
sales are largely due to one or several large firms, the potential entry will
make the industry completely competitive in price and output. There may
he sonic truth to the ease of entry argument. but this alternative theory
essentially says that the size distribution is irrelevant; the only relevant
dimension to the problem is the ease of entry. But this alternative theory
has nothing to say about the relevance of the size distribution if entry is
iwi extremely easy. Hence the entropy index seems to have highly unreasonable theoretical properties as a measure of industry concentration.
The Rosenhiuth index R suIrers all of the theoretical defects of (he
entropy index, except that its behavior seems to be dominated even more
strongly by small ilrni
in summarizing empirical size distributions. Its
usefulness in the analysis of highly concentrated industries for departures
from competition seems highly dubious.
The ClCl was developed in a highly intuitive way by }Ior'aih
(1970), in order to obtain a measure that refined what he considered to be
the desirable characteristics of the H index. The CICI does indeed safisfy
the first criterion of being larger than the I-I index if there are two or more
firms, since it is larger term-by-terni than H. However, it fails to satisfy the
important conditions 2, 3a, and 3b. It exaggerates the role of the first firm
if only one of the firms is "large" relative to the Cournot theory, since it
takes the share of the fIrm, and not its square (condition 2). Its order behaves according to the Cournot theory, although it converges to a value
three times the size of the I-I index as the share of the largest firm approaches zero, and as the number of firms increases without bound. The
asymmetry of its leading term generates several bizarre characteristics,
since in a two-firm industry, the index minimum occurs when the shares
are (2/3, 1/3) and not when the firms are of equal size, each with sales of
1/2. Furthermore, the index is decreasing, but eoicave, for values of ,i
in the interval from one to two.4° There is no underlying economic theory
,n+I
s;t (k/rn)5,
I
-=
F
I
s'(k/rn)"
and
Jim
siIk/,n5
=
40Marfels (1972) also notes the nonconvexity and odd behavior
of this index.
102
to justify these odd properties, they just happen to he among the iniplica-
tions of the formal definition of the index. They do suggest the logical
hazards of an intuitive approach to the delinitioji of mcasuies, since it is
easy to include by accident anomalous properties.
On balance, all of the indices of industrial COflcentfitjoii discussed in
this appendix have significant theoretical deficiencies. The II index seems
most satisfactory from most standpoints, except for the way it ignores
collusion. The concentration ratio (' is larger than H, as SCCI1IS reasonable, but completely ignores the relative size of the large firms. The multi-
plicatively-niodified Cournot measure Fl(o;
) and the additivelyadjusted Cournot measure Ha(fl; lsi ) defined in the main text satisk' all
of the theoretical criteria presented here. Section IV shows that the
statistical behavior of the Ha index differs very little from the H index in a
sample of.Swedish manufacturing industries. However the H(() faniilv
may prove to he a useful point of departure for theorizing about the site
distribution and collusion and for empirical experimentation, to see
whether it performs signicantl better than C4 or H in studies of industrial concentration and the degree of' competition.
APPENuIX B
A lierna live Concentration Measures for Siedish Manu/àcturinça industries
Table VII presents various concentration measures for 45 Swedish
manufacturing industries. A more detailed description of the data on
which they are based is provided in Du Rietz (1975). Several industries
were modified somewhat from the official industry classifIcations used for
Swedish manufacturing statistics.
t!nirersiti' 0/ Minnesota
National Bureau ot !ConOmic' Re.rt'arc/i
Suh,nitted Marc/i /976
Revised September /976
103
22
23
24
25
26
27
28
21
20
19
8
7
16
IS
14
13
12
ii
9
0
8
7
6
5
4
3
2
I
382991
382992
382510
382590
32490
351310
351320
356000
371010
371020
371030
372030
372040
381100
381200
381300
381910
381920
381930
381940
381950
381990
382100
382200
382310
382320
382410
382420
Industry
Code
Number
24
4
37
53
276
119
38
27
74
81
3
628
83
59
149
46
75
164
87
722
34
18
73
6
19
234
51
13
Firms
of
Nurnbr
TABLE VII
1.00000
.60682
.34642
.57106
.72490
.50612
.33348
1.00000
.87285
.41002
.70675
.65979
.57645
.66343
.37264
.42312
.15646
.09431
.86227
.62629
.27108
.54219
.94976
.53737
.90633
.46169
.44404
.47172
C4
.25024
.04867
.06482
.01063
.98413
.17189
.04140
.09838
.25500
.08500
.04023
.66086
.31028
.05813
.14812
.07811
.07151
.07860
.00578
.22106
.09740
.02854
.10594
.31826
.09072
.41535
.13041
.41094
H
.02314
.10465
.51171
.18391
.26686
.04523
.25897
.03443
.02256
.03501
.00276
.09578
.06376
.09218
.02219
.03265
.00391
.94971
.06494
.02224
.06230
.12282
.04748
.01271
.08941
.24237
.06685
.01166
E
.28647
.07645
.01230
.11643
.33546
.05026
.30843
.03725
.02030
.03805
.00323
.09929
.08237
.08071
.02516
.03827
.00417
.97916
.05945
.02718
.08018
.12749
.05645
.01182
.61905
.20575
.02405
.13734
R
.53569
.27514
.18317
.83562
.61320
.22520
.40554
.30571
.67166
.37768
.13622
.31300
.66402
.29697
.69055
.27813
.26427
.27638
.03957
.48907
.29625
.51702
.19670
.24264
.07077
.99209
.43124
.16217
CICI
1968 CONCFTRATJON MEASURES FOR 45 SwEDISh
.61568
.31405
.08279
.31181
.64080
.24255
.64800
.19818
.17406
.20460
.01622
.39713
.27605
.41526
.13820
.17966
.02999
.98618
.33764
.12387
.27160
.46107
.23782
.10804
.80668
.57103
.15660
.38439
Hm(.15)
.75128
.47592
.10124
.27866
.06771
.18404
.37353
.15795
.07431
.00852
.31388
.18576
.33817
.08603
.11566
.01650
98485
.25744
.13721
.53319
.22582
.04872
.20892
.53547
.16478
.56417
.13303
.11827
EI,(.25)
.45149
.15585
.03154
.13155
.41061
.10847
.47175
.08986
.08137
.09156
.00598
.24589
.11906
.27391
.05552
.07514
.01122
.98420
.19395
.04699
.11950
.29263
.10184
.04495
.69192
.37265
.06671
.18692
H,(.5)
MAUFACTIRJC iNDUSTRIES
.42556
.07892
.07217
.07956
.00578
2347
.09937
.25253
.04904
.06548
.01064
.98413
.17392
04166
.10032
.26007
.08638
04045
.66560
32149
.05865
.15302
(J9225
.41674
.13302
.02865
.10835
33896
H,,,(I .0)
14714
33170
.07042
09474
.01437
.98869
.23764
06025
.14757
.34854
2652
.05673
.76669
43104
.08367
22445
.00767
30030
1259
10119
11404
.52723
.19018
.04041
.16321
.46644
.13375
.54265
H(i.l)
*
61
Ferro-Alloy Works
4
S
IS
14
IS
12
I
9
10
8
7
6
3
Other Metal Goods for
Construction
Lines and Cables
Nails, Scrcsss and Bolts
Metal Furniture
Metal Construction
Metal Containers
Metal Wire. Screen.
Iron and Steel Castings
Non-Ferrous Products
Non-Ferrous Castings
Tools
17
2
I
.31311
.23028
.07338
.03161)
.03389
.40456
.19588
.14392
.27843
.32006
.22148
.04929
.15430
.01593
.19991
.45877
.34877
.30431
.02722
.23291
.19920
.13699
.36056
.38147
.26360
.05515
.19056
.01946
.24866
.53395
.40591
.02001
.23828
.09622
.75437
.31441
.84630
.63656
.59157
.68705
.69809
.64561
.31648
.52898
.11689
.53832
.83120
.71885
.36188
.68024
.28470
Household Metal Goods
Misc. Metal Goods
Stationary Motors & Turbines
18
Agricultural Machinery
19
20 Metal-Working Machinery
Wood-Working Machinery
21
Pulp and Paper Machinery
22
Cranes & Excavation Machinery
23
Other ProductioC Machincr3
24
25 Computers
Other Office Machines
2
Lifts and Hoists
27
Fluid Pumps
28
Machinery Parts
29
30 Other Machinery
Electric Motors. Generators.
SI
Electric Apparatus
.38247
.12366
.44075
.09021
.50071
16
.89153
1 00000
.99847
.53881
.82136
.49141
.84551
.24827
Semi-Fabricated Plastics
Plastic Goods
Iron and Steel Works
The industries in this table are
20
16
244
7
4
6
106
70
36
16
5
8
49
.92286
.83830
.79602
.93756
.99765
.88417
.55160
Basic Plastics
44
45
42
43
4!
40
:49
32
33
34
35
36
37
38
31
58
.51480
.09574
.70568
.36847
.27188
.37026
.34955
.36875
.10012
.22039
.02660
.22425
.48961
.93019
32
212
382993
382999
383100
383200
383300
383910
383920
383930
383990
384110
384120
384130
384210
384310
384320
384400
384900
29
30
.1481..
.70675
24
382992
28
.70946
.23192
.78482
.57586
.53446
.65890
.69214
.59674
.25973
.50692
.08133
.53454
.80700
.71535
.27314
.60240
.26922
4S
44
43
42
4!
40
39
37
38
36
35
32
33
34
.10986
.46581
.14646
.02931
.29195
.62093
.48487
.56436
.11063
.71704
40712
.34094
.45275
.45412
.41634
.12023
.28425
.23569
53717
.40650
.12617
.44330
.09182
.10201
.23136
.02669
.37598
.28581
.38857
.37405
.52678
.09688
.10674
.37358
Bicycles and Motorcycles
Other Transportation
Auto Motor Parts. Trailers
Telephone Products
Electric Household Appliances
Electric Filaments & Cables
Batteries (includtng Storage)
Incandescent and Fluorescent
Lumps
Misc. Electric Industry
Shipyards
Boat Construction
Boat and Ship Motors
RailwaY Cars
Automobiles
.20434
.53007
.17563
.73171
.61401
.74934
.48986
.44327
.56455
.58687
.50735
.17999
.39739
.04610
.41470
.16091
.64096
.51549
.51288
.48726
.14725
.32998
.03815
.34142
.68615
.54810
.17385
.54590
.13787
.38781
.13684
.77842
.47922
.63841
I
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fl?t-&E -:*e