The Critical Choice Between the Cr and the H-Index

8/3/2019 The Critical Choice Between the Cr and the H-Index

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The Critical Choice Between the Concentration Ratio and the H-Index in Assessing IndustryPerformanceAuthor(s): Leo Sleuwaegen and Wim DehandschutterSource: The Journal of Industrial Economics, Vol. 35, No. 2 (Dec., 1986), pp. 193-208Published by: Blackwell PublishingStable URL: http://www.jstor.org/stable/2098358 .Accessed: 12/05/2011 18:46

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THE JOURNAL OF INDUSTRIAL ECONOMICS 0022-1821 $2.00Volume XXXV December 1986 No. 2

THE CRITICAL CHOICE BETWEEN THECONCENTRATION RATIO AND THE H-INDEX IN

ASSESSING INDUSTRY PERFORMANCE*

LEO SLEUWAEGEN AND WiM DEHANDSCHUTTER

The paper shows that in addition to the theoretical relevance of choosingbetween the Hirschman-Herfindahl index and the concentration ratio,both measures may provide empirically very different information toassess industry performance. It is shown that the correspondence betweenthe two measures can be represented by a horn-shaped figure. The impli-cations of this horn-shaped relationship are investigated in the context ofspecifying empirical profitability-concentration models. Among differentspecification effects tested, it is shown that the neglect of the horn-relationship may bias the results towards the identification of a criticalconcentration ratio.

INTRODUCTION

Tus PAPER analyzes some of the implications of the relationship between theHirschman-Herfindahl index (from now H-index) and the k-firm concen-tration ratio (Ck) within the context of empirical models dealing withprice-cost margins and concentration.

In the first section of the paper, it is shown that the mapping of all H-indexvalues for given Ck values does not correspond to a simple function, butfollows the pattern of a horn. From using several properties of this horn-shaped relationship, alternative hypotheses about the choice between the twoconcentration measures in specifying empirical profit-concentration relation-ships are tested on a sample of US industries in section II. With the H-indexconsidered as the appropriate (Ck encompassing) concentration measure, wefurther investigate the specification effects of using several k-firm concen-tration ratios as surrogate variables in profitability regressions. Among thedifferent effects tested, the empirical identification of a critical concentrationratio in section III shows up as a remarkable (artifact) result. The last section

of the paper summarizes the results in view of the current practice of usingconcentration measures to assess industry performance.

*We would like to thank the members of the INdustry and Company Analysis Program(INCAP) for the very stimulating discussion from which the present paper originated. Wegratefully acknowledge the helpful comments from Raymond D. Bondt, Denis de Crombrugghe,Charles Baden Fuller. Also the constructive comments from an anonymous referee and theEditor of this Journal were very helpful in improving the final text. The usual disclaimer applies.

193


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194 LEO SLEUWAEGEN AND WIM DEHANDSCHUTTER

I. THE HORN RELATIONSHIP BETWEEN CkAND H CONCENTRATION NDICES

It is well known from theory that price-cost margins may be directly relatedto different measures of concentration, depending on the prevailing conductof firms within the industry. Within this framework, the k-firm concentration

kratio Ck= Z si (si denoting the market share of firm i, belonging to the

i = 1k largest firms) is consistent with collusive price-leadership oligopoly (Saving

n[1970]) while the H-index, H= E si2 (n, the number of firms in the

industry) is related to noncooperative Cournot-Nash behaviour (Cowling

and Waterson [1976]).In view of the theoretical relevance of choosing between the H-index andthe concentration ratio, several studies have explored the extent to which theconcentration ratio and the H-index provide different nformation (see Curryand George [1983] for a recent survey). The analyses in these studies are noteasy to interpret and sometimes produced conflicting results (see, for instance,Hause [1977] and Schmalensee [1977]). The conflict seems to be due to thestatistical nature of the studies, whereby the results are heavily dependentupon the particular samples of industries used in the analyses. In order toillustrate this problem we have plotted in Figure 1 and Figure 2 the C4 ratioagainst the H-index, for Belgian and US industry data, respectively. ForBelgium we can show the plot for a continuous range of C4 from 0 to 1.Unfortunately for the US data, presumably because of confidentiality reasons,the range stops at C4 = 0.87. This is very regrettable, since it is especially in

H 1.0-

0.9 -

0.8-

0.7- /

0.6-

/ o z~0.4- / o 0.5- q 0

00 00.4-

o% b 0-00.3- ~0.2~~~~~~~

0.1

0-0 0.2 0.4 0.6 0.8 1.0

C4Figure 1

The Horn Relationship or Belgian ndustries 1981)


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THE CONCENTRATION RATIO AND THE H-INDEX 195

H 1.0-

0.9-

0.8-

0.7-

0.6-

0.5-

0.4-

0.3-

0.2~~~~~~~~~~0.1

0-0 0.2 0.4 0.6 0.8 1.0

Figure 2 C4The Horn Relationship or US Industries 1956)

the top range that the relationship between C4 and H becomes veryinteresting and (possibly) divergent. Conclusions drawn for the particular USsample (Schmalensee [1977]) may therefore be very different from thosebased on the Belgian sample (or a Japanese sample, as in Hause [1977]).

The statistical problem may be understood better by exploiting some of themathematical properties of the two concentration indices in order to find theset of possible combinations between the two measures. From studying theseproperties we find that the ranges of possible H-index values correspondingto given Ck values follow the pattern of a horn. As proven in the appendix,the limiting boundaries are given by the following formulae:

min H = Ce/k k being the number of firms included in theconcentration ratio

max H = Ck2 if Ck > l/k

max H = Ck/k if Ck < 1/k (the linear piece in the graphs,except for a small fraction if k(l - Ck)Ck isnot an integer number (see appendix)).

This suggests that the relationship between C4 and H cannot, contrary towhat has been suggested before, be linearly approximated. Given the possibleranges of the H-index in the horn, a concentration ratio is likely to be(although it need not be) a bad proxy for the H-index in concentratedindustries. Therefore, the horn-shaped relationship provides an answer to thevery different degrees of correlations and kinds of relationships between Ckand H, found in studies where different samples of industries were used.


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196 LEO SLEUWAEGEN AND WIM DEHANDSCHUTTERH-1.0-

0.9-

0.8-

0.7-

0.6-

0.5- =

0.4-

0.3-

0.2-

0.1 k=4

0-0 0.2 0.4 0.6 0.8 1.0

Figure 3 CKThe Horn Relationship with a Varying Number of Firms k

Several interesting properties of the horn emerge. First, from the boundaryformulae it is possible to analyze the effects of varying k, the number ofleading companies included in the concentration ratio, on the shape of thehorn. This is done in Figure 3. It shows the narrowing of the horn fordecreasing values of k. The lower bound tilts up, as does the linear part of theupper bound (shown for C2 and C4).

Second, the maximum value of H in the region Ck > l/k as well as itsminimum value over the whole range assume an infinite number of firms.Considering a finite number of firms does not fundamentally affect the horn

shape of the relationship. Figure 4 shows the upper and lower bound forn = 10.1 The figure illustrates how the finite (actual) number of firms helps toexplain why empty regions occur in the actual Herfindahl distribution withinthe horn.

Third, a conditional (upon Ck) numbers equivalent H-index appears,which provides information to analyze the degree of concentration withinindustries (Adelman [1969]). This conditional numbers equivalent H is givenby the straight line, which is also the maximum (except for the eventualfraction) for values of Ck lower than l/k. The fact that many industries fallbelow it implies that the number of firms in these industries is larger than the(conditional) strict minimum number and that the variance of the sizes is notable to tilt their H-index above the line. Given their Ck-ratio these industriesare relatively unconcentrated in terms of the H-index measure.

I Using the transfer principle (see appendix), in this case the upper bound is calculated fromletting nine (n-1) firms hold an equal share, (1- Ck)/(n-k), and the tenth (dominant) one theremainder. The lower bound is given by (Ck,/k)+(1 -Ck)2/(n-k).


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H 1.0-

0.9-

0.8-

0.7-

0.6-

0.5-

0.4-

0.3-

0.2-

0.1

0-0 0.2 0.4 0.6 0.8 1.0

Figure 4 C4The Horn Relationship with a Finite and Fixed Number of Firms, Equal o 10

Fourth, the horn-shaped relationship can be related to the concept ofiso-concentration curves, introduced by Davies [1979]. Assuming lognormalfirm size distributions, Davies related curves representing fixed values ofsome concentration measure to n, the number of firms in the industry, and a2,the variance of their sizes. This property makes it easy to find for each iso-Ckcurve the implied range of H (comparable to a vertical slice in the hornsshown above). The interested reader can verify that from the definitions of theH-index ( = exp (2I2/n) under lognormality) Davies' results imply a wideningrange of H as Ck increases, confirming the horn-like pattern (see also Hart[1979]).

H. THE CHOICE OF CONCENTRATION MEASURE IN EMPRICAL MODEL SELECTION

In the previous section it was argued that a distinction between the C4 andthe H-index is justified on both theoretical and empirical grounds. This raisesthe question how to specify the empirical relationship between profitabilityand industrial concentration.

One possible approach to tackle the specification problem is to assumethat the choice problem is simple with only one of the two concentrationmeasures being relevant to include in empirical (cross-section) industryprofitability models. Assume this measure to be the H-index. It then becomespossible to predict what results will be obtained when other concentrationmeasures are substituted for the H-index. If the assumption holds this wouldimply the testable hypothesis that, for the sample considered, the resultsobtained for the H-index model can account for those obtained with some Ck


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ratio as explanatory variable and not vice versa. This "encompassing"property, as it is known in the econometrics literature (see Hendry and

Richard [1983]), may be explicited as follows. Consider the two "competing"hypotheses with alternatively H and Ck as regressors for explaining industryprofitability:

J(1) (r/R)i = f6o + IHCi + E |jzji +8Hi

(2) (n;/R), = bo + &1Ck,i+ = 2

where index i denotes the industry and Zji denotes other regressors thanconcentration measures. The cHi and ECi are error terms assumed tobe independently and normally distributed. However, since high profitindustries are often characterised by some "extreme" firms entailing lessaveraging-out of errors, we expect eHi and ici to be subject to hetero-scedasticity.

Describe similarly the H - Ck relationship by

(3) Hi = xO+OlCki+vi

where, following the horn-shaped relationship, vi is not expected to bestationarily distributed.

Substituting (3) in (1) yields:J

(4) (n/R)i = fio + #I oo+ #1al Cki + , jZji + (H1i + 1 vi)

Hence, if (1) and (3) are true, someone estimating (2) is in fact estimating (4)and is able to deduce from (1) and (3) what results will be obtained fromestimating (4).

Thus, if (3) is a "true" relation, we are led to the following testablehypotheses:

(i) Nesting both Ck and H as regressors in one equation should result in acoefficient of Ck nsignificantly different from zero. In terms of (2) and (3) thishypothesis implies (requiring no additional testing, see Hendry and Richard[1983, p. 140]) 51 = clx1/l.Similarly, 5j= lj orj = 2,..., J and 60 = PBo lpo.

(ii) The smaller k, the number of leading companies in Ck, the more theestimation results of model (2) using Ck will approach those obtained using

the H-index as concentration measure. This follows from the results in theprevious section where it was shown that the horn becomes narrower and alinear approximation of the H-index by the Ck-ratio better for decreasingvalues of k. Furthermore, we expect the non-stationarity of the error term in(3) to show up in the profitability regression where Ck is used as independentvariable. Although the non-stationarity applies to other moments than thevariance, and unless eHi and vi have strong negative correlation, we expect itto augment the measured degree of heteroscedasticity.


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(iii) Given the horn-shaped relationship, it follows that the choice betweenH and Ck in profit regressions should not matter too much for low

concentration industries (C4 < 0.50) but may become very important forhigh concentration industries (C4 > 0.50). Moreover (unless the conditionH < Ck/k s satisfied over the whole sample (cfr. section I)), we would expecta significant shift in magnitude of the estimated Ck coefficient for the groupsof high and low concentration industries.

These points are tested against US data, which enables us to relate ourfindings to previous work, where the same or similar data were used.

In line with many other studies the following empirical specification isposited to test the above hypothesis:

(5) (r/R) = f(CON, H, KO, GRO, ADV, CDR)

The dependent variable (r/R) is defined as the percentage gross return beforetaxes on sales for the industry in 1958. The reason why we took the year 1958is explained by the absence of H-index data for more recent years (seehereafter) (source: Collins and Preston [1968]). CON is one of the followingconcentration measures: C4, C8, C20 for the industry. H is the correspondingHirschman-Herfindahl index. The data were taken from Nelson [1963] and

cover H-indices and Ck ratios computed from the same US census data. Itappears this is the only data set available containing such information for alarge sample of US industries. Unfortunately, these data relate to the year1956. As already mentioned, another problem with the published data is thatthe most concentrated industry in the sample has a four firm concentrationratio equal to 0.87. As we argued, it is especially in the missing, very concen-trated industries that the choice problem becomes very important (compareFigures 1 and 2 in section I). Nevertheless, we assume the data set to containenough information for tests on it to be of interest. KO is the capital-outputratio for the industry in 1958 (source: Collins and Preston [1968]). GROmeasures industry growth in sales between 1954 and 1958. ADV stands forthe advertising to sales ratio of the industry (source: Ornstein [1977]). CDR isa cost disadvantage ratio measured as the value added per worker-hour of thefour largest firms divided by the average value added per worker-hour in theindustry (source: Nelson [1963]).

Matching the different data sources resulted in a sample of fifty-fourcomparable four-digit SIC industries.

The tests, corrected for heteroscedasticity,2 are presented in Tables I-III.The "nesting" regression (column (6) in Table I) clearly supports our basic

2In estimating the different equations, we found support for the heteroscedasticity assumption.The absolute values of the residuals correlated strongly with (ir/R)i. the fitted profitability (takenas the best estimate of E((7r/R)i I ), the conditional mathematical expectation of industryprofitability). We correspondingly assumed the error term variance to be Uj2 = U2E(ir/R)f andused 7r'R as weight factor in the weighted least squares regressions (see Kmenta [1971,pp. 261-264]).


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TABLEREGRESSIONS OF INDUSTRY PROFITABILITY ON VARIOUS MEASURES OF CONCENTRATION: ALL

INDUSTRIES

(6) (7) (8) (9) (10)

CONSTANT -0.055 -0.062 -0.139 -0.166 0.170(-0.528) (-0.786) (-1.475) (-1.574) (1.386)

H 0.557 0.519(1.671) (4.062)

C4 -0.021 0.208(-0.141) (3.546)

C8 0.194(3.153)

C20 -0.077(- 1.232)CDR 0.112 0.114 0.137 0.143 0.025

(1.892) (2.059) (2.321) (2.323) (0.345)GRO 0.276 0.279 0.298 0.351 0.249

(1.364) (1.358) (1.403) (1.604) (1.059)ADV 0.844 0.837 0.838 0.912 1.143

(4.305) (4.274) (4.112) (4.476) (5.046)KO 0.131 0.127 0.133 0.124 0.122

(2.615) (2.553) (2.593) (2.363) (2.140)

R2 0.480 0.480 0.432 0.400 0.209

S.E.R. 0.071 0.070 0.073 0.075 0.086

(Values n parentheses re t-ratios, .E.R. = standard rror fregression.)

H-encompassing hypothesis (i). The coefficient of C4 is not significantlydifferent from zero and the inclusion of C4 leaves, in comparison with theH-only regression (column (7)), the goodness of fit, as good as unchanged.Similar results obtain for C8 and C20 (not shown). In line with the en-compassing property of the H-index model, the estimated coefficients ofthe concentration ratio variables closely resemble the products of theH-coefficient in regression (7) with their respective coefficients in the H on Ckregressions of Table II (except for C20, which regression displays a largestandard error). Regarding the second testable point, the specification ofdifferent Ck ratios, a look at the R2 series (calculated from unweightedresiduals) reveals that of all the concentration ratio models, the model withC4 has the highest explanatory power. Completely in line with our expecta-

tions, the C8 and C20 models display an increasingly deteriorating perfor-mance. The correlation of H with C20 seems to be too small for theconcentration ratio to be able to serve as a surrogate variable for the H-indexat all, which regression (10) clearly points out. These results compare verywell with the recent work by Kwoka [1981] who, using recent data, finds asimilar decrease in the explanatory power of Ck ratio profitability models fork increasing from I to 10.

The non-stationarity hypothesis involving a widening of the residual term


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TABLE IIREGRESSION F THE H INDEX ON THE FOUR-,EIGHT- AND TWENTY-FIRM

CONCENTRATION ATIO: ALL INDUSTRIES

(11) (12) (13)

CONSTANT -0.076 -0.098 -0.022(-7.286) (-5.360) (-0.615)

C4 0.365(18.840)

C8 0.317(11.747)

C20 0.165(3.700)

R2 0.872 0.726 0.208S.E.R. 0.029 0.042 0.072

(Values n parentheses re t-ratios, .E.R. standard rror of regression.)

for high profit industries which becomes stronger when Ck is used assurrogate variable finds support in an estimated widening of the absolutevalue of the residuals of equations (8) and (9) versus equation (7).3

With C20 the linear approximation was too poor to yield any significantresult, consistent with the widening of the horn.

Finally, in order to test hypothesis (iii), stating that the choice between theCk ratio and the H-index is less important for low concentration industries,the sample of industries has been split up into two groups: one group ofindustries with a C4 smaller than 0.50 and another group of industries with aC4 greater than or equal to 0.50. These groups contain about the samenumber of observations.4 The main results from the profitability regressionsfor both groups are given in Table III. This table shows that there is indeed

no real difference in goodness of fit between the models including either the3The estimated relationship for the profitability model with the H-index (regression (7)) s

I HI -0.01 +0.297r/R.(-0.63) (3.38)

With C4 as regressor (regression (8)) the corresponding result is

I c41= -0.04 + 0.33 Ir/R.(-0.84) (3.76)

This result can be related to the heteroscedasticity test for the H-C4 relationship, yielding

101= 0.Ol+0.09H,(2.79) (2.93)

and the estimated value of f1P f 0.52. Similar results obtain for C8. A similar check giving strongsupport to the H-index encompassing property (excluding the reverse case with C4) can beconducted with the residual variances (the square of S.E.R. in the tables), given that

corr (H, 0) = 0.08: var ;t4) var (tH)+ l2var (e) + 2p1 corr (tH, 0) var (H)var(e)4These numbers equal 28 and 26 for the unconcentrated and concentrated industries

respectively. The threshold point of 50 percent is chosen arbitrarily (see next section).


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TABLE IIREGRESSIONS F INDUSTRY PROFITABILITY N CONcENTRA110N: LOW AND HIGH CONCENTRATION

INDUSTRIES

Low Concentration C4 < 0.50) High Concentration C4 > 0.50)

Estimate of Estimate ofConcentration Corresp. R2 Concentration Corresp. R2

Coefficient R2 (H on Ck) Coefficient R2 (H on Ck)

H 0.812 0.585 0.505 0.465 -(1.607) (1.993)

C4 0.184 0.589 0.873 0.321 0.419 0.766(1.650) (2.200)

C8 0.097 0.577 0.772 0.378 0.359 0.517(1.254) (2.089)

C20 0.057 0.567 0.622 -0.107 0.347 0.038(0.807) (-1.161)

(Values in parentheses are t-ratios.)

Ck or H-indices for the sample of low concentration industries. This contiastssharply with the results for high concentration industries, where the H-indexmodel shows up as the superior performer and the Ck models gradually

deteriorate in goodness of fit. The change in magnitude of the estimatedcoefficient of Ck is also in line with the formulated hypothesis and may againbe related to the H-Ck regressions of which the corresponding R2 arepresented in a separate column. This point will be developed more extensivelyin the next section.

HI. CRITICAL CONCENTRATION RATIOS

A remarkable empirical finding that has been repeatedly discussed inthe literature is the emergence of a critical concentration ratio in therelationship between industry profitability and concentration. Scherer [1980,pp. 279-280] reports it as follows: "The results for a diversity of profitabilityindices and U.S. industry samples are remarkably uniform: virtually all showa distinct upsurge in profit rates as the four-firm concentration ratio passesthrough a range somewhere between 45 and 59 percent. They lend supportto Chamberlin's hypothesis that respect for mutual oligopolistic inter-dependence tends to coalesce at some critical level of seller concentration."

Clearly, the studies Scherer refers to did not look at the problem to whatextent this discontinuity might be the result of misspecification. Again, giventhe H-index encompassing property and from looking at the existing horn-shaped relationship between Ck and H it is reasonable to hypothesize that abetter performance of a discontinuous relationship between profitability andthe concentration ratio is likely to show up as a specification artifact.

Various types of critical concentration ratios have been proposed in theliterature. The best known are those which introduced a dichotomous


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variable which takes a value equal to zero up to a critical level of the four(eight) firm concentration ratio and then becomes equal to one after that level

(see for example White [1976]). Other studies, which found a systematicrelationship of profitability with concentration, have, in addition to intro-ducing the dichotomous variable, investigated whether a significant change inslope at the critical concentration ratio occurs in the estimated model (see forexample Dalton and Penn [1976]).

Similarly, Rhoades and Cleaver [1973] have identified a critical ratio afterwhich a systematic positive association of profitability with concentrationoccurred which did not show up before that level. In a recent study, Bradburdand Over [1982] have identified two critical concentration ratios in therelationship between profitability and concentration.5

By making different scatter diagrams of possible H-C4 associations withinthe horn, it can be verified that these different proposed critical concentrationratio models are easily reconcilable with the specification problem of usingC4 instead of H as concentration variable. Figures (5a-5d) show thesedifferent plots, for E(r/R) = rHi. Obviously, from these figures alone wecannot conclude that the studies which considered these different types ofcritical concentration ratios have used a misspecified model. Ideally, the

H-index encompassing hypothesis should be tested separately on a number ofdifferent data sets before more general conclusions can be drawn. Given theH-index encompassing property we would interpret the introduction ofcritical concentration ratios as a means to mitigate some of the difficultiescaused by misspecification. Particularly the evidence presented in Table III ofthe foregoing section for the groups of low concentration and high concen-tration industries would help to understand the empirical success of adiscontinuous model as presented in Figure 5b.6

In searching whether a discontinuous relationship of profitability withconcentration is superior it may be more appropriate to consider differentregimes of the model with respect to all explanatory variables (see also White[1976]). Such an approach has been followed with the estimated models inTable IV where all the coefficients of the variables are allowed to change if theindustry has a C4 higher than the critical level. The variables with suffix Ddenote the products of the original variable with D, a dummy variable equalto one when the industry has a C4 greater than 0.59. Conformly with previous

5In this connection it is also interesting to look at William Shepherd's work who derived fromhis empirical results relating to the years 1963-1969 a horn-shaped relationship betweenprofitability and the four firm concentration ratio (see Shepherd [1972, Figure 2, p. 33]).

6 In connection with the split up of industries, it should be noticed that for low concentrationindustries (C4 < 0.50) the possible range of the H-index (0-0.25) is only half of the C4 counter-part. For a discrete (actual) number of firms, this maximum value becomes even much smaller,which might help to explain why, for the sample of low concentration industries considered inthis analysis, H has a maximum of only 0.08 (and hence a small variance), while the maximum ofC4 equals 0.49. This implies that relative to the range of C4 the low concentration industries areindeed very unconcentrated in terms of their corresponding H-index.


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Etszr/R)j

E

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tir/R);

THi

THi

o

o_X

*L

0

0.2

0.4

0.6

0.8

1

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0.2

0.4

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C4;

CRIT

C4;

Figure5a

Figure5b

The

Dichotomous

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The

Discontinuous

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THR

THi

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0

0.2

0.4

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CRIT

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Figure5c

Figure5d

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Figure5

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THE CONCENTRATION ATIOAND THE H-INDEX 205

TAnu IVREGRESSIONS FINDUSTRY PROFITABILITY N CONCENTRATION: HIEDISCONTINUOUSMODEL

(14) (15) (14) (15)

CONSTANT 0.055 -0.074 D -0.235 -0.569(0.570) (-0.739) (- 1.479) (-2.885)

H 0.514 HD -0.179(1.626) (-0.394)

C4 0.138 C4D 0.351(1.603) (1.823)

GRO 0.715 0.707 GROD -0.716 -0.659(3.048) (3.237) (-2.197) (-2.166)

ADV 2.338 2.298 ADVD -1.472 -1.598

(4.724) (5.018) (-2.665) (-3.122)KO 0.202 0.202 KOD -0.155 -0.093(4.626) (4.964) (- 1.489) (-0.993)

CDR 0.035 0.035 CDRD 0.365 0.396(0.575) (0.601) (2.885) (3.742)

R2 0.626 0.632S.E.R. 0.063 0.059

(Valuesnparenthesesre -ratios, .E.R. standard rror f regression.)

work, the choice of 0.59 as critical level of concentration among possibleother critical concentration ratios in the range indicated by Scherer wasbased on the superior goodness of fit of the model with this particular criticalC4 level.

Table IV indicates, relative to the continuous linear model, a substantialgain in goodness of fit for the discontinuous model. The goodness of fit of the

C4 and the H-index model is about the same. As expected, for the C4 model astatistically significant change in concentration coefficients occurs which doesnot show up for the H-index model.

As argued in the beginning of this section, this finding can be related to thesuperior goodness of fit of a discontinuous model of the H-Ck relationship:7

1t = -0.039 - 0.103 D + 0.262 C4 + 0.205C4D(-3.044) (-1.711) (8.517) (2.469)

(R2 = 0.904 and S.E.R. = 0.0254)

It might be noticed that the estimated change in C4 coefficients of

7Restricting his model to a linear spline function, yielding a continuous broken lineas relationship, oes not really change the goodness of fit. The SPLINE variable, putting arestriction n the model s defined s max {C4-0.59, 0}

R = -0.043 +0.275 C4+0.261 SPLINE(-3.463) (9.576) (3.939)

(R2= 0.902and S.E.R.= 0.0255)


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the discontinuous profitability model is larger than implied by the H-Ckregression results. However, this difference is statistically insignificant and is

dependent on the choice of the critical concentration level (cfr. Table III).More importantly, conform with our a priori hypotheses, the H-index doesnot show any significant sign of a shift in behaviour. For the H-index model a"critical" C4 value seems to have more meaning (using the goodness of fitcriterion) in delineating different regimes of the model with respect to theother regressors.

IV. CONCLUSION

The range of all possible H-indices for a given Ck ratio widens for increasingvalues of the last variable. The resulting horn-shaped relationship impliesthat for highly concentrated industries the H-index and the Ck ratio cannoteasily be substituted for each other. This knowledge has been used in someeconometric model tests presented in this paper. With the H-index as thesuperior measure it was shown for a sample of US industries how the neglectof the horn relationship between the H-index and the Ck ratio may bias theresults towards the empirical identification of a critical concentration ratio in

the relationship between profitability and concentration.The tests presented in this paper are based on US data from one particular(unfortunately old and truncated) data set. Ideally, if more recent H-indexdata would become available, these tests should be repeated on a largernumber of data sets.

LEO SLEUWAEGEN AND WIM DEHANDSCHUTTER, ACCEPTED FEBRUARY 1986INCAP, Department of Applied Economics,K. U. Leuven,

Dekenstraat 2,3000 Leuven,Belgium.

APPENDIX

From the definition of the H-index, which satisfies he transfer principle,8 t followsthat, or a given Ck, ts conditional maximum alue must correspond o

(A.1) H(m) (Ck-(k- 1)m)2 +(k- 1)m2 +((1 - Ck)/m)m2 +(D2-D)m2where m is the share held by the k + Ith firm in the industry and D is the fractionimpliedby the division 1- Ck)/m.9We may rewrite quation A.1) as follows:

8 This principle (also Dalton Principle) implies that the transfer of a part of the share of a firmto a bigger one increases concentration (see Hannah and Kay [1977, p. 48]).

9Strictly, m can only be interpreted as the market share of the k+ Ith firm, provided that(1 -Ck)/m > 1. This problem becomes relevant for k = 1 (C1 > 0.5) for which (A.1) still applies.Notice that in this last case a maximum always implies m = Ck, stressing the importance of thefraction in (A.1) fork _ 1.


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H(m) = Ck+S(m)+(D 2-D)m2

where

S(m) -2(k- 1)mCk+((k- 1)2 +(k- 1))m2 +(1 -Ck)m

To prove that Ck is a maximum, given that D2 < D, it suffices to prove that S < 0. Toprove this we study the shape of the function S with respect to m. Taking the derivativeyields the following condition:

-Dm 0 if m < w(Ck), where w(Ck) = Ck(2k-1)-iam < ~~~~~~~2(k2-)

It follows that the maximum value of S given Ck is either 0 at m = 0, or positive atm = Ck/k (the maximum possible value of m given Ck). Thus, the maximum depends

on the problem whether for a given m = Ck/k a value of S > 0 can be found. Aftersome algebra, this condition boils down to the following very simple condition:

Ck < l/kThis condition states that for a maximal H-index, given Ck, he minimum value m = 0applies in the range Ck > 1/k yielding a maximum H value equal to C2. Themaximum value m = Ck/k applies in the range Ck 1/k implying a maximum Hvalue equal to Ck/k, if k (1- Ck)/Ck s taken as an integer number, disregarding thefraction D in A(1). The latter maximum value corresponds to the straight part of theupper bound line in the figures presented in section I. The minimum value limiting

boundary poses no particular problem. Just as the maximum, it requires an infinitenumber of firms where each firm outside the k largest group holds a negligible marketshare, while the k largest firms are equally distributed, each with a market share equalto Ck/k.

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