Entropy and Other Measures of Con Cent Ratio

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Entropy and Other Measures of Concentration

Author(s): P. E. HartReviewed work(s):Source: Journal of the Royal Statistical Society. Series A (General), Vol. 134, No. 1 (1971), pp.73-85Published by: Blackwell Publishing for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2343975 .

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1971] 73

Entropy nd OtherMeasuresofConcentration

ByP. E. HART

UniversityfReading

[Received ecember 969.RevisedApril1970]

SUMMARY

In recent ears, conomistsavebegun ouse the ntropy,rredundancy,ofa sizedistributiono measure he xtent o which usinesssconcentratedin thecontrol f giantfirms.Thispapercompares hesenewmeasuresderived rom nformationheorywith heclassical tatistical easures fdispersionndwith raditional easures fbusiness oncentrationerivedfromhe umulativeoncentrationurve. tshows hatwhen henumber ffirmss arge noughouse tatisticalistributionheory,he lassicaltatisti-calmeasuresre uperiorothe ntropyr he edundancy. hen henumberof firmss small, heentropys superioro theredundancy,utbothareinferior o the traditionalmeasures f concentrationerived rom hecumulative oncentrationurve.Consequently,here s littlepoint inusing he nformationheorymeasures o measure usiness oncentration.

1. INTRODUCTION

ENGINEERShave made important ontributionso economics and to economicstatistics,hough longtime ag has often ended o elapsebefore conomists averecognizedheir alue. This s certainlyruefor hestudy fbusiness oncentration,forthe work of theFrench ngineerGibrat 1931)was neglected yeconomists ormanyyears. In recentyears, conomists avebegunto use theconceptofentropy,originallyormulatedn statisticalhermodynamicsnd subsequentlysed in infor-mation heory y conmmunicationsngineers. n particular heentropy unction asbeenused to measure he extent o which heoutput r employmentf an industryis concentratednthecontrol fa few argefirmsn an industrycf.Hildenbrand nd

Paschen, 1964; Finkelstein nd Friedberg,1967; Theil, 1967; Horowitz, 1968;Stigler, 968; Hexter nd Snow,1970). However, twillbe arguedherethat, n thecontext f business oncentration,heuse ofentropyheorys unjustifiedtecauseitaddsnothingotraditionaltatisticalmeasures fconcentration.When henumberofobservationss large nough o use a theoreticaltatistical istribution,heclassicalstatistical arameterserived rom hemomentsresuperior otheentropymeasure.Alternatively,hen henumber fobservationss small, or xample n an oligopoly,thetraditionalumulative oncentrationurve s superior.

2. CLASSICAL STATISTICAL MEASURES

Let thevariatez have a continuous istribution(z) witharithmeticmean u'.In the context f theeconomics f business oncentrationhevariate might e the

t Ofcourse, his oes not mplyhat ntropysnot usefulmeasurenother ontexts,conomicandnon-economic.ndeed, f he xioms nderlyinghe oncept fentropyreregarded s neces-sarythen heentropymeasuremustbe used for t is theonlyfunctionatisfyinghese xioms(cf.Theil,1967,p. 6).

4



74 HART- Entropy nd OtherMeasuresof Concentration [Part 1,

size f firms measuredy ts mployment.e maywritehe istributionunctionof

F(z)= f(z) dz. (1)

Thefirst omentistributionunctionf may ewritten

Fl(z) = zf z) dzff zf z) dz

1 z

= , | zf z) dz. (2)

Kendall ndStuart1958)usethe erm incompleteirst oment"or 2) and alsointegrateetweenhe imits oo and theparticularalue of z. However,n thepresentontexthevariates positivend t seems ettero followhe xample fAitchisonnd Brown1957, . 12) and usethe imitseroand z. Their ermfirstmomentistribution"s also usedfor 2).

TheLorenz urve f isthe ocus fpoints ithbscissa (z) andordinate&(z),with lope tanyvalue f , given y

dF1(z) zf z) dz z

dF(z) plf z) dz Pi

Kendall nd Stuart1958)showthat heLorenz urve s convex o the F axis.Moreover,tfollows rom3)that he oint n the orenz urvetwhich he angentisparallelothediagonal (z) = F&(z),orrespondsothe alue = p. Theabscissaof his oint iveshe roportionffirmselowmean ize ndthe rdinateivesheproportionfemploymentnfirmselowmean ize. Frequently,orenz urves resymmetricalboutthediagonal (z) = 1- F(z), so that heabscissa f the nter-sectionf theLorenz urve ndthediagonal (z) = 1- Fl(z) gives heproportionoffirmselowmean ize.This ymmetrysderivedrom he ropertiesf he nder-lyingizedistributionnd Kendall 1956)hadargued hat heres a large lass ofdistributionshich enerateymmetricalorenz urves.

It is customaryo argue hat ince hediagonal (z) = Fl(z) reflectsompleteequalityf size, movementf theLorenz urve wayfromhisdiagonal eflectsincreasingnequality.urthermore,s shown yKendall ndStuart1958), he reabetweenheLorenz urvend thediagonal (z) = Fl(z) ishalf fGini's oefficientof oncentrationl/2p4 hichsclearlymeasurefdispersioninceA1 sthemeandifferencewith epetition)etweenll possible airs fvalues fz. But n the aseofsymmetricalorenz urves, movementwayfrom hediagonal (z) = Fl(z)implies ot only n increasen inequality,utalso a decreasen theproportionffrequenciesbove mean ize. Thuswhen orenz urvesresymmetricalhe Ginicoefficientnd theLorenz urve re more hanmeasuresfdispersion;hey lsoreflectkewnessnd kurtosisnd come losetomeasuringhe conomists'oncept

ofthe xtentowhichhe mploymentf a particularndustrys concentratednthelargestompanies. heirmajor conomicimitationoncernshe ffectfchangesinN,thenumberfcompanies,ndwillbe consideredater.Theirmajor tatisticallimitationsre theirmathematicalntractabilitynd theabsenceof appropriatesamplingistributionsorthem,o it is preferableo use theparametersf theunderlyingizedistributionhich overnherelationshipetween (z) andFl(z)



1971] HART - Entropy ndOtherMeasuresofConcentration 75

and hencedeterminehe Gini coefficientnd Lorenz urves.For example,ftheunderlyingize distributionf firmss lognormal,hen r2 (the variance f thelogarithmsf ize) s the ppropriateeasure f oncentration;tgovernshe thers,is easilydecomposednd thewell-known-testmaybe used to see whetherheobservedhangesna2are ignificant.f t s desired o probe ehind he hangesnvariancehenStudent's-test, sing he standard rror f the regressionf thelogarithmf size t time on sizeat t 1, will atisfyherequirementsf samplingtheory. he derivationf the standardrror f 1(bJ1b2)y Horowitz 1964) sunnecessary,here 1 s theregressionf og ton logzt-1 and b2 s theregressionof og t-i on og t.

These tatisticalropertiesre mportantnd explainwhy he lassical tatistical

parametersmean, ariance,tc.) repreferredo Gini oefficientsfmean ifference,or Lorenzmeasuresfconcentration,o summarizesizedistributionffirms. utwhile statisticalnalysisf sizedistributionss appropriateor hemeasurementof business oncentrationn the conomys a whole, ecause henumberf firmsisveryarge,tmaynotbeappropriateor hemeasurementfconcentrationithinan individualndustry,ecause henumberffirms-particularlynthenearmono-polisticndustrieshich nterestconomists-maye too small o apply tatisticaldistributionheory. ndeed, t the ndustryevel t maybe the case that d hocmeasuresf oncentration,uch s the oncentrationatio repreferableomeasuressuch sthe ariancef heogarithms.hese d hocmeasuresre onsideredndetailinthenext ection fthis aper.

3. CONCENTRATION CURVE MEASURES

Inpractice e dealwithamplesffirms ith finite aximumize. Moreover,theobservedizedistributionsre notcontinuous. evertheless,t shelpfulousethe oncepts f Section and we estimate ' byY 1fgZz/Ek.=lf where here re k sizeclasses, herehe lassmark fthe th lass szi andwhere he lassfrequencysfi.Again, he umulativerequencyistribution,

if k

1 11

gives heproportionfthe otal requenciesqualto or below heupper imitftheithclass, say i', and correspondso F(z). Thismustbe distinguishedrom hecumulativeirst-momentistributionhich ives heproportionf otal mploymentbelow heupperimit fthe th lass,namely

it kEfi zi/Efi zi1 11

since i irepresentshe otal mploymentn the th lass. That s, fwe arestudyingthe ize distributionffirmsetweenizeclasses femployment,herelevantirstmoment istributions thedistributionfemploymentmong hose ize classes.Thefirst omentistributionasparametershichwe canestimate. orexample,

the rithmeticean fthefirst omentistributionay' e estimatedyweightingeach classmark i byemployment,nstead ffrequencies,nd then ividingythesum fweights,amely,

k /k

E zifiZI/ Efi z = z Z,. (4)i=1 Ii=1



76 HART- EntropyndOtherMeasuresof Concentration [Part 1,

Of course,fwehaveungroupedata, he rithmeticean and thefirstmomentmean lmay eestimatedy N zi/Nndby i zi/Eizi respectively,here =

is the otal umberffirms,i s the izeof he th irm,ndEly zi= Ek.lfi i sthetotal mploymentf all thefirmsnthe amplecf.Prais, 961, . 87, footnote).Clearly,

,l = (s2/2) , (5)

where2 estimateshevariancefthe sizedistributionf firms,o 2 >2. This ssimply special ase ofthegeneralifferenceetweenweightedndan unweightedmean, escribed,or xample,n Yule andKendall 1957,pp. 332-334),when hemeans ndthe tandardeviationsf thevariablesnd theweightsre equal, nd

where heres perfectositiveorrelationetweenhevariablend tsweight.Themost ommon easuref oncentrationt ndustryevels the oncentrationratiowhichs simplyheproportionf an industry'smploymentoroutput,ales,etc.)controlledy the argest business nits.Formally,t is simplyn upperquantilef hefirst omentistributionfz,namely,

Cr= 1-Fl(zr-7) = 1- Z{ zf(z) dz (6)

or n thediscretease,with ngroupedata,'r1 IN

Cr= 1- zZ/2zi). (7)

Alternatively,f he argest irms given ank andthe mallests given ankN,r IN r

Cr= z I z/ = 9i, (8)i=l JI=1 i=l

where ZIzj/iN zi and is the shareof the ithfirmn total employmentn theindustry.

EconomistsseCr o measureoncentrationecauset s ess ikelyo beaffectedby hangesnthenumberffirms.orexample,f everalmall irms ith egligibleemploymentnterr eave hendustry,hen r isunlikelyochange;he mall irms

below ank do not ffecthenumeratorn 8) and have verymall ffectnthedenominator.ince hese mall irmsreunlikelyo have ny ffectnthe ntensityof ompetition,r sa better easuref he egreef oncentrationhan,or xample,thevariancerGini oefficienthichwouldprobablyesignificantlynfluencedychangesn N. To take n extremease, fN fell o5equalfirms,he arianceecomeszero,F(z) = F1(z) andthemeasuref concentrationasedon thewhole izedistri-butionwould how eroconcentration.et the ndustryouldbe an oligopoly.TheCrmeasure, ith ayr= 4,wouldbe 0-80, nd would eflectheoligopolisticnature f the ndustry.n practicehevalueofrdepends ponthe onfidentialityrulesof the relevant overnumenttatisticalffice.n the UnitedKingdom isgenerally,thoughheBoard fTradehas ncreasedt to5 for 958-63.

Problemsrisewhen t s desiredocompareoncentrationn differentndustries,differentountries,r at differentoints f time. deally,conomistsould ike ohave ompletenformationnthe pper uantilesfF&(z) efore akingomparisons.Thecumulativeoncentrationurve or achsample acilitatesomparisonsf con-centration.ut sincedifferentoncentrationurves requentlyross, summarymeasure s required.



1971] HART - Entropy nd OtherMeasuresof Concentration 77

One possiblemeasuresthe verage alueofthe umulativeoncentrationurve

Br A{9 +(9A+92)+(9A+92+Y3)+ ..+(91+92+...+9r)}

rA

=,(r- i+l)9ilr

= -NE (r-i+ 1) z/i). (9)

This s a type fweightedveragefthe stimatedlopes ftheLorenz urve,ince

2 is an estimatororp4n 3), withweightsr i+ 1)/Nwhich iminishith ighervalues f , i.e. with ecreasesnsizeoffirm.Alternatively9) mayberegardedsthemean izeofthe argestfirms, ith achfirm eightedy r i+ 1)/Ni

1 rBr = -N (r-i+1)zi (10)

This ummary easuresunlikelyo be significantlyffectedythe ntryr exit fsmall irms,ecause hey ave ittlenfluencenNi = ,Zi.

Analternative easurefthis ypes simplyhe umulativeum fthe i,givenbyrBr D where

r N

D =Ed(r- i+ 1) i/Ezzi.i=l1I i=l

For the whole concentrationurve,withr= N, D is readily omparedwithz1= Ez,2Ezj, themean f thefirstmomentistribution:heonly ifferences thatin D the ank f i is substitutedor ne iinthenumeratorf 1. Thedisadvantageofthismeasure ith = N is that n increasenN increases : theuseofrank nthenumerator eans hat ven nemore mall irmncreases(r- i+ 1) Ezi.

Thesamedrawbacks presentn anothermeasure, amelyhearea under heconcentrationurve, iven y

J= 9/2 (N- 1) +Y2/2} (N- 22)92 +93/2}+N

= z9i+ E(N-i)4Ii=1

= +N-Ei9, since 9i= 1, (12)

which learlyncreases ithN. To overcomehis imitation,osenbluth1961)suggestshe seof he eciprocalf wice he rea bove he umulativeoncentrationcurve.Theareaabove sgiven y

K=N-J= ,i9A (13)

andRosenbluth'seasure,lso

proposed yHall and Tideman

1967),s

2K 2 9l-fs (14)

But snoted yFinkelsteinndFriedberg1967)K issimplyhe reaunder Lorenzcurvewith n absolutebscissa.Their roof lsoholdsfor heusualLorenz urve



78 HART - Entropynd OtherMeasuresof Concentration [Part1,

in which achfirmontributes/N o F(z) on the bscissa nd the otal reaundertheLorenz urves (1/N) Ei9- i) = K/N. Consequently,here eems o reason ouse 1/2K atherhan heGinicoefficientr Lorenzmeasure rovidinghenumberoffirmssknown.

Otherconomistsaveproposed serles tf easurestfoncentrationhichmaybe reduced o relationshipsetween1 and 2. Niehans 1958) uses21,Adelman 1969)and Herfindahl1950)use 21/Nz ndHirschman1945)uses 21/N)i to measureconcentration.hereas iehans sesa weighted ean, heothersse a ratio f aweightedo an unweightedean. In the contextf concentrationn thewholeeconomy,r n an industryhere henumberffirmss large nougho use distri-bution heory,hesemeasuresf concentrationre superfluousecause hey anbe

derived romheparametersftheunderlyingizedistribution.orexample,f hesizedistributions a two-parameterognormal,(p, 2), hen

= exp 2 + 3&2/2) and zJ1Nz exp a2)/N. (15)

In thecontext f concentrationn an individualndustry ith relativelyewfirms,hesemeasures ay e useful,ut hey re ikelyobe sensitiveo changesnN.The Herfindahl easurelearly ecreases ith ncreasesn N, for ffirm + 1 withemploymentN+l entershendustryt ncreaseshenumeratorzi by N+ whereasthe enominatorEZ,)2 s ncreasedy N+1 andthe ross-producterms. ometimes,butnotalways, uch decrease reateshefalsempressionhat here asbeen

reductionn the ompetitiveower f he eading irmsnthe ndustry.or example,ifseveral mall irmsnter highlyoncentratedndustryhey re likely o havelittle r noeffectn the ompetitiveower f he ew iant irms,ndthe oncentra-tion urvendB,measurestaymuchhe ame, ut heHerfindahleasure ecessarilyfalls. On theother and, f henew ntrantsre mportantnd have share ftheindustry'sutput hichignificantlyeduceshat f hegiant irms,hen hedecreasein theHerfindahleasures notmisleading.ut nsuch case,the oncentrationcurvend theBrmeasureslso fall. Thus tseems rudentouseBrin 9), or theconcentrationurve, or he mallN casebecauseheyre ess ikelyo bemisleadingfromn economic oint fview.

4. ENTROPY AND REDUNDANCY

A more ecentpproacho themeasurementfconcentrations based on theentropyonceptn nformationheory. et theobservedhare f the th ompanyin total mploymente \, where herere N companies

/N

9=ziz. (16)

Theentropy easurefthis istributionfcompaniesyemploymentsgiven yN

H(9) = - 9ilog9, 0 H(9) < logN. (17)i=1

Complete quality ccurswhen9i= 1/Nforall i and H(9) = logN. Completeinequality ccurswhenyi= 1, and y; = 0 forall j # so thatH(y) = 0, usingtheconventionhat i og9i 0 when i= 0. To obtain measure hich aries irectlywithhedegree f nequality,emayusetheredundancyr thedifferenceetween



1971] HART - EntropyndOtherMeasuresofConcentration 79

the bservedntropynd tsmaximumalue:log N- H(y) = logN+ E9tlog9i

= z 9q log N9t), since I9 = 1i

= E (zi/E zi) log Nz1/Nf)i i

= E zi og zi/zi)/Ezi

P.'

(18)

That s,theredundancys a weightedverage fthe ogarithmsftheestimatedslopes f he orenzurve teach iwhereheweightsre mployment.hedifferencebetweenhemean fthe irst omentistribution

ff=Ez Ezi (19)

and the edundancyn 18) ssimplyhatnthenumeratorf 18)the ermog zi/2),measuringheogarithmf he elativeeviationf fromtsmean, eplaceshe ermzi, measuringheabsolute eviation f z from ero. A furtherimplificationspossible,ince18)may ewritten

E zi og i/ - z log / *)

( z log i/ ) - log . (20)

The firsterms a weighted eanofthe ogarithmsfz, where heweightsreemployment.f theweights ere hefrequenciest wouldbe the ogarithmfthegeometric ean,whichs readilyompared ith he econd erm,he ogarithmfthe rithmeticean f .

It spossible ouse nformationheoryo obtain measuref oncentrationhichissimplyhedifferenceetweenogarithmsf he rithmeticndgeometriceans f

thedistributionfz. Theexpectednformationfanindirect essages definedsN

I(y: x) = E yi ogyi/xi), (21)i=1

where i sthe osteriorrobabilityf i,estimatedy 16), ndwhere i sthe riorprobabilityfzi. Wemay xpressheprior robabilitys 1/N, nthe ssumptionthatwe should xpectachoftheN companieso have1/N fthe otal mployment,i.e.completequality. hus 21)may eestimated:

N

1(y: x) = yqogN9j, (22)i=1

whichs the edundancyf i in 18).Similarly,emay efinehe xpectednforma-tion ontentfthe ndirect essage hich ransformsheprior robabilitiesi intotheposteriorrobabilitiesias

N

I(x: y) = E xi ogxi/yi), (23)i



80 HART - EntropyndOtherMeasuresofConcentration [Part1,

whichsestimatedyN II

, (IIN) log

N

E (1/N) og z/z*)i

N= logz-(1/N) E logzi

i

= logAM-log GM.

Clearly,hisnformationheory easure fconcentrationaybe obtained irectlyfrom hefrequencyistributionf z using lassical tatisticalnalysis. ndeed, sTheil 1967)reminds s,for he ognormalistributionhich pproximatesanysizedistributionsffirms,(x: y) reduces o cr2/2 here 2 is thevariance f thelogarithmsffirmsizes, ince he rithmeticean sexp/+ JU2)andthegeometricmean sexpp),where is themean fthe ogarithmsf . There eems ittle ointin replacing2,which asbeenusedextensivelyomeasure usinessoncentration,byan entropy easure hich ecomes r2/2.heil 1967,p. 124)argues hat hevariance fthe ogarithmss an inconvenienteasurefconcentrationecause t sthe econdmomentxtensionfthegeometric eanwhereas otal mployments

directlyelatedothe rithmeticean, amely z. Thisobjectionoses tspointfthedistributions known.For example,n a lognormalistribution,he totalemployments directlyelatedo thegeometric ean. Totalemploymentaybeobtainedymultiplyinghegeometricean yNexpJa2),where2 iS thevarianceofthe ogarithms.

5. NUMBERS-EQUIVALENTSStigler1968) nd FinkelsteinndFriedberg1967)have ransformedhe oncen-

tration easuresnSections and4 into he quivalentumberffirmsf qual ize,whichsyet nother ay fmeasuringusinessoncentration.heHerfindahlndexvaries etween/N, hen llN firmsave he ame ize, nd1,when hethfirm asall the mploymentnd i, #j, szero or ll . Thus he eciprocalf heHerfindahlindex, E= Nf/zl,may eregardeds the quivalentumberffirmsfequalsize.It variesbetween , reflectingompleteompetition,nd 1, reflectingompletemonopoly.However,his ndexnreciprocalorms simplyhe ctualnumber ffirms, ,multipliedy he atio f hemean tothemean f he irst omentistri-bution,f. Equation 5) shows hat <z2, exceptnthe ase ofcomplete onopolywhen = 2f, husN is reduced ya fraction hich epends n thedispersionffirms'izes. TheequivalentumberffirmsfequalsizebasedontheHerfindahlindex ssimply/(v2 1),where s the oefficientfvariation,ndthismeasureanbeobtainedirectlyrom and 2/22.For the ognormalistribution,ogNE U2=

logN,sothatheogarithmf he umberf irms inushe ariancef heogarithmsgives ogNE. In anydistribution,knowledgefN (whichs directlyelevantoconcentration,nanycase)and ofS2/22whichmeasuresnequality)s sufficientocalculateNE.

Theequivalentumberfequalfirmsmay lso be obtained rom heentropyin 17). If allfirmsere he ame ize s zithere ould e l/9i irmsnthe ndustry.



1971] HART - Entropynd OtherMeasuresofConcentration 81

A measure fthenumberfequivalentirms,e,basedoneach

9fsgiven yN

Ne flI( 9s)( 0, KY i (24)i=l

orN

logNe E y log1/9i) H(U). (25)i=1

Thus theredundancyn (18)maybe viewed s the ogarithmf theratioof theactualnumber ffirmso the quivalentumberfequalfirms. owever,his anbealsoobtainedirectlyromhe arametersf he nderlyingistributionffirms.

Forexample, heil 1967,p. 97) shows hat heredundancyeduces oU2/2nthecase of heognormalistribution,nd ogN- (U2/2)iveshe ogarithmfNe.Notealsothatnthe ognormalaseNe/NE expu2/2) ives he elationshipetweenhetwomeasuresf quivalentumberffirms.

6. COMPARISONSFMEASURESTheentropyrredundancyeasure fconcentrationay ecompared ith he

more raditionaleasuresfconcentrationor he arge roup nd the mall roupcases. In the ormerase thenumberffirmss arge nough o usestatisticalistri-butionheory. rom neconomicoint fview,hediscussionfbusinessoncen-

trationn this ontexts associated ith comparisonfgrowthates f arge ndsmall irms. enerallyndustryoundariesre gnorednd oncentrationsmeasuredfor hewhole orporateectorrthewhole fmanufacturing.his arge roup asemust e distinguishedrom he asewhere henumberffirmss toosmall ousestatisticalistributionheory,herendustrialoundariesre mportantndwherethedegreefmonopolyroligopolys themajor nterestf conomists.here s noreasonwhy he amemeasure fconcentrationhould eappropriaten both ases.

Informationnsizes fcompaniesnthe arge roup asetends obepublishedintheformffrequencyistributions.heil 1967,pp.99,128-134) stimatesheredundancyf ncome istributionsn theassumptionhat requenciesn the thclassearn he rithmeticid-pointncome fthat lass,Zi. Foropen ize-classes,he uses a Paretodistributiono estimatehemid-point. he arithmeticean sestimated,,andtotalncomesgiven yNY, ndequations16) and 18) areusedtoestimateheredundancy,. Similarechniquesere sedtoestimate for hedistributionsfcompaniesy profitn theUnitedKingdom or1949 nd 1964.tThefirst omentistributionsregiven nd twaspossible ocalculatehe rithmeticmean rofitneach ize-class,f,he osteriorrobabilityas stimatedsY'= Z/Eziandthe requenciesnthe th lasswere ssumedohave he amey9.TheestimatedredundanciesregivennTable1 nnaturalogarithmsothat heymay ecomparedwith he ariance f heogarithmsf i. The atter ere stimatedromraphsf hecumulativerequencyistributionsndcumulativeirst omentistributionsrawnonlogarithmic

robabilityaper. n the ognormalistribution=JU2.

For 19492A = 5-46which ompareswellwith 2 = 5-3, hrough or19642A = 7415 omparedwith2 = 8-7.Bothmeasuresndicatehat oncentrationncreased949-64,ut hea measuressuperioroAfor he ollowingeasons.

t This procedure nderestimatesheentropy ecause it excludesvariation f sizes withinincome-bracketsf.Theil, 1967),pp. 99,128-134.



82 HART - EntropyndOtherMeasuresofConcentration [Part1,

First,t s easier ocompute2 thanR. Graphicmethodsreavailable, nd soarethe hort-cutethods fcalculationroupingata nto lass ntervalsqual tounityn ogarithmsothebase 2. Moreover,t s necessaryoknow z3,which snot lwaysvailable,norder ocompute . Secondly,he fficienciesf hevarious

TABLE1

Size distributionf companies y trading rofit ssessedunder cheduleD, UnitedKingdom, 949and1964

1949 1964

Upper imit f Numbers Profit Numbers Profitprofit?) ('000) (?m) ('000) (?m)

250 62.8 6-0 1705 4.3500 168 59 166 5.8

1,000 21.1 143 193 13.31,500 120 141 114 13.92,000 7.9 136 85 1473,000 109 260 107 2594,000 7 1 24 2 6.7 23*35,000 4.7 208 49 21V8

10,000 122 84*1 134 95615,000 52 62.1 62 75.220,000 3.1 52-1 3 7 63.725,000 1.9 42*4 25 55 730,000 1P5 409 1 8 50-640,000 2.1 725 24 83650,000 1P3 56-5 1-6 69.875,000 1.9 110.6 2 3 142 5

100,000 1.0 83.8 1 3 110 1200,000 1*4 196.4 2.1 298.8

1,000,000 1.1 4248 1.9 7730> 1.000,000 0.2 466*7 0*4 1478.2

1763 18176 288.2 34198R (loge) 2 729 3 576&2 (loge) 5.3 8.7

Sources:Boardof nlandRevenue, nnualReports.1949relates oprofitsssessed1950/1.1964relates oprofitsssessed1965/6.

methodsfestimatingfromroupedndungroupedataareknown.But ittlesknown bout hedegreefreliabilityfestimatesffAromroupedata. Thirdly,

wecanapply he tandard-testfthe ignificancef he ifferenceetweenhe wovariances,ecause he bservationsre ampledromopulationshich re pproxi-matelyormalfterogarithmicransformation.ourthly,hedecompositionf 2

is elegantnd wellknown,o thatwe are able tostudyhe ffectsfbirths,eathsandamalgamationsfcompaniesn concentration.ifthly,t spossibleo usetheGalton egressionodel nd attributehangesn J2 o thedifferentverage rowth



1971] HART- Entropynd OtherMeasuresof Concentration 83

rates f firms f differentizes measured ybl) and to the ize mobilityffirms(measuredyr), as explainedn Hart nd Prais 1956), ince

r/OAr-_= b2/r2, (26)

where1 s the egressionf og i(t)on og i(t 1) andr s the orrelationoefficient.For allthese easons,r2is superioroA nthe arge roup ase.

TABLE 2

Estimated edundancynd Br-small group ase, United ingdom

Industry Date N mrnin Amax Br Hmin Hmax

Distilling 1952 7 2-1213 2x1213 96 00 0 6861 0 68611957 9 2-2389 2-2563 94 00 0-9136 0 9310

Cement 1952 6 1 0403 1-0403 88x17 1x5447 1x54471957 5 0-8685 0-8685 90 50 1 4534 1 4534

Tobacco 1952 13 2.1533 2x1931 88x50 1-5073 1-54711957 9 1-6722 1x6896 89 83 1-4803 1-4977

Cocoa, chocolate 1952 24 1x6617 1x8458 70 33 2x7392 2 9233and sugar confectionery 1957 27 1P9933 2.2052 73-33 2 5497 2 7616

Units: R and H in log2.

Table2 gives stimatesf theredundancy,, andofBrfor he mall roup roligopolyase. A range fA sgiven henndividualbservationsi i = 8,9, .., N)werenotavailable. ndividualbservationsf9 (i = 1,2, ..,6) are givenn theAppendix nd whenN= 7 itwas possible o obtain 7 bysubtraction.utwhenN> 7, two extreme ssumptionswere made. First, hatall 9i (i = 7, 8, .., N) wereequaltogive heminimumedundancy.econd, hat 7 I6, 8equalled he emain-ing proportion f assetsand 9i (i = 9, 10, .., N) equalledzero, to givethe maximumestimatefredundancy.hese stimatesay ecompared ith r, s nequation9).

Theresno correlationetween ndBr, nd nthe istilling,ementnd obaccoindustriesifferentirectionsfmovementnconcentration952-57 re ndicatedyA andBr. An inspectionfthedata ntheAppendixhows hat n thedistillingindustryhe concentrationurve or 1952 s abovethatfor 1957 at every oint,whichndicates decreasen concentration.heBrmeasureuggests decrease,butboth edundancyeasuresuggestn increasenconcentration.

In the ementndustryhe oncentrationurve or1952 s below hat or1957,indicatinghatbusiness oncentrationncreasedver hisperiod.TheBrmeasureshows n increaserom8417o9050,but heAmeasurehows decreasen con-centration.n the ocoa,chocolate nd sugar onfectioneryndustry,heBrandAmeasuresreconsistentith achother ndwithhe oncentrationurve ata n the

Appendix. he nconsistenciesnthedistillingndthe ementndustriesuggesthatA sinferioroBras a measure fconcentration.he main eason or hissthatAincreases ithogN so that larger tends oproduce higher egree fconcen-tration,hichscontraryo the tandardconomicnterpretationf n ncreasen Ninthe mall roup aseasreducinghe egree fmonopoly.hus n the mall roupcase Br smore ppropriatehanA.



84 HART - Entropynd OtherMeasuresofConcentration [Part1,

However,heuseofthe ntropys an inverse easure fconcentrationsmoreappropriatehanA, because t is lessdependentn N andthus hows he amedirectionf movementfthedegree f concentrations doesBr. Thisconsistencybetween (y) andBris comfortingutBris neverthelessreferableoH(y). Firstbecause venwhen ll therequired ata areavailable,t is easier o calculate r,whichsmerely simplemean.Secondly,ecauset sunnecessaryo have rangeof estimatesfBrfor nyone industryt anyone time.H(y) is indeterminate,because herelativeizesof small irmsregenerallynknown,hought sknownthat ollectivelyhey ontributesmall roportionf n ndustry'sutput remploy-ment.Thirdlyecause hangesnthenumberf small irmsver ime ave ittleeffectn the hares fthe opfirmsnd t s undesirableor measure fconcen-

trationo be nfluencedy ll the irms. heH(y)measureuffershis isadvantage,whereasrdoesnot. For these easons,t spreferableo useBr nthe mall roupcase.

Thegeneralonclusions that he edundancyndentropy easuresfbusinessconcentrationre nappropriate.orthe arge roup ase,we hould sea parameterofthedistributionuch s a2, thevariancef the ogarithms.or the mall roupcasewe should se the raditionalumulativeoncentrationurve rthemean f tsfirstvalues, enoted yBr

ACKNOWLEDGEMENTS

Theresearchesultseportedn this aper ollow romnenquiryntomergersnd

businessoncentrationhichsbeingonductedttheNationalnstitutefEconomicand SocialResearch. hisresearchroject as aunched na grant rom he ocialScience esearch ouncil ndwas subsequentlyssistedy grantrom heDepart-ment fTrade nd ndustry.

REFERENCES

ADELMAN, M. A. (1969). Comment on the H concentration measure as a numbersequivalent.Rev. Econ. Statist., 1, 99-101.

AITCHISON, J. andBROWN, J. A. C. (1957). TheLognormal istribution.ambridge: ambridgeUniversity ress.

FINKELSTEIN, M. 0. and FRIEDBERG, R. M. (1967). The application of an entropy theory ofconcentration o the Clayton Act. Yale Law J., 76, 677-721.

GIBRAT, R. (1931). Les Inegalites conomiques. aris: Sirey.HALL, M. and TIDEMAN, N. (1967). Measures of concentration.J. Amer. Statist. Assoc., 62,162-168

HART, P. E. and PRAIS, . J. 1956). The analysisof business concentration: a statistical pproach.J. R. Statist. oc. A, 119,2, 150-181.

HERFINDAHL, 0. C. (1950). Concentrationin the steel industry. Ph.D. dissertation, ColumbiaUniversity.

HEXTER, J.L. and SNOW, J. W. (1970). An entropymeasure of relative aggregate concentration.South. con.J.,36, 239-243.

HILDENBRAND, W. and PASCHEN, H. (1964). Ein axiomatischbegrundetesKonzentrationsmass.Statist. nform. urop. Wirtschafts.,r.3, 53-61.

HIRSCHMAN,A.O. (1945). NationalPowernd he tructurefForeign rade.Berkeley: Universityof California Press.

HOROWITZ, I. (1964). A note on the Hart-Prais measure of concentration. J. R. Statist.Soc. A,127,234-237.HOROWITZ, A. and I. (1968). Entropy,Markov processesand competition nthebrewing ndustry.

J. ndust. con., 16, 196-211.KENDALL, M. G. (1956). Discussion on Hart and Prais (1956), p. 185.KENDALL, M. G. and STUART, A. (1958). The Advanced heory fStatistics, ol. I. London:

Griffin.



1971] HART EntropyndOtherMeasuresofConcentration 85

NIEHANS, J. 1950). Anindex fthe izeof ndustrialstablishments.nt.Econ.Papers, ,122-132.PRAIS, . J. 1961). A methodologicalfterthoughtnbusinessoncentration.ev.Econ.Statist.,

43, 87.ROSENBLUTH, G. (1961). RoundTable-GesprachlberMessung er ndustriellenonzentration.

In Die Konzentrationn derWirtschaftAmndt,d.), p. 391. Berlin:Duncker& Humbolt.STIGLER,G. J. 1968). TheOrganisationf ndustry. omewood,ll.: Irwin.THEIL, H. (1967). Economicsnd nformationheory.Amsterdam: North-Holland.YULE, G. U. andKENDALL, M. G. (1957). An ntroductiono theTheory fStatistics. ondon:

Griffin.

APPENDIX

CumulativeoncentrationatiosIndustry Year forthe ixlargest usiness nits%)

1 2 3 4 5 6 N

Distilling 1952 90 93 96 98 99 100 71957 86 91 94 96 98 99 9

Cement 1952 60 83 90 97 99 100 61957 68 85 93 98 99 100 5

Tobacco 1952 73 82 89 94 96 97 131957 70 84 92 96 98 99 9

Cocoa, chocolate nd 1952 43 60 74 79 82 84 24

sugar onfectionery 1957 49 64 75 82 84 86 27

Sources:1952HART, P. E. (1958). Concentrationnselectedndustries.cot.J. Pol. Econ.,5,185-201.

1957HART, P. E. (1961). Concentrationnd tsmeasurementntheUnitedKingdom.In Die Konzentrationn der WirtschaftArndt, d.), pp. 653-674, Berlin:Duncker& Humbolt.

Note:The Board ofTradeconfidentialityulesmake t mpossibleo useCensus fProductiondata to estimate umulativeoncentrationurves.Theabovedata were ompiled rom ublishedcompanyccounts ndzi ismeasured ynettangiblessets.Unfortunately,heBoard ofTradeno longer tandardizes he accountsof companieswith ess than?0s5million ssets so con-centrationurves or ecent ears re notcomparablewith hose or1957.

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Entropy and Other Measures of Con Cent Ratio

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