基尼系數(shù)、泰爾指數(shù)等幾個公平性評價的介紹(英文)

TheTheoreticalBasicsofPopularInequalityMeasures

TravisHale,UniversityofTexasInequalityProject

Thisdocumentexploresseveralinequalitymeasuresusedbroadlyintheliterature,withaspecialemphasisonhowtocomputeTheil'sTstatistic.Inequalityisrelatedtoseveralmathematicalconcepts,includingdispersion,skewness,andvariance.Asaresult,therearemanywaystomeasureinequality,whichitselfarisesfromvarioussocialandphysicalphenomena.Whilethisisnotanexhaustivediscussionofinequalitymeasures,itdoesdealwithseveralofthemostpopularstatistics.Severalexamplesareincludedthatpertaintoinequalityofsalarieswithintwofictionalcompanies—UniversalWidgetandWorldwideWidget—butalloftheinequalitymeasuresdiscussedapplytoabroadsetofresearchquestions.Thesalaryschedulesfortheexampleproblemsarebelow,followedbydiscussionsofrange,rangeratios,theMcLooneIndex,thecoefficientofvariation,andtheGiniCoefficient.FollowingthesebriefintroductionsisanextendeddescriptionofTheil'sTstatistic.

UniversalWidgetSalarySchedule

Position

#ofEmployeesinPosition

ExactAnnualSalary

CustodialStaff

7

$18,000.00

OfficeStaff

10

$22,000.00

EquipmentOperators

280

$25,000.00

EquipmentTechnicians

15

$35,000.00

Foremen

15

$40,000.00

Salespersons

50

$60,000.00

Engineers

10

$75,000.00

Managers

6

$80,000.00

VicePresidents

4

$120,000.00

SeniorVicePresidents

2

$200,000.00

CEO

1

$1,000,000.00

WorldwideWidgetSalarySchedule

Position

#ofEmployeesinPosition

ExactAnnualSalary

CustodialStaff

12

$15,000.00

OfficeStaff

25

$20,000.00

EquipmentOperators

1000

$30,000.00

EquipmentTechnicians

35

$35,000.00

Foremen

100

$45,000.00

Salespersons

80

$50,000.00

Managers

10

$60,000.00

Engineers

25

$80,000.00

VicePresidents

8

$175,000.00

SeniorVicePresidents

4

$250,000.00

CEO

1

$5,000,000.00

Range

Perhapsthesimplestmeasureofdispersion,therangemerelycalculatesthedifferencebetweenthehighestandlowestobservationsofaparticularvariableofinterest.Strengthsoftherangeincludeitsmathematicalsimplicityandeaseofunderstanding.However,itisaverylimitedmeasure.Therangeonlyusestwoobservationsfromtheoverallset,itdoesnotweightobservationsbyimportantunderlyingcharacteristics(likethepopulationofastate,theexperienceofanemployee,etc.),anditissensitivetoinflationarypressures.Inthecaseofacompany,therangebetweenthesalariesofthehighestandlowestpaidemployeesmaynotgivemuchinformation.ForUniversalWidget,therangeinsalariesis$982,000($1,000,000-$18,000),whileforWorldwideWidgettherangeis$4,985,000($5,000,000-$15,000).DoesthismeanthatWorldwideWidgethasamuchmoreunequalwagestructurethanUniversalWidget?Notwithoutfurtherevidence.

RangeRatios

Tofindtherangeratioforacertainvariable,dividethevalueatacertainpercentile(usuallyabovethemedian)bythevalueatalowerpercentile(usuallybelowthemedian).OnerangeratiooftenusedinthestudyofinequalityineducationalexpendituresistheFederalRangeRatio,whichdividesthedifferencebetweentherevenueforthestudentatthe95thpercentileandthe5thpercentilebytherevenueforthestudentatthe95thpercentile.1Anotherpopularrangeratioistheinter-quartilerangeratio.Subtractingtheobservationatthe25thpercentilebytheobservationatthe75thpercentileresultsinaquantityknownastheinter-quartilerange,anddividingtheobservationatthe75thpercentilebythe25thpercentilecalculatestheinter-quartilerangeratio.Rangeratioscanmeasureallsortsofinequalitiesandthepercentilescanbeconstructedinanymanner.Arangeratiocantakeonanyvaluebetweenoneandinfinity,andsmallervaluesreflectlowerinequality.

Usingtheexampledata,onecancomputea90:10rangeratioforthetwowidgetcompanies.ForUniversalWidget,the90thpercentilefallsatasalaryof$60,000andthe10thpercentileis$25,000.Thus,the90:10rangeratiois$60,000/$25,000or2.4.ForWorldwideWidget,the90thpercentilefallsatasalaryof$35,000andthe10thpercentileis$30,000.Therefore,the90:10rangeratiois$35,000/$30,000or1.17.Giventhisinformation,WorldwideWidgethasamoreequalpaystructure,theoppositeconclusiongleanedfromtherange.

Rangeratiosareeasytounderstandandsimpletocompute.Theycandirectlycomparesthe“haves”-observationsatthe90thpercentileorelsewhereabovethemedianvalue-withthe“have-nots”-observationsatthe10thpercentileorelsewherebelowthemedian,withoutbeingsensitivetooutliersattheverytoporverybottomofthedistribution.However,liketherange,rangeratiosonlylookattwodistinctdatapoints,throwingawaythegreatmajorityofthedata.Becauseofthissignificantlimitation,researchersoftenemploymoresophisticatedinequalitymeasures.

McLooneIndex

TheMcLooneIndexisanotherexampleofameasurethatcomparesonepartofadistributiontoanother.However,theMcLooneIndextakesamuchlargerproportionofthedataintoaccount.Itcompareshowmuchofaresourceisconcentratedinthebottomhalfofadistributiontothemedianamount.TocomputetheMcLooneIndexvalue,dividethesumofalloftheobservationsatorbelowthemedianlevelbytheproductofthenumberofobservationsatorbelowthemedianlevelandthevalueofthemedianlevel.ValuesoftheMcLooneIndexareboundbelowbyzero-ifthelowerhalfofthedistributionreceivesnoneoftheresource-andabovebyone-iftherearenoobservationsbelowthemedian.Thelattercasewouldoccurifthelowestvalueissharedbyatleasthalfoftheobservations.Unlikemostinequalitymeasures,ahighervaluefortheMcLooneIndexdescribesamoreequitabledistribution.

Forexample,theUniversalWidgetCompanyhas400employees.Themediansalaryvalueisapproximatelythatofthe200thleastcompensatedemployee.ThatemployeeisanEquipmentOperatorwhomakes$25,000.TheMcLooneIndexistheratiooftheactualsalariesoftheleastpaidhalfoftheUniversalWidgetworkforcetothecounterfactualdenominatorof$25,000*200=5,000,000.ThustheMcLooneIndexforUniversalWidgetequals(7*18,000+10*22,000+183*25,000)/5,000,000or.9842.ParallelcomputationsrevealthatWorldwideWidgethasaMcLooneIndexvalueof.9595.ThisleadstoaconclusionthatUniversalhasamoreequalpaystructure.

TheMcLooneIndexisrelativelyeasytounderstand,andmightbeanappropriatemeasureifresearchersareprimarilyinterestedinthebottomofadistribution.Ifthemedianobservationreflectsan“adequatee”nthlevMeclL,otohneIndexgivessomesenseofhowthebottomhalfofthedistributionisdoingcomparedthemiddle.However,theMcLooneIndexhassomepotentiallyobjectionableproperties.First,itdoesnotuseallinformation,throwingawaytheobservationsabovethemedian.Certainlythereisasubstantialdifferencebetweenadistributionwherethehighervaluesliejustabovethemedianandonewheresomeobservationsliefarbeyondthemedian.TheMcLooneIndexcomparesrealitywithacounterfactualmodel,sotheresearchermaybeaskedtojustifythecomparisonofrealitytoanalternativewheretheentirebottomhalfofthedistributionsharesthemedianvalue.WhiletheMcLooneIndexhasthusfarbeenconcernedprimarilywithschoolfinanceinequalitymeasurement,therearesimilarmeasureswithbroaderapplication,andthereisnoreasonthattheMcLooneIndexitselfcouldnotbeappliedtootherphenomena.

TheCoefficientofVariation

Thecoefficientofvariationissimplythestandarddeviationofavariabledividedbythemean.2Graphically,thecoefficientofvariationdescribesthepeakednessofaunimodalfrequencydistribution.Foradatasetthatiscloselybunchedaroundthemean,thepeakwillbehigh,andthecoefficientofvariationsmall.Datathatismoredispersedwillhaveashorterpeakandahighercoefficientofvariation.Ceterisparibus,thesmallerthecoefficientofvariation,themoreequitablethedistribution.

Thefirststepincomputingcoefficientsofvariationforthesampledataistofindthemeanandstandarddeviationofeachset.Thisisfairlyeasytodowithstatisticssoftware,oraspreadsheetprogramsuchasMicrosoftExcel.UniversalWidgethasanaveragesalaryof$36,452.50andastandarddeviationof52,630.52.WorldwideWidgethasanaveragesalaryof$38,773.08andastandarddeviationof138,990.96.Thisleadstocoefficientsofvariationof1.44and3.58forUniversalandWorldwide,respectively,concludingthatUniversalhasthemoreequitablesalarystructure.

Thecoefficientofvariationhassomeattractiveproperties.Ifgroupdataisused,butweightedbypopulationsize,smalloutlyingobservationsdonotskewthedistributiongreatly.Individualswithevenalimitedstatisticalbackgroundarelikelytobefamiliarwiththestandarddeviationandsamplemean,makingthecoefficientofvariationeasytoexplaintoanon-technicalaudience.Furthermore,byconstruction,inflationdoesnotaffectthecoefficientofvariation.Adisadvantageofthemeasureisthat,theoretically,thecoefficientofvariationcantakeanyvaluebetweenzeroandinfinity,andthereisnouniversalstandardthatdefinesareasonablevalueofthemeasureforparticularphenomena.

TheGiniCoefficient

TheGinicoefficientderivesfromtheLorenzCurve.ToplotaLorenzcurve,ranktheobservationsfromlowesttohighestonthevariableofinterest,andthenplotthecumulativeproportionofthepopulationontheX-axisandthecumulativeproportionofthevariableofinterestontheY-axis.3TheGinicoefficientcomparesthiscumulativefrequencyandsizecurvetotheuniformdistributionthatrepresentsequality.Inthegraphicaldepictionbelow,adiagonallinerepresentsperfectequality,andthegreaterthedeviationoftheLorenzcurvefromthisline,thegreatertheinequality.TheGinicoefficientisdoubletheareabetweentheequalitydiagonalandtheLorenzcurve,boundedbelowbyzero(perfectequality)andabovebyone(thecasewhenasinglememberofthepopulationholdsallofaresource).

GraphicalRepresentaionoftheGiniCoefficient

1.2-i

■1

00.81

CumulativeProportionofPopulation

4>-q?」E>'5u.2'todo4>>4E-nluno

ThereareseveralwaystocomputetheGinicoefficientforadataset.ResearcherswhoarecomfortablewithCalculusandspreadsheetanalysisandhavealargeamountofdatathatresultsinsmoothplotscanestimateahighorderpolynomialfortheLorenzCurve(MicrosoftExcelwilladduptoa6thdegreepolynomialasatrendlineforanXYgraph),andthentakeanappropriateintegraltocomputethesizeoftheshadedarea.Likewise,otherestimationtechniques,suchasthemethodofrectangles,themethodoftrapezoids,orMonteCarlointegrationwillprovidereasonableestimates.Anotherwayto

computetheGiniisdirectlyfromanalgebraicformula.Giventhatthedataisorderedfromsmallesttolargestvaluesofthevariableofinterest,theformulais:

蘭(2i-n一1)x'

whereiistheindividual'srankordernumber,nisthe

G=

n2卩

numberoftotalindividuals,x'.istheindividual'svariablevalue,and卩isthepopulationaverage.4

TocomputetheGinicoefficientsforthesampledata,itiseasiesttoorganizethedatasuchthateachindividualisgivenhisorherownrecord(suchthatthesalaryscheduleforUniversalWidgethas400rows,oneforeachemployee).Aftersplittingthedatainthismanner,itisfairlystraightforwardtoapplytheformulaabove.ForUniversalWidget,theGinicoefficientis0.279625369,whileforWorldwideWidget,theGinicoefficientis0.227509252.

TheGinicoefficientisafull-informationmeasure,lookingatallpartsofthedistribution.Itisprobablythemostwell-knownandbroadlyusedmeasureofinequalityusedineconomicliterature.TheGinicoefficientfacilitatesdirectcomparisonoftwopopulations,regardlessoftheirsizes.Inotherwords,withtheGinicoefficientonecandirectlycomparetheinequalityinaclassroomtotheinequalityinacountry.WhiletheactualcomputationoftheGinicoefficientmayincludetakinganintegralorusingaslightlycomplexformula,thevisualdescriptioniselegantandeasytounderstand.TheGinicoefficientdoessufferfromthelackofatruezero,andtheneedforacontext.Whileadistributionalpolicy,likegivingeveryonebelowthepovertyline$1,000,hasrealimplications,therepercussionsofa5%reductionoftheGinicoefficientaremuchlessclear.

Theil'sTStatistic

Theinequalitymeasuresdiscussedaboveareeachappropriateincertaincircumstances.TherationaleforpreferringTheil'sTstatisticisnotthatthereissomeinherentflawintheothermeasures,butthatTheil'sThasamoreflexiblestructurethatoftenmakesitmoreappropriate.Ifaresearcheralwayshadaccesstocomplete,individualleveldataforthepopulationofinterest,thenmeasureslikethecoefficientofvariationortheGinicoefficientwouldusuallybesufficientfordescribinginequality.However,inpractice,individualdataisrarelyavailable,andresearchersareaskedtomakeduewithaggregateddata.Returningtotheexampleproblemillustratesthepoint.WhatiftheUniversalWidgetsalaryscheduledidnotreflecttheexactsalaryforeachemployeebuttheaveragesalaryovereachjobcategory?ItwouldbepossibletocomputevaluesforthecoefficientofvariationortheGinicoefficientundertheassumptionthateachemployeereceivesexactlytheaveragesalary,buttheresultswouldonlygiveanupperorlowerboundofeachinequalitymeasure,becausevariancewithineachjobcategorywillcontributetototalinequality.Formostpracticaldata,datathathassomedegreeofaggregationoranunderlyinghierarchy(e.g.citieswithinregionswithinnations),Theil'sTstatisticisoftenamoreappropriateandtheoreticallysoundtool.5

ThefollowingformulaegivethealgebrabehindTheil'sTstatistic.Whiletheseparticularequationsuseincomeasthevariableofinterest,Theil'sTcanaddressanynumberofquantifiablephenomena.Whenhouseholddataisavailable,Theil'sTstatistic

is6:

n

=x<

*

(、

*ln

(、

p=1

ln丿

冷yJ

冷y丿

[1]

wherenisthenumberofindividualsinthepopulation,ypistheincomeofthepersonindexedbyp,and巴isthepopulation'saverageincome.Ifeveryindividualhasexactlythesameincome,Twillbezero;thisrepresentsperfectequalityandistheminimumvalueofTheil'sT.Ifoneindividualhasalloftheincome,Twillequallnn;thisrepresentsutmostinequalityandisthemaximumvalueofTheil'sTstatistic.

Ifmembersofapopulationcanbeclassifiedintomutuallyexclusiveandcompletelyexhaustivegroups,thenTheil'sTstatisticismadeupoftwocomponents,thebetweengroupelement(T'g)andthewithingroupelement(Twg).

gg

[2]T=T'g+Twg

Whenaggregateddataisavailableinsteadofindividualdata,T'gcanbeusedasalowerboundforthepopulation'svalueofTheil'sTstatistic.ThebetweengroupelementofTheil'sTcanbewrittenas:

[3]

i=1

顯丿J

whereiindexesthegroups,piisthepopulationofgroupi,Pisthetotalpopulation,yiis

theaverageincomeingroupi,and卩istheaverageincomeacrosstheentirepopulation.

T'gisboundedabovebyln(P/pi(min)),thenaturallogarithmofthetotalpopulationgi

dividedbythesizeofthesmallestgroup.Thisvalueisattainedwhenthesmallestgroupholdsalltheresource.Whendataishierarchicallynested(i.e.everymunicipalityisinaprovinceandeachprovinceisinacountry)Theil'sTstatisticmustincreaseorstaythesameasthelevelofaggregationbecomessmaller(i.e.T’.>T'.>T',、n

populationg(district)g(county)

T'g(region)).Theil'sTstatisticforthepopulationequalsthelimitofthebetweengroupTheilcomponentasthenumberofgroupsapproachesthesizeofthepopulation.

BecausethecentralpurposeofthisdocumentistoshowhowtouseTheil'sTstatistic,theexamplesrelatingtoitsusewillbealittlemoreinvolved.

Example1:ConstructTheil'sTstatisticforUniversalWidgetandWorldwideWidgetwiththedataasgiven.

First,considerUniversalWidget.TofollowalonginExcel,openthespreadsheet“ExampleProblemswithTheil'sTStatistic”andselecttheworksheet“TheilExample1A”.Sinceindividualleveldataisavailable,Equation1isrelevant.Thefirststepistosumthenumberofemployeesandthetotalpayrollandtodividetotalpayrollbythenumberofemployeestogettheaveragesalary.Next,computethesalary/averagesalaryquotientforeachsalarylevel.Then,takethenaturallogarithmofthesamequotient.Anindividual's“Theilelement”isthecontributionthatheorshemakestoTheil'sTstatistic.Thisvalueiscomputedas[1/n]*[salary/averagesalary]*[ln(salary/averagesalary)].AftercomputingtheTheilelementsforeachjobposition,multiplybythenumberofemployeesintheposition.AddingupthesevaluesyieldstheTheilIndex,whichinthecaseofUniversalWidgetis0.28615395.

TocomputeTheil'sTstatisticforWorldwideWidget,followtheexactsamestepsasinPartA.ThecomputationscanbefoundintheExcelspreadsheet“Example

ProblemswithTheil'sTStatistic”undertheworksheet“TheilExampleIB”.Theresult

isaTheil'sTStatisticvalueof0.463162658.

Analysis:ComputingvaluesforTheil'sTstatisticisarelativelysimpleprocessofpluggingvaluesintoaformula.Therealconcernistomakesomeconclusionaboutinequality.IsitpossibletoconcludethatWorldwideWidgethasamoreunequalsalarystructurethanUniversalWidgetbecauseWorldwidehasahighervalueofTheil'sTstatistic?Notnecessarily.Asdiscussedabove,withindividualdata,thevalueofTheil'sTstatisticisboundedbyln(n),sowhileUniversalWidgethasanupperboundofln(400)=5.991464547,WorldwideWidgethasanupperboundln(1300)=7.170119543.BecauseWorldwideWidgethasmoreemployees,ceterisparibusitwillhaveagreatervalueofTheil'sTstatistic(infactifthecompanieshadidenticalTheil'sTstatisticvalues,onecouldconcludethatthelargercompanyhadlessinequality).Generallyspeaking,valuesofTheil'sTstatisticneedacontexttomakesense.Giventhatlastyear'sTheil'sTstatisticforsalariesatUniversalWidgetwas.1000,theTheil'sTstatisticforsalariesatWorldwideWidgetwas.5000,andbothcompanieshadworkforcesofsimilarsizetotheircurrentlevels,onecouldconcludethatsalaryinequalityincreasedatUniversalanddecreasedatWorldwideoverthelastyear.Knowingonlythisyear'sinformationandthatthetwocompanieshavesignificantlydifferentsizedworkforces,itisdifficulttomakemanysubstantiveconclusions.Ifonlyoneyear'sworthofdataisavailable,thenanotherinequalitymeasure,suchastheGinicoefficientorcoefficientofvariationmaybemoreappropriate.

Example2:WhatistheinterpretationofTheil'sTstatisticifthesalaryschedulesgivenrepresenttheaveragesalaryacrosspositions,nottheexactsalaries?

Inotherwords,forUniversalWidget,the7membersoftheCustodialStaffhaveanaveragesalaryof$18,000peryear,butthismayfluctuateamongindividuals.

Analysis:LookingatEquation2,Theil'sTstatisticiscomposedofabetweengrouppartandwithingrouppart.Undertheassumptionsofthisproblemstatement,thereisnowaytocomputethewithingroupcomponent,becausethereisnoknowledgeofindividualsalaries,onlyaveragesalaries.However,itispossibletocomputethebetweengroupcomponentandnotethatthisisthelowerboundfortotalinequality.Forthistask,thepropermathematicalrelationisEquation3,whichbynoaccidentbearsastrikingresemblancetoEquation1.Becausethesalaryfiguresarethesame,thenumericalvaluesofTheil'sTstatisticdonotchangeforeithercompany,buttheinterpretationdoes.Now0.28615395representsthebetweengroupcomponentofTheil'sTstatisticforUniversalWidgetandthelowerboundoftotalinequality.Thespreadsheetanalysisforboth

UniversalandWorldwidecanbefoundintheExcelSpreadsheet“ExampleProblemswithTheil'sTStatistic”undertheworksheets“TheilExample2A”and“TheilExample2B.”Noticehowthecolumnheadingschange,whichchangestheunderlyinginterpretationofthecalculations.

Example3:

Considerthefollowingdata:

UniveralWidgetSalarySchedule

JobType

Experience

#ofEmployeesinPosition

ExactAnnualSalary

CustodialStaff

Entry

2

$16,000.00

Mid

3

$18,000.00

Senior

2

$20,000.00

OfficeStaff

Entry

2

$18,000.00

Mid

6

$22,000.00

Senior

2

$26,000.00

EquipmentOperators

Entry

70

$20,000.00

Mid

140

$25,000.00

Senior

70

$30,000.00

EquipmentTechnicians

Entry

5

$29,000.00

Mid

5

$35,000.00

Senior

5

$41,000.00

Foremen

Entry

2

$25,000.00

Mid

10

$40,000.00

Senior

3

$50,000.00

Salespersons

Entry

10

$47,000.00

Mid

30

$60,000.00

Senior

10

$73,000.00

Engineers

Entry

3

$70,000.00

Mid

4

$75,000.00

Senior

3

$80,000.00

Managers

Entry

2

$60,000.00

Mid

2

$80,000.00

Senior

2

$100,000.00

VicePresidents

Entry

1

$100,000.00

Mid

2

$120,000.00

Senior

1

$140,000.00

SeniorVicePresidents

Entry

1

$160,000.00

Mid

1

$240,000.00

CEO

Senior

1

$1,000,000.00

Unlikeexamples1Aand1B,employeesdrawdifferentsalariesbasedonboththeirlevelofseniority(entry,mid,senior)andtheirjobposition.Example3resumestheassumptionfromthefirstexamplethatthedatarepresentsexactsalaryinformationforeachindividual.

Giventhisnewdata,whatistheTheilIndexforUniversalWidget?

Answer:Thereareseveralwaystodothisproblem,andfoursolutionsareworkedoutinthespreadsheet.Thefirstsolution(TheilExample3A)startsbycomputingthewithin-groupinequalityforeachjobposition(custodialstaff,engineers,etc.).ATheilcomponentiscomputedforeachexperiencelevelwithineachjobposition,thesummationofwhichgiveswithingroupinequality.However,beforeconcludinghowmuchthejobpositioninequalitycontributestototalcompany-wideinequality,wemustre-weightbytheproportionofsalarieswithinthejobposition.(Inotherwords,inequalitywithintheequipmentoperatorgrouptakesongreaterweightthanamongthecustodialstaff,because70%ofworkersoperateequipmentwhilelessthan2%performcustodialservices.)ComputingtheTheilIndexinthismannerhelpsustoparsetotalinequalityintowithin-groupandbetween-groupcomponents.ThetotalvalueoftheTheilIndexisnow0.12860521,ofwhich0.124275081isbetween-groupinequalityand0.004330129.Thesubstantivelessonhereisthatthedifferenceinaveragesalariesbetweenjobpositionscausesthevastmajorityoftheinequality,andthedifferencesamongsenioritylevelswithinjobpositionscontributeverylittletototalinequality.

TheilExample3Bcalculatestotalinequalitybycomparingeachjobpositionexperiencelevelcombinationtotheaveragesalary.ThevalueofthetotalTheilIndexisthesame,butthismethoddoesnotnaturallyparsetheIndexintowithin-groupand

between-groupportions.Fullenumeration-TheilExample3Cmakeseachemployeeaseparaterecord,which,ye