Centre for Efficiency and Productivity Analysis
Working Paper Series
No. WP05/2016
A Unifying Framework for Farrell Efficiency Measurement Coherent with Profit-maximizing
Principle
Rolf Färe, Xinju Heb, Sungko Lib, Valentin Zelenyuk
Date: May 2016
School of Economics
University of Queensland
St. Lucia, Qld. 4072
Australia
ISSN No. 1932 - 4398
AUnifyingFrameworkforFarrellEfficiencyMeasurement CoherentwithProfit-maximizingPrinciple
RolfFärea,XinjuHeb,SungkoLib,ValentinZelenyukc
Abstract
Measuringprofitefficiencyisachallengingtask.Thispapersynthesizesexistingapproachestoforma
general Farrell-type model of profit efficiency. Our derivations help us unveil new and interesting
relationship between existing profit efficiency measures and the Farrell-type profit efficiency
measures.Inturn,thishelpsusestablishingacompleteframeworkofstudyingefficiencybehaviorof
firms, where the profit efficiency measure satisfies some desirable properties and contains Farrell
output-oriented or input-oriented measures of technical efficiency and allocative efficiency as
multiplicativeelements.Thenewcomponent,revenueefficientallocativeefficiency,introducedinthis
papercanhelpfirmstomakedecisionandhasnotbeenstudiedintheliteraturebefore.
Subject
classification
: Organizationalstudies:productivity.Profitefficiency.Dataenvelopment
analysis
Areaofreview
: Optimization
JELCodes:C44,D24
a
OregonStateUniversity
HongKongBaptistUniversity
c
UniversityofQueensland
b
1. Introduction In measuring technical efficiency of production two types of scaling inputs and outputs are
frequentlyused:(i)scalingeachcomponentofavector;and(ii)scalingthevectorbyitself.Thesetwo
approachesareoftenreferredtoastheRussellmeasureandtheFarrellmeasure,respectively.Inthis
paper we apply the above scaling methods to the profit function and introduce some Farrell-type
efficiencymeasuresinthissetting.
TheFarrelltechnicalefficiencymeasurewasproposedbyFarrell(1957)andthengeneralizedand
popularizedbyCharnes,CooperandRhodes(1978),FäreandLovell(1978),andBanker,Charnesand
Cooper(1984),justtomentionafew.Tostrengthentheabilityofmeasuresoftechnicalefficiencyto
differentiatefirmperformance,researchersimposeconditionstotheempiricalmodel.Forexamples,
“assuranceregion”(Thompsonetal.1986),“factorweights”(Ruggiero2000),“performancestandards”
(Cook and Zhu 2006) and “input allocation” (Cherchye et al. 2013) are added to the model of
measuring technical efficiency. When both price and quantity data are available, investigating the
profitefficiencywouldbeanaturalwaytostudytheperformanceofafirm,andthisistheapproach
wetakeinthispaper.
Farrell (1957) also introduced the concepts of overall (cost or revenue) efficiency and its
decomposition into technical and allocative efficiencies. These efficiency measures can be
characterizedbythescalingmethod(ii)statedabove.HerewerefertothemasFarrell-typeefficiency
measures. The dominating popularity of Farrell-type efficiency measures in practice is reflected by
thousands of empirical works in the past several decades. It is desirable to include them as
componentswhentheperformanceofafirmisanalyzed.However,allFarrelloutput-orientedand
input-orientedmeasureshavenotbeenrelated(atleastexplicitly)toanyprofitefficiencymeasureas
multiplicativecomponentsandfillingthisgapisoneofthemaincontributionsofthispaper.
Itisworthnotingthatmeasuringprofitefficiencyisaparticularlychallengingtask.Forexample,
themaximalprofitlevelcanbezero,positive,orevenundefined.Meanwhile,theactualprofitlevel,
besidespossibilitytobepositive,canalsobezeroorevennegative.Thiscomplicatestheproblemof
definingasuitablemeasureofprofitefficiencysothatitiswelldefinedmathematicallyaswellashas
convenient(e.g.,percentage)interpretation.Variousmeasuresofprofitefficiencyweresuggestedto
overcomethesedifficulties. Here we will synthesize existing ideas in the literature (some of which were proposed in a
differentcontext)intoaFarrell-typeframeworkofstudyingefficiencybehavioroffirms.Importantly,
wewillexplicitlyrelatetoprofitmaximizationprincipleandderiveseveralFarrell-typeprofitefficiency
measuresthathaveusefulinterpretationsandsomeadvantagesoverexistingmeasures.Relatedideas
oftheseprofit-efficiencymeasuresgobacktotheworksofBankerandMaindiratta(1988),Chavas
andCox(1994),FäreandPrimont(1995),Chambersetal.(1998),andRay(2004).Thegeneralmodel
introduced in this paper helps us unveil new and interesting relationship between existing profit
efficiency measures and the Farrell-type profit efficiency measures. Further, the output-oriented
profit efficiency measure contains Farrell output-oriented measures of technical efficiency and
allocativeefficiencyasmultiplicativeelements.
We illustrate our approach with a help of data on US banks (from Ray (2004)). By employing
DataEnvelopment Analysis (DEA), a new component, revenue efficient allocative efficiency, is
identifiedtohelpthefirmtodetermineherproductionscaleandinputmix.Ourapproachcanalsobe
adaptedtoothervariants,suchastheIDEA(Cooper,ParkandYu(2001)andZhu(2004)),CAR-DEA
(Cook and Zhu (2008)), DEA with non-homogeneous firms (Cook et al. (2013)), and based onDEA
nonparametricmodelsofoptimizingbehavior(Cherchyeetal(2008)),tomentionjustafew.
Inthenextsection,weintroduceageneralframeworkofprofitefficiencymeasure.Thensection
3 discusses the Farrell output-oriented profit efficiency measure specifically. In section 4, some
interpretations of this measure are provided. In section 5, we show that the relationship among
different measures in Farrell-type framework, technical efficiency, revenue efficiency and cost
efficiency are just some special case of profit efficiency under different constraints. Section 6
expressesmultiplicativedecompositionoftheprofitefficiencymeasure.Section7linksthisFarrelltypeprofitefficiencymeasuretosomeexistingprofitefficiencymeasures.Section8illustratesthe
Farrell-typeprofitefficiencymeasureanditsdecomposition.Finally,weconcludeourmainresultsin
section9.
2. AGeneralFrameworkforFarrell-typeMeasuresofProfitEfficiency
(
Let ! ∈ ℝ&
% and ' ∈ ℝ% beinputandoutputvectors.Thecorrespondingpricesaredenoted
(
withrow-vectors ) ∈ ℝ&
% and * ∈ ℝ% ,respectively.Theproductiontechnologyisgivenby ℑ =
{(!, '):!234*567829'} whichweassumesatisfyingstandardregularityconditions(seeFäreand
Primont (1995), Table 2.1, p. 27). Let (! ; , ' ; ) be an observed input-output vector and let the
observedtotalrevenue(orsales)atoutputprices * ; andobservedtotalcostsatinputprices ) ; be
denotedby < ; = * ; ' ; and = ; = ) ; ! ; respectively.Further,let > ; betheobservedprofit,i.e.,
> ; = * ; ' ; − ) ; ! ; . The maximal profit a firm can attain facing prices * ; , ) ; and using
technology ℑ isgivenby > * ; , ) ; = @8*{* ; '–) ; !: !, ' ∈ ℑ}(1)
A,B
Inthefollowingdiscussions,afirmwillbereferredtoasprofitefficientifanonlyifitachievesmaximal
profit,i.e.,when> * ; , ) ; = > ; .
Beforegoingfurtherwithadefinitionofprofitefficiency,itisimportanttonotethatthereare
quite many cases where using > *, ) is problematic. This includes the case of global increasing
returnstoscale(IRS),where> *, ) = +
(CRS),where > *, ) = +
.Similarsituationiswiththeconstantreturnstoscale
or> *, ) = 0,dependingontheprices *, ) relativetotheshape
of frontier of technology setℑ . One potential remedy to such cases is to allow for additional
constraints because in practice, firms indeed often face various constraints: e.g., fixed inputs or
endowments at a given period, minimal employment requirement, maximal amount of inputs
available,budgetlimits,etc. Wewilldenotesuchconstraintswithaconstraintset G and,sothe
constrained profit will be denoted with > *, )|G , where G = {(!, '):<IJ ≦ *' ≦ <LJ , 'IJ ≦
' ≦ 'LJ , =IJ ≦ )! ≦ =LJ , 347!IJ ≦ ! ≦ !LJ } . One could also add many other types of
constraints. Such constraint set has been adopted extensively in the measurement of technical
efficiency. For example, when there are some nondiscretionary input variables (Kopp 1981, and
BankerandMorey1986),thelowerandupperboundsforsomeinputsareequal.Ifeachinput 4 and
output N is associated with a “weight” )O or *P , the assurance region method introduced by
Thompsonetal.(1986)imposesconstraintslike Q ≦ )R )S ≦ T where Q and T aretwopositive
numbers.Inthespecialcasewheneachlowerboundiszeroandeachupperboundispositiveinfinity,
the constraint set becomes G = ℝ(%&
and > ), *|G will be the usual profit function. We will
%
refer to > ), * and > *, )|G as unconstrained and constrained maximal profit function,
respectively. Anexampleofsuchgeneralizedprofitfunctionisgivenby
> *, )|G = sup *' − )! ∶ !, ' ∈ ℑ ∩ G (2)
A,B
Furthermore, let [ =
(
&
(
&
(
!, '; ), * ∈ ℝ&
% ×ℝ% ×ℝ% ×ℝ% : !, ' ∈ ℑ, ) ∈ ℝ%% , * ∈ ℝ%% . A
general measure of profit efficiency for the observed quantity vector (! ; , ' ; ) and price vector
() ; , * ; ) isafunction ^:[ → ℝ% ∪ {+
},andsubjectto G,definedas
^ ! ; , ' ; ; ) ; , * ; G = sup {c(de , … , d( ; ge , … , g& ) ∶
a,b,A,B
(
&
;
;
*P
gP 'P
−
Phe
)O; dO !O; ≦ * ; ' − ) ; !,
Ohe
!, ' ∈ ℑ ∩ G}(3)
forall ! ; , ' ; ∈ ℑ and c(. ) isboundedfromaboveundertheconstraints, while > * ; , ) ; G =
+∞ whenthereisnoupperboundforthethevalueof c(. ) undertherestrictions.Differentforms
of c(de , … , d& ; ge , … , g( ) canbechosen.Inthispaperwewillfocusonthecasewhere gP = gfor
N = 1, … , k, and dO = d for4 = 1, … , l. Thus,(3)becomes
^ ! ; , ' ; ; ) ; , * ; G = sup c d, g : * ; g' ; − ) ; (d! ; ) ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G .(4)
a,b,A,B
Incidentally,ifinadditionwehave d = 1 g and c(. ) = g,then(4)becomes
^ ! ; , ' ; ; ) ; , * ; G = sup g: * ; g' ; − ) ; (! ; g) ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G .
a,A,B
ThisisthereciprocaloftheoverallgraphefficiencymeasureintroducedbyFäre,GrosskopfandLovell
(1985,p.119).Ontheotherhand,ifwelet g = d = Tand c(. ) = T instead,then(4)becomes
^ ! ; , ' ; ; ) ; , * ; G = sup T: T * ; ' ; − ) ; ! ; ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G a,A,B
=
> ! ; , ' ; ; *, )|G
.
>;
ThisisthereciprocaloftheprofitefficiencymeasuredefinedinBankerandMaindiratta(1988). Theattentionofthispaperisonc g, d = g − d + 1.Themeasureofprofitefficiency(4)
becomes
^ ! ; , ' ; ; ) ; , * ; G = @8* g − d + 1: * g' ; − ) d! ; ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G .(5)
a,b,A,B
Thisisaformulationthatcontainsmanyways.Specifically,let d = 1andusesubscript“o”todenote
themeasurein(5),thenwehave
^o ! ; , ' ; ; ) ; , * ; G = @8* g: * ; g' ; − ) ; ! ; ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G .
a,A,B
Wecallittheoutput-orientedFarrellprofitefficiencymeasure.Notethat ^o ! ; , ' ; ; ) ; , * ; G = @8* g: g * ; ' ; − ) ; ! ; ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G a,A,B
= @8* g: g< ; − = ; ≦ > * ; , ) ; |G a,A,B
=
> * ; , ) ; |G + = ;
> * ; , ) ; |G − > ;
=
1
+
.(6)
<;
<;
Since> * ; , ) ; |G − > ; ≧ 0,wehave^o ! ; , ' ; ; ) ; , * ; G ≧ 1,andthemeasuretellsusthatthe
firmcanraiseprofitlevelby (^o − 1)×100% oftheobservedtotalrevenue.
Similarly,if g = 1andusesubscript“i”todenotethemeasurein(5),onlyinputsideisconsidered
andwecalltheexpressioninput-orientedFarrellprofitefficiencymeasure,then ^R ! ; , ' ; ; ) ; , * ; G = @8* 2 − d: * ; ' ; − ) ; d! ; ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G b,A,B
= 2 − s4c d: < ; − d= ; ≦ > * ; , ) ; G b,A,B
=2−
< ; − >(* ; , ) ; |G)
=;
=1+
>(* ; , ) ; |G) − > ;
.(7)
=;
Again,^R ! ; , ' ; ; ) ; , * ; G ≧ 1.Thismeasurehasameaningofraisingtheprofitlevelby ^R – 1 ×
100% oftheobservedtotalcosts. Incorporatingbothoutputandinputsidessimultaneouslyispossible.Oneexampleistoletd =
2 − g in(5).WecallthisthejointlyorientedFarrellprofitefficiencymeasureanduseasubscript“io”
todenoteit.Then
^Ro ! ; , ' ; ; ) ; , * ; G = @8* g − 2 − g + 1: * ; g' ; − ) ; 2 − g ! ; ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G .
a,A,B
= @8* 2g − 1: g < ; + = o − 2= ; ≦ > * ; , ) ; G a,A,B
= @8* 2g − 1: g ≦
a,A,B
=1+
> * ; , ) ; |G + 2= o > * ; , ) ; |G − > ;
=
+1 <; + = o
<; + = o
> * ; , ) ; |G − > ;
.(8)
< ; + = o /2
If < ; + = o /2 istreatedasanestimateofthefirmsize,thentheprofitefficiencymeasure ^Ro canbeinterpretedasthefirmcanincreaseprofitlevelby ^Rw – 1 ×100% ofthefirmsize. In summary, we have introduced a Farrell measure of profit efficiency (5) that includes radial
changesinoutputsin(6)orinputsin(7)asspecialcases.Further,italsoallowsforsimultaneousradial
changesofinputsandoutputsin(8).Eachofthesethreemeasuresin(6),(7),and(8)aregreaterthan
orequaltoone.Iftotalrevenue,totalcostsandtheaverageofrevenueandcostsareregardedas
three different indicators of firm size, then^R , ^w , 347^Rw all have the interpretation that the
potentialincreaseinprofitlevelasapercentageoffirmsize.
Clearly,itisbeyondthespacepermittedbyanyjournaltoconsiderallthese(andother)casesin
reasonabledetailsinonepaperandsowewillfocusonlyononeofthem—theoutput-orientedFarrell
Profit Efficiency measure. We develop a framework of profit efficiency measure that encompasses
existingFarrelloutput-orientedefficiencymeasuresasspecialcasesofthemodelandasmultiplicative
componentsofthemeasure.Alldiscussionsoftheoutput-orientedFarrellprofitefficiencymeasure
apply, with some modifications, to the other two measures: the input-oriented Farrell efficiency
measureandthejointlyorientedFarrellefficiencymeasure.
3. TheFarrellOutput-orientedMeasureofProfitEfficiencyanditsProperties
Formally,wedefinetheFarrelloutput-orientedmeasureofprofitefficiency, ^o : [ → ℝ% ∪ {+
},as
^o ! ; , ' ; ; ) ; , * ; G = @8* g > 0: * ; g' ; − ) ; ! ; ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G (9)
a,A,B
forall ! ; , ' ; ∈ ℑ ∩ G and *' ; ≠ 0,while ^o ! ; , ' ; *, ) = +∞ for *' ; = 0.1
Thismeasurecanberestatedintermsofthemaxi-max(orsup-sup)principle—asatwo-stage
optimization strategy, first optimizing for profit and then optimizing for measuring the distance
betweentheprofitefficientallocationandtheactualallocation,i.e.,
^o ! ; , ' ; ; ) ; , * ; G = sup sup g > 0:* ; g' ; − ) ; ! ; ≦ * ; ' − ) ; !, !, ' ∈ ℑ ∩ G a
A,B
= sup sup
a
A,B
*; ' − ) ; ! + ) ; ! ;
: !, ' ∈ ℑ ∩ G ≧ g > 0 (10)
*; ';
forall ! ; , ' ; ∈ ℑ ∩ G and * ; ' ; ≠ 0,while ^o ! ; , ' ; ; ) ; , * ; G = +∞ for * ; '; = 0.
Intuitively,thismeasureexpandsobservedoutputsequi-proportionallyorradially,inthespiritof
theFarrellmeasure,toraisetheprofitleveltothemaximalpossibleprofitlevelunderthereference
technology.TheconstraintsZ(e.g.,onrevenue,cost,outputsandinputs)areincludedforthesakeof
generality,toencompassvariouspracticalcaseswhereafirmisrequiredtosatisfysomeconditions.
Inthefollowingdiscussions,let g ∗ beasolutionin(9).Notethat (! ; , g ∗ ' ; ) maybeaninfeasible
input-outputvector. A geometric illustration of the Farrell profit efficiency measure is in order. Consider Figure 1,
where “abcde” denotes the production frontier and f1 is the observed input-output vector. The
output-oriented technical efficiency is measured by 'e ’/'e . **′ is the iso-profit line. We can
hypotheticallyexpandtheoutputtoraiseprofituntilitcoincidestheoptimaliso-profitline **’. Then
'e "/'e canberegardedasameasureofprofitefficiency,whilethegapbetweentechnicalefficiency
and the profit efficiency can be regarded as the (profit-oriented) allocative efficiency, which is
measuredvia 'e "/'e inFigure1.
Itisalsoworthremindingagainthatwithoutsomeupperboundson !, ' or )!, *',therewill
be no maximum profit level if technology exhibits IRS and so, as can be shown,
^o ! ; , ' ; ; ) ; , * ; G = +
. Similar situation may occur under CRS. In a sense, the upper
constraints we have in the definition help regularizing the profit function and, in turn, the related
Farrellprofitefficiencymeasureforsuchcases,besideshelpingtotailortheseconceptsclosertothe
reality.
[InsertFigure1here] Many researchers have stated, explicitly or implicitly, some desirable properties of the profit
efficiency measure. To adapt them to our context, let  be a profit efficiency measure. Here we
summarize P.1
 isawell-definedfunctionforallmaximalandobservedprofitlevels,andforall ! ; , ' ; ∈
ℑ.
P.2
Thereexistsavaluecsuchthatafirmisidentifiedasprofitefficientifandonlyif  = 2.
P.3
 ishomogeneousofdegreezeroininputandoutputprices.
P.4
 isindependentofunitsofmeasurement,i.e.,commensurable.
Thesepropertieshavebeendiscussedbymanyauthorsbefore,seeNahmandVu(2013,p.46),
Asmildetal.(2007,p.318),amongothers.Theintuitionbehindthemandtheirimportanceareobvious.
Itisnaturaltoacceptthemasaxiomsthatonewouldexpectanyprofitefficiencymeasuretosatisfy.
Ourmeasuredoessatisfyalltheabovefourproperties.Inparticular,supposetheobservedtotal
revenueispositiveandthemaximumprofitlevelisafinitenumber.I.e., < ; > 0 andπ * ; , ) ; |G >
0.In(6),
> * ; , ) ; |G − > ;
^o ! , ' ; ) , * G = 1 +
.(11)
<;
;
;
;
;
Sincethedenominatorispositiveandthedifferencebetweenthemaximumandobservedprofitsis
finite,theFarrelloutput-orientedprofitefficiencymeasureiswell-definedforallobservedinputand
outputquantitiesandpricesaslongasthetotalrevenueispositive.Further,inviewof(11),itisclear
that ^o ! ; , ' ; ; ) ; , * ; G = 1 if and only if > * ; , ) ; |G − > ; = 0 which means profit efficient.
Property P.2 is satisfied, too. Also one expression in (6) is ^o ! ; , ' ; ; ) ; , * ; G =
> * ; , ) ; |G + ) ; ! ; * ; ' ; .Sincetheprofitfunctionishomogeneousofdegreeoneininputand
output prices, we have, for t > 0, ^o ! ; , ' ; ; Å) ; , Å* ; G = > Å* ; , Å) ; |G + Å) ; ! ; Å* ; ' ; =
> * ; , ) ; |G + ) ; ! ; * ; ' ; = ^o ! ; , ' ; ; ) ; , * ; G . The Farrell output-oriented profit efficiency
measureishomogeneousofdegreezeroininputandoutputprices.Finally,let ! ∗ , ' ∗ beaprofitmaximizing input-output vector. Then ^o ! ; , ' ; ; ) ; , * ; G =
* ; ' ∗ − ) ; ! ∗ + ) ; ! ; /* ; ' ; .
Since after changes of units of measurement, prices and quantities will adjust so that *P 'P and
)O !O remainunchangedforall N and4,thecommensurabilityof ^o ! ; , ' ; ; ) ; , * ; G follows.
HencetheFarrelloutput-orientedprofitefficiencymeasuresatisfiesallthefourpropertiesP.1toP.4.
Inadditiontotheabovepropertiesforanyprofitefficiencymeasure,theoutput-orientedprofit
efficiencymeasureishomogeneousofdegree-1intheobservedoutputvectorq0.Itisalsocontinuous
ininputandoutputpricesaswellasininputsandoutputs(for*' ; > 0).
4. IntuitionoftheFarrellOutput-orientedProfitEfficiencyMeasure
Tosimplifyfurtherdiscussionsandnotations,weassumethatthemaximumprofitlevelexistsat
theinputandoutputpricesunderconsiderationand* ; ' ; > 0.ManyinterpretationsoftheFarrell
output-orientedprofitefficiencymeasurecanbefoundbyrearrangingitsformula.
Firstly,from(6), ^o (!; , ' ; ; * ; , ) ; |G) = (>(* ; , ) ; |G) + = o )/< o .Itfollowsthat
^o ! o , ' o ; ) ; , * ; |G =
= o >(* ; , ) ; |G)
+
.(12)
<o
<o
Theinterpretationof(12)isveryusefulbecause ^o (∙) isexpressedintermsoftwofactors:(i)realized
cost-benefitratio,and(ii)thebestpossibleprofitmarginforthefirm.Bothofthesefactorsarewell
understoodbypractitionersinbusinessandinvestmentcommunity.Indeed,thecost-benefitratiois
among the most important key performance indicators (KPI) in business practice, as is the profit
marginindicator.
Secondly,subtractingandaddingR0tothenumeratorof(6),weget
^o ! o , ' o ; ) ; , * ; |G =
>(* ; , ) ; |G) − > o
+ 1.(13)
<o
TheFarrelloutput-orientedprofitefficiencymeasure ^o (∙) intheformof(13)tellsaboutunrealized
orlostprofitinpercentagetoactualtotalrevenueorsalesandsocanbeunderstoodbypractitioners
asthe‘lostprofitmargin’(additivelynormalizedby1).
Thirdly,let < ∗ and = ∗ betheprofit-maximizingtotalrevenueandtotalcostsrespectivelyand
notingthat> * ; , ) ; |G = < ∗ − = ∗ ,onecanrewrite(6)asfollows:
^o ! o , ' o ; ) ; , * ; |G =
<∗
=; − =∗
+
(14)
<;
<;
Thisinterpretationisveryusefulbecauseitshowsdecompositioninto(i)ratioofbestpossiblerevenue
toactualtotalrevenue(sales)and(ii)excessivecost(beyondthepossibleminimum)relativetothe
actualsales. Notethat < ∗ /< ; mightbelarger,equalorsmallthanunity,whichcanbeinterpreted
as undersell, efficient sale and oversell, all relative to the profit maximizing efficiency criterion.
Similarly, (= ; − = ∗ )/< ; might be larger, equal or small than unity, which can be interpreted as
overspend,efficientcostandunderspend,allwithrespecttotheprofitmaximizingefficiencycriterion. 5. SomeImportantSpecialCases
Manyspecialcasescanbederivedfromtheproposed Farrelloutput-orientedprofitefficiency
measurein(9)bysettingvariousrestrictionstocatertoparticularsituationsafirmmaybefacingin
practice.Wewillpresentjustafewhere,whichareperhapsthemostinterestingtogeneralaudience.
Whenthemaximumprofitleveliszero(whichcanhappeninlong-runcompetitiveequilibrium),
theprofitefficiencymeasurewillcoincidewiththeveryintuitiveandfrequentlyusedinpracticecostbenefitratio(reciprocaltothe‘returntodollar’efficiencymeasure),i.e., ^; = = o < o .
Whenweimposerestrictionsonquantitiesandprices,wefindthattheFarrellprofitefficiency
measure is closely related to other Farrell measures. First, when !IJ = !LJ = ! ; , i.e., the firm is
required(e.g.,intheshort-run)toutilizethealreadyavailableresourcesandtherearenoboundson
*' and ' . Let G ! = ! ; =
!, ' : !, ' ∈ G, ! = ! ; . Then the profit efficiency measure
coincideswiththestandardrevenueefficiencymeasure,i.e.,from(10),
^o ! o , ' o ; ) ; , * ; |G ! = ! ; = sup sup
*; ' − ) ; ! ; + ) ; ! ;
: !, ' ∈ ℑ ∩ G ! = ! ;
*; ';
= sup sup
*; '
: !, ' ∈ ℑ ∩ G ! = ! ;
*; ';
a
A,B
a
=
A,B
≧g>0 ≧g>0 < ! ; , *;
= <É ! ; , ' ; , * ; .(15)
*; ';
forall ! ; , ' ; ∈ ℑand * ; ' ; > 0,where <(! ; , * ; ) isthemaximumrevenueat (! ; , * ; ).Second,
inadditionto !IJ = !LJ = !; considerthesupremumwhentheoutputpricescantakeanypositive
values,andtheconstraintsinZ,otherthantheinputvector,arenon-binding.Then,(10)becomes
^o ! o , ' o ; ) ; , *|G ! = ! ; , * > 0 = sup sup
a
A,B
*' − ) ; ! ; + ) ; ! ;
: !, ' ∈ ℑ ∩ G ! = ! ;
*; ';
= sup g: g ≦
a,B
*'
, !, ' ∈ ℑ ∩ G ! = ! ; , * > 0 *' ;
≧ g > 0:* > 0 Let the solution to this optimization problem beg ∗ and' ∗ . For the case where the output set is
convex,duetotheSupportingHyperplanetheoreminmathematics,thereexistssomenonzeroprice
vector * Ñ ≥ 0 suchthat * Ñ ' ∗ ≧ * Ñ ' for ! ; , ' ∈ ℑ.AsprovedsimilarlyinDebreu(1951), sup g: g ≦
a,B
= inf sup
âä; B∈ℑ
*'
, !, ' ∈ ℑ ∩ G ! = ! ; , * > 0 *' ;
*'
* Ñ '∗
=
*' ;
* Ñ ';
= @8* g: ! ; , g' ; ∈ ℑ = ã; ! ; , ' ; .
a
where ã; ! ; , ' ; istheFarrelloutput-orientedtechnicalefficiencymeasure.Thuswehave
^o ! o , ' o ; * Ñ , )|G ! = ! ;
= ã; ! ; , ' ; .å (16)
ThereforetheFarrelloutput-orientedprofitefficiencymeasureencompassestheFarrelloutputorientedmeasuresofrevenueefficiencyandtechnicalefficiencyasspecialcases.Recallthattheprofit
efficiency measure can be expressed in the form of equation (13) and the value of ã; ! ; , ' ; is
independentofoutputprices,alltheprofitefficiency,revenueefficiencyandtechnicalefficiencycan
beinterpretedasthepercentageofinefficiencylossesinprofittothetotalrevenue. 6. DecompositionoftheFarrellProfitEfficiencyMeasure
Therelationshipsin(15)and(16)provideaunifiedframeworkthatunitestheprofitefficiency
with revenue efficiency and Farrell technical efficiency, as well as help establishing various
decompositionsoftheformer.InviewoftheincreasingrestrictionsimposedontheFarrelloutputorientedprofitefficiencymeasure,wehave
^o ! o , ' o ; ) ; , * ; |G ≧ ^o ! o , ' o ; ) ; , * ; |G ! = ! ;
≧ ^o ! o , ' o ; ) ; , * Ñ |G ! = ! ; .(17)
Applying(15)and(16)to(17),itfollowsthat
^o ! o , ' o ; ) ; , * ; |G ≧ <É ! o , ' o , * ; ≧ ão ! o , ' o .ç (18)
Wecandefinetheprofitallocativeefficiencyas
éÉè ! o , ' o ; ) ; , * ; |G =
^o ! o , ' o ; ) ; , * ; |G
<É ! o , ' o , * ;
≧
≧ 1.(19)
ã; ! o , ' o
ã; ! o , ' o
Rewrite(19),theoutput-orientedprofitefficiencymeasurecanbedecomposedintotwo
components:profitallocativeefficiencyandFarrelloutput-orientedtechnicalefficiency:
^o ! o , ' o ; ) ; , * ; |G = éÉè ! o , ' o ; ) ; , * ; |G ∙ ã; ! o , ' o .(20)
Denotetheoutput-orientedallocativeefficiencyby éÉo ! o , ' o , * ; . Recallthat éÉo ! o , ' o , * ; =
<É ! o , ' o , * ;
ã; ! o , ' o .Itisclearfrom(19)that éÉè ≧ éÉ; . Thus ^o = éÉè ×ã; ≧ éÉ; ×ã; .
Define éÉêë ! o , ' o ; ) ; , * ; |G =
éÉè ! o , ' o ; ) ; , * ; |G
^o ! o , ' o ; ) ; , * ; |G
=
.(21)
éÉo ! o , ' o , * ;
<É ! o , ' o , * ;
éÉêë shows the improvement of profits from the maximum revenue for a given input vector to
maximum profits when all inputs and outputs are variable. We call éÉêë the revenue efficient
allocativeefficiency.So(20)canbefurtherwrittenas
^o ! o , ' o ; ) ; , * ; |G = éÉêë ! o , ' o ; ) ; , * ; |G ×éÉo ! o , ' o , * ; ×ã; ! o , ' o . (22) Decompositionofprofitefficiencyisingeneraladditiveintheliteratureofefficiencyanalysis.A
decompositionthatincludedmultiplicativecomponentswassuggestedbyAparicio,PastorandRay
(2013).However,theirdecompositionofprofitefficiencyconsistsofbothadditiveandmultiplicative
elements.NoneofthosedecompositionsincludedexplicitlyFarrelltechnicalandallocativeefficiency
measures as components. Our decomposition in (22) fills in the gap of the literature. It provides
decomposition of the profit efficiency that explicitly includes the Farrell technical and allocative
efficiencymeasuresasmultiplicativecomponentswithmeaningfulinterpretationineachitem.
7. Relationshiptoothermeasures
Aswementionedintheintroduction,somerootstotheframeworkofprofitefficiencydiscussed
here, can be found in various works in different contexts. The most prominent and perhaps most
closelyrelatedformulaisfoundinChavaxandCox(1994).Specifically,intheirinsightfulpaper,Chavax
andCox(1994)usedthefollowingformula
ío ! ; , ' ; = min g ∶ * ;
a
';
–); !; ≦ >(* ; , ) ; ) .(23)
g
Afterfixingsomedetails(e.g.,changingmintoinf,re-definingthedomainandrangeofthefunction
tocircumventpeculiarcases,etc.),onecanseethatthe ío (! ; , ' ; ) in(23)isthereciprocalofthe
Farrelloutput-orientedmeasureofprofitefficiencyintheunrestrictedcase.Itisalsoimportantto
notethattheattentionofChavaxandCox(1994)wasnottheprofitefficiencymeasurementsbutthe
outerboundsoftheinputandoutputdistancefunctionsandthey(andtothebestofourknowledge
others), have not explored the possibility of using this formula as a profit efficiency measure. In a
relatedwork,Asmildetal.(2007,Theorem4,p.316)showedthatthesameformulaisthelimitofa
linkedconemodel.
Anotherveryimportantandperhapsthemostgeneralmodelofprofitefficiencymeasureisthe
directionaltechnologymeasureofprofitefficiency,introducedbyChambers,ChungandFäre(1998).
^ îï ! o , ' o ; ) ; , * ; ; ñA , ñB =
>(* ; , ) ; ) − > ;
.(24)
*ñB + )ñA
Themeasure ^ îï actuallycontainsmanyexistingmeasuresasspecialcases.Itiscloselyrelatedtothe
measuresproposedinthispaper.Indeed,notethatifwelet ñB = ' ; and ñA = 0 in(24),thentheir
measure is equivalent to the unrestricted version of the output oriented Farrell profit efficiency
measure.Specifically,wehave
^ îï ! o , ' o ; ) ; , * ; ; ñA , ñB =
> *; , ) ; − > ;
> *; , ) ; − > ;
=
+ 1 − 1 = ^o − 1.
*; ';
<;
ThevitalityofthisresultfortheFarrellprofitefficiencymeasureisthatonedoesnotneedtoderivea
newdualitytheoryforthismeasureaswellasothertechnicalproperties—itisalreadyprovidedin
Chambers,ChungandFäre(1998),fortheunrestrictedcaseandcanbeadaptedaccordinglywhen
somerestrictionsarenecessary.
Furthermore,Ray(2004,pp.233–234)proposedthefollowingprofitefficiencymeasure ^óòô ! o , ' o ; ) ; , * ; = (> * ; , ) ; − > ; )/< ; .
Clearly,thismeasureisequivalenttotheFarrellprofitefficiencyintheunrestrictedcase(i.e., ^o =
^óòô + 1).ItisalsoworthnotingthatbothChambers,ChungandFäre(1998)andRay(2004)treated
thedenominatorintheirformulationasanarbitrarynormalization.Inthispaper,weexplicitlyshow
thatsuchnormalizationcanbejustifiedbyacertainorientationasin(6).Thenormalizationcanbe
used to relate the profit efficiency measure to the output-oriented Farrell measure of technical
efficiencyfordecomposingtheprofitefficiencymeasureintovarioussources,aswedointheprevious
section.
ItisalsoimportanttonotethatRay(2004)alsomentionednormalizingthedifferencebytotal
costs,whichhasbeenappliedbyFäre,GrosskopfandWeber(2004)andDasandGhoshb(2009).4 It
canbeshownbysimilarreasoningandderivationsaswehavedoneabovethatnormalizationbytotal
costcanbejustifiedifoneinsteadusetheinputorientedFarrellprofitefficiencymeasure. 8
NumericalIllustration
Thetheoreticaldevelopmentswepresentedabovearegeneralratherthanspecificallyrelatedto
aparticularestimationapproach. Here,inthissectionwepresentnumericalillustrationusingone
ofthemostpopularmethodsofestimatingefficiencymeasuresinpractice—thedataenvelopment
analysis(DEA)approach,whileotherpopularapproaches(e.g.,stochasticfrontieranalysisapproach)
canalsobeused.
Specifically, suppose there are K firms. It is observed that firm k uses N inputs to produce M
outputs. The observed input and output vectors are ! ö = !öe , ⋯ , !ö& and' ö = 'öe , ⋯ , 'ö( .
Forillustration,noexplicitboundsareimposed.Letempiricaltechnologysetusedinthecomputation
be
ü
ú
ùêÑ
=
!, ' :
ü
ûö 'öP ≧ 'P ,
öhe
ü
ûö !öO ≦ !O ,
öhe
ûö = 1 ,
öhe
ûS ≧ 0, † = 1, ⋯ , °, N = 1, ⋯ k, 4 = 1, ⋯ , l .
Thevariousefficiencymeasuresmentionedinprevioussectionsarecomputedasfollows:
Farrelloutput-orientedprofitefficiency
^o ! ; , ' ; ; ) ; , * ; = max g: * ; g' ; − ) ; ! ; ≦ * ; ' − ) ; !, !, ' ∈ ú ùêÑ , ! ≧ 0, ' ≧ 0 a,§,A,B
Revenueefficiency
<É ! ; , ' ; , * ; = max g: * ; g' ; ≦ * ; ', ! ; , ' ∈ ú ùêÑ , ' ≧ 0 .
a,§,B
Farrelloutputorientedtechnicalefficiency
ã; ! ; , ' ; = max g: ! ; , g' ∈ ú ùêÑ .
a,§
Outputorientedallocativeefficiency
éÉo (! ; , ' ; , * ; ) = <É(! ; , ' ; , * ; )/ã; ! ; , ' ; Revenueefficientallocativeefficiency
éÉêë ! ; , ' ; ; * ; , ) ; = ^o ! ; , ' ; ; * ; , ) ; /<É ! ; , ' ; , * ; WecomputethevaluesofthesemeasuresusingthedatasetinRay(2004).Therearepriceand
quantity data of five outputs and four inputs. Each observation is a bank in the US in 1996. For
illustration, although all 50 observations are used for computation, only the results of the first 20
banksarereportedinthefollowingTable. [InsertTable1here]
InTable1,thegeometricmeanoftheFarrelloutput-orientedprofitefficiencymeasure(^o )is2.85.
This means that, on average, the banks could increase their profit levels by about 185% of the
observedtotalrevenue.Onaverage,thevalueoftheoutput-orientedtechnicalefficiencymeasure(ão )
is1.02whichmeansmanybanksareoperatingonorclosetotheproductionfrontier.Althoughthe
banks are fairly allocatively inefficient (éÉo = 1.15), the value of the revenue efficient allocative
efficiency is the biggest (éÉêë = 2.42). On average, the sources of profit inefficiency come from
technical inefficiency (2.2%), output-oriented allocative inefficiency (15.1%), and revenue efficient
allocativeefficiency(142.4%).Thusmostprofitinefficiencycomesfromrevenueefficientallocative
inefficiency.Whenfirmsinthissamplearerevenueefficient,theycanstill,onaverage,raisetheprofit
levelby142%bychanginginputmixandinputquantities.Thiscomponentofallocativeinefficiency
afterthefirmisrevenueefficienthasnotbeendiscussedintheliteraturebefore.
9. Conclusion
Bysynthesizingvariousapproachesintheliterature,thispaperpresentsacohesiveframeworkof
profitefficiencymeasure.TheFarrelloutput-orientedmeasureofprofitefficiencyderivedfromthis
modelsatisfiessomenicepropertiesdiscussedbyotherresearchersintheliterature.Thismeasure
has intuitive interpretation and includes Farrell output-oriented measure of technical efficiency,
revenue efficiency as special cases. This is the first time in the literature that a profit efficiency
measureisexpressedasmultiplicativeelementscontainingFarrelltechnicalefficiencyandrevenue
efficiency. Notes
1.
Theframeworkfortheinputorientedversionisanalogousandsoisomittedforthesakeof
space.
2.
FäreandPrimont(1995,p.129)hasasimilarexpressionthat 1 D¶ x ¶ , q¶ =
sup θ: p θq¶ − wx ; ≦ π p, w , p, w ≥ 0 .Theirderivationrequiresconvextechnologyset
®
whereasthetechnologysetinourformulationisnotassumedbutinputsarefixed.Whenthe
technologysetisnotconvex,itispossiblethat sup θ: p θq¶ − wx ; ≦ π p, w , p, w ≥
®
0 > 1 D¶ x ¶ , q¶ .
3.
Theleft-andright-handsideof(18)canberewrittenas π p¶ , w ¶ + w ¶ x ; /p¶ q; ≧
1/D¶ x ; , q; where D¶ x ; , q; istheoutputdistancefunction.FäreandPrimont(1995,p.131)
havementionedthisinequalityfromdualityrelation.FäreandGrosskopf(2004,p.39)havestated
thesamethroughtherelationbetweentheprofitfunctionandthedirectionaloutputdistance
function.
4.
Aparicio,Pastor,andRay(2013)alsoproposedanothernormalization--usingtheaveragetotalcosts
ofallfirms.
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ℑ
Figure1.Theoutput-orientedFarrellprofitefficiencymeasure
Table1: Farrelloutput-orientedprofitefficiencyanditscomponents
Firm
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
f11
f12
f13
f14
f15
f16
f17
f18
f19
f20
Geometric
Mean
Technical efficiency
1.00
1.00
1.00
1.12
1.00
1.06
1.00
1.10
1.00
1.00
1.00
1.00
1.00
1.09
1.00
1.06
1.00
1.00
1.00
1.00
1.02
Allocative
efficiency
1.09
1.60
1.00
1.12
1.00
1.13
1.17
1.17
1.06
1.46
1.00
1.00
1.00
1.09
1.23
1.43
1.27
1.14
1.28
1.00
1.15
Revenue
efficient
allocative
efficiency
1.48 2.62 2.79 2.49 2.99 2.44 2.24 2.30 2.39 1.91 2.89 3.03 2.86 2.34 2.31 2.14 2.11 2.42 3.00 2.43 Profit efficiency
2.42
2.85
1.61
4.18
2.79
3.14
2.99
2.93
2.62
2.96
2.52
2.80
2.89
3.03
2.86
2.78
2.85
3.23
2.68
2.76
3.83
2.43