Who's afraid of aggregating money metrics?

20 Pages·2017·0.68 MB·English

TheoreticalEconomics13(2018),467–484 1555-7561/20180467
Who’safraidofaggregatingmoneymetrics?
KristofBosmans
DepartmentofEconomics,MaastrichtUniversity
KoenDecancq
HermanDeleeckCentreforSocialPolicy,UniversityofAntwerp
ErwinOoghe
DepartmentofEconomics,KULeuven
We provide an axiomatic justification to aggregate money metrics. The key ax-
iom requires the approval of richer-to-poorer transfers that preserve the over-
all efficiency of the distribution. This transfer principle—together with the ba-
sicaxiomsofanonymity,continuity,monotonicity,andaversionofwelfarism—
characterizes a standard social welfare function defined over money metric
utilities.
Keywords.Moneymetricutility,transferprinciple,efficiency.
JELclassification.D61,D63,D71,I31.
1. Introduction
Themoneymetricutilityofanindividualistheminimumincome,computedatrefer-
enceprices,thatsheneedstoreachabundlethatisatleastasgoodasheractualbun-
dle(McKenzie1957,Samuelson1974). Moneymetricutilityformsthebasisofapplied
welfareanalysis. Itis, forexample,standardpracticetoevaluatepolicyreformsbythe
changeinmoneymetricutilityusingasreferencepricesthepre-reformprices(theHick-
sianequivalentvariation)orpost-reformprices(theHicksiancompensatingvariation).1
However, several theoretical objections have been raised against the aggregation
of money metrics.2 The most powerful critique came from Blackorby and Donaldson
(1988). They show that the money metric utility function is, in general, not concave.
Thisimpliesthatastandard(quasiconcave)socialwelfarefunctiondefinedovermoney
metricsmayfailtoapprovetransfersfromrichertopoorerindividuals.
KristofBosmans:[email protected]
KoenDecancq:[email protected]
ErwinOoghe:[email protected]
WethankAntoineBommier,AndréDecoster,MarcFleurbaey,NicolasGravel,FrançoisManiquet,Paolo
Piacquadio, Erik Schokkaert, Alain Trannoy, and audiences in Bari (Fifth Meeting of ECINEQ), Gent
(GhentUniversity),Glasgow(UniversityofGlasgow),Leuven(KULeuven),Louvain-la-Neuve(Université
CatholiquedeLouvain),Lund(ThirteenthMeetingoftheSocietyforSocialChoiceandWelfare),Luxem-
bourg(SixthMeetingofECINEQ),Marseille(Aix-MarseilleUniversity),andZürich(ETHZürich)foruseful
comments.Theauthorsareresponsibleforremainingshortcomings.
1SeeSlesnick(1998)foranoverview.
2SeeFleurbaey(2009,pp.1052–1055)foradiscussion.
Copyright©2018TheAuthors. ThisisanopenaccessarticleunderthetermsoftheCreativeCommons
Attribution-NonCommercialLicense,availableathttp://econtheory.org.https://doi.org/10.3982/TE2825
468 Bosmans,Decancq,andOoghe TheoreticalEconomics13(2018)
Figure1. Atransferingoodsthatisleakyinmoneymetrics.
Figure1illustratestheproblem. Individuals1and2haveidenticalpreferencesover
thegoodsaandb. Abundleδistransferredfromthericherindividual2tothepoorer
individual1.3 Thedistancesbetweenthestraightlinesrepresentthechangesinmoney
metric utility (for some reference price vector). Clearly, the transfer is leaky: the gain
in money metric utility of the poorer individual is smaller than the loss of the richer
individual.Therefore,onlyasocialwelfarefunctionexhibitingasufficientlyhighdegree
ofinequalityaversionwouldapprovethedepictedtransfer. Moreover,bychangingthe
shapeoftheindifferencecurves,theleakcanbemadearbitrarilylarge.Thismeansthat
nosocialwelfarefunctionapprovesallricher-to-poorertransfers,withtheexceptionof
Rawlsian social welfare functions—such as maximin or leximin—that assign absolute
prioritytothepoorerofthetwoindividuals.
Thisobservationhasgivenrisetotwofar-reachingandopposingresponses. Black-
orbyandDonaldson(1988,p.129)concludenegatively,statingthat“socialwelfareanal-
ysisbasedonmoneymetricsisflawed.” FleurbaeyandManiquet(2011,p.21),bycon-
trast, conclude that “this observation (cid:3)(cid:3)(cid:3), instead of undermining the approach, can
serve to justify the maximin or the leximin as aggregation criteria.” Although the two
responsesarediametricallyopposed,theysharethepremisethattheapprovalofricher-
to-poorertransfersisanessentialrequirementforallsocialwelfarerankings.
We question this premise. We argue that not every richer-to-poorer transfer is an
unequivocalimprovement. Suchatransfer,whileimprovingequity,mayhavetheside
effect of worsening the overall efficiency of the distribution. To see this, note that the
transfer in Figure 1 transforms an efficient distribution—with equal marginal rates of
3ThetransferdepictedinFigure1yieldspost-transferbundlesthatareconvexcombinationsofthepre-
transferbundles.BlackorbyandDonaldson(1988)imposethisasarestrictiononricher-to-poorertransfers.
Thisrestrictionisnotessential,however,andwewillnotimposeitinouranalysis.
TheoreticalEconomics13(2018) Aggregatingmoneymetrics 469
substitution—intoaninefficientdistribution.Hence,thejudgmentofwhetherapartic-
ulartransferimprovessocialwelfaredependsonthepositiononetakeswithrespectto
theequity–efficiencytrade-off. Byinsistingthatalltransfersmustbeapproved,regard-
lessoftheassociatedefficiencylosses,BlackorbyandDonaldson(1988)andFleurbaey
and Maniquet (2011) implicitly take the extreme stance that gives absolute priority to
equity over efficiency. In this light, it is not surprising that they arrive at such strong
conclusions.
Weintroduceatransferprinciplethatrequirestheapprovalofonlythosetransfers
thatpreserveefficiency.Thisobviouslyrequiresawaytomeasureefficiency.Ratherthan
choosingamongthemanyefficiencymeasuresthathavebeenproposedintheliterature
(seeDiewert1985foranoverview), wefocusonwhattheyhaveincommon. Allthese
measuresquantifyefficiencybywhatcouldbedisposedofwithoutloweringanyindi-
vidual’sutility. Formally,theymeasurethedistancebetweentheactualsocietalbundle
(listingthetotalamountsofallgoods)andtheScitovksyboundary(collectingthemini-
mumsocietalbundlesthatcandelivertoeachindividualthesameutilitylevelasherac-
tualutilitylevel). Wedefineanefficiency-preservingtransferasatransferthatchanges
neitherthesocietalbundlenortheScitovksyboundary. Allefficiencymeasuresunani-
mouslyagreethatsuchatransferpreservesefficiency. Ourtransferprincipledemands
thatonlyefficiency-preservingtransfershavetobeapproved.4
We combine the efficiency-preserving transfer principle with the basic axioms of
anonymity, continuity, monotonicity, and a version of welfarism. Our main result has
twoimplications.First,acontinuous,strictlyincreasing,andSchur-concavesocialwel-
farefunctiondefinedovermoneymetricutilitiessatisfiesallfiveaxioms.5 Contraryto
theconclusionofBlackorbyandDonaldson(1988),theuseofmoneymetricsinsocial
welfareanalysiscanbejustified. Inparticular, sinceanystandardsocialwelfarefunc-
tionovermoneymetricsisadmissible,itisnotnecessary—contrarytotheconclusionof
FleurbaeyandManiquet(2011)—toadoptaRawlsiansocialwelfarefunction. Second,
and more strikingly, the opposite is also true: only if the social ranking can be repre-
sentedinthisparticularform,doesitsatisfyallaxioms. Insum,weshownotonlythat
onecan,butalsothatonemust,aggregatemoneymetrics.
Weproceedasfollows. Section2introducesnotationandthefiveaxioms. Section3
presentsanddiscussesthemainresult.Section4concludes.
2. Axioms
2.1 Preliminaries
Thesetofindividuals in societyis N ={1(cid:4)2(cid:4)(cid:3)(cid:3)(cid:3)(cid:4)n} with n≥2. Eachindividual i hasa
bundlexi inX =Rm+ withm≥2. Fortwobundlesxandy inX,wewritex≥y ifxk≥yk
4ChambersandHayashi(2012)alsomakeuseoftheScitovskyboundarytostudysocialwelfarerankings,
butforaverydifferentpurpose.Theirinterestliesininformationalparsimony.Theyrequirethesocialrank-
ingoftwodistributionstodependsolelyontheaggregatedatacontainedinthecorrespondingScitovsky
boundaries.Thisexcludesanyconcernforequity,whichispreciselyourfocus.
5Schur-concavityisaweakversionofconcavitythatisstandardintheliteratureoninequalitymeasure-
ment.See,e.g.,Dasguptaetal.(1973).
470 Bosmans,Decancq,andOoghe TheoreticalEconomics13(2018)
foreachk=1(cid:4)2(cid:4)(cid:3)(cid:3)(cid:3)(cid:4)m,wewritex>y ifx≥y andx(cid:3)=y,andwewritex(cid:4)y ifx >y
k k
foreachk=1(cid:4)2(cid:4)(cid:3)(cid:3)(cid:3)(cid:4)m. WedenotetheboundaryofthesetA⊆X by∂A.6 Thesumof
twosubsetsAandBofX isdefinedtobethesetofallsumsofanelementofAandan
elementofB.Thatis,A+B={z∈X |z=x+ywithxinAandyinB}.
Each individual i has a preference relation R over bundles in X. As usual, xR y
i i
meansthatbundlex isatleastasgoodasbundle y accordingtoindividual i, whereas
P and I denote the corresponding strict preference and indifference relations. We
i i
write x R A to denote that bundle x is at least as good as all bundles in set A ac-
i
cording to individual i. For a bundle-preference pair (x (cid:4)R ), the better-than set is
i i
B(x (cid:4)R )={y ∈X |y R x }. We sometimes use B as shorthand for B(x (cid:4)R ). Individ-
i i i i i i i
ualpreferencesbelongtoR,thesetofcomplete,transitive,continuous,monotone,and
convexpreferencerelations.7
AdistributionX=(x (cid:4)x (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)x )inXncontainsabundleforeachindividualinN.
1 2 n
Werefertothesumofallbundlesx +x +···+x asthesocietalbundle.Apreference
1 2 n
profileR=(R (cid:4)R (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)R )inRncontainsapreferencerelationforeachindividualinN.
1 2 n
Asocialrankingspecifiesforeachpreferenceprofileasocialpreferencerelationover
all distributions.8 Formally, a social ranking (cid:2) maps each preference profile R in Rn
intoacompleteandtransitivesocialpreferencerelation(cid:2) onXn. WeuseX (cid:2) Y to
R R
denotethatdistributionXisatleastasgoodasdistributionY intermsofsocialwelfare.
The relations (cid:7) and ∼ denote the corresponding strict social preference and social
R R
indifferencerelations.
2.2 Threebasicaxioms
We define three basic axioms. Anonymity requires that switching the bundles of two
individualswiththesamepreferencesdoesnotchangesocialwelfare.
Anonymity. For each preference profile R in Rn, for each distribution X in Xn, and
forallindividualsiandj inN suchthatR =R ,wehave(x (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)x (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)x (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)x )∼
i j 1 i j n R
(x (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)x (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)x (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)x ).
1 j i n
Continuityensuresthatsmallchangesindistributionsdonotleadtolargechanges
intheirsocialranking.
Continuity. ForeachpreferenceprofileRinRn,foralldistributionsX andY inXn,
andforeachsequenceofdistributions{Xk}k∈NthatconvergestoX,ifXk(cid:2)RY foreach
kinN,thenX(cid:2) Y,andifY (cid:2) XkforeachkinN,thenY (cid:2) X.
R R R
6Let(cid:9)x−y(cid:9)betheEuclideandistancebetweenbundlesxandy. TheboundaryofthesetA(relative
toX)isdefinedas∂A={x∈A|foreachε>0,thereisabundle yinX\Asuchthat(cid:9)x−y(cid:9)<ε}.
7ApreferencerelationRiiscompleteifxRiyoryRixforallxandyinX. ItistransitiveifxRiyand
yRizimplyxRizforallx,y,andzinX. Itiscontinuousifeachbetter-thansetandeachworse-thanset
isclosed. Itismonotoneifx(cid:4)yimpliesxPiy forallxandyinX. Itisconvexifeachbetter-thansetis
convex.
8Thus, we focus on social comparisons for a fixed population with a given preference profile. It is
straightforwardtoextendtheanalysistocomparisonsacrosssocietieswithdifferentpopulationsizesand
preferenceprofiles.
TheoreticalEconomics13(2018) Aggregatingmoneymetrics 471
Although continuity excludes leximin, the axiom is compatible with social prefer-
encerelationsarbitrarilyclosetoleximin.
Monotonicityimposesthatincreasingallamountsinsomeindividual’sbundleim-
provessocialwelfare.
Monotonicity. ForeachpreferenceprofileRinRn andforalldistributionsX andY
in Xn, if x ≥y foreachindividual i in N and x (cid:4)y forsomeindividual i in N, then
i i i i
X(cid:7) Y.
R
Individualpreferencesaremonotone.Therefore,monotonicityofthesocialranking
isimpliedbytheParetoprinciple(obtainedbyreplacing≥inthemonotonicityaxiom
byR and(cid:4)byP ).Notethat,conversely,thecombinationofallouraxiomsimpliesthe
i i
Paretoprinciple(seeLemma1below).
2.3 Referencesetwelfarism
Weimposereferencesetwelfarismasourfourthaxiom.9 Tothebestofourknowledge,
thisversionofwelfarismunderliesallexistingapproachesthatbasesocialwelfarerank-
ingsonordinalandnoncomparableindividualpreferences(seeFleurbaeyandBlanchet
2013,Chapter4,foranoverview).Forelementaryaxiomaticunderpinningsofreference
setwelfarism,seeCato(2016)andPiacquadio(2017).
Reference set welfarism prescribes two steps to rank distributions. The first step
usesalistofreferencesetstocardinalizeeachindividualsituation—abundle-preference
pair—intoautilityvalue.Thesecondstepusestheresultingvectorsofindividualutility
valuestorankdistributions. Wenowturntoadiscussionofreferencesetsandtheiruse
inreferencesetwelfarism.
Referencesetsaresetsofbundles.Consideralistofnestedreferencesets.Eachrefer-
encesetislabelledusingarealnumber,withlargersetsreceivinglargernumbers.These
realnumbersareusedtocardinalizebundle-preferencepairs. Eachbundle-preference
pairisassignedautilityvalueequaltothenumberofthereferencesetthatisjusttangent
totheindifferencecurvethroughthebundle.Figure2showsthreenestedreferencesets,
labelledbythenonnegativerealnumbersα,β,andγ,withα<β<γ. Forthedepicted
bundle-preferencepair(x (cid:4)R ),theassignedutilityvalueisβ.
i i
Westressthattheobtainedutilityvaluesaretreatedasinterpersonallycomparable.
Twoindividualswhoseindifferencecurvesaretangenttothesamereferencesetareas-
signedthesameutilityvalueand,hence,areregardedasequallywelloff. Thechoiceof
alistofreferencesetsdetermineshowinterpersonalcomparisonsaremadeandmust
thereforebebasedonvaluejudgments. Ouraxioms,andespeciallythetransferprinci-
ple,makethesevaluejudgmentsexplicitand,asTheorem1demonstrates,putconsid-
erablestructureontheshapeofthereferencesets.
WenowformalizethepropertiesofalistofreferencesetsS=(Sλ)λ∈R+. Alistcon-
tainsacompactreferencesetSλ⊆X foreachλinR+,startsfromtheorigin(S0={0}),
expandsinastrictlynestedway(λ<μimpliesS ⊂S and∂S ∩∂S =∅),andhasno
λ μ λ μ
9Fleurbaey(2009)referstoreferencesetwelfarismastheequivalenceapproach.
472 Bosmans,Decancq,andOoghe TheoreticalEconomics13(2018)
Figure2. Ageneralexample.
(cid:2)
gaps(theunionofallboundaries λ∈R+∂Sλ isequaltothesetofbundlesX). Because
individual preferencesaremonotone, we can, without loss of generality, requireaddi-
tionallythateachlistsatisfiesfreedisposal(if x belongsto S , theneachbundle y ≤x
λ
belongstoS ).LetS bethesetofalllistsofreferencesetsthatsatisfytheseproperties.
λ
For a given list of reference sets S in S, the utility value assigned to a bundle-
preference pair (x (cid:4)R ) is the greatest number λ for which x R S . Accordingly, the
i i i i λ
referencesetutilityfunctionu isdefinedas
S
u (x (cid:4)R )=max{λ|x R S } foreachx inX andeachR inR. (1)
S i i i i λ i i
ThepropertiesofthepreferencesinRandofthelistsofreferencesetsinS ensurethat
the reference set utility function u in (1) is well defined, unique, and continuous (in
S
bundles). Thisutilityfunctionrepresentsthepreferencerelation,i.e.,forallbundlesx
andy inX,wehaveu (x(cid:4)R )≥u (y(cid:4)R )ifandonlyifxR y.
S i S i i
Beforewestatetheaxiomreferencesetwelfarism, wedefinetwoprominentrefer-
encesetutilityfunctions.10 Thequantitymetricutilityfunction, illustratedintheleft-
handpanelofFigure3,isdefinedby(1)withS ={x∈X |x≤λr}forafixedreference
λ
bundle r (cid:4) 0.11 Quantity metric utilities were introduced by Samuelson (1977) and
Pazner and Schmeidler (1978) in welfare economics. The money metric utility func-
tion, which we denote by u , is illustrated in the right-hand panel of Figure 3. The
p (cid:3)
function up is defined by (1) with Sλ ={x∈X | mk=1pkxk ≤λ} for a fixed reference
price vector p(cid:4)0. Money metric utilities were introduced by McKenzie (1957) and
Samuelson (1974), and applied in welfare economics by Deaton (1980), King (1983),
RavallionandvandeWalle(1991),CreedyandHérault(2012),andChiapporiandMeghir
(2014),amongothers.
Reference set welfarism requires that welfare comparisons are based on reference
set utility valuesonly. Fora list of referencesets S, a distribution X, and a preference
10SeethediscussioninDeatonandMuellbauer(1980,pp.179–182).
11Weuse0todenoteavectorofzeroesofappropriatelength.
TheoreticalEconomics13(2018) Aggregatingmoneymetrics 473
Figure3. Quantitymetricandmoneymetricutility.
profile R, weabbreviatethevectorofreferencesetutilities (u (x (cid:4)R )(cid:4)u (x (cid:4)R )(cid:4)(cid:3)(cid:3)(cid:3)(cid:4)
S 1 1 S 2 2
u (x (cid:4)R ))byu (X(cid:4)R).
S n n S
Reference set welfarism. There exists a list of reference sets S in S and a binary
relation(cid:2)∗definedoverreferencesetutilityvectorsinRn+suchthat,foreachpreference
profileRinRnandforalldistributionsX andY inXn,wehave
X(cid:2) Y ifandonlyif u (X(cid:4)R)(cid:2)∗u (Y(cid:4)R),
R S S
whereu isthereferencesetutilityfunctiondefinedin(1).
S
Weconcludethissectionbycombiningreferencesetwelfarismwithanonymity,con-
tinuity,andmonotonicity. Asocialrankingsatisfiesthesefouraxiomsifandonlyifthe
social ranking can be represented by a continuous, strictly increasing, and symmetric
socialwelfarefunctiondefinedoverreferencesetutilities. Westatethisstraightforward
resultwithoutproof.
Lemma1. Asocialranking(cid:2)satisfiesanonymity,continuity,monotonicity,andreference
set welfarism if and only if there exists a list of reference sets S in S and a continuous,
strictlyincreasing,andsymmetricsocialwelfarefunctionW :Rn+→Rsuchthat,foreach
preferenceprofileRinRnandforalldistributionsX andY inXn,wehave
(cid:4) (cid:5) (cid:4) (cid:5)
X(cid:2) Y ifandonlyif W u (X(cid:4)R) ≥W u (Y(cid:4)R) ,
R S S
whereu isthereferencesetutilityfunctiondefinedin(1).
S
ThefouraxiomsinLemma1leaveopenthequestionofwhichlistofreferencesets
touse.12 Ourfinalaxiom,theefficiency-preservingtransferprinciple,willdeterminethe
shapeofthereferencesets.
12ThesocialrankinginLemma1satisfiestheParetoprinciple. Indeed,thesocialwelfarefunctionW is
strictlyincreasingandtheutilityfunctionuSisarepresentationofindividualpreferences.
474 Bosmans,Decancq,andOoghe TheoreticalEconomics13(2018)
2.4 Anefficiency-preservingtransferprinciple
Underlying the conclusions of Blackorby and Donaldson (1988) and Fleurbaey and
Maniquet(2011),whichwediscussedintheIntroduction,istheiracceptanceofastrong
transferprinciplethatwedefineasfollows.13
Societal-bundle-preservingtransferprinciple. ForeachpreferenceprofileRin
Rn, for all distributions X and Y in Xn, and for all individuals i and j in N such that
R =R , if X is obtained from Y by a richer-to-poorer transfer from j to i (y (cid:15)x (cid:15)
i j i i
x (cid:15)y andx =y fork(cid:3)=i(cid:4)j)thatpreservesthesocietalbundle(x +x =y +y ),then
j j k k i j i j
X(cid:2) Y.
R
The transfer in this principle preserves the societal bundle, but may considerably
worsentheefficiencyofhowthissocietalbundleisdistributed. Thetransferdepicted
inFigure1illustratesthispoint: itpreservesthesocietalbundle, buttakesusfroman
efficient distribution (where the marginal rates of substitution are equal) to an ineffi-
cient distribution (where they are unequal). We introduce a weaker transfer principle
thatrequiresonlytheapprovalofthosetransfersthatpreserveefficiency.
We develop a concept of efficiency preservation based on the two building blocks
of the efficiency measurement literature: the actual societal bundle and the Scitovsky
boundary. TheScitovskyboundarycollectstheminimumsocietalbundlesabletode-
livertoeachindividualtheutilitylevelsheobtainsintheactualdistribution(Scitovsky
1942). Formally,foradistributionX andapreferenceprofileR,theScitovskysetisde-
finedasthesumofthebetter-thansetsB +B +···+B ,andtheScitovskyboundaryis
1 2 n
∂(B +B +···+B ). AdistributionisefficientonlyifitssocietalbundleliesontheSc-
1 2 n
itovskyboundary: forthegivensocietalresources,noindividualcanbemadebetteroff
withoutmakinganotherindividualworseoff.Commontoallefficiencymeasuresinthe
literatureisthattheyquantifyinefficiencyasthedistancebetweenthesocietalbundle
andtheScitovskyboundary. Whatdistinguishestheseefficiencymeasuresishowthey
definedistance.Diewert(1985)providesageneraloverview.
Figure4givesanexampleforthecaseoftwoindividualsandtwogoods.Bothpanels
ofthefigureshowdistribution X =(x (cid:4)x ) andthecorrespondingbetter-thansets B
1 2 1
and B . Thesocietalbundleis x +x andtheScitovsky boundaryis theboundaryof
2 1 2
theScitovskysetB +B . NotethatdistributionX isnotefficient: thesocietalbundle
1 2
does not lie on the Scitovsky boundary, but rather in the interior of the Scitovsky set.
Thetwopanelsillustratethetwodominantapproachesintheefficiencymeasurement
literature,referredtobyDiewert(1985)asthequantity-orientedandprice-orientedap-
proaches.Theleft-handpanelillustratesthequantity-orientedmeasuresofAllais(1943)
andDebreu(1951). Allais(1943)measuresinefficiencyasAC/BC,therelativedistance
between the societal bundle and the efficient bundle A that contains less only of a
numéraire good (here good a). Debreu (1951) measures inefficiency as DC/OC, the
13This transfer principle only considers transfers among individuals with the same preferences.
FleurbaeyandTrannoy(2003)showthatwithoutthisrestriction,thetransferprincipledirectlyclasheswith
theParetoprinciple.SeeWeymark(2017)foranoverviewofimpossibilityresultsinthisvein.
TheoreticalEconomics13(2018) Aggregatingmoneymetrics 475
Figure4. Quantity-orientedandprice-orientedefficiencymeasures.
relative distance between the societal bundle and the efficient bundle D that is pro-
portional to the societal bundle. The right-hand panel illustrates the price-oriented
approach, proposedbyHicks(1942)andBoiteux (1951). Forthegivenreferenceprice
vectorp=(p (cid:4)p ),inefficiencyequalsp ×EF,orthedistance,expressedinexpendi-
a b a
tureterms,betweenthesocietalbundleandthecheapestbundleHB ontheScitovsky
boundary.
Our efficiency-preserving transfer principle requires the approval of each richer-
to-poorer transfer that keeps the societal bundle and the Scitovsky boundary fixed.14
These restrictions on the transfer guarantee, as shown above, that all efficiency mea-
suresunanimouslyagreethatthetransferpreservesefficiency.
Efficiency-preservingtransferprinciple. ForeachpreferenceprofileRinRn,for
alldistributions X and Y in Xn, andforallindividuals i and j inN suchthatR =R ,
i j
ifX isobtainedfromY byaricher-to-poorertransferfromj toi(y (cid:15)x (cid:15)x (cid:15)y and
i i j j
x =y fork(cid:3)=i(cid:4)j)thatpreservesefficiency(x +x =y +y andB(x (cid:4)R )+B(x (cid:4)R )=
k k i j i j i i j j
B(y(cid:4)R )+B(y (cid:4)R )),thenX(cid:2) Y.
i i j j R
Both the efficiency-preserving and the societal-bundle-preserving transfer princi-
plesgeneralizetheunidimensionalPigou–Daltontransferprinciple. ThePigou–Dalton
transferprinciplerequirestheapprovalofricher-to-poorertransfersinincome(thesin-
glegood)thatpreservetotalincome(thesocietalbundle). Becausepreservingtheeffi-
ciency of the distribution reduces to preserving the societal bundle in the unidimen-
sional setting, the Pigou–Dalton transfer principle is also efficiency-preserving. We
claimthattheefficiency-preservingtransferprinciplecapturesanaspectoftheunidi-
mensionalPigou–Daltontransferprinciplethatthesocietal-bundle-preservingtransfer
14Thisaxiomdoesnotcoverallcaseswherethedistributionsbeforeandafterthetransferarebotheffi-
cient(equalmarginalratesofsubstitution). Indeed,insomesuchcases,theScitovskyboundariesdonot
coincide,butratherintersectatthesocietalbundle. Asiseasytoshow,astrongeraxiomthatwouldalso
coverthesecasesclasheswiththeotheraxiomsinTheorem1.
476 Bosmans,Decancq,andOoghe TheoreticalEconomics13(2018)
principledoesnot.Theformertwotransferprinciplesaresilentontheequity–efficiency
trade-offbecausetheconsideredtransfersimproveequitywithoutchangingefficiency.
Thesocietal-bundle-preservingtransferprinciple,incontrast,doestakeastanceregard-
ingtheequity–efficiencytrade-offand,aswehaveargued,anextremestance.Transfers
that only preserve the societal bundle improve equity, but may cause arbitrarily large
efficiencylosses. Ourefficiency-preservingtransferprincipledoesnotexcludesuchan
extremestance,butis,moreover,compatiblewithmoremoderateethicalpositions.
3. Result
RecallthatLemma1leavesopenthechoiceofthereferencesetutilityfunction. Theo-
rem1singlesoutthemoneymetricutilityfunctionbyaddingourefficiency-preserving
transferprincipletothefouraxiomsinLemma1. Anaturaladditionalconsequenceis
that the social welfare function must be Schur-concave.15 The proof of Theorem 1 is
givenintheAppendix.
Theorem 1. Asocialranking(cid:2)satisfiesanonymity,continuity,monotonicity,reference
setwelfarism,andtheefficiency-preservingtransferprincipleifandonlyifthereexistsa
vectorpinRm++ andacontinuous,strictlyincreasing,andSchur-concavesocialwelfare
functionW :Rn+→Rsuchthat,foreachpreferenceprofileRinRn andforalldistribu-
tionsX andY inXn,wehave
(cid:4) (cid:5) (cid:4) (cid:5)
X(cid:2) Y ifandonlyif W u (X(cid:4)R) ≥W u (Y(cid:4)R) ,
R p p
whereu isthemoneymetricutilityfunctionusingpasthereferencepricevector.
p
Theorem 1 gives necessary and sufficient conditions for a social ranking to satisfy
the five axioms. The sufficiency part states that any standard social welfare function
definedovermoneymetricutilitiessatisfiestheaxioms. Morestrikingly, thenecessity
partstatesthattheaxiomsaresatisfiedbythisparticularsocialrankingexclusively. We
discussinturnthesufficiencyandnecessitypartsofthetheorem.
Tounderstandthesufficiencypart,notethatifatransferpreservesefficiency,then
it also preserves the sum of money metric utilities. We show this in the first part of
theproof. ItfollowsimmediatelythatanySchur-concavewelfarefunctiondefinedover
moneymetricutilitieswillapproveefficiency-preservingtransfers.
The sufficiency part stands in sharp contrast to Blackorby and Donaldson (1988).
They show that a standard social welfare function defined over money metric utili-
tiesfailstosatisfythestronger—andintheirviewessential—societal-bundle-preserving
transfer principle.16 Theorem 1 demonstrates that aggregating money metrics is per-

fectlyjustifiedifoneaddsthesensiblerequirementofpreservingefficiencywhiletrans-
ferringgoods.ThesufficiencypartofthetheoremfurthermorecontrastswithFleurbaey
15AfunctionW :Rn+→RisSchur-concaveifW(Qx)≥W(x)foreachvectorxinRn+ andforeachbis-
tochasticmatrixQ. AmatrixQ inRn+×n isbistochasticifeachrowsumandeachcolumnsumisequal
to1.
16Itiseasytoshowthat,moregenerally,nosocialrankingsatisfiesthesocietal-bundle-preservingtrans-
ferprincipletogetherwithanonymity,continuity,monotonicity,andreferencesetwelfarism.