Giving gifts to groups : how altruism depends on the size on the number of recipients

19 Pages·2007·0.74 MB·English

JournalofPublicEconomics91(2007)1731–1749
www.elsevier.com/locate/econbase
Giving gifts to groups: How altruism depends on the
☆
number of recipients
James Andreoni
UniversityofCalifornia,SanDiego
Received27January2007;receivedinrevisedform22June2007;accepted28June2007
Availableonline10July2007
Abstract
Whenasinglegiftgoestoagroupofrecipients,howdoesgivingdependonthesizeofthegroup?This
questionisimportantforunderstandingcharitablegivingandfund-raising,publicgoodsprovision,family
altruism,andmore.Ifwethinkofthegiftasgivingupadollartocreateasocialsurplus,thenwewanttoknow
howthenumberofrecipientsofthatsurplusaffectsitsvaluetothegiver.Inotherwords,howcongestibleis
altruism?Thispaperbuildsatheoreticalframeworkforthisquestionandbeginstoansweritwithacontrolled
experiment.Thefindingisthatformostsubjectsaltruismiscongestible.Fortheaveragesubject,agiftthat
resultsinonepersonreceivingxisequivalenttooneinwhichnpeoplereceivex/n0.68each.
©2007Elsevier B.V. All rightsreserved.
Keywords:Altruism;Publicgoods;Charitablegiving;Revealedpreference
1. Introduction
Howmuch wouldyou give upif each dollar youforfeitedbenefited anotherperson by$10?
Whatifinsteadyourdollargave$5eachtotwoothers,or$2eachtofiveothers,or$1eachtoten
others?Ineachcasetheprivatecostsandsocialbenefitsarethesame,butwouldweviewtheseas
thesamesituation?Similarly,whatifeachdollaryougaveupbenefitedoneotherby$1each,or
benefited2othersby$1each,or3others,andsoon?Ifthesameprivatecostgeneratesbenefits
for more people,do you spend more or less on giving?
Thesequestionsareimportantforunderstandinggiftgivingwhentherearegroupsofrecipients.
Considertwopublicgoods,oneatalocallevelandoneatanationallevel.Supposethelocalgood
☆ IamextremelygratefultoJosephGuseforhisoutstandingresearchassistance,andtoBillHarbaugh,MartinSefton,
and Megan Ritz for their helpful comments. I also acknowledge the financial support from the National Science
FoundationandtheRussellSageFoundation.
E-mailaddress:[email protected].
0047-2727/$-seefrontmatter©2007ElsevierB.V.Allrightsreserved.
doi:10.1016/j.jpubeco.2007.06.002
1732 J.Andreoni/JournalofPublicEconomics91(2007)1731–1749
provideshighvaluetoafewpeople,thenationalgoodprovideslowvaluetomany,andthatthetotal
valueoftheseisidentical.Whichwouldanaltruistprefer?Anotherexamplewouldbecharitable
solicitations from both the local chapter and the national organization of a charity. A particular
donationmightaffectmorepeopleatahigherlevel.Woulddonorsgivemoreorlessasaresult?
Supposeinsteadthegiftistoafoodbank.Thereisafiniteamountoffood,andmorerecipients
means each gets less. Do you give more or less as the number of recipients increases? Another
exampleisarestaurantstaffthatpoolsitstips.Agiventipwillnowonlygopartlytoyourwaiter,
and partly to the others. Does that make people more or less generous? Or consider a parent
thinking of buying a new television for the family. Will she spend more, all else equal, if the
householdhas4 members rather than 3?
Thesequestionsarealsorelevantforexperimentalresearch.Ifagenerousactinanexperiment,
suchasgivingtoapublicgoodorstrikingafairbargain,issharedwithmorepeopleatthesame
cost, do people value it more orless?
Economictheorycangiveussomeguidanceinansweringthesequestions.Considerthefirst
counterfactualabove:Thesamesocialsurplusisdividedamongmoreindividuals.Ifpeoplecare
onlyaboutthetotalamountreceived,thenincreasingthenumberofothers,n,willhavenoeffect.
If, at the opposite extreme, they care about the average amount received, then increasing n
increases the effective price of giving, which means they may reduce giving.
Considerthesecondcounterfactual:Increasingthegroupsizeexpandsthesurplus,butkeeps
theaveragebenefitsthesame.Increasingnheremeansthatanydollargivenupwillgeneratemore
benefits, indicating people should be more generous as n increases—an income effect. On the
otherhand,thesamesocialbenefitcanbeattainedatlowercost.Thismeanspeoplemaygiveless
away—asubstitutioneffect.Whicheffectdominateswilldependonpreferencesofindividuals.
Aunifyingapproachtothisistoframetheissueintermsofcongestion,asintheliteratureon
congestiblepublicgoods.Thecongestionhere,however,isintheheartsofthealtruists,ratherthan
inthetechnologyofthepublicgood.Thinkfirstofthetotalsocialsurplusgeneratedforothersby
the dollars forfeited. Call this total surplus Π . Then let π =Π /n be the average surplus for
o o o
others.Iftheindividualgiverviewsthegift,g,asbeingthetotalsocialsurplus,independentofn,
thenthisisagoodthatisnotcongestedatallandg =Π =nπ .Ontheotherextreme,ifthegiver
0 o o
thinksofthegiftastheaveragesocialsurplus,thenthegoodiscompletelycongested,g =Π /
1 o
n=π . As with congestible public goods, actual congestion may be better described as a
o
combinationofthesetwo,g=gbg1−b=nbπ ,withb∈[0,1].Thispaperwillbeaboutidentifyingb.
0 1 o
There have been no field studies and only a few experimental studies of altruism and the
effectsofgroupsize.Comparinggroupsofsizetwoandfourinpublicgoodsprovision,Goeree,
Holt and Laury (2002) found no clear effect. Isaac and Walker (1988) and Isaac, Walker and
Williams (1994) compared public goods games with groups of size 4, 10, 40 and 100, with
varying marginal returns to giving. They found no effect of group size when the price was low
(0.33),butwhenthepriceofgivingwashigh(2.33)groupsof4werelessgenerousthangroupsof
10—the income effect dominates the substitution effect—but the effect largely vanished when
comparinggroupsof40and100.TheexperimentsofKahnemanandKnetsch(1992)findsimilar
effects.Whensubjects were asked theirwillingnesstopayfor alocal publicgood, followedby
askingabouta regional ornational public good,giving rose.1
1 Othersstudygroupsizeandgivingtodiscreteorsteplevelpublicgoods.Thesearequitedifferentfromwhatisin
mindhere,astheseexperimentsusuallyrequireaminimumtotalcontributionornumberofcontributorsinordertomove
theprovisionofapublicgoodfromzerotoasinglepositivevalue.SeeBagnoliandMcKee(1991),CrosonandMarks
(1998),forinstance.
J.Andreoni/JournalofPublicEconomics91(2007)1731–1749 1733
Although themain interest instudyingcongestionin giving is, obviously, real world giving,
webeginthestudyofitherewithacontrolledlaboratoryexperiment.Theeffectswefindcould
help shape the field studies necessary todemonstrate the validityof thelaboratory findings.
Theexperimentreported inthispaper offerssubjectsaseries of24decisions whichvarythe
number of recipients and the “price” of generosity. The experiment builds on earlier work by
AndreoniandMiller(2002)whousedsimilartechniquestouncoverpreferencesforaltruism,but
who kept thenumber ofrecipients fixed at one.
We will use three approaches to uncover preferences. First is a non-parametric revealed pref-
erence approach that holds n fixed. Next is a semi-parametric approach. This begins by hypoth-
esizingaparametric“giving”functionthatentersutility,andthenbyusingnon-parametricanalysis
ofutilitytofindthebestparametersonthegivingfunctions.Thatis,wefindtheparameterinthe
givingfunctionthatresultsinnoviolations(orthesmallestviolations)ofrevealedpreference.Third
isafullyparametricapproachtotheestimationofutilityfunctions.Witheachapproachwedescribe
andcataloguethevarietyofpreferencesexhibitedbyoursubjects.
The findings are, first, that there is a remarkable degree of consistency within subjects, and
surprisingly few violations of the axioms of revealed preference. Second, while there are many
examplesofindividualswhobehaveasthoughb=0orb=1,themajorityofsubjectsarebestdescribed
byabbetween0 and1.Combining the entiresample andestimatingasinglerepresentative utility
function, we estimate the best aggregate b to be b=0.68. This means that, whether looking at the
problemnon-parametrically,semi-parametricallyorfullyparametrically,orwhetheritisindividualor
aggregateanalysis,wemustconcludethataltruismisacongestiblegood,butthecongestionispartial—
doublingthenumberofrecipientsincreasesbutdoesnotdoublethevalueofaltruismtothegiver.
The paper is organized as follows. The next section will present the formal framework of
congestiblealtruism.Section3willpresenttheexperimentaldesign.Wepresentthegeneralresultsin
Section 4, followed by non-parametric and semi-parametric analysis of the data using revealed
preferencetechniquesinSection5.Sections6and7turntoparametricanalysis,presentingestimates
ofutilityforeachindividualsubjectaswellasforarepresentativealtruist.Section8concludes.
2. Model
Letπ bethemarginalpayofftoindividualiandletπ betheadditionalpayofftoeachofthe
i o
others as a consequence of i's choice. Again, let n be the number of others affected. Then the
maintained hypothesis is that individuals have utility u =u(π, π ; n), where preferences are
i i i o
convexinπ andπ ,and(weakly)increasinginn.Itisalsonaturaltoexpectthatuisincreasingin
i o
both π and π , as found byAndreoni and Miller (2002).2
i o
Supposeutilityisseparableon(π ,n).Thenthereissomefunctiong(π ,n)suchthatutilitycanbe
o o
written as u(π, π , n)=u(π, g(π , n)). We can think of g abstractly as an individual's “giving”
i i o i i o
function, that is, the particular aggregation of π that this person cares about.3 The question for
o
2 SeealsoFisman,KarivandMarkovits(inpress).Withdownwardslopingbudgetsitisimpossibletoidentifypreferences
whereπ maybea“bad”atthemargin.However,offeringsomeupwardslopingbudgets,AndreoniandMiller(2002)foundthat
o
therewasasignificantminorityofsubjectsforwhomtheindifferencecurvemayactually“bendback”forsomeextremeinequality.
Andreoni,CastilloandPetrie(2003)confirmthisforstrategicgamesaswell.Sinceweareconcernedwithvoluntaryaltruismhere,
wedonotconcernourselveswithsuch“jealous”or“resentful”preferences,butthisisaninterestingareaforfutureresearch.
3 Themainrestrictionimposedbytheassumptionofseparabilityisthathowpeopleaggregategivingisindependentof
theirownlevelofconsumption,thatis,themarginalrateofsubstitutionbetweenπ andnisindependentofx.Inour
o
experiments,asubject'sincomevariesonlyslightly,sothisisunlikelytobeacostlyassumption.Whetheritisavalid
assumptionforfielddataisaninterestingquestion.
1734 J.Andreoni/JournalofPublicEconomics91(2007)1731–1749
researchistoidentifytheseaggregations,g(π ,n),aswellaslearnhowutilitychangesasthevalueof
o
thisaggregationchanges.4Thinkingfirstaboutg(π ,n),atoneextremewecouldhaveg(π ,n)=π ,
o o o
thatis,peoplecareonlyabouttheaveragereceived.Attheotherextremewecouldhaveg(π ,n)=nπ ,
o o
sopeoplecareaboutthetotalamountreceived.Whataboutsomethingintermediate?Intheliterature
oncongestioninpublicgoods,thereisahistoryofestimating“crowding”withaparameter,b,anda
function,g(π ,n)=nbπ .Thusforb=0orb=1wehavetheextremecases,whileforb∈[0,1]we
o o
allowintermediatecases.5
Considerapersonfacingalinearbudgetoverpayoffs.Forsimplicitywewillalwaystreatπ as
i
thenumeraire, anduse p asthe price of π . Thus, write the optimization problemas
o
maxuðp;nbp Þ
i i o
p;p
s:ti:po þpp ¼m
i o
Itisconvenienttothinkofthepersonaschoosingg=g(π ,n)ratherthanπ .Thus,substitute
o o
π =g/nb andlet the price be p/nb. Thenwe canrewrite this optimization as
o
maxuðp;gÞ
pi;g ip i ð1Þ
s:t:p þ g ¼m
i nb
Letp′=p/nb.SolvingEq.(1)wegetademandfunctiong(p′,m).Asusual,assume∂g/∂p′b0.
Thenwecanmeasuretotalgivingaspπ =p′g.Howdoesthischangewithn?Takingthederivative
o
wesee
(cid:1) (cid:3)
App Ag ApV
0 ¼ pV þg a0:
An ApV An
We can now see the income and substitution effects of n on giving, π . As n increases, the
o
priceofgfallsbutthetotalamountofbenefitrises,andtheeffectontotalspendingisambiguous.
With this as our setup, we use the axioms of revealed preference to identify whether, for a
given b, preferences are consistent with maximizing behavior. We do this by applying the
Generalize Axiom, GARP. We say a bundle (π , g ) is directly revealed preferred to a bundle
ia a
(π , g ) if π +p′ g ≥π +p′ g . The relation is strict if the inequality is strict. We then say a
ib b ia a a ib a b
bundle(π ,g )isrevealedpreferredtoabundle(π ,g )ifthereisachainofdirectlyrevealed
ia a ib b
preferred relations that links the two. That is, the revealed preferred relation is the transitive
closureof thedirectly revealedpreferred relation. Then define GARP this way (Varian,1982):
Definition 1. Generalized Axiom of Revealed Preference (GARP): If (π , g ) is revealed
ia a
preferredto (π ,g ) then (π , g ) isnever strictly directly revealed preferredto (π ,g ).
ib b ib b ia a
4 Notethatwearegeneralizingthemodelof“warm-glowgiving”(Andreoni,1989,1990)tomeanthatutilityofgiving
dependsonthenumberofrecipients.Withnconstant(orasweseelater,nlarge)thedistinctionisirrelevant.Theanalysis
could be generalized to a strategic environment where altruism also depends on the contributions of others, but the
fundamentalinsightsofthismodelwouldbeunaffected.
5 Suppose,forinstance,thatg(π ,n)combinesthetwoextremenotionsinaCobb–Douglasproductionfunctionfor
o
giving,sog(π,n)=(nπ)bπ1−b=nbπ .Onthegenesisoftheliteratureoncongestiblepublicgoods,seeBergstromand
o o o o
Goodman(1973),forinstance.Forageneraltreatmentofpublicgoodsandgroupsize,seeBergstrom,BlumeandVarian
(1986)andAndreoni(1988a).
J.Andreoni/JournalofPublicEconomics91(2007)1731–1749 1735
AsVarian(1982)shows,ifthedataisconsistentwithGARP,thenthereexistsautilityfunction
that could have generated thedata. CheckingGARP will beattheheart ofour analysis.
2.1. Interpreting the price
How do we think about price p in this setting? Recall, we constructed π as the value per
o
recipient.Thus,ifgivingistoapurepublicgood,theaveragevaluecansimplybemeasuredasthe
cost to thegiver.In this case, absent any subsidies orother distortions, we can assume p=1.
Whatifinsteadthegiftisapureprivategood,asintheexampleofpoolingoftips?Ifπ isthe
o
tip received byeach of thewaitstaff,then total spending on tips is nπ .This means p=n.
o
Itispossible,aswell,thatthegiftprovidedisitselfacongestiblepublicgood.Perhapsthegift
providesforaseminarseriesintheeconomicsdepartment.Ifseminarsarecrowdedthepleasure
oftheexperienceisdecreasedforall.Followingtheliteratureoncongestioninpublicgoods,we
wouldsaythetotalsurplusisΠ=ncπ ,where0≤c≤1.Inordertogenerateagivenπ inaverage
o o
value, therefore, a person will need to spend ncπ dollars. Thus, p=nc, meaning that techno-
o
logicalcongestionhas theeffect ofraisingthe price ofaltruism.
Intheexperimentsbelowweenforcegivingasapurepublicgood,butweartificiallyvarythe
priceinordertoallowustoidentifypreferences.Whenapplyingtheseresultstotherealworld,
however, we need tobe thoughtful about what isdeterminingthe price.
3. Experimental design
TherestatementoftheprobleminEq.(1)indicatesthatwewillneedvariationinn,m,pand
p/n in order to identify both the utility function and the parameter b. With this in mind, we
designed a task with 24 decisions. The decisions were presented to the subjects on a computer
screen like that shown in Fig. 1. Subjects filled in the blanks in sentences like this: “Divide 60
tokens: Hold _ _ _ @ 20 cents, and Pass _ _ _ @ 10 cents, to each of 2 other people”. The
computerautomaticallymadesure thatchoiceswereonthebudgetconstraint. Italsocalculated
Fig.1.Sampledecisionscreen.
1736 J.Andreoni/JournalofPublicEconomics91(2007)1731–1749
theearningsforthesubjectsinthestudy,andreportedtheminthecolumnsontheright,asseenin
Fig.1.Itcalculatedearningsintwoways.Firstisaverageearnings(“Eachofnothersearns”)and
second istotal earnings(“In Total the nothers earn”), ascan be seenin thefigure.
Thedecisionswerepresentedonpagesorganizedbyn.Subjects,however,werefreetotabbackand
forthacrosspagesandcouldfillinblanksinanyorder.Theycouldnotsubmittheirdecisionsuntilall
blankswerefilled.Tocheckforpossibleordereffectswepresentedthegameintwoorders,forward,as
showninthefigure,andreverse,wherethetabsandthedecisionsoneachtabwereallinreverseorder.
Foranychoice,wecancalculatepandmasfollows.ThetaskistodivideMtokenswithahold
value of H cents and a pass value of P cents. Let X be the amount held and Y be the amount
passed,adheringtothetokenconstraintX+Y=M.Thenπ =HXandπ =PY.Substitutetheseinto
i o
thetoken constraint andrearrange toget
H
p þ p ¼HM:
i P o
Finally,substitutingg(π ,n)=nbπ ,anddefining p=H/Pandm=HM,werewritethisasthe
o o
budget constraint
p
p þ g ¼m;
i nb
asinEq.(1)above.Thus,mhastheinterpretationoftheearningsbyplayeriifhepasseszeroand
keeps M,and p isthe cost toperson iof increasing thepayoffof each other person.
ThefullsetofexperimentalparametersisshowninTable1.Therearesixpossiblevaluesforthe
numberofothers,n∈{1,2,3,4,5,9}.Likewiseptakeson10possiblevalues,andp/ntakeson
12.Carewastakentoassurethattherewouldbeseveralsetsofbudgetsthatwouldbeidenticalonp
andmbutdifferonp/n(budgets4and12,5and9,10and21),andthatotherswouldbethesame
onp/nandmbutdifferonp(budgets6and12,and11and22).Thiswillhelpwithidentifyingb
apartfromutilityparameters.Wealsoassuredthatallbudgetscrossedwithinagivenn.
Whilesubjectsmade24choices,theywereonlypaidforonechoiceselectedatrandom.6Each
choice was equally likely tobe chosen. Subjects were told this prior totheir decisions.
We were also interested in the possible effect of reciprocal altruism. For instance, is there a
differenceingivingwhenitcanbereciprocated,asinapublicgood,andgivingwhenitisexplicitly
non-reciprocal? We explored this with two matching conditions. Under reciprocal matching
peopleareexplicitlyputintogroupsandthesamedecisioniscarriedoutforeachmemberofthe
group.Wetellsubjectsthatthosewhoreceivetheirpassamountsarealsoinapositiontopassto
them.7Weweightedtheprobabilitiesofchoosingdifferentgroupssothateachdecisionisequally
likely to be chosen. We compare this to circular matching, which only allows for unilateral
altruism.8Afterchoicesweremade,thecomputerprogramrandomlyplacedsubjectsina“circle”.
6 HeyandLee(2005)presentevidencetosuggestthatrandomlyselectingoneofasubject'schoicesissuccessfulin
gettingthesubjecttotreatthemasseparateindependentevents,ratherthanasaportfolioofrandomevents.
7 Forthereciprocalmatchingtheinstructionsread:“Youwillalsoberandomlyselectedtobearecipientofothers'Pass
allocations.TherecipientsofyourPassportionwillbethosewhowillbepassingtoyou.Thatis,anypassedtokensyou
receive will be from the same other participants who may receive tokens passed by you. Moreover, these other
participantswillhavethesamedecisionchosenbythecomputerthatyoudo.Forexample,supposethecomputerselects
thisdecisionforyouand4others:Divide80tokens:Hold____@20cents,andPass____@30centstoeachof4
otherpeople.Thenyouwillbeinagroupof5people.Eachofthe5hasthesamedecisionchosen,andeachispotentially
passingtokenstothesameothermembersinthatgroup”.
8 Forthecircularmatchingtheinstructionsread:“Youwillalsoberandomlyselectedtobearecipientofothers'Pass
allocations.TherecipientsofyourPassportionwillnotbethesameparticipantswhowillbepassingtoyou.Thatis,you
willnotreceiveanypassedtokensfromthoseotherparticipantswhomayreceivetokenspassedbyyou”.
J.Andreoni/JournalofPublicEconomics91(2007)1731–1749 1737
Table1
Experimentalparametersandaveragepassed
Experimentalparametersa Averagepassedb
Budget n M Holdvalue Passvalue p/n p m Circulartokens Reciprocaltokens
1 1 40 10 40 0.25 0.25 400 14.6 15.9
2 48 10 30 0.33 0.33 480 17.3 18.6
3 60 10 20 0.50 0.50 600 17.1 22.9
4 90 10 10 1 1 900 19.9 24.2
5 60 20 10 2 2 1200 12.1 14.3
6 2 45 20 30 0.33 0.67 900 15.1 18.1
7 81 10 20 0.25 0.50 810 27.7 32.2
8 100 10 10 0.50 1 1000 27.5 32.2
9 60 20 10 1 2 1200 12.8 16.7
10 40 40 10 2 4 1600 7.0 9.3
11 3 40 10 30 0.11 0.33 400 15.8 16.6
12 90 10 10 0.33 1 900 29.9 31.4
13 40 30 20 0.50 1.5 1200 10.9 12.1
14 52 30 10 1 3 1560 10.8 12.6
15 4 80 10 10 4 0.25 800 28.7 28.0
16 72 20 10 2 0.50 1440 22.3 23.5
17 42 10 20 0.5 0.125 420 17.3 17.9
18 50 40 10 4 1 2000 11.4 12.3
19 5 60 10 10 0.20 1 600 20.7 21.3
20 51 20 10 0.40 2 1020 15.5 17.5
21 40 40 10 0.80 4 1600 10.3 11.2
22 9 40 10 10 0.11 1 400 16.6 13.5
23 30 20 10 0.22 2 600 10.6 10.0
24 20 40 10 0.44 4 800 6.3 6.6
a Misreportedintokens.Holdvalue,Passvalueandmarereportedincents.
b Averageacrossallsubjects,pooledacrossthetwoorders.
Foreachsubjectiachoiceamongthe24madewasselectedatrandom,andthenpeopletotheright
ofiweretherecipients.Hence,thisgameassuressubjectsthatnoneofthepeoplewhoreceivetheir
generositywillhaveanoptiontogiveanythingback,andthisisexplainedtoallthesubjects.
Weran6sessionsof20subjectseach,foratotalof120subjectsinthisstudy.Threesessionswererun
withcircularmatchingandthreewithreciprocalmatching.Withineachmatchingtype,twosessions
wererunwiththeforwardpresentationandonewiththereversepresentation.Eachsessionlastedless
than1h,andsubjectsearnedanaverageof$14.23.Completeinstructionsareavailablefromtheauthor.9
Weshouldnotethatthisdesigndiffersfrompriorpublicgoodsexperiments.Earlierexperiments,
typifiedbyIsaacandWalker(1988),usuallyexamineasinglegroupsizeandasinglepriceduringany
session,plustheyusuallyhaverepeatedinteractionseitherwiththesamegrouporwith“strangers”
rematching (Andreoni, 1988b, 1995). This experiment, by contrast, asks for a series of decisions
withoutanyfeedback,andcompareschoicesoveranumberofdifferentpricesandgroupsizes.
4. Overall results
Table 1showstheaverageamountpassedoneachbudgetforboththecircularandreciprocal
treatments.TheseresultsarealsoshowninFigs.2and3.Bothfiguresshowthebudgetsfor(π,g).
i
9 Gotohttp://econ.ucsd.edu/~jandreon/.
1738 J.Andreoni/JournalofPublicEconomics91(2007)1731–1749
Fig.2.Budgetsassumingg(π,n)=π .
o o
Fig.2showstheaveragechoiceundertheassumptionofb=0,whileFig.3assumesb=1.Ascan
beseen,bothshowsensitivitytopriceandincomeinthepredictedway.10
Nextaskwhethertherewereeffectsofeithertheorderofdecisionsorthematchingofsubjects.
WeconductedMann–Whitneytestsonthedistributionsofaverageholdamountsacrosssubjects.
Keepingthematchingconstant,wefoundnodifferencefororderforeitherthecircularorreciprocal
matching, or for the whole sample combined. Combining all 120 subjects, for instance, we find
z=1.22 (α≤0.22).11 This gives us confidence that the data is not biased by the presentation of
thedecisiontask.Comparingthematchingschemes,weagainemployMann–Whitneytestsand,
perhapssurprisingly,weagainfoundnosignificantdifferencebetweenmatchingrules.Comparing
allsubjectsincircularmatchingtothosereciprocalmatching,wefindz=0.33(α≤.74).12 Thisis
evidence that there is no real effect of reciprocal versus unilateral altruism. These two findings
indicatethatwecan safelypoolall120 subjectsintoasinglesample. We willnonethelessreport
10 AlthoughaggregatedataneednotsatisfyGARP,therearenoviolationsofrevealedpreferencesforb=0,whileforthe
b=1aggregationthereisoneviolationinvolvingbudgets17and22.Thesetwobudgetsarenearlyidenticalwhenb=1
andthechoicesnearlyoverlapattheintersection.
11 Forreciprocalmatching,z=0.679(α≤0.49)andforcircularmatchingz=0.88(α≤0.38).
12 Thisalsoholdswhenwelookattheforwardandreverseordersseparately.Forforwardorder,z=0.48.(α≤.63)and
forreverseorderz=0.13(α≤0.89).
J.Andreoni/JournalofPublicEconomics91(2007)1731–1749 1739
Fig.3.Budgetsassumingg(π,n)=nπ .
o o
resultsseparatedbycondition,whenconvenient.Aswewillsee,however,thebehaviorinallfour
conditionsisremarkablysimilarthroughouttheanalysis.
4.1. The power of the GARP tests
Thereareseveralmethodsforlookingatthepowerofrevealedpreferencetests.13Theeasiestarethe
Bronarsindex(Bronars,1987)andabootstrappingindex(AndreoniandHarbaugh,2006).Bronars
specifiesasthealternativehypothesisthatchoicesaredrawnrandomlyfromauniformdistributionon
eachbudget.Thebootstrappingindexalsodrawsrandomlyoneachbudget,butinsteadusestheactual
distribution ofchoicesmadeoneachbudgetbythe pooledsetofsubjects.Aftercreatingsynthetic
subjects,GARPtestsareconducted.Theprocessistheniteratedasufficientnumberoftimes.
Applyingthesepowerteststoourdatarequiresthatwealsospecifyagivingfunction,g(π ,n).
o
Choosingg(π ,n)=nbπ ,wecalculatedbothpowerindicesfor41valuesofbbetween0and1.In
o o
eachcase,using10,000randomiterations,over99.6%ofallrandomlygenerateddatahadviolations
ofGARP,forbothBronarsandbootstrappingindices.14Giventhesemethodsofrandomlygenerating
13 SeeAndreoniandHarbaugh(2006)foracompletediscussionofthepowerofrevealedpreferencetests.
14 Thisistruewhetherboostrappingwasconditionaloncircularmatching,reciprocalmatching,orpooled.
1740 J.Andreoni/JournalofPublicEconomics91(2007)1731–1749
alternativeorganizationsofthedata,thesetwotestsindicatethatwehavedesignedapowerfultestof
consistentpreferences.15
5. Non-parametric analysis
ThissectionbeginsbyaskingwhetherchoicesareconsistentwithGARP.Wefirstholdgroup
sizeconstantandcheckGARPconditionalonn.Wethencombineallthedataandtrytouncover
theoptimal b ing(π , n)=nbπ for each subject.
o o
5.1. Checking GARP within group size
TheadvantageofcheckingGARPwithinagroupsizeisthatweneednoassumptionsonhow
preferencesdependonn.Table2showsthenumberofviolationsofGARPconditionaloneachn
andthematching scheme.Within eachn only asmall numberof subjects had anyviolationsof
GARP.Ofthoseviolations,onlyafractionwere“significant”asmeasuredbytheAfriatcriticalcost
efficiencyindex,A.Thisindexisameasureofgoodnessoffit.Itindicateshowmuchabudgethas
i
tobe“relaxed”toavoidviolationsofGARP.Theindexhastheinterpretationthatapersoncould
havepurchasedarevealedpreferredbundleatafractionA ofwhatwasspent.16IfA ¼1thereare
i i
no violations, and the lower A the worse the violation. The standard proposed by Varian for a
i
significantviolationofGARPisAb0:95.Ofthe30violationslistedinTable2,12aresignificant
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bythisdefinition,whichcomefromonlynineindividualsubjects.Thismeans92%ofthesubjects
havechoicesthatareconsistentbasedonthesetests.Thisisaveryhighdegreeofrationalizability.
5.2. Checking GARP across group sizes
WenextturntoevaluatingGARPwhenwepoolacrossgroupsizes.Whetherdataisconsistent
withGARPwilldependonthedefinitionofthegivingfunctiong(π ,n)=nbπ .Hereweask,does
o o
there exist a b∈[0,1] for an individual isuch that there are no violationsof GARP?
Theanswertothisquestionisyesformostsubjects.Forboththecircularandreciprocalmatching,
exactly83subjects,69%,havechoicesthatdidnotviolateGARPasingletimeforatleastonevalue
ofb.ThoseforwhomtheanswerisnoaresummarizedinTables3and4.Herewereportthevalueof
b that resulted in the highest value of the Afriat index A for all the subjects that always violate
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GARP.Ofthe 37subjectswithviolations,16haveindicesabovetheAz0:95 threshold,and20
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below. By this measure, 16% of subjects have severe violations of GARP, and 84% are ratio-
nalizable.17Withinthecontextofrevealedpreferenceexperiments,thisisarelativelyhighdegreeof
agreementwiththemodel,especiallyforsuchalargenumberofbudgetconstraints.18
15 WealsoconductedBronars(1987)powerovereachsubsetofbudgetswiththesamen.Thiswillnecessarilyreveal
lowerpowersincetherearefewerbudgets.Forn=1,2,3,4,5,9theBronarspowerfindsthefractionofiterationswith
GARPviolationsis,respectively,0.7,0.60,0.50,0.36,0.30,0.36.Thiswasover10,000iterationeach.Togetaperspective
onhowhighthesenumbersare,notethatwithtwobudgetsthemaximumvalueaBronarsindexcantakeis0.25.
16 SeeAfriat(1972)andVarian(1982)fordiscussionsofthisindex.
17 We also conducted further tests of differences between circular and reciprocal matching and again found no
significantdifferences.Comparingtheb'sreportedinTables3and4withaMann–Whitneytest,z=0.127.Comparing
theA'sreported,z=0.427.Finally,comparingthenumberofGARPviolations,z=0.174.
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18 See Andreoni and Harbaugh (2006) for a discussion of recent experiments using revealed preference tests. An
intrestingrecentcomparisonisFisman,KarivandMarkovits(inpress)whoprovidesubjectswith50choicesofdividing
apiewithoneothersubject.Theyfindonly10.2%ofsubjectshadnoviolationsofGARP,and41%wereabovethe
Az0:95threshold.
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