Fillable Printable Social Cost-Benefit analysis in a nutshell
Fillable Printable Social Cost-Benefit analysis in a nutshell
Social Cost-Benefit analysis in a nutshell
(SOCIAL)COST-BENEFITANALYSISINANUTSHELL
RUFUSPOLLOCK
EMMANUELCOLLEGE,UNIVERSITYOFCAMBRIDGE
1.Introduction
Cost-benefitanalysisisaprocessforevaluatingthemeritsofaparticularprojector
courseofactioninasystematicandrigorousway.Socialcost-benefitanalysisrefersto
cases where the project has a broad impact across society – and, as such, is usually carried
outbythegovernment.
Whilethecostandbenefitsmayrelatetogoodsandservicesthathaveasimpleand
transparentmeasureinaconvenientunit(e.g.theirpriceinmoney),thisisfrequently
notso,especiallyinthesocialcase.Itshouldthereforebeemphasizedthatthecostsand
benefitsconsideredby(social)‘cost-benefit’analysisarenotlimitedtoeasilyquantifi-
ablechangesinmaterialgoods,butshouldbeconstruedintheirwidestsense,measuring
changesinindividual‘utility’andtotal‘socialwelfare’(thougheconomistsfrequentlyex-
pressthosemeasuresinmoney-metricterms).
Initsessencecost-benefitanalysisisextremely,indeedtrivially,simple:evaluatecosts
CandbenefitsBfortheprojectunderconsiderationandproceedwithitif,andonlyif,
benefitsmatchorexeedthecosts.Thatis:
B
|{z}
Benefits
≥C
|{z}
Costs
?
Sowhatmakesthingsmorecomplex?Thereareavarietyoffactors:
•Benefits and costs may accrue to differentsets ofpeople.If this is so we need some
waytoaggregateandcomparedifferentbenefitsandcostsacrosspeople.
•Benefitsandcostsmayoccuratdifferentpointsintime.Inthiscaseweneedto
comparethevalueofoutcomesatdifferentpointsintime.
Email:rufus[at] rufus[dot] pollock [dot]org.Thispaperis licensedunder Creative Commonsattribution
(by)licensev3.
1
2RUFUS POLLOCKEMMANUELCOLLEGE,UNIVERSITYOFCAMBRIDGE
•Benefitsandcostsmayrelatetodifferenttypesofgoodsanditmaybedifficult
tocomparetheirrelativevalues.Thisusuallyoccurswhenoneofthegoodsdoes
nothaveanobviousandagreeduponprice.Forexample,wemaybespending
standardcapitalgoodstoday inordertoobtainenvironmentalbenefitstomorrow.
•Benefitsandcostsmaybeuncertain.
•Benefitsandcostsmaybedifficulttocalculateand, asaresult,theremaybe widely
differingviewsabouttheirsizes.Onemightthinkthiscouldbesubsumedunder
uncertainty,howeverthetwopointsareratherdifferent:twopeopleagreeingthat
anoutcomefollowssomeprobabilitydistributionisdifferentfromthemarguing
aboutitsmeanandvariance.
Usually,inreal-worldcasesthedominantissueisusuallythelastone:thebasicjob
ofcalculatingestimatesfortheproject’scostsandbenefits.Thisespeciallytrueinthe
‘social’casewheretheprojectsunderconsiderationmayinvolvecostsandbenefitsthat
verydifficultto quantify –what isthe benefit ofthe nationalsecurityderivedfrom military
spending, howlargearethebenefits fromeducation,etcetc.Necessarilythisquantification
onlymakessenseonacase-by-casebasis.Hereweareconcernedwithgeneralprinciples
andwethereforefocusonlyontheprecedingfouritemsandlookathowtheycanbe
incorporatedintotheanalysisinageneralway.
2.TheBasicModel
Weareconsideringaprojectwithknown(thoughperhapsuncertain)benefitBand
costC.Ourtaskistodecidewhetheritisworthwhile.Asalreadydiscussed,ifBandC
weredenominatedinexactlythesameterms(i.e.thesamegood,atthesametime)for
asinglepersonandwithnouncertaintythingswouldbestraightforward:checkwhether
B≥C.However,thisisunlikelytobethecasesowewillneedtodomorework.All
ofthiswork,initsessenceinvolvesconvertingbenefitsandcostsintovaluesthatcanbe
easilycompared.Equivalentlyweneedtohavebenefitsandcostsdenominatedinterms
ofsomestandardgoodormeasureofvalue.Weshalltermthisgoodormeasureofvalue
thenumeraire.
In theory,this numeraire could beanything:apples, years of life,acresof rainforestetc.
However,giventhatmany(thoughbynomeansall)goodsarealreadydenominatedin
(SOCIAL)COST-BENEFITANALYSISINANUTSHELL3
termsofmoney, itisoftennaturaltouseanumeraire thatismoney-metric.Wealsoneed
tospecifymoneyinwhosehands(forexample,£1inthehandsofsomeoneonbreadline
islikelyofdifferentvalueto£1inthehandsofabillionaire).
Forthe purposesof socialcost benefitanalysisa very naturalnumeraire is‘(uncommit-
ted)governmentfunds’,thatismoneythegovernment hasbutisnotyetallocatedtoany
givenproject.Thisisanaturalnumerairesinceitislikelytobegovernmentfundswhich
areusedinpayingfortheprojectbeingconsidered.
Wewill alsoassume, tomake our lives straightforward,that,unlessspecifiedotherwise,
the cost of a given project is exactly one unit of government funds today.This allows us to
focusonlyonthebenefitswhichmakesthingssimplerwhilesacrificingnorealgenerality.
Webegin,inthesectionthatfollows,byfocusingsolelyonthedistributionalissues
andignoringtemporalanduncertainty.Wethenintroducetemporalconsiderationsand
discountingandconcludebydiscussinguncertainty.
3.DistributionalConcerns
Letus suppose thereareNpeople orgroups.Thebenefit togroupifrom theprojectun-
der considerationis b
i
of income/consumption.
1
Income/consumption isconstrued broadly
heretoincludenotonlynormaltradedgoodssuchasfoodordigitalmusicplayersbut
alsothingslikesecurity,astablebiosphereetc.Individualsexistingincome/consumption
isdenotedbyx
i
.Thoseoperatingtheproject(thegovernment)haveautilitarianwelfare
function:
2
W=
X
i
u(x
i
)
Thechangeinwelfare(assumingawayanychangesinbehaviour)arisingfromthe
individualbenefitsis:
∆W=
X
i
u(x
i
+b
i
)−u(x
i
)
Ifthegainsaresmallrelativetoexistingincomewemayapproximatethechangein
individual welfare usingthederivative:u(x
i
+b
i
)−u(x
i
) = u
0
(x
i
)b
i
.Defining,w
i
= u
0
(x
i
)
wethenhaveasetof‘distributionalweights’,thatisweightingsforindividualssuchthat
1
Wewon’tdistinguishbetweenincomeandconsumptionheresinceitwillnotmatterforourpurposes.
2
Wecouldavoidanyreferenceto individualutilityherebypositinga social welfare functiondefined directly
intermsoftheoutcomesbeingaffectedbethatmoney,education,securityetc.
4RUFUS POLLOCKEMMANUELCOLLEGE,UNIVERSITYOFCAMBRIDGE
thetotal(welfare)benefitisjustthesumoftheweightstimestheindividualbenefits:
∆W=
X
i
w
i
b
i
Onelaststepremains:weneedtoconvertutilityback intoournumeraire(government
funds)viamultiplicationbysomeconstantθ–theoverallbenefitBwillthenbeθ∆W.
TospecifythisconstantwechooseabenchmarkprojectandthendefineitsbenefitBto
exactlyoneunitofthenumeraire–sincecostsarealso1thisimpliesthisprojectisjust
worthwhile.Thestandardapproachisforthebenchmarktobeaprojectwhichgenerates
benefitsequivalenttooneunitequallydividedequallyamongallgroups,i.e.b
i
=1/N.
3
Thisimpliesthat:
θ
X
i
w
i
1
N
≡1 ⇒θ=
N
P
i
w
i
Notethatifincomewerealreadyequallydistributedsox
i
=xandutility(whichis
only definedupto aconstant) were normalized so thatthatthe marginalutility of income
atthereferenceincomexwereexactlyonewewouldhavew
i
=w=1⇒θ=1.To
summarize:
N=Numberofbeneficiaries
b
i
=Benefitstogroupi
x
i
=Incomeofgroupi
w
i
=Weights=MarginalUtilityofgroupi
∆W=Welfarebenefit =
X
i
w
i
b
i
θ=ConversionfactorfromSWtoNumeraire
B=Benefit = θ
X
i
w
i
b
i
3.1.CalculatingtheConversionFactorθ.Withafewassumptionsontheformof
the utility function and knowledge of the distribution of income we can calculate an actual
3
Clearlythechoiceofthebenchmarkprojectisimportant.Whythenchoosethisproject?Theanswer
isthat£equallydistributediswhatstandardgovernmentprojectslikeprovisionoffreeeducationorfree
healthcareactuallyamountto.Assuchsuchthis‘equaldistribution’projectisdefinitelyincludedinthe
governmentportfolioandfurthermorenoworseprojectshouldbeworthwhilebecausewecouldreallocate
fromthatprojecttothe‘equaldistribution’andimprovewell-being.
(SOCIAL)COST-BENEFITANALYSISINANUTSHELL5
figureforθ.AssumethatutilitytakesCESform:
u(x) =
x
1−α
−1
1−α
Thus,theweights(equaltomarginalutility)arew
i
=x
−α
i
andhenceθ
−1
=E(x
−α
).
Atthispoint,itisusefultomovetocontinuousratherthandiscretevariablesso:
1
θ
=
Z
w
i
(x)dF(x) =
Z
x
−α
dF(x) = E(x
−α
)
Now, the distribution of income xis (approximately) log-normal LN(ν,σ) in which case
usingtheformulaforthemomentgeneratingfunctionofthenormalwehave:
E(x
−α
) = e
−αµ+
α
2
σ
2
2
3.2.Example1:BenefitsinProportiontoIncome.Supposetheprojectgenerates
value Vwhich is then distributed in proportion to income.Let λbe the ratio of individual
benefittoincomesob
i
=λx
i
.Now
P
i
b
i
=V⇒λ=V/
P
i
x
i
.Thususingourformula
fromabovethetotalbenefitintermsofthenumeraireis:
B=θ
X
i
w
i
b
i
=θλ
X
i
w
i
x
i
=θ
NV
P
i
x
i
1
N
X
i
w
i
x
i
UsingaCESutilityfunctionsow
i
x
i
= x
1−α
i
andusingexpectationswehave:
B=
VE(x
1−α
)
E(x
−α
)E(x)
Usingalog-normaldistributionforincomeandtheexpressionfortheMGFasbefore
this further reducesto B= e
−ασ
2
.Forlog-utility,α= 1, anda reasonableestimateof σis
0.47(Newbery2008)whichimpliesB=0.8V.Thuseachpound/euro/dollardistributed
generates a benefitin termsofthe numeraire of 0.8 and,if theproject is tobe worthwhile,
itmusthaveadirectyieldofatleast25%(= 1/0.8).
3.3.Discounting.Wenowcometothetimefactor:benefitsofeffortorexpenditure
todaymaynotberealiseduntiltomorrow.Inthespiritofkeepingthingssimplelet
6RUFUS POLLOCKEMMANUELCOLLEGE,UNIVERSITYOFCAMBRIDGE
usfixeverythingabouttheproblemexceptthetemporalaspect.Inparticular,ignore
distributionalissues,uncertaintyandvariationsinthetypesofgoodsinvolved.
AssumethatbygivinguponeunitofexpendituretodaywegainVunitsattimeT.
LetexpendituretodaybexandattimeT(intheabsenceoftheproject)x
T
.Thereisa
utilityfunctionuwhichconverts expenditure into contemporaraneousutility(i.e.utilityin
thatperiod).Letthenumeraire bepresent periodutilitynormalizedsothatthemarginal
utilitytodayequals1,i.e.u
0
(x) = 1.
Thustherearetwomajorfactorstotakeintoaccount.First,howtoconvertutility
fromperiodTtonow.Humanstendtopreferthingssoonerratherthanlater.Hence,
evenwithallelseequal,utilitytodayispreferredtoutilitytomorrow.Themeasureof
thisistermedthelevelof‘puretime-preference’andwewilldenoteitbyρ(t).Second,
thereferencesituationtomorrow (intermsofresources,consumptionetc)may notbethe
sameastodayandmarginalutilityfromgainingorlosingaunitwilldifferacrosstime,
quiteapartfromtime-preference.
Assumingchangesinexpenditurearerelativelysmallwecanapproximateutilitychanges
usingderivativeswehave:
C=Cost = −u
0
(x) = −1
b=BenefitatT= Vu
0
(x
T
)
B=Benefitintoday’sutility = ρ(T)b
Sotheprojectisworthwhileif:
Vρ(T)u
0
(x
T
) ≥1
This implicitly defines a discount factor δ(T) = ρ(T)u
0
(x
T
) with the payoff of Vat time
Tvaluedatδ(T)Vtoday.
3.4.Example2:DiscountRatesandClimateChange.Supposewecanspend
resources(orequivalentlyforgoconsumption)todaytomitigatetheeffectsofclimatein
theformofbenefitsrelativetothedo-nothingscenario,atsomepointTinthefuture.
Suppose,intheabsenceofthisproject,theeconomywouldgrowatrategperyearso
(SOCIAL)COST-BENEFITANALYSISINANUTSHELL7
consumptionattimeTise
gT
timesconsumptiontoday.Supposepuretimepreference
takesanexponentialformsoρ(T) = e
−ρT
andwehaveCESutilityasbefore.Then:
B= Ve
−ρT
e
−αgT
= Ve
−(ρ+αg)T
Comparingthiswithastandardexponentialdiscountratee
−δT
givesanimplieddis-
countrateδ= ρ+αg.
4
3.5.Uncertainty.Themodelsdiscussedaboveinvolvenouncertainty:allrelevantval-
ues,e.g.theproject’spayoff,futureconsumptionlevelsetc,areknownwithcomplete
certainty.This isclearly unrealisticanditis usefulto beable toconsidersituationswhich
doinvolve(known)uncertainty.
Thenaturalapproachhereissimplytoreplacecostsandbenefitswiththeirexpected
values.Ifthosereceivingbenefits(orbearingcosts)areriskaverse–asislikely–uncer-
taintywillactto reducebenefitsand increasecosts(thecertainty of£1m isworth more to
mostpeoplethananevens chance of0or£2m).Weillustratebyreturningtoourclimate
changeexample.
3.6.Example3:UncertaintyandClimateChange.Considerour previousexample
regardingclimatechange.Wewillincorporateuncertaintyalittleindirectlyherebyas-
sumingthatdirectcostsandbenefitsremaincertainbutthatweareuncertainastothe
growthrategintheabsenceofaction(i.e.whattheeffectsofclimatechangewillifwe
donothing).
5
Uncertaintyin the growth rate will affect our calculation of benefits because it will alter
thelevelofconsumption,andhencemarginalutility,atwhichbenefitsinthefutureare
evaluated.Remember,thebenefitsofmitigationaregreaterthemoreterribletheworld
whenweofdoingnothing(i.e.inthecaseofunmitigatedclimatechange).
Formally,supposethegrowthrategfollowssomeprobabilitydistributiongivenbyG.
Then:
4
Notethatthis is,notcoincidentally, thesameas therealinterestratefoundin theRamseyKassCoopmans
model.
5
Analternativeformulationoftheclimatechangequestionwouldbetochangethedefaultscenarioto
fullmitigationandtheactionscenariobeingdonothing.Inthisframeworkthebenefitwouldbegained
‘consumption’ today while the cost would be the reduction in ‘consumption’in the future.This costwould
thenbedirectlyrelatedtothegrowthrateinourmainformulationofthisproblem.
8RUFUS POLLOCKEMMANUELCOLLEGE,UNIVERSITYOFCAMBRIDGE
C=−E(u
0
(x)) = −1
B=ρ(T)VE(u
0
(e
gT
x))
E(u
0
(e
gT
x))isexpectedmarginalutilityattimeT.Withriskaversionthiswillbe
increasingin‘uncertainty’(e.g.a increaseinthevariance ofthegrowthratethat preserved
themean).TosaymoreweneedtospecifyafunctionalformforuandG.Takeu
asourpreviousCESformandsupposegrowthisnormallydistributed:g∼N(µ,σ
2
).
Then,recallingthatthemomentgeneratingfunctionforX,anormalrandomvariable,is
E(e
tX
) = e
tµ+
1
2
t
2
σ
2
,wehavethat:
E(u
0
(e
gT
x)) = E(e
−αgT
)E(x
−α
) = E(e
−αgT
)E(u
0
(x)) = e
−αµT+
1
2
α
2
T
2
σ
2
Thus under uncertainty the discount rate is ρ+αµ−
1
2
α
2
σ
2
.Recall that under certainty
itwasρ+αg.Takingthecasewherethegrowthrateundercertainty,g,isequaltothe
expectedgrowthrateunderuncertaintyµ,weseethattheimpactofuncertainty(in
theformofvarianceinthegrowthrate)actstoreducethediscountrate.
Toillustrate,takeasabenchmarkcaseρ= 0.5,α= 1.0,g/µ= 2.5sothediscountrate
is3%inthecaseofcertaintyoverthegrowthrate.Supposehoweverwearen’tsurehow
bad(unmitigated)climatechangewillbesowereplaceourcertaingrowthrateof2.5%
withanormallydistributedonewiththesamemeanbutastandarddeviation(σ)of2%.
Thiswouldreducethediscountrateto1%.Moreover,astandarddeviationgreaterthan
2.5% would result in an negativediscount rate – in expectation,consumption in the future
is more valuable than consumption today(even though on averagewe expect consumption
togrowat2.5%peryear)!
Thecrucialpointhereisthatuncertaintyactsasymmetricallybecauseofdiminishing
returns(equivalentlyriskaversion)intheutilityfunction.Increasinguncertaintyinthe
growth rate increases thechances bothof beingin a ‘greatworld’ where we are (relatively)
well-off andof being in a ‘terrible world’ where we are (relatively) poor.In the former case
ourgainfromextraconsumptioninthefuture(theresultofclimatechangemitigationin
this model) is reduced because we arealready in a good situation but in the latter case the
(SOCIAL)COST-BENEFITANALYSISINANUTSHELL9
gainsaregreatlyincreasedbecausewearesobadlyoff.Moreover,thankstodiminishing
returnstoconsumptionthe gainsinthebad casegreatlyoutweigh thereductioninbenefits
inthegoodcase.Asaresultincreasinguncertaintyincreasesthevalueofeachunitof
benefitinthefutureandthusreducesthediscountrate.
