The Suntory and Toyota International Centres for Economics and Related Disciplines
Externality
Author(s): James M. Buchanan and Wm. Craig Stubblebine
Source: Economica, New Series, Vol. 29, No. 116 (Nov., 1962), pp. 371-384
Published by: Wiley on behalf of The London School of Economics and Political Science and The Suntory
and Toyota International Centres for Economics and Related Disciplines
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1962]
Externality
M. BuCHANAN
and WM. CRAIGSTUBBLEBINE
By JAMES
Externalityhas been, and is, centralto the neo-classicalcritiqueof
market organisation.In its various forms-external economies and
diseconomies, divergenciesbetween marginal social and marginal
privatecost or product,spilloverand neighbourhoodeffects,collective
or publicgoods-externality dominatestheoreticalwelfareeconomics,
and, in one sense, the theory of economicpolicy generally.Despite
this importanceand emphasis, rigorous definitionsof the concept
itselfarenot readilyavailablein the literature.As Scitovoskyhas noted,
" definitionsof externaleconomiesare few and unsatisfactory".' The
followingseemstypical:
External effects exist in consumption whenever the shape or position
of a man's indifference curve depends on the consumption of other men.
[External effects] are present whenever a firm's production function
depends in some way on the amounts of the inputs or outputs of anothex
firm.2
It seemsclear that operationaland usabledefinitionsare required.
In this paper, we propose to clarify the notion of externalityby
definingit rigorouslyand precisely.When this is done, severalimportant, and often overlooked, conceptual distinctionsfollow more or
less automatically.Specifically,we shalldistinguishmarginaland inframarginalexternalities,potentiallyrelevantand irrelevantexternalities,
and Pareto-relevantand Pareto-irrelevantexternalities.These distinctionsare formallydevelopedin SectionI. As we shall demonstrate,
the term," externality", as generallyused by economists,corresponds
only to our definitionof Pareto-relevantexternality.Therefollows, in
Section II, an illustrationof the basic points describedin terms of a
simpledescriptiveexample.In SectionIII, some of the implicationsof
our approachare discussed.
It is useful to limit the scope of the analysisat the outset. Much of
the discussionin the literaturehas been concernedwith the distinction
andpecuniaryexternaleffects.We do not propose
betweentechnological
to enter this discussionsince it is not relevantfor our purposes.We
note only that, if desired,the whole analysiscan be takento applyonly
to technologicalexternalities.Secondly,we shall find no causefor discussing productionand consumptionexternalitiesseparately.Essentiallythe sameanalysisappliesin eithercase.In whatfollows, " firms"
may be substitutedfor " individuals" and " productionfunctions"
1 Tibor Scitovosky," Two Concepts of ExternalEconomies", Journalof Political
Economy,vol. LXII (1954), p. 143.
2 J. de V. Graaf, TheoreticalWelfareEconomics,Cambridge,1957,p. 43 and p. 18.
371
c
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ECONOMICA
372
[NOVEMER
for " utilityfunctions" withoutmodifyingthe centralconclusions.For
expositionalsimplicityonly, we limit the explicit discussionto consumptionexternalities.
I
We definean externaleffect,an externality, to be presentwhen,
(1)
UA = UA (X1, X2, ...
e,X
M, Y).
This states that the utility of an individual,A, is dependentupon the
" activities", (X1, X2, .-X Xm), that are exclusivelyunder his own
control or authority,but also upon anothersingle activity, Y1,which
is, by definition,underthe control of a second individual,B, who is
presumedto be a memberof the same social group. We define an
humanactionthatmaybe measured,
activity hercas any distinguishable
such as eating bread, drinkingmilk, spewing smoke into the air,
dumpinglitter on the highways,givingto the poor, etc. Note that A's
utilitymay, and will in the normalcase, dependon other activitiesof
B in additionto Y1,and also upon the activitiesof otherparties.That
is, A's utility functionmay, in more generalterms,include such variables as (Y2, Y3, ...., Ym;Zl Z2. *- -, Zm). For analyticalsimplicity,
however,we shall confineour attentionto the effectsof one particular
activity,Y1,as it affectsthe utilityof A.
We assumethat A will behaveso as to maximiseutilityin the ordinary way, subjectto the externallydeterminedvalues for Y1,and that
he will modifythe valuesfor the X's, as Y1changes,so as to maintain
a state of " equilibrium ".
A marginalexternalityexistswhen,
* 0.
(2) uOy,
Here, small u's are employedto representthe " partial derivatives"
of the utilityfunction of the individualdesignatedby the super-script
with respect to the variables designatedby the subscript. Hence,
uAl,=auAI Yl, assumingthat the variationin Y1 is evaluated with
respectto a set of " equilibrium"values for the X's, adjustedto the
givenvaluefor Y1.
An infra-marginal
externalityholds at those points where,
(3) uyl = 0,
and (1) holds.
These classificationscan be broken down into economiesand diseconomies:a marginalexternaleconomyexistingwhen,
(2A) zy4>0,
that is, a smallchangein the activityundertakenby B will changethe
utility of A in the same direction; a marginalexternaldiseconomy
existingwhen,
(2B)
Uy,A<
0
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1962]
EXTERNALITY
373
An infra-marginalexternaleconomy exists when for any given set
of values for (X1, X2, ....,
X.), say, (C1, C2, **., Cm),
Yl
(3A)
uAy
= O, and fuwyltdyl> O.
0
This conditionstates that, while incrementalchangesin the extent of
B's activity, Y1, have no affect on A's utility, the total effect of B's
action has increasedA's utility. An infra-marginaldiseconomyexists
when (1) holds, and, for any given set of valuesfor (X1, X2, ..., Ki),
say, (C1, C2,..., C.), then,
(3B)
uAy,
Y1
= O, and IuA
0
dyl< O.
Thus, small changesin B's activity do not change A's level of satisfaction,but the total effect of B's undertakingthe activityin question
is harmfulto A.
We are able to classifythe effectsof B's action, or potentialaction,
on A's utility by evaluatingthe " partial derivative" of A's utility
functionwith respectto Y1over all possiblevaluesfor Y1.In orderto
introduce the further distinctions between relevant and irrelevant
externalities,however,it is necessaryto go beyond considerationof
A's utilityfunction.Whetheror not a relevantexternalityexistsdepends
uponthe extentto whichthe activityinvolvingthe externalityis carried
out by the personempoweredto take action,to make decisions.Since
we wish to considera single externalityin isolation,we shall assume
that B's utility function includes only variables(activities)that are
withinhis control, including Y1.Hence, B's utility functiontakes the
form,
(4) U" = U" VDl YV, . .,Ym).
Necessaryconditionsfor utilitymaximisationby B are,
= f y Ilf By
(5) uBylluBy,
whereY, is usedto designatethe activityof B in consumingor utilising
some numerairecommodityor servicewhich is, by hypothesis,available on equaltermsto A. The right-handtermrepresentsthe marginal
rate of substitutionin " production" or " exchange" confrontedby
B, the partytakingactionon Y1,his productionfunctionbeingdefined
as,
(6
f
B= f
(Y11Y2,
. .,
YM)1,
whereinputs are included as activitiesalong with outputs. In other
words,the right-handtermrepresentsthe marginalcost of the activity,
Y1,to B. The equilibriumvaluesfor the Yi'swill be designatedas Yi's.
An externalityis definedas potentially relevant when the activity,to
the extentthat it is actuallyperformed,generatesany desireon the part
of the externallybenefited(damaged)party (A) to modify the behaviour of the party empoweredto take action (B) through trade,
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374
ECONOMICA
[NOVEMER
persuasion, compromise, agreement, convention, collective action,
etc. An externalitywhich, to the extentthat it is performed,exertsno
such influenceis definedas irrelevant.Note that, so long as (1) holds,
an externalityremains;utility functionsremaininterdependent.
A potentiallyrelevantmarginalexternalityexistswhen,
(7) UA4
+ O.
Y,=Y,
This is a potentiallyrelevantmarginalexternaleconomywhen (7) is
greaterthan zero, a diseconomywhen (7) is less than zero. In either
case, A is motivated,by B's performanceof the activity,to make some
effortto modifythis performance,to increasethe resourcesdevotedto
the activitywhen (7) is positive,to decreasethe quantityof resources
devotedto the activitywhen (7) is negative.
Infra-marginalexternalitiesare, by definition,irrelevantfor small
changesin the scope of B's activity, Y1.However,when large or discrete changesare considered,A is motivatedto changeB's behaviour
with respectto Y1in all cases except that for which,
(8) uji1
=
0, and
uA (C1, C2, *.*
ylt
CM
IX) ? uA (C1, C2,
...,
Cm Y1), for all
Y1
When (8) holds, A has achievedan absolutemaximumof utility with
respectto changesover Y1,given any set of valuesfor the X's. In more
prosaicterms,A is satiatedwith respectto Y1.1In all other cases,where
infra-marginalexternal economiesor diseconomiesexist, A will have
some desire to modify B's performance;the externalityis potentially
relevant.Whetheror not this motivationwill lead A to seek an expansion or contractionin the extent of B's performanceof the activity
will dependon the location of the infra-marginal
regionrelativeto the
absolutemaximumfor any given valuesof the X's.2
Pareto relevance and irrelevancemay now -be introduced. The
existenceof a simpledesireto modifythe behaviourof another,defined
as potential relevance,need not imply the ability to implementthis
when the extent
desire.An externalityis definedto be Pareto-relevant
of the activitymaybe modifiedin sucha way thatthe externallyaffected
party,A, can be madebetteroff withoutthe actingparty,B, beingmade
worseoff. That is to say, " gainsfrom trade" characterisethe Paretorelevantexternality,trade that takes the form of some change in the
activityof B as his part of the bargain.
1Note that,
uA
=0, is a necessary, but not a sufficient, condition for
irrelevance.
2 In this analysis of the relevanceof externalities,we have assumed that B will
act in such a manneras to maximisehis own utility subjectto the constraintswithin
whichhe must operate.If, for any reason,B does not attainthe equilibriumposition
definedin (5) above,the classificationof his activityfor A may, of course,be modified.
A potentiallyrelevantexternalitymay become irrelevantand vice versa.
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1962]
XTRNALITY
375
when'
A marginalexternalityis Pareto-relevant
and when IujJx4<0, and
(9) (-) uy.,/z> (urJu2 -fy'Jf,1Y1=F
fyl/frsy
swhen ul/z4> 0.
uiljux> ( -) u2lByl/2rJu
In (9), X, and Yj are used to designate,respectively,the activitiesof
A and B in consumingor in utilisingsome numerairecommodityor
servicethat, by hypothesis,is availableon identicalterms to each of
them.As is indicatedby the transpositionof signsin (9), the conditions
for Pareto relevance differ as between external diseconomies and
economies.This is becausethe " direction" of changedesiredby A on
the part of B is differentin the two cases.In statingthe conditionsfor
Pareto relevanceunderordinarytwo-persontrade, this point is of no
significancesincetradein one good flows only in one direction.Hence,
absolutevalues can be used.
The condition, (9), states that A's marginalrate of substitution
betweenthe activity, Yl, and the numeraireactivitymust be greater
than the " net " marginalrate of substitutionbetweenthe activityand
the numeraireactivity for B. Otherwise," gains from trade" would
not exist betweenA and B.
Note, however,that when B has achievedutility-maximisingequilibrium,
=f/B IfB
(10) U1/uYBj
That is to say, the marginalrate of substitutionin consumptionor
utilisationis equatedto the marginalrate of substitutionin production
or exchange,i.e., to marginalcost. When (10) holds, the termsin the
bracketsin (9) mutuallycancel. Thus, potentiallyrelevantmarginal
externalitiesare also Pareto-relevantwhen B is in utility-maximisimg
equilibrium.Some trade is possible.
Paretoequilibriumis definedto be presentwhen,
<0, and
(11) (- )ujI/UX = [u1/uB _fB/If"], and when uA /uAx
ujtjz4 =(-)[uB I/u _f/fB
f
] whenuAju,> 0.
Condition(11) demonstratesthat marginalexternalitiesmay continue
to exist, evenin Paretoequilibrium,as heredefined.This point may be
shownby referenceto the specialcase in whichthe activityin question
may be undertakenat zero costs. Here Pareto equilibriumis attained
when the marginalrates of substitutionin consumptionor utilisation
for the two personsare preciselyoffsetting,that is, wheretheirinterests
are strictlyopposed,and not wherethe left-handtermvanishes.
What vanishesin Pareto equilibriumare the Pareto-relevantexternalities.It seems clear that, normally,economistshave been referring
only to what we have here called Pareto-relevantexternalitieswhen
1 We are indebted to Mr. M. McManus of the University of Birminghamfor
pointing out to us an error in an earlier formulation of this and the following
similarconditions.
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376
(NOVEMER
ECONOMICA
they have, implicitlyor explicitly,stated that externaleffectsare not
presentwhen a position on the Paretooptimalitysurfaceis attained.1
For completeness,we must also considerthose potentiallyrelevant
infra-marginalexternalities.Refer to the discussionof these as summarisedin (8) above.The questionis now to determinewhetheror not,
A, the externallyaffectedparty,can reach some mutuallysatisfactory
agreementwith B, the acting party, that will involve some discrete
(non-marginal)change in the scope of the activity, Y1.If, over some
range,any range,of the activity,whichwe shall designateby AY1, the
rate of substitutionbetween Y1and Xj for A exceedsthe " net" rate
of substitutionfor B, the externalityis Pareto-relevant.The associated
changesin the utilisationof the numerairecommoditymust be equal
for the two parties.Thus,for externaleconomies,we have
(12)
/j
f\>()[/\Y
//\
-/\Af7A
,
and the
samewith the sign in parenthesistransposedfor externaldiseconomies.
The differenceto be noted between (12) and (9) is that, with inframarginal externalities,potential relevance need not imply Pareto
relevance.The bracketedterms in (12) need not sum to zero when B
equilibrium.
is in his privateutility-maximising
We have remainedin a two-personworld,with one person affected
by the single activity of a second. However,the analysiscan readily
be modifiedto incorporatethe effectsof this activityon a multi-person
group.That is to say, B's activity, Y1,may be allowedto affectseveral
parties simultaneously,several A's, so to speak. In each case, the
activitycan thenbe evaluatedin termsof its effectson the utilityof each
person. Nothing in the constructionneed be changed.The only stage
in the analysisrequiringmodificationexplicitlyto take accountof the
possibilitiesof multi-persongroups being externallyaffectedis that
which involves the condition for Pareto relevanceand Pareto equilibrium.
For a multi-persongroup(A1, A2, ...., A), any one or all of whom
may be externallyaffectedby the activity, Y1,of the single person,B,
the conditionfor Pareto relevanceis,
n
(9A)
(- ) 27 ujl/u?> turl/ur -fy1/frJ, y when uAi/ux< 0, and,
E
url/2Xs>
(-
Y,/r-y/
Yi
rsy
when uAy/lu>x.
evaluationof the activity by B. Again, in prvate equilibriumfor B,
1This applies to the authors of this paper. For recent discussion of external
effectswhenwe have clearlyintendedonly what we heredesignateas Pareto-relevant,
see James M. Buchanan," Politics, Policy, and the Pigovian Margins", Economica,
vol. xxvix (1962), pp. 17-28, and, also, James M. Buchananand Gordon Tullock,
The Calculus of Consent, Ann Arbor, 1962.
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1962]
EXRNALITY
377
marginal externalitiesare Pareto-relevant,provided that we neglect
the importantelementinvolvedin the costs of organisinggroup decisions. In the real world, these costs of organisinggroup decisions
(togetherwith uncertaintyand ignorance)will preventrealisationof
some "6gainsfrom trade"-just as they do in organisedmarkets.This
is as truefor two-persongroupsas it is for largergroups.But this does
not invalidatethe point that potential" gains from trade" are available. The conditionfor Paretoequilibriumand for the infra-marginal
case summarisedin (11) and (12) for the two-personmodel can readily
be modifiedto allow for the externallyaffectedmulti-persongroup.
II
The distinctionsdevelopedformallyin SectionI may be illustrated
diagrammaticallyand discussed in terms of a simple descriptive
example.Considertwo persons,A and B, who own adjoiningunits of
residentialproperty.Within limits to be noted, each person values
privacy,which may be measuredquantitativelyin terms of a single
criterion,the heightof a fence that can be constructedalongthe common boundaryline. We shall assumethat B's desirefor privacyholds
over rather wide limits. His utility increaseswith the height of the
fence up to a reasonablyhigh level. Up to a certainminimumheight,
A's utility also is increasedas the fence is made higher. Once this
minimumheightis attained,however,A's desirefor privacyis assumed
to be fully satiated.Thus,over a secondrange,A's total utilitydoes not
changewith a change in the height of the fence. However,beyond a
certainlimit, A's view of a mountainbehindB's propertyis progressively obscuredas the fence goes higher.Overthis third range,therefore, A's utility is reducedas the fence is constructedto higherlevels.
Finally, A will once again become wholly indifferentto marginal
changesin the fence'sheightwhenhis view is totallyblockedout.
We specifythat B possessesthe sole authority,the only legal right,
to constructthe fence betweenthe two properties.
The preferencepatternsfor A andfor B are shownin Figure1, which
box diagram.Note, however,
is drawnin the formof an Edgeworth-like
that the originfor B is shown at the upperleft ratherthan the upper
rightcornerof the diagramas in the morenormalusage.Thismodification is necessaryhere because only the numerairegood, measured
along the ordinate,is strictlydivisiblebetweenA and B. Both must
adjust to the same height of fence, that is, to the same level of the
activitycreatingthe externality.
As describedabove, the indifferencecontoursfor A take the general
shapeshownby the curvesaa, a'a',whilethosefor B assumethe shapes,
bb, b'b'. Note that these contoursreflectthe relativeevaluations,for
A and B, betweenmoneyand the activity,Y1.Sincethe costs of undertaking the activity, for B, are not incorporatedin the diagram,the
" contractlocus " that might be derived from tangencypoints will
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rNOVEMER
ECONOMICA
378
havelittle relevanceexceptin the specialcase wherethe activitycan be
undertakenat zero costs.
Figure 2 depicts the marginalevaluationcurves for A and B, as
derivedfrom the preferencefieldsshownin Figure 1, along with some
incorporationof costs. These curvesare derivedas follows: Assume
Figure 1
O
ab
A
HI
H2
Height of Fence
(B's Activity)
H4
H5
->
an initialdistributionof" money" betweenA and B, say, that shown
at M on Figure 1. The marginalevaluationof the activityfor A is then
derivedby plottingthe negatives(i.e., the mirrorimage)of the slopes
of successiveindifferencecurves attained by A as B is assumed to
increasethe.height of the fence from zero. These values remainpositive for a range, become zero over a second range, become negative
for a third. and. finally.returnto zero again."
I For an early use of marginal evaluation curves, see J. R. Hicks, " The Four
Consumer'sSurpluses", Reviewof EconomicStudies, vol. xi (1943), pp. 3141.
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1962]
379
EXRNALITY
B's curvesof marginalevaluationare measureddownwardfrom the
upperhorizontalaxisor baseline,for reasonsthatwill becomeapparent.
The derivationof B's marginalevaluationcurve is somewhatmore
complexthan that for A. This is becauseB, who is the personauthorised to undertakethe action, in this case the buildingof the fence,
must also bear the full costs. Thus, as B increasesthe scope of the
activity,his realincome,measuredin termsof his remaininggoods and
services,is reduced.This changein the amountof remaininggoods and
t<
Figure 2
ME
A
S
'4
.%
ME
a~~~~m
MEB
,
X
+S
-
NMEB= MSCA
MC
serviceswill, of course, affect his marginalevaluationof the activity
in question.Thus,the marginalcost of buildingthe fencewill determine,
to some degree,the marginalevaluationof the fence. This necessary
interdependencebetween marginal evaluation and marginal cost
complicates the use of simple diagrammaticmodels in finding or
locatinga solution.It need not, however,deterus from presentingthe
if we postulatethat the marginalevaluation
solutiondiagrammatically,
curve,as drawn,is based on a singlepresumedcost relationship.This
done, we may plot B's marginalevaluationof the activity from the
negatives of the slopes of his indifferencecontours attained as he
constructsthe fence to higherand higherlevels. B's marginalevaluation, shown in Figure2, remainspositivethroughoutthe rangeto the
point H6, whereit becomeszero.
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380
ECONOMICA
[NOVEMBER
The distinctionsnoted in Section I are easily related to the constructionin Figure2. To A, the partyexternallyaffected,B's potential
activityin constructingthe fence can be assessedindependentlyof any
predictionof B's actualbehaviour.Thus,the activityof B would,
(1) exertmarginalexternaleconomieswhich are potentiallyrelevant
over the range OH1;
(2) exert infra-marginalexternaleconomiesover the range HH12,
which are clearly irrelevantsince no change in B's behaviourwith
respectto the extent of the activitywould increaseA's utility;
(3) exertmarginalexternaldiseconomiesover the rangeH2H4which
are potentiallyrelevantto A; and,
externaleconomiesor diseconomiesbeyond
(4) exertinfra-marginal
H4, the directionof the effect being dependenton the ratio between
the total utility derivedfrom privacyand the total reductionin utility
derivedfromthe obstructedview. In any case, the externalityis potentially relevant.
To determineParetorelevance,the extentof B's predictedperformance must be determined.The necessaryconditionfor B's attainment
of " private" utility-maximisingequilibriumis that marginalcosts,
which he must incur, be equal to his own marginalevaluation.For
simplicityin Figure2, we assumethat marginalcosts are constant,as
shown by the curve, MC. Thus, B's position of equilibriumis shown
at HB,withinthe rangeof marginalexternaldiseconomiesfor A. Here
the externalityimposedby.B's behaviouris clearlyPareto-relevant:A
can surelywork out some means of compensatingB in exchangefor
B's agreementto reducethe scope of the activity-in this example,to
reducethe height of the fence betweenthe two properties.Diagrammatically,the position of Pareto equilibriumis shown at H3 where
the marginalevaluationof A is equalin absolutevalue,but negatively,
to the "net" marginalevaluationof B, drawn as the curve NMEB.
Only in this position are the conditions specified in (11), above,
satisfied.'
m
Asidefromthe generalclassificationof externalitiesthatis developed,
the approachhereallowscertainimplicationsto be drawn,implications
that have not, perhaps,been sufficientlyrecognisedby some welfare
economists.
The analysismakes it quite clear that externalities,externaleffects,
may remaineven in full Paretoequilibrium.That is to say, a position
1 This diagrammaticanalysis is necessarilyoversimplifiedin the sense that the
Paretoequilibriumposition is representedas a uniquepoint. Overthe rangebetween
the " private" equilibriumfor B and the point of Pareto equilibrium,the sort of
bargainsstruckbetweenA and B will affect the marginalevaluationcurvesof both
individuals within this range. Thus, the more accurate analysis would suggest a
" contract locusV"of equilibrium points. At Pareto equilibrium, however, the
condition'shownin the diagrammaticpresentationholds, and the demonstrationof
this fact ratherthan the location of the solution is the aim of this diagrammatics.
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1962]
EXTERNALITY
381
may be classifiedas Pareto-optimalor efficientdespitethe fact that, at
the marginal,the activityof one individualexternallyaffectsthe utility
of anotherindividual.Figure2 demonstratesthis point clearly.Pareto
equilibriumis attained at H3, yet B is imposing marginalexternal
diseconomieson A.
This point has significantpolicy implicationsfor it suggeststhat the
observationof externaleffects,takenalone, cannotprovidea basis for
judgmentconcerningthe desirabilityof some modificationin an existing stateof affairs.Thereis not a primafacie casefor interventionin all
cases where an externalityis observedto exist.' The internalbenefits
from carryingout the activity,net of costs, may be greaterthan the
externaldamagethat is imposedon otherparties.
In full Pareto equilibrium,of course, these internalbenefits,measuredin termsof somenumerairegood, net of costs, mustbe just equal,
at the margin,to the externaldamagethat is imposedon otherparties.
Thisequilibriumwill alwaysbe characterised
by the strictoppositionof
interestsof the two parties,one of whichmaybe a multi-persongroup.
In the generalcase, we may say that, at full Paretoequilibrium,the
presence of a marginal external diseconomy implies an offsetting
marginalinternaleconomy,whereasthe presenceof a marginalexternal
economy implies an offsetting marginal internal diseconomy. In
" private" equilibrium,as opposed to Pareto equilibrium,these net
internaleconomiesanddiseconomieswould,of course,be eliminatedby
the utility-maximising
actingparty.In Paretoequilibrium,theseremain
becausethe actingpartyis beingcompensatedfor " suffering"internal
economiesand diseconomies,that is, divergenciesbetween" private"
marginal costs and benefits, measured in the absence of compensation.
As a second point, it is useful to relate the whole analysishere to
the more familiar Pigovian discussion concerning the divergence
between marginal social cost (product) and marginal private cost
(product).By sayingthat such a divergenceexists,we are, in the terms
of this paper, saying that a marginalexternalityexists. The Pigovian
terminologytends to be misleading,however,in that it deals with the
acting party to the exclusionof the externallyaffectedparty. It fails
to takeinto accountthe fact that thereare alwaystwo partiesinvolved
in a singleexternalityrelationship.2As we have suggested,a marginal
externalityis Pareto-relevantexcept in the position of Pareto equilibrium; gains from trade can arise. But there must be two partiesto
any trading arrangement.The externallyaffected party must compensate the acting party for modifying his behaviour.The Pigovian
terminology,throughits concentrationon the decision-makingof the
acting party alone, tends to obscurethe two-sidednessof the bargain
that must be made.
1Cf. Paul A. Samuelson,Foundationsof EconomicAnalysis, Cambridge,Mass.,
1948, p. 208, for a discussionof the views of variouswriters.
2This criticism of the Pigovian analysis has recently been developed by R. H.
Coase; see his " The Problem of Social Cost ", Journal of Law and Economics,
vol. m (1960), pp. 1-44.
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382
ECONOMCA
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To illustratethis point, assumethat A, the externallyaffectedparty
in ourmodel,successfullysecures,throughthe auspicesof the " state",
the levy of a marginaltax on B's performanceof the activity, Y1.
Assume furtherthat A is able to securethis changewithout cost to
himself. The tax will increase the marginalcost of performingthe
activityfor B, and, hence,will reducethe extentof the activityattained
in B's " private" equilibrium.Let us now presumethat this marginal
tax is levied "correctly" on the basis of a Pigoviancalculus;the rate
of tax at the marginis madeequalto the negativemarginalevaluation
of the activity to A. Under these modifiedconditions,the effective
marginalcost, as confrontedby B, may be shownby the curvedesignated as MSCB in Figure 2. A new " private" equilibriumfor B is
shown at the quantity,H3, the same level designatedas Paretoequilibrium in our earlier discussion, if we neglect the disturbinginterdependencebetweenmarginalevaluationand marginalcosts. Attention
solely to the decisioncalculusof B here would suggest,perhaps,that
undertheserevisedcircumstances,
thispositionremainsPareto-optimal
and that it continuesto qualify as a position of Pareto equilibrium.
There is no divergencebetweenmarginalprivate cost and marginal
social cost in the usual sense.However,the position,if attainedin this
manner,is clearlyneitherone of Paretooptimality,nor one thatmay be
classifiedas Paretoequilibrium.
In this new " private" equilibriumfor B,
X
(13) uYu- By BY Y _
whereuy,/uA,representsthe marginaltax imposedon B as he performs
the activity, Y1. Recall the necessarycondition for Pareto relevance
definedin (9) above,which can now be modifiedto read,
y when AJUA
y+Ul/ux]
(9B) (-) UAL/XA,> [uR/z4ju -fi/f
whenuj juX>O.
andUAUX (,
y4
In (9B), Y1representsthe " private" equilibriumvalue for Y1,determined by B, after the ideal Pigovian tax is imposed.As before, the
bracketedterms represent the " net" marginal evaluation of the
activityfor the acting party, B, and these sum to zero when equilibriumis reached.So long as the left-handterm in the inequalityremains non-zero, a Pareto-relevant marginal externality remains,
despitethe fact that the full " Pigoviansolution" is attained.
The apparentparadox here is not difficultto explain. Since, as
postulated,A is not incurringany cost in securingthe changein B's
behaviour,and, since there remains,by hypothesis,a marginaldiseconomy,further" trade" can be workedout betweenthe two parties.
Specifically,Paretoequilibriumis reachedwhen,
l
(JJA) (-)U4,1/U4 =[UyIIUPf _f_ /f ?ullUAU] when uA1JuA<O, and
uI Iux = (-)[UIlU/u _-f 1/f yj +uU4/uA] whenuluxAJ>
O.
Diagrammatically,this point may be made with reference to
Figure2. If a unilaterallyimposedtax, correspondingto the marginal
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1962]
ETLmNALrrY
383
evaluationof A, is placed on B's performanceof the activity,the new
position of Pareto equilibriummay be shown by first subtractingthe
new marginalcost curve, drawnas MSCB, from B's marginalevaluation curve.Wherethis new " net" marginalevaluationcurve, shown
as the dottedcurvebetweenpointsH3 andK, cuts the marginalevaluation curvefor A, a new position of Paretoequilibriumfallingbetween
H2 and H3 is located, neglectingthe qualifyingpoint discussedin
Footnote 1, page 380.
Theimportantimplicationto be drawnis thatfull Paretoequilibrium
can neverbe attainedvia the impositionof unilaterallyimposedtaxes
and subsidiesuntil all marginalexternalitiesare eliminated.If a taxsubsidymethod, ratherthan " trade", is to be introduced,it should
involve bi-lateraltaxes (subsidies).Not only must B's behaviourbe
modifiedso as to insurethat he will take the costs externallyimposed
on A into account,but A's behaviourmust be modifiedso as to insure
that he will take the costs " internally" imposedon B into account.
In such a doubletax-subsidyscheme,the necessaryParetoconditions
would be readilysatisfied.'
In summary,Paretoequilibriumin the case of marginalexternalities
cannot be attainedso long as marginalexternalitiesremain,until and
unlessthosebenefitingfromchangesarerequiredto pay some " price"
for securingthe benefits.
A third point worthy of brief note is that our analysisallows the
whole treatmentof externalitiesto encompassthe considerationof
purelycollectivegoods. As studentsof publicfinancetheorywill have
recognised,the Pareto equilibriumsolution discussedin this paper is
similar,indeed is identical, with that which was presentedby Paul
Samuelsonin his theoryof publicexpenditures.2
The summedmarginal
rates of substitution(marginalevaluation)must be equal to marginal
costs. Note, however, that marginalcosts may include the negative
marginalevaluationof otherparties,if viewedin one way. Note, also,
that there is nothing in the analysiswhich suggestsits limitationsto
purely collective goods or even to goods that are characterisedby
significantexternalitiesin theiruse.
Ouranalysisalso lends itself to the more explicitpoint developedin
Coase'srecentpaper.3He arguesthat the same " solution" will tend
to emergeout of any externalityrelationship,regardlessof the structure
of property rights, provided only that the market process works
smoothly. Strictly speaking, Coase's analysis is applicableonly to
inter-firmexternalityrelationships,and the identicalsolution emerges
only becausefirmsadjustto pricesthat are competitivelydetermined.
In ourtermsof reference,this identityof solutioncannotapplybecause
of the incomparabilityof utility functions.It remainstrue, however,
1Although developedin rather differentterminology,this seems to be closely in
accordwith Coase's analysis. Cf. R. H. Coase, loc. cit.
2 Paul A. Samuelson, " The Pure Theory of Public Expenditure", Review of
Economicsand Statistics, vol. xxxvi (1954), pp. 386-9.
8 R. H. Coase, loc. cit.
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384
ECONOMCA
rNOVEMBER
that the basic characteristics of the Pareto equilibrium position remain
unchanged regardless of the authority undertaking the action. This
point can be readily demonstrated, again with reference to Figure 2.
Let is assume that Figure 2 is now redrawn on the basis of a different
legal relationship in which A now possesses full authority to construct
the fence, whereas B can no longer take any action in this respect. A
will, under these conditions, " privately " construct a fence only to the
height Ho, where the activity clearly exerts a Pareto-relevant marginal
external economy on B. Pareto equilibrium will be reached, as before,
at H3, determined, in this case, by the intersection of the " net "
marginal evaluation curve for A (which is identical to the previously
defined marginal social cost curve, MSC, when B is the acting party)
and the marginal evaluation curve for B.' Note that, in this model, A
will allow himself to suffer an internal marginal diseconomy, at equilibrium, provided that he is compensated by B, who continues, in Pareto
equilibrium, to enjoy a marginal external economy.
Throughout this paper, we have deliberately chosen to introduce and
to discuss only a single externality. Much of the confusion in the literature seems to have arisen because two or more externalities have been
handled simultaneously. The standard example is that in which the
output of one firm affects the production function of the second firm
while, at the same time, the output of the second firm affects the
production function of the first. Clearly, there are two externalities to
be analysed in such cases. In many instances, these can be treated as
separate and handled independently. In other situations, this step
cannot be taken and additional tools become necessary.2
Universityof Virginia.
1The Ha position, in this presumablyredrawn figure, should not be precisely
comparedwith the same position in the other model. We are using here the same
diagramfor two models, and, especially over wide ranges, the dependenceof the
marginalevaluationcurves on income effects cannot be left out of account.
2 For a treatmentof the dual externalityproblemthat clearlyshows the important
differencebetween the separableand the non-separablecases, see Otto Davis and
Andrew Whinston, " Externalities,Welfare, and the Theory of Games ", Journal
of Political Economy,vol. LXX (1962), pp. 241-62. As the title suggests, Davis and
Whinstonutilise the tools of game theoryfor the inseparablecase.
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