The Review of Economic Studies, Ltd.
The General Theory of Second Best
Author(s): R. G. Lipsey and Kelvin Lancaster
Source: The Review of Economic Studies, Vol. 24, No. 1 (1956 - 1957), pp. 11-32
Published by: Oxford University Press
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The
General
Theory
of
Second
est'
Thereis an importantbasic similarityunderlyinga numberof recentworksin apparently widely separatedfields of economictheory. Upon examination,it would appear
that the authorshave been rediscovering,in some of the many guises given it by various
specificproblems,a singlegeneraltheorem. This theoremformsthe core of whatmay be
called The General Theory of Second Best. Although the main principles of the theory
of secondbesthaveundoubtedlygainedwideacceptance,no generalstatementof themseems
to exist. Furthermore,the principlesoften seemto be forgottenin the contextof specific
problemsand,whentheyarerediscoveredandstatedin the formpertinentto someproblem,
this seemsto evoke expressionsof surpriseand doubtratherthan of immediateagreement
and satisfactionat the discoveryof yet anotherapplicationof the alreadyacceptedgeneralizations.
In this paper, an attemptis made to develop a generaltheory of second best. In
SectionI there is given, by way of introduction,a verbalstatementof the theory'smain
generaltheorem,togetherwith two importantnegativecorollaries. Section II outlines
the scope of the generaltheory of secondbest. Next, a brief surveyis given of some of
the recentliteratureon the subject. This surveybringstogethera numberof casesin which
the generaltheory has been appliedto variousproblemsin theoreticaleconomics. The
implicationsof the generaltheory of second best for piecemealpolicy recommendations,
especiallyin welfareeconomics,are consideredin Section IV. This general discussion
is followed by two sections giving examplesof the applicationof the theory in specific
models. These exampleslead up to the general statementand rigorous proof of the
centraltheoremgiven in Section VII. A brief considerationof the existenceof second
best solutionsis followed by a classificatorydiscussionof the natureof these solutions.
This taxonomy serves to illustratesome of the importantnegative corollariesof the
theorem. Thepaperis concludedwitha briefdiscussionof the difficultproblemof multiplelayer second best optima.
I A GENERAL THEOREMIN THE THEORY OF SECOND BEST2
It is well knownthat the attainmentof a Paretianoptimumrequiresthe simultaneous
fulfillmentof all the optimum conditions. The general theorem for the second best
optimumstates that if there is introducedinto a generalequilibriumsystema constraint
which preventsthe attainmentof one of the Paretianconditions,the other Paretianconditions, although still attainable,are, in general,no longer desirable. In other words,
given that one of the Paretianoptimumconditionscannot be fulfilled,then an optimum
situationcan be achievedonly by departingfrom all the other Paretianconditions. The
optimumsituationfinally attainedmay be termeda second best optimumbecauseit is
achievedsubjectto a constraintwhich,by definition,preventsthe attainmentof a Paretian
optimum.
From this theoremthere follows the importantnegativecorollarythat there is no
a prioriway to judgeas betweenvarioussituationsin whichsome of the Paretianoptimum
1 The authorsare indebtedto ProfessorHarryG. Johnsonfor a numberof helpful suggestionsrelating
to this paper. The appelation," Theoryof SecondBest," is derivedfrom the writingsof ProfessorMeade a
See Meade, J. E., Tradeand Welfare,London, Oxford UniversityPress, 1955. Meade has given, in Trade
and Welfare,what seems to be the only attemptto date to deal systematicallywith a numberof problems
in the theory of second best. His treatment,however,is concernedwith the detailedcase study of several
problems,ratherthan with the developmentof a generaltheory of second best.
2 See section VII for formal proofs of the statementsmade in this section.
11
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12
REVIEWOF ECONOMICSTUDIES
conditionsare fulfilledwhile othersare not. Specifically,it is not true that a situationin
whichmore, but not all, of the optimumconditionsare fulfilledis necessarily,or is even
likelyto be, superiorto a situationin whichfewerare fulfilled. It follows, therefore,that
in a situationin which there exist many constraintswhich preventthe fulfillmentof the
Parteianoptimumconditions,the removalof any one constraintmay affect welfareor
efficiencyeitherby raisingit, by loweringit, or by leavingit unchanged.
The generaltheoremof the second best states that if one of the Paretianoptimum
conditionscannotbe fulfilleda secondbest optimumsituationis achievedonly by departing
fromall otheroptimumconditions. It is importantto note that in general,nothingcan be
said about the directionor the magnitudeof the secondarydeparturesfrom optimum
conditionsmadenecessaryby the originalnon-fulfil)ment
of one condition. Consider,for
example,a casein whichthe centralauthorityleviesa tax on the purchaseof one commodity
and returnsthe revenueto the purchasersin the form of a gift so that the sole effectof the
tax is to distortrelativeprices. Thenall thatcanbe saidin generalis thatgiventhe existence
andinvariabilityof this tax, a secondbest optimumcan be achievedby levyingsomesystem
of taxes and subsidieson all othercommodities. The requiredtax on some commodities
may exceedthe given tax, on other commoditiesit may be less than the given tax, while
on still othersa subsidy,ratherthan a tax, may be required.'
It follows from the above that there is no a priori way to judge as betweenvarious
situationsin which none of the Paretianoptimumconditionsare fulfilled. In particular,
it is not true that a situationin which all departuresfrom the optimumconditionsare of
the same directionand magnitudeis necessarilysuperiorto one in which the deviations
vary in directionand magnitude. For example,there is no reason to believe that a
situationin which thereis the same degreeof monopolyin all industrieswill necessarily
be in any sensesuperiorto a situationin whichthe degreeof monopolyvariesas between
industries.
II THE SCOPE OF THE THEORY OF SECOND BEST
Perhapsthe best way to approachthe problemof definingthe scope of the theoryof
secondbest is to considerthe role of constraintsin economictheory. In the generaleconomic problemof maximizationa functionis maximisedsubjectto at least one constraint.
For example,in the simplestwelfaretheorya welfarefunctionis maximizedsubjectto the
constraintexercisedby a transformationfunction. The theory of the Paretianoptimum
is concernedwith the conditionsthat must be fulfilledin orderto maximizesome function
subjectto a set of constraintswhicharegenerallyconsideredto be " in the natureof things".
Thereare, of course,a wholehost of possibleconstraintsbeyondthose assumedto operate
in the Paretianoptimizationproblem. These furtherconstraintsvary from the " naturedictated" ones, such as indivisibilitiesand boundariesto productionfunctions,to the
obviously" policy created" ones such as taxes and subsidies. In general,there would
seemto be no logicaldivisionbetweenthoseconstraintswhichoccurin theParetianoptimum
theory and those which occur only in the theory of second best. All that can be said is
that,in the theoryof the Paretianoptimum,certainconstraintsareassumedto be operative
and the conditions necessaryfor the maximizationof some function subject to these
constraintsare examined. In the theory of second best there is admittedat least one
constraintadditionalto the ones existingin Paretianoptimumtheoryand it is in the nature
of this constraintthat it preventsthe satisfactionof at least one of the Paretianoptimum
conditions. Considerationis then givento the natureof the conditionsthat mustbe satisfied in orderto maximizesome functionsubjectto this new set of constraints.2
1 See Section V.
The generaltheory of second best is, thus, concernedwith all maximizationproblems not just with
welfaretheory. See Section III for examplesof non-welfareapplications.
2
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THE GENERAL THEORY OF SECOND BEST
13
It is important to note that even in a single general equilibrium system where there is
only one Paretian optimum, there will be a multiplicity of second best optimum positions.
This is so because there are many possible combinations of constraints with a second
best solution for each combination.' For this reason one may speak of the existence
of the Paretian optimum but should, strictly speaking, refer to a second best optimum.
It is possible to approach problems in the theory of second best from two quite different
directions. On the one hand, the approach used in this paper is to assume the existence
of one constraint additional to those in the Paretian optimum problem (e.g., one tax, one
tariff, one subsidy, or one monopoly) and then to investigate the nature of the conditions
that must be satisfied in order to achieve a second best optimum and, where possible,
to compare these conditions with those necessary for the attainment of a Paretian optimum.
On the other hand, the approach used by Professor Meade is to assume the existence of a
large number of taxes, tariffs, monopolies, etcetera, and then to inquire into the effect
of changing any one of them. Meade, therefore, deals with a system containing many
constraints and investigates the optimum (second best) level for one of them, assuming
the invariability of all the others.2 It would be futile to argue that one of these approaches
was superior to the other. Meade's is probably the appropriate one when considering
problems of actual policy in a world where many imperfections exist and only a few can
be removed at any one time. On the other hand, the approach used in the present paper
would seem to be the more appropriate one for a systematic study of the general principles
of the theory of second best.
III THE THEORY OF SECOND BEST IN THE LITERATURE OF ECONOMICS
The Theory of second best has been, in one form or another, a constantly recurring
theme in the post-war literature on the discriminatory reduction of trade barriers. There
can be no doubt that the theory of customs unions provides an important case study in
the application of the general theory of second best. Until customs union theory was
subjected to searching analysis, the 'free trader'3 often seemed ready to argue that any
reduction in tariffs would necessarily lead to an improvement in world productive efficiency
and welfare. In his path-breaking work oIn the theory of customs unions4 Professor
Viner has shown that the removal of tariffs from some imports may cause a decrease in the
efficiency of world production.
One important reason for the shifts in the location of production which would follow
the creation of a customs union was described by Viner as follows :5
There will be commodities which one of the members of the customs union will
now newly import from the other, whereas before the customs union it imported them
from a third country, because that was the cheapest possible source of supply even
after payment of the duty. The shift in the locus of production is now not as between
the two member countries but as between a low-cost third country and the other,
high-cost, member country.
I
Theremay be more than one second best optimumfor any given set of constraints. See Section VIII.
Meade, J. E., Tradeand Welfare,op. cit., especially p. 96.
3 i.e., one who believes that trade carried on in the absence of any restraintsnecessarilyleads to an
optimum situation. Of course, as soon as there exist restrictionspreventingthe satisfaction of at least
one of the Paretianoptimumconditions in the domesticmarketof any of the tradingcountries,there is no
longer a case for perfectlyfree trade " . . . the generalcase for free trade rests on the contention that in a
world of utopiandomesticpolicies (i.e., where domesticeconomic policies ensure the satisfactionof all the
Paretianoptimumconditions)it sets internationallythe propermarginalconditionsfor economicefficiency".
2
Meade, J. E., Trade and Welfare, op. cit., p. 139.
4 Viner,Jacob,TheCustomsUnionIssue,New York, CarnegieEndowmentfor InternationalPeace, 1950.
6
Ibid., p. 43.
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14
REVIEWOF ECONOMICSTUDIES
Vinerused the termtradediversionto describeproductionshifts of this sort and he took
it as self-evidentthat they would reducethe efficiencyof world production. Since it is
quite possible to conceive of a customs union having only trade divertingproduction
effects, it follows, in Viner's analysis,that the discriminatoryreductionof tariffs may
reduce, ratherthan raise, the efficiencyof world production.
Viner emphasisedthe productioneffects of customs unions,' directinghis attention
to changesin the location,and hencethe cost, of world production. RecentlyProfessor
Meade has shown that a customsunion has exactlyparalleleffectson the location, and
hence the " utility" of world consumption.2 Meadeisolatesthe " consumptioneffects"
of customsunionsby consideringan examplein whichworldproductionis fixed. In this
case Viner'sproblemof the effects of a union on the cost of world productioncannot
arise. Meade arguesthat, underthese circumstances,a customsunion will tend to raise
welfareby encouragingtradebetweenthe membercountriesbut that, at the same time, it
will tend to lower welfareby discouragingthe alreadyhamperedtradebetweenthe union
area and the rest of the world. In the final analysisa customsunion will raise welfare,
lower it, or leave it unchanged,dependingon the relativestrengthof these two opposing
tendencies.3 TheViner-Meadeconclusionsprovidean applicationof the generaltheorem's
negativecorollarythat nothingcan be said a priori about the welfareand efficiencyeffects
of a changewhich permitsthe satisfactionof some but not all of the Paretianoptimum
conditions.
Anotherapplicationof second best theoryto the theoryof tariffshas been provided
by S. A. Ozgawho has shownthat a non-preferential
reductionof tariffsby a singlecountry
may lead " away from the free trade position"4 In other words, the adoption of a
freetradepolicyby one country,in a multi-countrytariffriddenworld,may actuallylower
the realincomeof that countryand of the world. Ozgademonstratesthe existenceof this
possibilityby assumingthatall commoditiesare,in consumption,rigidlycomplementary,
so
that theirproductioneitherincreasesor decreasessimultaneously. He then showsthat in
a threecountryworldwith tariffsall around,one countrymay adopt a policy of freetrade
and, as a result,the worldproductionof all commoditiesmay decrease. This is one way
of demonstratinga resultwhich follows directlyfrom the generaltheory of second best.
In the field of Public Finance,the problemsof second best seem to have found a
particularlyperplexingguisein the long controversyon the relativemeritsof directversus
indirecttaxation. It would be tedious to reviewall the literatureon the subjectat this
time. In his 1951article,I. M. D. Little5has shown that becauseof the existenceof the
" commodity" leisure,the priceof whichcannotbe directlytaxed,both directand indirect
taxes must preventthe satisfactionof some of the conditionsnecessaryfor the attainment
of a Paretianoptimum. An indirecttax on one good disturbsratesof substitutionbetween
that good and all otherswhilean incometax6disturbsratesof substitutionbetweenleisure
and all other goods. Littlethen arguesthat thereis no a priori way to judge as between
1 His neglect of the demand side of the problem allowed him to reach the erroneousconclusion that
trade diversionnecessarilyled to a decreasein welfare. It is quite possible for an increasein welfare to
follow from the formationof a customsunion whose sole effect is to diverttradefrom lower-to higher-cost
sources of supply. Furthermore,this welfare gain may be enjoyed by the country whose import trade is
divertedto the higher-costsource, by the customs union area consideredas a unit and by the world as a
whole.
See: Lipsey, R. G., " The Theory of Customs Unions: Trade Diversion and Welfare" in a forthcoming issue of Economica.
2 Meade, J. E., The Theoryof CustomsUnions,Amsterdam,the North Holland PublishingCo., 1955.
3 Ibid., Chapter III.
4 Ozga, S. A., " An Essay in the Theory of Tariffs", Journalof Political Economy,December, 1955,
p. 489.
5 Little, I. M. D., " Direct versusIndirectTaxes ", TheEconomicJournal,September,1951.
6 In this analysisan income tax may be treatedas a uniformad valoremrate of tax on all commodities
except leisure.
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THE GENERAL THEORY OF SECOND BEST
15
these two positionswheresome Paretianoptimumconditionsare satisfiedwhile othersare
not. This is undoubtedlycorrect. However,Little might have gone on to suggestthat
there is an a prioricase in favour of raisinga given amount of revenueby some system
of unequalindirecttaxes ratherthan by either an income tax or an indirecttax on only
one commodity. This interestingconclusionwas first statedby W. J. Corlettand D. C.
thatthe optimumwayto raiseanygivenamount
Hague'. Theseauthorshavedemonstrated
of revenueis by a systemof unequalindirecttaxes in whichcommodities" most complementary" to leisurehave the highest tax rates while commodities" most competitive"
with leisurehave the lowest rates.2 The reasonfor this generalarrangementof tax rates
shouldbe intuitivelyobvious. Whenan equalad valoremrateof tax is placedon all goods
the consumptionof leisurewill be too high while the consumptionof all other goods will
The consumptionof untaxedleisuremaybe discouragedby placingespecially
be too low.3
high rates of tax on commoditieswhich are complementaryin consumptionto leisure
andby placingespeciallylow ratesof tax on commoditieswhicharecompetitivein consumption with leisure.
ProfessorMeadehas recentlygiven an alternateanalysisof the same problem.4 His
conclusions,however,supportthose of Corlettand Hague. In theoryat least, the tables
have been completelyturnedand the indirecttax is provedto be superiorto the income
tax, providedthat the optimumsystemof indirecttaxes is levied.5 This conclusionis but
anotherexample of an applicationof the generaltheorem that if one of the Paretian
optimumconditionscannot be fulfilledthen a second best optimumsituationcan be obtained by departingfrom all the other optimumconditions.
Whatis perhapsnot so obviousis that the problemof directversusindirecttaxes and
that of the " consumptioneffects" of customs unions are analyticallyidentical. The
Little analysisdeals with a problemin which some commoditiescan be taxed at various
rateswhile othersmust be taxed at a fixedrate. (It is not necessarythat the fixedrate of
tax should be zero). In the theory of customsunions one is concernedwith the welfare
and efficiencyeffects of varyingsome tariff rates while leaving others unchanged. In
Little'sanalysisthere are three commodities,X, Y and Z; commodityZ being leisure.
By renamingZ home goods and X and Y importsfrom two differentcountriesone passes
immediatelyto the theoryof customsunions. An incometax in Little'sanalysisbecomes
inmportduties while a single indirect tax becomes the
a system of non-discriminatory
tariffintroducedafterthe formationof a customsunion with the producers
discriminatory
of the now untaxedimport. A model of this sort is consideredfurtherin SectionV.
An applicationof the generaltheoryof secondbest to yet anotherfield of economic
theory is provided by A. Smithies in his article, The Boundariesof the Productionand Utility
Function.6 Smithiesconsidersthe case of a multi-inputfirmseekingto maximizeits profits.
Thiswill be donewhenfor eachfactorthe firmequatesmarginalcost withmarginalrevenue
productivity. Smithiesthen suggeststhat there may exist boundariesto the production
function. These boundarieswould take the form of irreducibleminimumamounts of
I Corlett, W. J. and Hague, D. C., " Complementarityand the Excess Burden of Taxation", Review
of EconomicStudies, Vol. XXI, No. 54, 1953-54.
2
Ibid., p. 24.
3 Too high and too low in the sense that a decreasein the consumptionof leisure combined with an
increasein the consumptionof all other goods would raise the welfareof any consumer.
4 Meade, J. E., Tradeand Welfare,MathematicalSupplement,
London, Oxford UniversityPress, 1955,
ChapterIII.
6 Of course two special cases are always possible. In the first the optimum rates of tax will be equal
for all commodities(i.e., the income tax is the optimum tax). This will occur if the supply of effort (the
demandfor leisure)is perfectlyinelastic. In the second case the optimumrates will be zero for all but one
commodity (i.e., an indirecttax on one commodity is the optimum tax). This will occur if the demand
for one commodity is perfectlyinelastic.
Explorationin
6 Smithies, A., " The Boundaries of the Production and Utility Function ", in:
Economics,London, McGraw-Hill,1936.
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16
REVIEW OF ECONOMIC STUDIES
certaininputs,it beingpossibleto employmorebut not less thanthese minimumamounts.
It mighthappen,however,thatprofitmaximizationcalledfor the employmentof an amount
of one factorless than the minimumtechnicallypossibleamount. In this case production
would take place " on the boundary" and the minimumpossible amount of the input
wouldbe used. However,in the caseof thisinput,marginalcost wouldno longerbe equated
with marginalproductivity,the boundaryconditionsforcingits employmentbeyondthe
optimumlevel. Smithiesthen shows that given the constraint,marginalcost does not
equal marginalproductivityfor this input, profits will be maximisedonly by departing
fromthe conditionmarginalcost equalsmarginalproductivityfor all otherinputs. Furthermore,thereis no a priorireasonfor thinkingthat the natureof the inequalitywill be the
same for all factors. Profit maximizationmay requirethat some factors be employed
only to a point wheremarginalproductivityexceedsmarginalcost while other factorsare
usedup to a pointwheremarginalproductivityfallsbelowmarginalcost.
Problemsof the " mixed economy" provide an applicationof second best theory
frequentlyencounteredin populardiscussion. Consider,for example,a case where one
section of an economyis rigidlycontrolledby the centralauthoritywhile anothersection
is virtuallyuncontrolled. It is generallyagreedthat the economyis not functioningefficias to the appropriateremedy. Onefactionarguesthat more
entlybut thereis disagreement
controloverthe uncontrolledsectoris needed,whileanotherfactionpleadsfor a relaxation
of the degreeof controlexercisedin the publicsector. The principlesof the generaltheory
of secondbestsuggestthatbothsidesin the controversymaybe advocatinga policyappropriate to the desiredends. Given the high degreeof control in one sector and the almost
completeabsenceof controlin another,it is unlikelythat anythinglike a secondbest optimum position has been reached. If this is so, then it follows that efficiencywould be
increasedeitherby increasingthe degreeof controlexercisedover the uncontrolledsector
or by relaxingthe controlexercisedoverthe controlledsector. Both of these policieswill
move the economyin the directionof some secondbest optimumposition.
FinaLlymentionmay be made of the problemof " degreesof monopoly". It is not
intendedto reviewthe voluminousliteratureon this controversy. It may be mentionedin
passingthat,in all but the simplestmodels,a Paretianoptimumrequiresthat marginalcosts
equalmarginalrevenuesthroughoutthe entireeconomy.' If this equalityis not established
in one firm,then the secondbest conditionsrequirethat the equalitybe departedfrom in
all otherfirms. However,as is usualin secondbest casesthereis no presumptionin favour
of the samedegreeof inequalityin all firms. In general,the secondbestpositionmaywell be
one in whichmarginalrevenuesgreatlyexceedmarginalcosts in some firms,only slightly
exceed marginalcosts in others, while, in still other firms, marginalrevenuesactually
fall short of marginalcosts.
A similarproblemis consideredby LionelW. McKenziein his article" Ideal Output
of Firms."2 He dealswith the problemof increasingthe money
and the Interdependence
value of outputin situationsin whichmarginalcosts do not equalpricesin all firms. The
analysisis not conductedin a generalequilibriumsettingand manysimplifyingassumptions
are made such as the one that resourcescan be shifted betweenoccupationsas desired
without affectingtheir supplies. McKenzieshows that even in this partialequilibrium
setting if allowanceis made for inter-firmsales of intermediateproducts,the condition
that marginalcosts should bear the same relationto pricesin all firmsdoes not provide
a sufficientcondition for an increasein the value of output. Given that the optimum
condition,marginalcosts equalspricecannotbe achieved,McKenzieshows that a second
1 For example,if the supplyof effortis not perfectlyinelastican equal degreeof monopoly throughout
the entire economy has the same effect as an income tax on wage earners. Following Little's analysis it
is obvious that this tax will preventthe attainmentof a Paretianoptimum.
2 McKenzie,Lionel W., " Ideal Output and the Interdependenceof Firns ", EconomicJournal,1951.
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THE GENERAL THEORY OF SECOND BEST
17
best optimumwould requirea complex set of relationsin which the ratio of marginal
cost to pricewould vary as betweenfirms. Althoughthe analysisis not of a full general
equilibrium,the conclusionsfollow the now familiarpattern: (1) If a Paretianoptimum
cannot be achieveda second best optimum requiresa general departurefrom all the
Paretianoptimumconditionsand(2) thereareunlikelyto be any simplesufficientconditions
for an increasewhen a maximumcannotbe obtained.
IV THE THEORY OF SECOND BEST AND " PIECEMEAL" POLICY
RECOMMENDATIONS
It shouldbe obviousfrom the discussionin the precedingsectionsthat the principles
Ofthe generaltheoryof secondbest show the futilityof " piecemealwelfareeconomics".11
To applyto only a smallpart of an economywelfareruleswhichwould lead to a Paretian
optimumif they wereappliedeverywhere,may movethe economyaway from,not toward,
a second best optimumposition. A nationalizedindustryconductingits price-output
policy accordingto the Lerner-Lange" Rule " in an imperfectlycompetitiveeconomy
may well diminishboth the generalproductiveefficiencyof the economyand the welfare
of its members.
Theproblemof sufficientconditionsforan increasein welfare,as comparedto necessary
conditionsfor a welfaremaximum,is obviouslyimportantif policy recommendations
are
to be made in the real world. Piecemealwelfareeconomicsis often based on the belief
that a study of the necessaryconditionsfor a Paretianwelfareoptimummay lead to the
discoveryof sufficientconditionsfor an increasein welfare.2 In his Critiqueof Welfare
Economics,I. M. D. Littlediscussesthe optimumconditionsfor exchangeand production
both as necessaryconditionsfor a maximum,and as sufficientconditionsfor a
desirable economic change ".3
Later on in his discussion Little says " . . . necessary con-
ditionsare not veryinteresting. It is sufficientconditionsfor improvementsthat we really
want ... ."4 But the theoryof secondbest leadsto the conclusionthat thereare in general
no such sufficientconditionsfor an increasein welfare. Thereare necessaryconditions
for a Paretianoptimum. In a simplesituationtheremayexista conditionthatis necessary
and sufficient. But in a generalequilibriumsituationthere will be no conditionswhich
in generalare sufficientfor an increasein welfarewithoutalso beingsufficientfor a welfare
maximum.5
The precedinggeneralizations
maybe illustratedby consideringthe followingoptimum
conditionfor exchange: " The marginalrate of substitutionbetweenany two 'goods'
must be the same for everyindividualwho consumesthem both."8 Littleconcludesthat
this conditiongivesa sufficientconditionfor an increasein welfareprovidedonly that when
it is put into effect," . . . the distributionof welfareis not therebymadeworse."7However,
the whole discussionof this optimumcondition occurs only after Little has postulated
" . . . a fixed stock of ' goods ' to be distributedbetweena numberof ' individuals'."8
The optimumconditionthat all consumersshould be faced with the same set of prices
becomesin this-case a sufficientconditionfor an increasein welfare,becausethe problem
at handis merelyhow to distributeefficientlya fixedstock of goods. But in this case the
1 For a descriptionof this type of welfareeconomicssee I. M. D. Little,A Critiqueof WelfareEconomics,
Oxford,The ClarendonPress, 1950, p. 89.
2 Indeed any economics that attemptspiecemealpolicy recommendationsmust be based on the belief
that there can be discoveredsufficientconditions for an increasein, as distinct from necessaryconditions
for a maximumof, whateverit is that is being considered.
3 Little, op cit., p. 120.
4 Ibid., p. 129.
'This conclusionfollows directlyfrom the negativecorollarystatedin the secondparagraphof SectionI.
'Little, I. M. D., op. cit., p. 121.
7 Ibid., p. 122.
8 Ibid., 121.
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18
REVIEWOF ECONOMICSTUDIES
condition is a necessaryand sufficientcondition for a Paretianoptimum. As soon as
variationsin outputareadmitted,the conditionis no longersufficientfor a welfaremaximum
and it is also no longersufficientfor increasein welfare.
The aboveconclusionmay be illustratedby a simpleexample. Considera community
of two individualshavingdifferenttaste patterns. The " government" of the community
desiresto raisea certainsumwhichit will giveawayto a foreigncountry. The community
has made its valuejudgementabout the distributionof incomeby decidingthat each individualmustcontributehalfof the requiredrevenue. It has also beendecidedthatthe funds
are to be raisedby meansof indirecttaxes. It followsfromthe Corlettand Hagueanalysis
that the best way to raise the revenueis by a system of unequal indirecttaxes in which
" to leisure are -taxed at the highest rates while
commodities"most complementary
commodities"most substitutable"for leisureare taxedat the lowest rates. But the two
individualshave differenttastes so that commodityX is substitutablefor leisurefor individual I and complementaryto leisurefor individualII, while commodity Y and leisure
are complementsfor individualI and substitutesfor II. The optimumway to raise the
revenue,therefore,is to tax commodityX at a low ratewhenit is sold to individualI and at
a high rate when it is sold to individualII, while Y is taxed at a high rate when sold to I
but a low rate whensold to II. A secondbest optimumthus requiresthat the two individuals be faced with differentsets of relativeprices.
Assumethat the optimumtax ratesare charged. The governmentthen changesthe
tax system to make it non-discriminatory
as betweenpersons while adjustingthe rates
to keep revenueunchanged. Now the Paretianoptimumexchangeconditionis fulfilled,
but welfarehas beendecreased,for both individualshavebeen movedto lowerindifference
curves. Therefore,in the assumedcircumstances,this Paretianoptimumconditionis a
sufficientconditionfor a decreasein welfare.
V A PROBLEMIN THE THEORY OF TARIFFS
In this sectionthe simpletype of model used in the analysisof direct versusindirect
taxesis appliedto a problemin the theoryof tariffs. In the Little-Meade-Corlett
& Hague
analysisit is assumedthat the governmentraisesa fixedamountof revenuewhichit spends
in some specifiedmanner. The optimumway of raisingthis revenueis then investigated.
A somewhatdifferentproblemis createdby changingthis assumptionaboutthe disposition
of the tax revenue. In the presentanalysisit is assumedthat the governmentreturnsthe
tax revenueto the consumersin the form of a gift so that the only effectof the tax is to
change relativeprices.1
A simplethreecommoditymodel is used, there being one domesticcommodityand
two imports. It is assumedthat the domesticcommodityis un-taxedand that a fixedrate
of tariffis levied on one of the imports. The optimumlevel for the tariffon the other
importis theninvestigated. Thisis an obviousproblemin the theoryof secondbest. Also
it is interestingto note that the conclusionsreachedhave immediateapplicationsto the
theory of customs unions. In the second part of this section the conclusionsof part A
are appliedto the problemof the welfareeffectsof a customsunion whichcausesneither
tradecreationnor tradediversion,but only the expansionand contractionof the volumes
of alreadyexistingtrade.
A.
SECOND BEST OPTIMUM TARIFF SYSTEMS WITH FIXED TERMS OF TRADE:
The conditionsof the model are as follows: CountryA is a small countryspecializing in the productionof one commodity(Z). Some of Z is consumedat home and the
1 If consumers have different utility functions then each consumer must receive from the government
an amount equal to what he pays in taxes. However, if all consumers have identical homogeneous utility
functions then all that is required is that the tax revenue be returned to some consumer or consumers.
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THE GENERAL THEORY OF SECOND BEST
19
remainderis exportedin returnfor two imports,X fromcountryB and Y fromcountryC.
The pricesof X and Y in termsof Z are unaffectedby any taxesor tariffsleviedin country
A. It is furtherassumedthat none of the tariffsactuallylevied by A are high enoughto
protect domesticindustriesproducingeither X or Y,1 that countryB does not produce
commodity Y and that country C does not produce commodity x.2 The welfare of
countryA is definedby a communitywelfarefunctionwhich is of the same form as the
welfarefunctionsof the identicalindividualswho inhabitA.
It is assumedthat A levies some fixed tariff on importsof commodity Y and that
commodityZ is not taxed. It is then asked: Whattariff(50) on importsof commodity
X will maximizewelfarein countryA ? This tariffwill be termedthe optimumX tariff.
The model may be set out as follows: Let therebe threecommodities,X, Y and Z.
Let Px and py be the prices of X and Y in terms of Z. Let the rate of ad valoremtariff
charged on X and Y be tx u
=
au
1 and ty -1.4
(5.1)
u(x, y, z)
ax =a
au
(5.2-a)
Xx(.<
au
au
(5.2-b)
az
(5.3)
XPX+ Ypy+ Z = C
Equation(5.1) expressescountryA's communitywelfarefunction. Equations(5.2-a and
-b) are the demand equilibriumconditions. Equation (5.3) gives the condition that
A's internationalpaymentsbe in balance.5
These equationswill yield a solution in generalfor any tx and ty, in X, Y and Z.
Hence, for givenPx,py, C and whateverparametersenterinto (5.1):
(5.4-a)
X = f(txQty)
(5.4-b)
y = g Q, ty)
(5.4-c)
Z = h (tx, ty)
of
in
when
with
>
changes
the
the
U
change
to
sign
Attentionis directed
ty 1 kept
t,
constant. From equations(5.1) and (5.4):
ay
au
au
ax
au ay
au az
(5 5)
ax
atx
atX
Substitute (5.2-a and -b) into (5.5):
au ax
t -px tx g:*aX+
au
-t
aidxat
a
ax
au ay
au az
ty vz* rtx+ AZ* t
PY
txrtx+ pytyay~~
aat- +
zu(px
az_~
atZ-
(5.6)
(5.6)
s6
therecan be no questionof a reductionof A's tariffscausingtradecreation.
Therefore,
2Therefore, there can be no question of a preferentialreductionof A's tariffscausing trade diversion.
Obviously,this is a problemin the theory of second best. The initial tariffon Y causesthe consumption of Yto be too low relativeto both X andZ. If the consumptionof Ycan be encouragedat the expense
of X, welfare will be increased. However, if the consumptionof Z is encouragedat the expense of X,
welfarewill be lowered. A tariffon X is likely to cause both sorts of consumptionshift and the optimum
X tariffwill be that one where,at the margin,the harmfuleffect of the shift from X to Z just balancesthe
beneficialeffect of the shift from X to Y.
4We are greatly indebted to Dr. George Morton for suggestingthe following mathematicaldemonstration. It now replacesa much more cumbersomedemonstration.
6 i.e. The value of imports(Xp. + Ypy)plus the value of domesticproductionconsumedat home (Z)
equals the total value of domesticproduction(C).
8
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20
REVIEW OF ECONOMIC STUDIES
Next, take the partial derivative of (5.3) with respect to tx.
ay
ax
az
°
px -tx + Ppy a + tx
or:
ax
ay
px x + py =t_
-
az
(5.7)
xa
Substitute (5.7) into (5.6) :
Ax
ax
azu/
au
\
ay
+ pyty
-pX a-t - PY etJ
Pxtx
T
tYt~-- a-z
ax
=au
ay
= az [ px 23tX
(X-1) + py aYi(ty-1)
1
(5.8)
It is assumed, first, that some tariff is levied on Y but that X is imported duty free. Therefore, tx = 1 and ty > 1. Equation (5.8) reduces to:
au
au
ay (t)
In (5.9)
a-
(5 9)
ay
ty
a
aax
takes the same sign as Y .1 It follows that the introduction of a margin-
C~atx
edatx~
al tariff on Xwill raise welfare if it causes an increase in imports of commodity Y, will leave
welfare unchanged if it causes no change in imports of Y and will lower welfare if it causes
a decrease in imports of Y. Therefore, the optimum tariff on X is, in fact, a subsidy, if
imports of Y fall when a tariff is placed on X, it is zero if the X tariff has no effect on
imports of Y and it is positive if imports of Y rise when the tariff is placed on X.
It is now assumed that a uniform rate of tariff is charged on X and Y. Therefore,
tx = ty = T and equation (5.8) becomes:
au
au
(
a_l x
(T- 1) (pX a
ay
+ PY aT)
Substituting from (5.7) :
az(Tl)I
au =
[au
it-
O@z Atx(T-1]
au
(5-10)
az
In (5.10) the sign of at will be opposite to the sign of -t. It follows that a marginal
increase in the tariff on X will increase welfare if it causes a decrease in the consumption
of Z, will leave welfare unchanged if it causes no change in the consumption of Z and will
lower welfare if it causes an increase in the consumption of Z. It may be concluded,
therefore, that the optimum tariff on X exceeds the given tariff on Y if an increase in the
X tariff reduces the consumption of Z, that the optimum X tariff equals the given Y tariff
if there is no relation between the X tariff and the consumption of Z and that the optimum
X tariff is less than the given Y tariff if an increase in the X tariff causes an increase in
consumption of Z.
In the case where an increase in the tariff on X causes an increase in the consumption
of Y and of Z the optimum X tariff is greater than zero but less than the given tariff on Y.
1 These relationships are not as simple as they might appear.
If worked out ay and
found to be of the same order of complexity as are the Qi's in Section IX.
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aZ
would
found
sinSection
IX.
toxthe
be
THE GENERAL THEORY OF SECOND BEST
21
ANDTRADE
ONLY
TRADE
EXPANSION
UNIONCAUSING
B. WELFARE
OFA CUSTOMS
EFFECTS
CONTRACTION:
It is assumedthat countryA initiallychargesa uniformad valoremrate of tariffon
importsof X and Y. A then formsa customsunionwith countryB. Now X is imported
dutyfreewhilethe pre-uniontariffstill appliesto Y. Whatis the effecton A's welfareof
such a customsunion ? Some answers'follow immediatelyfrom the previousanalysis:
Case 1 : Any increasein the tariffon X causesa fall in the consumptionof Y. The
optimumtariff on X is, in fact, a subsidy. Therefore,the customsunion must raiseA's
welfare.
Case 2: Variationsin the tariff on X have no effect on consumptionof Y. The
optimumtariffon X is now zero. The customsunion raiseswelfarein A. Furthermore,
it raisesit to a secondbest optimumlevel (assumingthat only the X tariffcan be varied).
Case 3: Variationsin the tariff on X have no effect on the purchasesof Z. The
optimumtariff on X is equal to the Y tariff. The customs union lowers A's welfare.
Furthermore,the union disturbsan alreadyachievedsecond best optimum.
Case4: An increase,in the tariffon X causesa fall in the consumptionof Z. In this
case the optimumtariff on X exceedsthe given Y tariff. Therefore,the customs union
lowers A's welfare.
Case5: An increasein the tariffon X causesan increasein the consumptionof both
Y and Z. The optimumX tariff is greaterthan zero but less than the given Y tariff.
The effect of the customs union on welfareis not known. Assume, however,that the
X tariffis removedby a seriesof stages. It followsthat the initialstagesof tariffreduction
mustraisewelfareand that the final stagesmust lowerit. Althoughnothingcan be said
aboutthe welfareeffectof a completeremovalof the X tariff,anotherimportantconclusion
is suggested. A small reductionin tariffs must raise welfare. A large reductionmay
raise or lower it. It follows, therefore,that a partialpreferentialreductionof tariffsis
morelikelyto raisewelfarethan is a completepreferentialeliminationof tariffs. Of course,
this conclusiondependsupon the specificassumptionsmade in the presentmodel but it
3
does providean interestingand suggestivehypothesisfor furtherinvestigation.2
VI NATIONALISED INDUSTRY IN AN ECONOMY WITH MONOPOLY:
A SIMPLE MODEL
An interesting,and not unlikely,situationin which a " second best" type of policy
mayhaveto be pursuedis that of a mixedeconomywhichincludesboth nationalisedindustries and industrieswhich are subjectto monopolycontrol.
The monopoly is assumedto be one of the data: for one reason or anotherthis
monopolycannot be removed,and the task of the nationalisedindustryis to determine
that pricingpolicy whichis most in " the publicinterest".
When there is full employmentof resourcesthen, if the monopoly is exercisingits
power,it will be producingless of the monopolisedproductthan is requiredto give an
optimum(in the Paretiansense)allocationof resources. Sincethereis less than the optimumproductionof the monopolisedgood, therewillbe morethanthe optimumproduction
of the non-monopolisedgoods as a group.
1 Only the most obvious applicationsof the conclusionsreachedin part A are given in this part. This
is not the place for a detailedreport on originalwork in the theory of customs unions.
2 This conclusion is also reached by Professor Meade, See: The Theory of Customs Unions, op. cit.,
p. 51.
a Another conclusion suggestedby the analysisis that the higher are the tariffsreducedby the union
relativeto all othertariffs,the more likelyis it that the union will raisewelfare,c.f. Meade,op. cit., pp. 108-9.
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22
REVIEWOF ECONOMICSTUDIES
Suppose that one of the non-monopolisedindustriesis now nationalised. What
shouldbe its price/outputpolicy ? If it behavescompetitivelythen it will tendto produce
more of its product,relativeto the monopolisedgood, than the Paretianoptimumwould
require. If, on the otherhand,it behavesmonopolisticallyitself,then it will cut down the
excess of its own productionrelativeto that of the monopolybut will increasethe excess
of theremaininggoodsrelativeto bothits ownproductandthatof the monopolisedindustry.
This is a typical" secondbest" situation: any policy will make some things worse and
some better.
It is clearthat no policyon the partof the nationalisedindustrycanrestorethe Paretian
optimum,for the existenceof the monopoly preventsthis. The nationalisedindustry
must aim at a secondbest policy, designedto achievethe best that still remainsopen to
the economy. In purelygeneraltermsit is impossibleto be moredefinitethanthis, as will
be shown in sectionIX. Intuitively,however,one might expect that, in some situations
at least, the best policy for the nationalisedindustrywould be to behavesomethinglike
the monopoly,but to a lesserextent. In the case of the simplemodelto be presentedin
this section, one's intuitionswould be correct.
There are assumedto be, in the present model, three industriesproducinggoods
x, y, z. Labouris the only input, costs are constant,and the total supply of labour is
fixed. These assumptionsdefine a unique linear transformationfunction relatingthe
quantitiesof the three goods:
ax + by + cz
=
(6.1)
L
The productionfunctionsfrom whichthis is derivedare:
bIy Z =
x = aIlx, Y-
Ilz;
(5.2)
llx + ly + z=L.
The marginalcosts are constant and proportionalto a, b, c.
The " publicinterest" is assumedto be definedby a communitypreferencefunction,
whichis of the sameformas the preferencefunctionsof the identicalindividualswho make
up the society. For simplicity,this preferencefunctionis assumedto take the logarithmic
form:
(6.3)
U - xa y zY, oc, , y > O
The partialderivativesof this are:
au
uau
_
ax
Mx' ay
u au
y'
u
az
-
so that the marginalutilitiesof x, y, z are proportional,respectively,to I- I z* For a
y'
and
utility function of this type, all goods are substitutesin both the Edgeworth-Pareto
Hicksiansenses.
If therewere no constraintsin the economy(otherthan the transformationfunction
itself), the Paretianoptimumwould be that found by maximisingthe expression,U),(ax + by + cz - L), whereX is the Lagrangianmultiplier.This would lead to the three
equations:
au
au
)a
0
_.
au JC
Z-0
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(6.4)
mc
THE GENERAL THEORY OF SECOND BEST
which can be expressedin the proportionalform:
b
a
x
-x = y = -z
23
(6.5)
Theseconditionsare of the familiarParetiantype, namelythat the marginalutilities
(or priceswhich, assumingthe ordinaryconsumerbehaviourequations,are proportional
to them)areproportionalto the marginalcosts. Therebeingno monetaryconditions,and
the supplyof labourbeingfixed,equalitybetweenpricesand marginalcosts is not necessarilyimplied.
Suppose now that the industryproducingx is a monopoly. The monopoly will
set the priceof x higher(in termsof somenumeraire,whichwill be takento be z) in relation
to marginalcost thanin the conditionsof the Paretianoptimum. A numeraireis necessary
since money, and money prices,are not being considered.
For the presentpurposes,the exact marginbetweenmarginalcost and price in the
monopolisedindustry(relativeto the numeraire)does not matter. It is necessaryonly
for the problemthat the monopolistset the pricesof x higher,relativeto the price of z,
than the ratio of the marginalcost of prodcuingx to the marginalcost of producingz.
In other words, the monopolist'sbehaviourcan be expressedby:
mcx
mcz
px
Pz
Substituting for
Px
aL 1W
au =- cz
- = - )
xx az
Pz\
oaz
yx/
and
mcx I
a\
, this gives:
mrcc(= c
a
c
yx
cc'z > ayx
= kayx where k > 1
(6.6)
The actualvalue of k (providedit is > 1) does not matterfor the analysis. It is not
necessaryfor the argumentthat k is constantas the monopolistfaces the changesbrought
about by the policiesof the nationalisedindustries,but it simplifiesthe algebrato assume
this.
The behaviourof the monopolist,assumedunalterable,becomesan additionalconstrainton the system. The best that can be done in the economyis to maximiseU subject
function(6.1)and the monopolybehaviourcondition
to two constraints,the transformation
(6.6). The conditionsfor attainingthe secondbest optimum(the Paretianoptimumbeing
no longerattainable)arefound,therefore,as the conditionsforthe maximumof the function
U - Vz(coz- kayx) - X'(ax + by + cz - L), where there are now two Lagrangean
multipliers V.,X'. Neither of these multipliers can be identified with the multiplier X in
the equations(6.4).
The conditionsfor attainingthe second best are, therefore:
a--kay
x
-X'a
-X'b
y(BU
y- ± ,ucoc-)X'c
0=
(6.7)
= 0
(6.8)
=0
(6.9)
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24
REVIEW OF ECONOMIC STUDIES
To appreciate these conditions, it is necessary to compute the ratio P-,
Pz
compare it
with the ratio -y, and relate the result to both the Paretian optimum conditions and the
mcz
mode of behaviour of the monopolist.
Although there are three equations (6.7), (6.8), (6.9) above, these involve the two
Lagrangean multipliers, so that there is actually only one degree of freedom. Hence,
the policy of the nationalised industry (that which produces y) is sufficient for attaining
the second best. If the nationalised industry sets its price, relative to its marginal cost, so
as to satisfy the above conditions, it will have done all that is within its power to further
the public interest.
To complete the solution it is necessary to determine ,u and X'.
From (6.7)
Vkayx = oU - X'ax
(6.10)
= yU - X'cz
and from (6.9)
-coz
(6.11)
=
Hence,
(a + y)U- X'(ax + cz)
{.(kayx cmz)
but, from (6.6), kayx - caz = 0
so that,
(a + y)U -X'(ax + cz) = 0
(a + y)U
(612)
ax + cz
Substituting for X'in (6.10)
a
,kayx
(o
+
ax + y)U
cz
yax
ax + cz
cacz-
(k k -i
k
tL
1) axyax cz U
+
U
ax + cz
[caz = kyax, from (6.6)]
(6.13)
The correct pricing policy for the nationalised industry is given from the ratio pPz
which is implicit in the equations (6.7), (6.8), (6.9).
au
py
aY
AU
Pz
8z
U
U
yz
- mcc ±Xb'c
b
-
a
c-7,cm
[From (6.8), (6.9)]
[From (6.12), (6.13)]
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THE GENERAL THEORY OF SECOND BEST
25
b
(6.14)
kk
Now
M+y
MCy' from (6.2), so that:
p
Pz
MCkMCz
(1/*- k-I
I
*
Considerthe expression( k-
k
-·
).
U+Y
\ )
Since k > 1,
(6.15)
<
k-
k
< 1, and
_+Y
< 1 sincey > 0. Thusthe bracketedexpressionon the righthandside of (6.15)is greater
thanunity.
In other words,P- > -C, so that, relative to the num6raire,the nationalised
Pz
MCz
industryshouldset its pricehigherthan its marginalcost and, to that extent, behavelike
the monopoly.
Butnow considerthe relationshipbetweenthe nationalisedindustryandthe monopoly.
U
y
U
x
Py
px
b
a
-, ky +
1
b
a
k-
.y
b
a+y
(6.16)
a oa+ ky
Inthiscase,
since
k>1,
the nationalised
In this case, since k > 1, o,,y > 0,
<1.MCySincethe nationalised
a
±
+ Cy
MCx
industryshould set its price less high, in relationto marginalcost, than the monopoly.
In short,in the particularmodel analysed,the correctpolicy for the nationalisedindustry,with monopoly entrenchedin one of the other industries,would be to take an
intermediatepath. On the one hand, it should set its price higher than marginalcost
(relativeto the numeraire)but, on the other hand,it shouldnot set its price so far above
marginalcost as is the case in the monopolisedindustry.
These conclusionsrefer,it should be emphasised,to the particularmodel which has
been analysedabove. This model has many simplifying(and thereforespecial)features,
includingthe existenceof only one input, constantmarginalcosts and a specialtype of
utility function. As is demonstratedlater,in SectionIX, there can be no a prioriexpectations about the natureof a second best solution in circumstanceswhere a generalised
utilityfunctionis all that can be specified.
,
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26
REVIEW OF ECONOMIC STUDIES
VII A GENERAL THEOREM OF THE SECOND BEST
Let there be some function F(x . . Xn)of the n variables xl .... xn, which is to be
maximised (minimised) subject to a constraint on the variables (D(xl .... Xn) = 0. This
is a formalisation of the typical choice situation in economic analysis.
Let the solution of this problem-the Paretian optimum-be the n-I condition
Then the following theorem, the theorem of the
fQ(xl .... xn) = 0, i = I .... n-1.
second best, can be given :
If there is an additional constraint imposed of the type fQ = 0 for i = j, then the
maximum (minimum) of F subject to both the constraint (D and the constraint Qf 0
will, in general, be such that none of the still attainable Paretian conditions £f = 0, i # j,
will be satisfied.
PROOF:
In the absence of the second constraint, the solution of the original maximum (minimum) problem is both simple and familiar. Using the Lagrange method, the Paretian
conditions are given by the n equations:
i = I .... n
Ft- X(--Di 0
(7.1)
Eliminating the multiplier, these reduce to the n-1 proportionality conditions:
i =
=
....
I --
I
(7.2)
where the n'th commodity is chosen as numeraire.
The equations (7.2) are the first order conditions for the attainment of the Paretian
optimum. Now let there be a constraint imposed which prevents the attainment of one
of the conditions (7.2). Such a constraint will be of the form (the numbering of the
commodities is, of course, arbitrary):
-
-
k
kI
(7.3)
It is not necessary that k be constant, but it is assumed to be so in the present analysis.
There is now an additional constraint in the system so that, using the Lagrangean method,
the function to be maximised (minimised) will be:
F-
-
,(- F_
k (D)
(7.4)
The multipliers X', ,t will both be different, in general, from the multiplier X in (7.1).
The conditions that the expression (7.4) shall be at a maximum (minimum) are as
follows :
Ft -
(FnFi-_FiFni
X"-si
If the expression Fn-l
Fn
FFn
1
-
k(n^Dif
-
10ni=
0
i = I ... n (7.5)
is denoted by Qi and the equivalent expression for the
P's by Rj, then the conditions (7.5) can be re-written in the following form:
I +
Fn
n
[
+
(Qt - kRt)
(Qn-
kRn)]
These are the conditions for the attainment of the second best position, given the
constraint (7.3), expressed in a form comparable with the Paretian conditions as set out
in (7.2).
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THE GENERAL THEORY OF SECOND BEST
27
Clearly,any one of the conditionsfor the secondbestwill be the sameas the equivalent
Paretian condition only if the expression:
1+
-(Qi
kRi)
is unity.
1 + xi (Qn - kRn)
Now this will only be the case if:
(i) t = 0
or
(ii) ,u=# 0, but Q -kR = Qn -kRn
The first of these cannot be true for, if it were,then, when i =- 1
Fn
would be equal to
in contradictionwith the constriantcondition(7.3).
It is clear from the natureof the expressionsQi, Q., Ri, R. that nothingis known,
in general,about their signs, let alone their magnitudes,and even the signs would not be
sufficientto determinewhether(ii) was satisfiedor not.
ConsiderQ. - FnF F
If F werea utilityfunctionthen it wouldbe known
*Fn
that F1, F. were positiveand Fnnnegative,but the sign of F,. may be eitherpositive or
negative.' Even if the sign of F,. were known to be negative,the sign of Qn would still
be indeterminate,since it would dependon whetherthe negativeor the positiveterm in
the expressionwas numericallythe greater. In the case of Qi, wherei : n, the indeterminacy is even greater,since there are two expressionsFi1 and Fi for which the signs
may be either positive or negative.
The same considerationsas apply for the Q's also apply for the R's of course. In
general,therefore,the conditions-for the secondbest optimum,given the constraint(7.3),
willall differfromthe correspondingconditionsfor the attainmentof the Paretianoptimum.
Conversely,given the constraint(7.3), the applicationof these rules of behaviourof the
Paretiantype whichare still attainablewill not lead, in general,to the best positionin the
circumstances.
The generalconditionsfor the achievementof the second best optimumin the type
of case with whichthis analysisis concernedwill be of the type
F
kjA1, so that F
D
~Fi
=
(i
F,and
Fk
=A
(Dk
-
=
k
ki
where ki#
the usual Paretian rules will be
brokenall round.
VIII THE EXISTENCEOF A SECOND BEST SOLUTION
The essentialconditionthat a truesecondbest solutionto a givenconstrainedsituation
shouldexist is that, if thereis a Paretianoptimumin whichF has a maximum(minimum)
whenthe constraintis removed,then the expression(7.4) must also have a true maximum
(mnimum). Thereis no reasonwhy this should,in general,be the case.
For one thing, whereaswell-behavedfunctionsF and D will alwayshave a solution
which satisfiesthe comparativelysimple first order conditionsfor a Paretianoptimum,
I The Hicksian definitionsof complementarityand substitutiongive no informationabout the signs
of the individualFij's, whereF is a utility function. The Edgeworth-Paretodefinitionsdo, and sectionIX
considersthe extent to which the knowledgeof these signs enablesa priori statementsto be made abouitthe
nature of second best solutions.
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28
REVIEW OF ECONOMIC STUDIES
it is by no means certain that the much more complex first order conditions (7.5) for a
second best solution will be satisfied, since these conditions involve second order derivatives
whose behaviour (subject only to convexity-concavity conditions of the functions) is
unknown.
If the first order conditions for the existence of second best solutions present difficulties,
the difficulties are quite insurmountable in the case of the second order conditions. Let it
be supposed, for concreteness, that the nature of the case is such that F is to be maximised.
Then the existence of a second best solution requires that the first order conditions (7.5)
shall give a maximum, not a minimum or a turning point. This requires that'the second
differential of the expression (7.4) shall be negative. But the second differential of (7.4)
involves the third order derivatives of F and (D. Absolutely nothing is known about these
in the general case, and their properties cannot be derived from the second order condition
that the Paretian optimum represents a true maximum for F.
IX
THE NATURE OF SECOND BEST SOLUTIONS
The extraordinary difficulty of making a priori judgments about the types of policy
likely to be required in situations where the Paretian optimum is unattainable, and the
second best must be aimed at, is well illustrated by examining the conditions (7.6) in the
light of possible knowledge about the signs of some of the expressions involved.
In order to simplify the problem, and to render it less abstract, the function F will be
supposed to be a utility function and 0, which will be supposed to be a transformation
function, will be assumed to be linear. The second derivatives of (D disappear, so that
R -= 0 for all i, and attention can be concentrated on the expressions Q.
With the problem in this form, the derivatives Fi are proportional to the prices pi,
and the derivatives (i are proportional to the marginal costs MCi. As an additional
simplification which assists verbal discussion but which does not affect the essentials of
the model, it will be supposed that price equals marginal cost for the n'th commodity,
which will be referred to as the num6raire.
From (7.6), with these additional assumptions, therefore:
1+
Q
Fi
Ppi
pn
+
i Fn_
OQi
(9.1))
MC
MCf
1 + OQn
(Di
1 + ,, Qn
<>n~ MCn
/ above \
Thus, for the i'th commodity, price is
equal to marginal cost, when the second best
\below /
optimum is attained, according as:
1
OQi
1 + OQn 2
The problem is reduced to that of discovering what can be said, apriori, about the magnitude
of this expression.
p _
n
F1Fn
Now Qi F=FnFlNQ1- Fn 2 A1ni At most, it may be possible to deduce the sign of Qi
but the order of its magnitude will remain unknown unless a specific utility function is
given.
With knowledge of signs, and no more, the most that can be said can be summarised
very simply :
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29
THE GENERAL THEORY OF SECOND BEST
(i) If 0 > 0, P > I if Qi > O0Qn < 0.
(9.2)
P< 1 if Q1 <0O Qn> 0.
0.
1
>
P
if
0,
<
If
0
>
0,
<
(ii)
Qi
Q.
P < 1 if Q, > 0, Qn < 0.
Nothing can be said aboutP if Qi, Qnare of the samesigns.
Now considerQi. The denominatoris alwayspositive,and F1,Fn are both positive,
so that the determiningfactorsarethe signsof the mixedpartialderivativesF1landFni It is
assumedthat goods are known to be substitutes(Fi; < 0) or complements(Fi; > 0) in
sense. Thereare four possiblecases
the Edgeworth-Pareto
(a) If F1 > O,Fni > 0, then Qi ; 0
(b) If Fli < 0, Fn1< 0, then Q- ! 0
(c) If F1i> 0, Fni < 0, then Qi > 0
(d) If F1i <0, Fni > 0, then Qi <0.
In cases (c) and (d), but not in cases (a) and (b), therefore,the sign of Qi is determinate.
To completethe picturethe sign of 0 is also needed. Wherethe sign of this can be
foundat all, it is foundby puttingi = 1 and substitutingin the constraintcondition(7.3).
For concreteness,let k be > 1 (the firstgood will be referredto as the monopolisedgood).
0Q1= k > 1, it can be deducedthat, if Q1< 0, Qn > 0, then 0 < 0,
Then, since I-+
1 + 0Qn
and if Q1> 0, Qn < 0, then 0 < 0. In all other cases the sign of 0 is indeterminate.
For Q1,Qn,it is knownthat Fll, Fnn < 0, and Fn1= Fln so that thereare only two
cases, Fnl1> 0 and Fn1 < 0. The informationconveyedin each of the two cases is as
follows:
I F1 > 0: Q1 < 0, Qn> 0, so that 0 < 0.
II Fnl < 0 : Q,
0Qn t 0, so that 0 < 0.
The combinationof cases I and II with the independentlydeterminedcases (a), (b),
(c), (d) gives a total of eight cases. These are tabulatedbelow, showingthe information
which can be derivedabout the signs of Qi, Qnand 0, and the consequentinformation
about P using the conditions(9.2).
TABLE I
Sign of
Case
1 Fnl>0 (a) Fij, Fni>O
(b)Fij,Fni<O
(c) F1i>0, Fni<O
(d) F1l<0, Fni> 0
Qi
-?
?
+
Qn
0
+
+
+
+
Relationship of Price
to marginal cost for xi
?
?
?
Price exceeds marginal
cost
II Fni<0 (a) Fli, Fni>O
(b) Fli, Fn <0
(c) Fpi>0, Fn<O
(d) F1l<0, Fn>0
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
Of the eightcasestabulated,the signsof Qi, Qnand 0 are simultaneouslydeterminate
in only two, I(c) and I(d),and in only one of thesetwo, I(d), does this leadto a determinate
relationshipbetweenprice and marginalcost. This sole case leads to the only a priori
statementthat can be made about the natureof secondbest solutionson the basis of the
signsof the mixedsecondorderpartialderivativesof the utilityfunction:
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30
REVIEWOF ECONOMICSTUDIES
If the monopolisedcommodityis complementary(in the Edgeworth-Paretosense)
to the numeraire,and the i'th commodityis also complementaryto the num6raire,but a
substitutefor the monopolisedgood, then,in orderto attaina secondbestsolution,the price
of the i'th commoditymust be set higherthan its marginalcost.
Since knowledgeof the sign alone of the derivativesF0 revealsonly one determinate
case, it would seem worth while to examinethe situationif more heroic assumptionscan
be madeaboutthe knowledgeof the utilityfunction. The additionalinformationwhichis
assumedis that two commoditiesmay be knownto be " weaklyrelated", that is, that the
derivativeFij is eitherzero or of the secondorderrelativeto otherquantities.
In the expressionQF =2
'
-
, for example,if the i'th commodityand the
numeraireare weaklyrelatedin this sense,then the term F1 Fji can be neglectedrelative
to the termF. F1l,andthe sign of Qi is whollydeterminedby the sign of Fli.
If the monopolisedgood andthe numeraireareweaklyrelated,then Q, < 0 and Q.> O.
This is similarto the case I, in which the two goods were complements,leadingto the
same conclusions. There are now, however, four additionalcases to add to (a), (b),
(c), (d), for variouscombinationsof weakrelatednesswithsubstitutionandcomplementarity
as betweenthe i'th commodityand the monopolisedgood and the numeraire. All the
cases whichcan be givenin termsof the threerelationships(weaklyrelated,complements,
substitutes)are tabulatedin Table II. Thereare now three determinatecases, which can
be summarisedas follows:
If the monopolisedgood and the numeraireare eithercomplementsor only weakly
related,then the second best solution will certainlyrequirethe price of the i'th good to
be set above its marginalcost eitherif the good is a substitutefor the monopolisedgood
or only weaklyrelatedto the num6raire,or if the good is weakly
and eithercomplementary
relatedto the monopolisedgood but complementaryto the num6raire.
Withany othercombinationsof relatednessamongthe goods,it cannotbe determined,
a priori,whetherthe secondbest solutionwill requirethe price of any particulargood to
be above or belowits marginalcost. In particular,if thereis no complementarity
between
TABLEII
Relationshipbetween
monopolisedgood
and numeraire
Complements,or
weak
Relationshipof i'th good to
Numeraire
Monopolised
Good
Complements Complements ? +
? +
Substitutes
Substitutes
Complements Substitutes
+ +
Substitutes
Substitutes
Signs of
Qi Qn 0
Complements
Price of i'th
good relative
to marginal
cost
-
?
?
?
Higher
-
+
-
Complements Weak
+
+
-?
Substitutes
Weak
Weak
Complements
-
+
+
-
Higher
Higher
Weak
Substitutes
+
+
-
?
Any
Any
?
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-
?
?
THE GENERAL THEORY OF SECOND BEST
31
any pairs of goods, and the relationshipbetweenthe monopolisedcommodityand the
num6raireis not weak, then there are no determinatecases.
As a matterof interestit is possibleto workout conditionsthat may be likelyto bring
aboutanyparticularresult. For example,a possiblecasein whichthe priceof a good might
be set belowits marginalcost wouldbe thatin whichthe nronopolisedgood, the num6raire,
and the other good were all substitutes,but the rate at whichmarginalutility diminished
was small in the case of the monopolisedgood (so that Q1,Q. would both be positive,
with Q1largecomparedwith Qn,givinga positivevaluefor 0), and the relationshipof the
good under discussionwas much strongerwith the monopolisedgood than with the
numeraire(so that Qi might be negative). Therecan be few real cases, however,where
such guessesabout the-magnitudesof the quantitiesinvolvedcould be made.
X THE PROBLEMOF MULTIPLE-LAYEROPTIMA
In all the precedinganalysis,the problemshave been conceivedin termsof a singlelayer optimum. It has been assumed that the constraintwhich defined the Paretian
optimum(the transformationfunction,for example)was a technicallyfixed datum, and
was not, itself,the resultof an optimisationprocessat a lowerlevel.
The characteristicof generaleconomicsystemsis, however,that they usuallyinvolve
severalsuccessiveprocessesof optimisation,of increasinggenerality. The transformation
function,for example,may havebeen derivedas the resultof competitivefirmsmaximising
theirprofits. Firmsare assumedto have minimisedtheircosts beforeprocedingto maximise theirprofits,and these costs are themselvesderivedfrom processesinvolvingoptimisationby the ownersof the variousfactorsof production.
It is of the natureof the economicprocess,therefore,that optimisationtakesplace at
successivelevels,andthatthe maximisationof a welfarefunctionsubjectto a transformation
functionis only the topmost of these. It is also of the natureof Paretianoptima (due
to the simpleproportionalityof the conditions)that the optimisationat the differentlevels
can be consideredas independentproblems.
In the case of a secondbest solution,however,the neatproportionalityof the Paretian
conditionsdisappears: this immediatelyposesthe questionwhethera secondbest solution
in the circumstancesof a multiple-layereconomicsystem will requirea breakingof the
Paretianconditionsat lowerlevelsof the system,as wellas at the levelat whichthe problem
was initiated.
The presentpaperdoes not proposeto examinethe problem,for it is a subjectthat
would seem to merit full-scaletreatmentof its own. There seems reason to suppose,
however,thattheremaywellbe casesin whicha breakingof the Paretianrulesat lowerlevels
of the process(movingoff the transformationfunction,for example)may enablea higher
level of welfareto be obtainedthanif the scope of policyis confinedto one level only.
A two-dimensionalgeometricillustrationthat is suggestive,althoughnot conclusive,
is set out in Figure 1. Ox, Oy representthe quantitiesof two goods x, y. The line AB
representsa transformationfunction(to be consideredas a boundarycondition)and CD
a constraintcondition. In the absenceof the constraintCD the optimumpositionwill be
some point, such as P, lying on the transformationline at the point of its tangencywith
one of the contoursof the welfarefunction.
If the constraintconditionmustbe satisfied,only pointsalong CD can be chosen,and
the optimumpoint P is no longer attainable. A point on the transformationline (Q) is
still attainable. Will the secondbest solutionbe at the point Q, or should the economy
move off the transformationline ? If the welfarecontoursand the constraintline are as
shownin the diagram,then the secondbest point will be at the point R, insidethe transformationline.
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REVIEW OF ECONOMIC STUDIES
32
y
A
0
B
D
FIo.
X
1
It is obvious,of course,thatthe secondbestwill neverbe at a pointwhichis technically
inefficient(has less of one commodityand no more of the other)relativeto any attainable
point. Althoughthereare points (the segmentMN) on the transformationline whichare
technicallymoreefficientthan R. these are not attainable. R is not technicallyinefficient
relativeto Q, even though R lies inside the transformationline.
If the line CD had a positiveslope (as have the types of constraintwhichhave been
exemplifiedin the precedinganalyses),the second best would always lie at its point of
intersectionwiththe transformation
line, sinceall otherpointson CD wouldbe technically
inefficientrelative to it.
London.
R. G. LIPSEY.
KELVIN LANCASTER.
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