Table Of ContentLecture Notes in Economics
and Mathematical Systems 661
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PierCarlo Nicola
Efficiency and Equity
in Welfare Economics
123
Prof.PierCarloNicola
Universita`diMilano
Dipto.Matematica
Milano
Italy
ISSN0075-8442
ISBN978-3-642-30070-7 ISBN978-3-642-30071-4(eBook)
DOI10.1007/978-3-642-30071-4
SpringerHeidelbergNewYorkDordrechtLondon
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(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Preface
The main stimulus for presenting this short monograph is linked to one of my
first conceptual interest which can be traced back to the beginning of my studies
of political economy and economic policy. A topic that caught my attention
considerably was welfare economics; yet during my career, I did not have the
opportunity to investigate the subject with the exception of Pareto efficiency and
Walrasianequilibrium.Now,theoccasiontofillthelacunainmyscientificinterest
hasfinallyarrivedwiththepresentwork.
This monograph links welfare functions to the study of efficiency in income
distributionandtoinequalityandpoverty,allveryimportantappliedtopicsinreal
economies.Thetopicsareconsideredinatheoreticalframework;noattemptismade
toapplythemodeltoanyrealeconomy,exceptforashortdigressiononItalyinthe
appendix.
Many theoretical and applied works study inequality and poverty. Some are
based on social welfare functions, while others are statistically based. Only when
assuming that a social welfare functionis obtainedthrougha democratic process,
itcanbeconsideredethicallyfounded.Ideally,thisgivesstrengthandcredibilityto
studies on inequality and poverty indexes based on social welfare functions. It is
well known, however, that Arrow’s impossibility theorem warns against seeking
a social welfare function at any cost, especially when there are two or more
individualswhoareallowedtofreelyexpresstheir(transitive)preferencesinaset
ofatleastthreealternatives.
Despite Arrow’s negativeresult, it is here assumed that inequality and poverty
indexesareproducedfromasocialwelfarefunction.Thejustificationforthisstrong
assumption,atthecommunitylevel,isthat,inpractice,duetoacertainuniformity
intheorderofindividualpreferences,whilenotconsideringtheastonishingeffects
ofglobalization,impossiblesituationsareveryrare.Infact,inagivencommunity,
themajorityofpeopleprioritizewithcommonlysharedcharacteristics:education,
culture, religion, political principles, ..., undoubtedly placing some measures of
constraintontheprioritygiventotheorderofpreferences.
v
vi Preface
ThemodelthatIpresentandimplementinthismonographassessesthetrade-off
betweenefficiencyandinequality,inordertomaximizesocialwelfare,assumedto
be an increasing function of individualincomes. When considering the economic
systemingeneral,greaterefficiencyisassociatedwithgreaterinequality.
Individualincomeisassumedtobeavaluableproxyforindividualutilitysince,
amongotherthings,incomeisacardinalquantitywhileutilityisanordinalconcept.
Interpersonal comparisons, thus, become meaningful when considering income,
while this is not the case for utility. Individual income is taken as an argument
in favour of the social welfare function, despite many economists’ claim to be
dissatisfied with assigning so difficult a task to it given that a number of other
variables need to be considered when measuring welfare, including individual
utility. In defence of the assumption made, let me point out that all magnitudes,
other than income,impact individualwelfare, for example education,good health
andhappiness.Theyareallstrictlycorrelatedtoincome(refertothe2010“Special
Christmas Double Issue” of The Economist, pp. 33–36, in particular the chart on
p.36whichplotssampledataon“TheGeographyofHappiness”formorethan90
countries,showingapositivecorrelationbetweenpercapitaincomeandsatisfaction
scores).
The monograph is divided as follows: Chapter 1 provides a general overview.
Chapter 2 gives a brief summary presentation of social welfare functions and
incomedistributions.Chapter3looksatinequalityandpovertyindexes.Chapters4
and 5 explicate new thoughts, as far as I am aware, at least at a formal level.
Theyassesstherelationbetweenefficiencyandinequality,showingthatundervery
simplehypotheses,adegreeofinequalityinindividualincomescanproduce,ceteris
paribus,greatersocialwelfare,byincreasingsocialproduction,hencetotalincome;
andthataconsiderablequotaofthelattercanbedistributedtothepoorestmembers
of the community. This result is not new, of course. Many scholars, for instance
Schumpeter (1954, p. 789), have said the same. I, however, would like to stress
thatreasoningon a specific questionin verbaltermsis one thing,formalizingand
analysing it is another. As John Stuart Mill pointed out in his 1848 Preface to
PrinciplesofPoliticalEconomyinreferringtoAdamSmith’sWealthofNations:
TheWealthofNationsisinmanypartsobsolete,andinall,imperfect.
This is tantamount to saying that anything can be improved.Therefore, I hope to
have formalized, and shown by means of numerical simulations, that in general,
higher individual income is associated with greater individual productivity, and
consequently to a better individual contribution to social output, or material,
the so-called GDP, or Gross Domestic Product. Yet, these results should not be
seen as confirming some form of economic Darwinism (an aspect I have never
acknowledged).
Chapter6brieflydiscussesproblemsthatcanarisefrominformationincentives
and performance, with the aim of compelling economic agents to show their true
characteristics. Chapter 7 presents a summary example of the results obtained
in Chaps.4 and 5 and introduces the specific functions for implementing the
simulations. Chapters 8 and 9 contain numerical implementations showing how
Preface vii
solutions can vary when the Public Authority changes parameters, over which it
hascontrol,withtheeconomicobjectiveofobtainingthe
greatestfelicityofthegreatestnumber,
asproposedlongagobyBentham(1789).
Moreover,Chap.9introducesthedynamiccounterpartofthemodel.Itconsiders
andappliessomesimplenonlineardifferenceequationstosimulateddata.
Chapter10providestheconcludingremarksandsomehintsforincludingpublic
goodsintheproposedmodel.Asimplemodelispresentedandelaborated.
The appendix to this monograph applies the model to Italian income data for
2008.Thisexercisehasbeenrelegatedto theendto underlinethatthemonograph
doesnotintendtobeasortofeconometricexercise.
Inclosing,letmeunderlinethatIhavenotdedicatedachaptertothemathematics
employed in the monograph since mathematical applications are very common
nowadays. The application used is that of maximization under constraints, i.e.
thewell-knownmethodofLagrangemultipliersforfunctionswithmanyvariables
underconstraints.
I am deeply grateful to Giovanni Marseguerra and Aldo Montesano, for their
supportandsuggestionsthathavebeendecisiveinimprovingthesubstanceandthe
presentationofthiswork.IalsowishtowarmlythankCRANEC,aresearchcentre
basedatUniversita`Cattolica(Milan)forallowingmetouseitsfacilitiesduringthe
preparationofthisbook.I wouldalsolike tothankMicaelaTavasaniforcarefully
correctingmyEnglish.
Milano-Lerici,Italy PierCarloNicola
June2012
•
Contents
1 Overview..................................................................... 1
1.1 Preliminary........................................................... 1
1.2 IndividualIncomeandUtility....................................... 3
1.3 SocialWelfareFunctions............................................ 3
1.4 InequalityIndexes ................................................... 4
1.5 PovertyIndexes...................................................... 5
1.6 SocialWelfareandEfficiency....................................... 5
1.7 TaxationandTransfers .............................................. 6
1.8 InformationandImplementation.................................... 6
1.9 SimulatingtheProposedModel..................................... 6
1.10 References ........................................................... 6
PartI Inequality,Poverty,andWelfare
2 SocialWelfareFunctionsandIncomeDistributions.................... 9
2.1 Introduction.......................................................... 9
2.2 SocialWelfareFunctions............................................ 10
2.3 SymmetricalSocialWelfareFunctions............................. 10
2.4 IncomeDistributionandSocialWelfare............................ 12
2.5 Optimal,EquivalentandEvenIncomeDistributions .............. 13
2.6 SocialWelfareandStatusQuo...................................... 14
3 InequalityandPovertyIndexes........................................... 17
3.1 Introduction.......................................................... 17
3.2 WelfareBasedInequalityIndexes .................................. 17
3.3 StatisticallyBasedInequalityIndexes.............................. 19
3.3.1 AxiomsforInequalityIndexes.............................. 19
3.3.2 InequalityIndexes........................................... 20
3.4 ParetoLawandLorenzCurve ...................................... 23
3.5 LognormalLawandIncomeDistribution.......................... 25
ix
x Contents
3.6 MeasuresofPoverty................................................. 25
3.6.1 AxiomsforPovertyIndexes ................................ 25
3.6.2 PovertyIndexes.............................................. 26
3.7 ComplementaritybetweenInequalityandPovertyIndexes ....... 27
PartII WelfareandEfficiency
4 Inequality,Efficiency,andtheProductionSphere...................... 31
4.1 Introduction.......................................................... 31
4.2 APreliminaryViewonSocialProduction.......................... 33
4.3 SocialProductionandOptimalIncomeDistribution .............. 35
4.4 EfficiencyandEquity................................................ 37
4.5 SolvingtheSecondBestProblem .................................. 38
5 EfficiencyandWelfare..................................................... 41
5.1 Preliminary........................................................... 41
5.2 CombiningWelfareandEfficiency ................................. 42
5.3 EquityConsidered................................................... 43
5.4 IndividualConsumptionsandSocialWelfare...................... 46
6 InformationIncentivesandPerformance................................ 49
6.1 OntheSocialWelfareFunction..................................... 49
6.2 OnSocialProduction................................................ 50
6.3 OnTaxationandTransfers .......................................... 50
PartIII SummaryExampleandNumericalSimulations
7 ASummaryExample ...................................................... 55
7.1 ChoiceofFunctions ................................................. 55
7.2 MaximumOutputDistribution...................................... 57
7.3 MaximizingSocialWelfare ......................................... 58
7.4 EquityIncluded...................................................... 60
7.5 DynamicsofMaximumSocialOutput ............................. 63
7.6 MaximizingSocialWelfare ......................................... 65
7.7 ImplementingOptimalChoices..................................... 67
7.8 IncomeInequalityConsidered...................................... 68
7.9 PovertyConsidered.................................................. 68
8 ImplementingtheExample:OneTimePeriod.......................... 69
8.1 InitialIncomeDistribution.......................................... 69
8.2 FirstSimulation:˛ D1:25.......................................... 71
8.2.1 GeneratingParetoDistribution ............................. 71
8.2.2 InitialIncomeDistributionandSocialIncome ............ 72
8.2.3 InequalityConsidered....................................... 74
8.2.4 PovertyConsidered.......................................... 76
8.2.5 MaximizingSocialProduction ............................. 76
