MSc Public Economics 2011/12

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10 October 2011. MSc Public Economics 2011/12 Equity, Social Welfare and Taxation Frank A. Cowell. Welfare and distributional analysis. We’ve seen welfare-economics basis for redistribution as a policy objective - PowerPoint PPT Presentation

Transcript of MSc Public Economics 2011/12

  • MSc Public Economics 2011/12, Social Welfare and Taxation

    Frank A. Cowell 10 October 2011

  • Welfare and distributional analysisWeve seen welfare-economics basis for redistribution as a policy objectivehow to assess the impact and effectiveness of such policy?need appropriate criteria for comparing distributions of incomeissues of distributional analysis (Cowell 2008, 2011)Distributional analysis covers broad class of economic problemsinequalitysocial welfarepovertySimilar techniquesrankingsmeasuresFour basic components need to be clarifiedincome conceptincome receiving unit concepta distributionmethod of assessment or comparison

  • 1: Irene and Janet approach0Irene's incomeJanet's incomeKaren's incomeray of equality

    A representation with 3 incomes Income distributions with given total Xxi + xj + xk = X Feasible income distributions given X Equal income distributionsparticularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria

  • 2: The parade x0.20.810x0.8q"income" (height)proportion of the populationx0.2especially useful in cases where it is appropriate to adopt a parametric model of income distributionPen (1971) Plot income against proportion of population Parade in ascending order of "income" / height Related to familiar statistical concept

  • Overview...Welfare axiomsRankingsEquity and social welfareTaxation and sacrificeEquity, Social Welfare, TaxationBasic principles of distributional comparisons

  • Social-welfare functionsA standard approach to a method of assessmentBasic tool is a social welfare function (SWF)Maps set of distributions into the real lineI.e. for each distribution we get one specific numberIn Irene-Janet notation W = W(x1, x2,,xn )Properties will depend on economic principlesSimple example of a SWF:Total income in the economy W = Si xiPerhaps not very interestingConsider principles on which SWF could be based

  • The approachThere is no such thing as a right or wrong axiomHowever axioms could be appropriate or inappropriateNeed some standard of reasonablenessFor example, how do people view income distribution comparisons?Use a simple framework to list some of the basic axiomsAssume a fixed population of size nAssume that individual utility can be measured by xIncome normalised by equivalence scalesRules out utility interdependenceWelfare is just a function of the vector x := (x1, x2,,xn )Follow the approach of Amiel-Cowell (1999) Appendix A

  • SWF axiomsAnonymity. Suppose x is a permutation of x. Then: W(x) = W(x)Population principle. W(x) W(y) W(x,x,,x) W(y,y,,y)Decomposability. Suppose x' is formed by joining x with z and y' is formed by joining y with z. Then :W(x) W(y) W(x') W(y')Monotonicity. W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn)Transfer principle. (Dalton 1920) Suppose xi< xj then for small : W(x1,x2..., xi+ ,..., xj ,..., xn) > W(x1,x2,..., xi,..., xn)Scale invariance. W(x) W(y) W(lx) W(ly)

  • Classes of SWFs (1)Use axioms to characterise important classes of SWFAnonymity and population principle imply we can write SWF in either Irene-Janet form or F formmost modern approaches use these assumptionsbut may need to standardise for needs etc Introduce decomposability and you get class of Additive SWFs W : W(x) = Si u(xi) or equivalently in F-form W(F) = u(x) dF(x)think of u() as social utility or social evaluation functionThe class W is of great importanceBut W excludes some well-known welfare criteria

  • Classes of SWFs (2)From W we get important subclassesIf we impose monotonicity we get W1 W : u() increasingsubclass where marginal social utility always positiveIf we further impose the transfer principle we get W2 W1: u() increasing and concavesubclass where marginal social utility is positive and decreasing We often need to use these special subclassesIllustrate their properties with a simple example

  • Overview...Welfare axiomsRankingsEquity and social welfareTaxation and sacrificeEquity, Social Welfare, TaxationClasses of SWF and distributional comparisons

  • Ranking and dominanceConsider problem of comparing distributionsIntroduce two simple conceptsfirst illustrate using the Irene-Janet representationtake income vectors x and y for a given nFirst-order dominance:y(1) > x(1), y(2) > x(2), y(3) > x(3)Each ordered income in y larger than that in xSecond-order dominance: y(1) > x(1), y(1)+y(2) > x(1)+x(2), y(1)+y(2) ++ y(n) > x(1)+x(2) + x(n)Each cumulated income sum in y larger than that in xWeaker than first-order dominanceNeed to generalise this a littlegiven anonymity, population principle can represent distributions in F-formq: population proportion (0 q 1)F(x): proportion of population with incomes xm(F): mean of distribution F

  • 1st-Order approachBasic tool is the quantile, expressed asQ(F; q) := inf {x | F(x) q} = xqsmallest income such that cumulative frequency is at least as great as qUse this to derive a number of intuitive conceptsinterquartile range, decile-ratios, semi-decile ratiosgraph of Q is Pens ParadeAlso to characterise 1st-order (quantile) dominance:G quantile-dominates F means: for every q, Q(G;q) Q(F;q), for some q, Q(G;q) > Q(F;q)A fundamental result:G quantile-dominates F iff W(G) > W(F) for all WW1Illustrate using Parade:

  • Parade and 1st-order dominanceFGQ(.; q)10q Plot quantiles against proportion of population Parade for distribution F again Parade for distribution GIn this case G clearly quantile-dominates FBut (as often happens) what if it doesnt?Try second-order method

  • 2nd-Order approachBasic tool is the income cumulant, expressed asC(F; q) := Q(F; q) x dF(x)The sum of incomes in the Parade, up to and including position qUse this to derive a number of intuitive conceptsthe shares ranking, Gini coefficientgraph of C is the generalised Lorenz curveAlso to characterise the idea of 2nd-order (cumulant) dominance:G cumulant-dominates F means: for every q, C(G;q) C(F;q), for some q, C(G;q) > C(F;q)A fundamental result (Shorrocks 1983):G cumulant-dominates F iff W(G) > W(F) for all WW2

  • GLC and 2nd-order dominance100C(G; . )C(F; . )C(.; q)m(F)m(G)qcumulative income Plot cumulations against proportion of population GLC for distribution F GLC for distribution GIntercept on vertical axis is at mean income

  • 2nd-Order approach (continued)A useful tool: the share of the proportion q of distribution F is L(F;q) := C(F;q) / m(F)income cumulation at divided q by total incomeYields Lorenz dominance, or the shares ranking:G Lorenz-dominates F means: for every q, L(G;q) L(F;q), for some q, L(G;q) > L(F;q)Another fundamental result (Atkinson 1970):For given m, G Lorenz-dominates F iff W(G) > W(F) for all W W2Illustrate using Lorenz curve:

  • Lorenz curve and ranking00. of incomeproportion of populationL(G;.)L(F;.)L(.; q)q Plot shares against proportion of populationLorenz curve for distribution FLorenz curve for distribution GIn this case G clearly Lorenz-dominates FSo F displays more inequality than GBut (as often happens) what if it doesnt?No clear statement about inequality (or welfare) is possible without further informationPerfect equality

  • Overview...Welfare axiomsRankingsEquity and social welfareTaxation and sacrificeEquity, Social Welfare, TaxationQuantifying social values

  • Equity and social welfareSo far we have just general principlesmay lead to ambiguous resultsneed a tighter description of social-welfare?Consider reduced form of social welfareW = W (m, I) m = m(F) is mean of distribution F I = I(F) is inequality of distribution F W embodies trade-off between objectivesI can be taken as an equity criterionfrom same roots as SWF?or independently determined?Consider a natural definition of inequality

  • Gini coefficientRedraw Lorenz diagramA natural inequality measure?

    normalised area above Lorenz curvecan express this also in I-J termsAlso (equivalently) represented as normalised difference between income pairs:

    In F-form:

    In Irene-Janet terms:Intuition is neat, butinequality index is clearly arbitrarycan we find one based on welfare criteria?

  • SWF and inequalityO The Irene &Janet diagram A given distribution Distributions with same mean Contours of the SWFEx(F)m(F)F Construct an equal distribution E such that W(E) = W(F) Equally-Distributed Equivalent incomeSocial waste from inequality Curvature of contour indicates societys willingness to tolerate efficiency loss in pursuit of greater equality contour: x values such that W(x) = constdo this for special case

  • An important family of SWFTake the W2 subclass and impose scale invariance. Get the family of SWFs where u is iso-elastic:

    x 1 e 1 u(x) = , e 0 1 ehas same form as CRRA utility function Parameter e captures societys inequality aversion.Similar interpretation to individual risk aversionSee Atkinson (1970)

  • Welfare-based inequalityFrom the concept of social waste Atkinson (1970) suggested an inequality measure: x(F)I(F) = 1 m(F)Atkinson further assumed additive SWF, W(F) = u(x) dF(x)isoelastic uSo inequality takes the form

    But what value of inequality aversion parameter e?

  • Values: the issuesSWF is central to public policy makingPractical example in HM Treasury (2003) pp 93-94We need to focus on two questionsFirst: do people care about distribution?Experiments suggest they do Carlsson et al (2005)Do social and economic factors make a difference?Second: What is the shape of u? (Cowell-Gardiner 2000)Direct estimates of inequality aver