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Session 7: Social Choice and Welfare Economics

Session 7: Social Choice and Welfare Economics. Subgroup Consistency and Decomposability Inequality Poverty Welfare Income Standards. Review. Inequality Four basic axioms Symmetry Replication invariance Scale invariance Transfer principle Unanimity is Lorenz

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Session 7: Social Choice and Welfare Economics

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  1. Session 7: Social Choice and Welfare Economics Subgroup Consistency and Decomposability Inequality Poverty Welfare Income Standards

  2. Review Inequality Four basic axioms Symmetry Replication invariance Scale invariance Transfer principle Unanimity is Lorenz SD2 over mean normalized distributions Additional axiom Transfer sensitivity Unanimity more complete than Lorenz SD3 over mean normalized distributions

  3. Review Welfare Four basic axioms Symmetry Replication invariance Monotonicity Transfer principle Unanimity is generalized Lorenz SD2 Fixing mean yields Atkinson’s Th Additional axiom Transfer sensitivity Unanimity more complete than generalized Lorenz SD3 Fixing the mean yields analog of Atkinson for transfer sensitive Unanimity for first three is SD1

  4. Review Poverty Five basic axioms Symmetry Replication invariance Focus Monotonicity Transfer principle Types of quasiorderings Variable poverty line Variable measure

  5. Review Poverty Variable Poverty Line unlimited range Headcount ordering P0 is SD1 Poverty gap ordering P1 is SD2 FGT ordering P2 is SD3 Variable Poverty Line up to z* Headcount ordering P*0 is SD*1 Poverty gap ordering P*1 is SD*2 FGT ordering P*2 is SD*3 Variable Measure for a given z* and continuous measures Unanimity for (sym, rep. inv., foc., mon..) is is SD*1 Unanimity for (above and trans.) is is SD*2 Unanimity for (above and trans. sens.) is is SD*3 Variable Measure up to z* and for continuous measures Same

  6. Review Recall Welfare conclusions identical if Consider only additive welfare functions Replace mon., trans., trans. sens. with respective deriv. cond’s. Entirely analogous to expected utility Same quasiordering with or without additivity assumption Note Sen’s broadening suggests generalizing expected utility Non-additive forms such as Sen measure S(x) Indeed decision theory took this step to address paradoxes Additivity assumption important here Restricts predicted behavior in meaningful ways

  7. Preview This Session Considers two axioms related to additive form Context of: inequality, poverty, welfare, income standards Subgroup consistency Conceptual axiom Requires coherence between overall and subgroup changes Decomposability Practical axiom for empirical applications Overall index is weighted additive sum of subgroup values Plus between group term in case of inequality Inequality is perhaps most restrictive application Starting point

  8. Inequality Decomposability Helps answer questions like: Is most of global inequality within countries or between countries? How much of total inequality in wages is due to gender inequality? How much of today’s inequality is due to purely demographic factors? Idea (Theil, 1967) Analysis of variance (ANOVA) Total variance V(.) is divided into: part that is ‘explained’ part that is ’unexplained’

  9. ANOVA An example A program is made available to a randomly selected population (the treatment group). Outcomes are A second randomly selected group does not have access to the program. Outcomes are Q/ Did the program have an impact?

  10. ANOVA Notation xy(x,y) Population nx ny n Mean μx μy μ Variances V(x) V(y) V(x,y) Smoothed Decomposition V(x,y) =

  11. ANOVA Idea V(x,y) = overall variance = within group variance = between group variance the part of the variance ‘explained’ by the treatment = share of the variance ‘explained’ by treatment Q/ What makes this analysis possible? A/ Additive decomposition of variance

  12. Inequality Decompositions Theil’s entropy measure where sx = |x|/|(x,y)| is the income share of x Theil’s second measure mean log deviation where px = nx/n is the population share of x Note Weights > 0 and Depend on subgroup statistics: nx, ny, μx and μy

  13. Inequality Decompositions Axiom Additive Decomposability For any x, y we have where weights wx and wy and nonnegative and depend onnx, ny, μx and μy. Note Usually stated for arbitrary number of subgroups Ex Generalized entropy Half Squared coefficient of variation

  14. Example Decompositions Generalized Entropy with a = 2 half sq coef var Income Distributions x = (12,21,12) y = (15,32,10) (x,y) = (12,21,12,15,32,10) Populations and Means nx = 3 ny = 3 n = 6 μx = 15 μy = 19 μ = 17 Inequality Levels Iα(x) = 0.040 Iα(y) = 0.123 Iα(x,y) = 0.099 Weights wx = 0.389 wy = 0.625 Within Group wxIα(x) + wyIα(y) = (0.016 + 0.077) = 0.092 Between Group Iα(x,y) =Iα(15,15,15,19,19,19) = 0.00692 Note Adds to total inequality = 0.099 Betw group contr. 7.1%

  15. Example Decompositions Generalized Entropy with a = 1 Theil’s entropy Income Distributions x = (12,21,12) y = (15,32,10) (x,y) = (12,21,12,15,32,10) Populations and Means nx = 3 ny = 3 n = 6 μx = 15 μy = 19 μ = 17 Inequality Levels Iα(x) = 0.038 Iα(y) = 0.118 Iα(x,y) = 0.090 Weights wx = 0.441 wy = 0.559 Within Group wxIα(x) + wyIα(y) = (0.017 + 0.066) = 0.083 Between Group Iα(x,y) =Iα(15,15,15,19,19,19) = 0.00694 Note Adds to total inequality = 0.090 Betw group contr. 7.8%

  16. Example Decompositions Generalized Entropy with a = 1/2 Income Distributions x = (12,21,12) y = (15,32,10) (x,y) = (12,21,12,15,32,10) Populations and Means nx = 3 ny = 3 n = 6 μx = 15 μy = 19 μ = 17 Inequality Levels Iα(x) = 0.037 Iα(y) = 0.118 Iα(x,y) = 0.087 Weights wx = 0.470 wy = 0.529 Within Group wxIα(x) + wyIα(y) = (0.017 + 0.062) = 0.080 Between Group Iα(x,y) =Iα(15,15,15,19,19,19) = 0.00695 Note Adds to total inequality = 0.087 Betw group contr. 8.0%

  17. Example Decompositions Generalized Entropy with a = 0 Theil’s second Income Distributions x = (12,21,12) y = (15,32,10) (x,y) = (12,21,12,15,32,10) Populations and Means nx = 3 ny = 3 n = 6 μx = 15 μy = 19 μ = 17 Inequality Levels Iα(x) = 0.037 Iα(y) = 0.119 Iα(x,y) = 0.085 Weights wx = 0.500 wy = 0.500 Within Group wxIα(x) + wyIα(y) = (0.018 + 0.059) = 0.078 Between Group Iα(x,y) =Iα(15,15,15,19,19,19) = 0.00697 Note Adds to total inequality = 0.085 Betw group contr. 8.2%

  18. Example Decompositions Generalized Entropy with a = -1 transfer sens. Income Distributions x = (12,21,12) y = (15,32,10) (x,y) = (12,21,12,15,32,10) Populations and Means nx = 3 ny = 3 n = 6 μx = 15 μy = 19 μ = 17 Inequality Levels Iα(x) = 0.036 Iα(y) = 0.127 Iα(x,y) = 0.084 Weights wx = 0.567 wy = 0.447 Within Group wxIα(x) + wyIα(y) = (0.020 + 0.057) = 0.077 Between Group Iα(x,y) =Iα(15,15,15,19,19,19) = 0.00702 Note Adds to total inequality = 0.084 Betw group contr. 8.3%

  19. Example Decompositions Note Only Theil measures have weights summing to 1 Between group term smaller rose slightly as α fell contribution increased as α fell Within group term larger decreased as α fell contribution decreased as α fell Q/What about other additively decomposable measures? A/ Explored by Bourguignon (1979), Cowell and Kuga (1979), Shorrocks (1980, 1984), Foster (1984), and others.

  20. Axiomatic Characterizations Methods Use functional equations (Aczel, 1966, Aczel&Dhombres, 1989) Ex Cauchy equationsContinuous solutions f(a+b) = f(a) + f(b) f(s) = ks f(a+b) = f(a) f(b) f(s) = eks or 0 f(ab) = f(a) + f(b) f(s) = klns f(ab) = f(a) f(b) f(s) = sk or 0 Outline Assume I(x) satisfies various axioms (and reqularity condition) Derive a function f(s) from I(x) Show f must satisfy a particular functional equation Derive functional form for f Use f and axioms to reconstruct allowable forms for I Econ to math to econ

  21. Characterization of Theil • Axiom (Theil Decomposability) • For any x,y we have Q/ Is there any other relative measure that has this decomposition? Theorem Foster (1983) I is a Lorenz consistent inequality measure satisfying Theil decomposability if and only if there is some positive constant k such that I(x) = kT(x) for all x. IdeaOnly the Theil measure has its decomposition Proof (Key step) Set f(s) = I(s,1-s) for 0 < s < 1. Note f(s) = f(1-s) by sym. By TD and LC, f(s) + (1-s)f(t/(1-s)) = f(t) + (1-t)f(s/(1-t)) for all s,t with s+t < 1. This functional equation has the solution: f(t) = t ln t + (1-t)ln (1-t) which eventually yields T(x).

  22. Characterization of Generalized Entropy Key papers Bourguignon (1979), Shorrocks (1980) Characterize Theil measures and GE measures However – Assumed that I is (twice) differentiable Violated by Gini

  23. Characterization of Generalized Entropy Recall Additive decomposability For any x,y we have where weights wx and wy and nonnegative and depend onnx, ny, μx and μy. Q/ Other measures satisfying this axiom? Theorem Shorrocks (1984) I is a continuous, normalized, Lorenz consistent inequality measure satisfying additive decomposability if and only if there is some positive constant k and some α such that I(x) = kIα(x) for all x. IdeaOnly the generalized entropy measures Shorrocks proved more general result: aggregability

  24. Gini Breakdown Note Gini coefficient is not additively decomposable Ok for some subgroups not for others Consider Formula with weights wx = (nx/n)2(μx/μ)

  25. Exact Breakdown Example Income Distributions x = (10,12,12)y = (15,21,32) (x,y) = (10,12,12,15,21,32) Populations and Means nx = 3 ny = 3 n = 6 μx = 11.33μy = 22.67μ = 17 Inequality Levels G(x) = 0.039 G(y) = 0.167 G(x,y) = 0.229 Weights wx = 0.167wy = 0.333 Within Group wxG(x) + wyG(y) = (0.007 + 0.056) = 0.062 Between Group G(x,y) =G(11.3,11.3,11.3,22.7,22.7,22.7) = 0.167 Note Adds to total inequality = 0.229 Non-overlapping groups!

  26. Breakdown with Residual Exampleoverlapping groups Income Distributions x = (12,21,12)y = (15,32,10) (x,y) = (12,21,12,15,32,10) Populations and Means nx = 3 ny = 3 n = 6 μx = 15μy = 19μ = 17 Inequality Levels G(x) = 0.133 G(y) = 0.257 G(x,y) = 0.229 Weights wx = 0.221wy = 0.279 Within Group wxG(x)+ wyG(y) = (0.029 + 0.072) = 0.101 Between Group G(x,y) =G(15,15,15,19,19,19) = 0.059 Note Adds to 0.160 < 0.229 R = residual = 0.69 Why?

  27. Breakdown with Residual Answers Use rank order definition of Gini (def 3) Impact of transfer on Gini depends on: Size of transfer Number (or percent) of population between the two Use expected difference definition of Gini (def 1) Construct difference matrix Submatrices on diagonal relate to within group Submatrices off diagonal relate to between group So long as nonoverlapping Otherwise sum of entries is too large - residual in proportion to number of overlapping entries Try for example x = (2,6) y = (6,10) vs. x’ = (1,7), y’ = (5,11)

  28. Subgroup Consistency Now examine related axiom Subgroup consistency Idea: Helps answer questions like Will local inequality reductions decrease overall inequality? If gender inequality stays the same and inequality within the groups of men and women rises, must overall inequality rise? Sources Cowell (1988) “Inequality decomposition: three bad measures” FGT (1984) similar concept called subgroup monotonicity Shorrocks (1988) axiomatic result

  29. Subgroup Consistency Axiom (Subgroup Consistency) Suppose that x’ and x share means and population sizes, while y’ and y also share means and population sizes. If I(x’) > I(x) and I(y’) = I(y), then I(x’,y’) > I(x,y). Ex (from book) x = (1,3,8,8) y = (2,2) (x,y) = (1,3,8,8,2,2) x’ = (2,2,7,9) y’ = (2,2) (x’,y’) = (2,2,7,8,2,2) G(x) = G(x’), G(y) = G(y’), G(x,y) > G(x’,y’) ‘Almost’ a violation of SC IdeaResidual R fell Modified Ex x” = (2,2,6.9,9.1) y” = (2,2) (x”,y”) = (2,2,6.9,9.1,2,2) G(x) < G(x”), G(y) = G(y”), G(x,y) > G(x”,y”) Gini violates. How about the coeff of var? I2(x) = I2(x’), I2(y) = I2(y’), I2(x,y) = I2(x’,y’) I2(x) < I2(x”), I2(y) = I2(y”), I2(x,y) < I2(x”,y”)

  30. Subgroup Consistency Result If I is additively decomposable then I is subgroup consistent Proof Stare at this Implication All generalized entropy measures are SC Same true for increasing transformation of additively decomposable measure Example Atkinson’s measure (μ – eα)/μ An alternative Atkinson’s measure (μ – eα)/eα Others?

  31. Subgroup Consistency TheoremShorrocks (1988) I is a Lorenz consistent, continuous, normalized inequality measure satisfying subgroup consistency if and only if there is some α and a continuous, strictly increasing function f with f(0)=0 such that I(x) = f(Iα(x)) for all x. Implications Only the GE measures or monotonic transformations Limited scope for satisfying SC

  32. Subgroup Consistency and Additive Decomposability Discussion Decomposability is useful Allows analyses otherwise impossible Disentangle demographic changes from economic changes Not always needed Many analyses concern overall inequality – not broken down Represents a strong restriction Only generalized entropy Is that all there is to inequality? But can be generalized To restrict domain of applicability (say non-overlapping) To allow other income standards as basis of decomposition Wider classes of measures

  33. Subgroup Consistency and Additive Decomposability Discussion Subgroup consistency very compelling Especially in policy context Most policies rely on regional component Try explaining away the inconsistency Relativity run rampant But inequality centrally relative Why can’t our determination of greater or less inequality depend on all the incomes? (where do we stop) Ex (from book) x = (1,3,8,8) y = (2,2) (x,y) = (1,3,8,8,2,2) x’ = (2,2,7,9) y’ = (2,2) (x’,y’) = (2,2,7,8,2,2) Alt ex x = (1,3,8,8) y = (8,8) (x,y) = (1,3,8,8,8,8) x’ = (2,2,7,9) y’ = (8,8) (x’,y’) = (2,2,7,9,8,8) Must they go in same direction?

  34. Subgroup Consistency and Additive Decomposability Should comparison depend on whether a = 2 or a = 8?

  35. Poverty Now change context - Poverty AxiomDecomposability Decomposability: For any distributions x and y, we have P(x,y;z) = (nx/n) P(x;z) + (ny/n) P(y;z). Q/ Why useful? A/ Poverty Profiles Sudhir Anand (1983) Inequality and poverty in Malaysia : measurement and decomposition, Oxford University Press Idea Who are the poor? Where are the poor? See the following guide at the World Bank’s site http://info.worldbank.org/etools/docs/library/103073/ch7.pdf

  36. Decomposability Sudhir Anand, (1977) Rev Inc Wealth

  37. Decomposability and Poverty Profiles • Source: http://info.worldbank.org/etools/docs/library/103073/ch7.pdf

  38. Decomposable Poverty Measures Q/ Which measures are decomposable? A/ FGT, CHU, and an infinity of others Note Not like inequality gen. entropy unique class Consider Pf(x;z) = (1/n)Σif(xi;z) for any f Note Pfsatisfies decomposability for any f Prf (nx/n) Pf(x;z) + (ny/n) Pf(y;z) = (1/n)Σif(xi;z) + (1/n)Σif(yi;z) = Pf(x,y;z) Note Foster and Shorrocks (1991) show that Pf can be axiomatically characterized by additive decomposability Q/Other properties?

  39. Decomposable Poverty Measures A/ Every Pf satisfies symmetry, replication invariance, subgroup consistency Note Other properties of Pf depend on f focus: f(s;z) is a constant function in s for z ≤ s normalization: f(s;z) = 0 for z ≤ s continuity: f(s;z) is continuous in s Q/ what is f for H? monotonicity: f(s;z) is a decreasing function of s for s ≤ z transfer: f(s;z) is a strictly convex function of s for s ≤ z transfer sensitivity: the marginal impact of an extra dollar on individual poverty, or ∂f(s;z)/∂s, is a strictly concave function of s for s ≤ z scale invariance: f(s;z) = r(s/z) for some function r

  40. Subgroup Consistent Poverty Measures Axiom Subgroup Consistency Let x, x’, y, and y’ be distributions satisfying nx = nx’ and ny = ny’. If P(x;z) > P(x';z) and P(y;z) = P(y';z) then P(x,y;z) > P(x',y';z). Recall Shorrocks Result for inequality measures: Only subgroup consistent inequality measures are gen entropy and mon. transf. Note Here All Pf at least! Theorem Foster and Shorrocks (1991) If P is a continuous, symmetric, replication invariant, monotonic poverty measure satisfying subgroup consistency, then P is a continuous increasing transformation of Pf for some continuous f that is decreasing in income. Note Decomposable measure or some transformation

  41. Additive Decomposability Discussion Sudhir Anand, (1977) Rev Inc Wealth • Profiles are immensely important in empirical studies • Need decomposability to construct them • Used in targeting – basis of Mexico’s Progresa program • Not all evaluations require subgroup analysis • Property is needlessly restrictive

  42. Subgroup Consistency Discussion Absolutely no hope of policy discussion if violated Should hold for all partitions – since all can be relevant Existing critique based on differential links across individuals is actually a critique of symmetry, transfer, and the construction and relevance of the income variable itself – not SC per se If links are known, should be incorporated into variable Analogous to equivalence scales This axiom is contingent and should only be applied in certain cases

  43. Welfare Note Much overlap with poverty AxiomDecomposability Decomposability: For any distributions x and y, we have W(x,y) = (nx/n) W(x) + (ny/n) W(y). Note Per capita form – otherwise straight sum Q/ Which functions are decomposable? Consider Wu(x;z) = (1/n)Σiu(xi) for any increasing u Clearly Wusatisfies decomposability Note Can adapt standard arguments as in Foster and Shorrocks (1991) to axiomatically characterize Wu by additive decomposability (or see Hamada discus. p. 39) Q/Other properties?

  44. Decomposable Welfare Functions A/ Every Wu satisfies symmetry, replication invariance, monotonicity, subgroup consistency Note Other properties of Wu depend on u continuity: u(s) is continuous in s Q/ what is f for H? transfer: u(s) is a strictly concave function of s transfer sensitivity: the marginal utility is a strictly convex function of s

  45. Subgroup Consistent Welfare Functions Axiom Subgroup Consistency Let x, x’, y, and y’ be distributions satisfying nx = nx’ and ny = ny’. If W(x) > W(x’) and W(y) = W(y’) then W(x,y) > W(x',y’). Recall Shorrocks result for inequality measures: Only subgroup consistent inequality measures are gen entropy and mon. transf. Note All Wu at least! Others? Theorem If W is a continuous, symmetric, replication invariant, monotonic, welfare function satisfying subgroup consistency, then W is a continuous increasing transformation of Wu for some continuous and increasing u. Note Decomposable measure or some transformation

  46. Subgroup Consistency and Additive Decomposability Discussion Similar to above discussions.

  47. Income Standard One final context: Income standard s(x) Idea Income standard summarizes entire distribution x in a single ‘representative’ income s(x) Ex Mean, median, income at 90th percentile, mean of top 40%, Sen’s mean, Atkinson’s ede income… Measures ‘size’ of the distribution Can have normative interpretation Atkinson’s ede Related papers Foster (2006) “Inequality Measurement Foster and Shneyerov (1999, 2000) Foster and Szekely (2008)

  48. Income Standard Properties of s(x) Symmetry Replication invariance Linear homogeneity Normalization s(μ,μ,…, μ) = μ Continuity Weak monotonicity Examples Mean , median, 10th percentile income, 90th pc income, mean of the bottom fifth, mean of top 40%, Sen welfare function S(x), geometric mean 0, Euclidean mean 2, all the other general means (x) = [(x1+ … + xn)/n] 1/ for  ≠ 0 (where lower  implies greater emphasis on lower incomes) Income standards provide unifying framework for inequality, poverty

  49. Inequality What is inequality? Canonical case Two persons 1 and 2 Two incomes x1 and x2 Min income a = min(x1, x2) Max income b = max(x1, x2) Inequality I = (b - a)/b or some function of ratio a/b Caveats Cardinal variable Relative inequality

  50. Inequality between Groups Group Based Inequality Two groups 1 and 2 Two income distributions x1 and x2 Income standard s(x) “representative income” Lower income standard a = min(s(x1), s(x2)) Upper income standard b = max(s(x1), s(x2)) Inequality I = (b - a)/b or some function of ratio a/b Caveats What about group size? Not relevant if group is unit of analysis

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