Inequality, poverty and redistribution Frank Cowell: MSc Public Economics 2011/2

Issues Key questions about distributional tools Inequality measures what can they tell us about recent within-country trends? about trends in world inequality? Poverty measures how, if at all, related to inequality? what do they tell us about world “convergence”? Dominance criteria go beyond inequality welfare trends? understand tax progressivity and redistribution? extend to comparisons when needs differ? 30 January 2012Frank Cowell: EC January 2012Frank Cowell: EC426 2

Overview... Inequality and structure Poverty Welfare and needs Inequality, Poverty Redistribution The composition of inequality. Is there convergence? 3 Frank Cowell: EC January 2012

Approaches to Inequality 1: Intuition example: Gini coefficient but intuition may be unreliable guide 2 Inequality as welfare loss example: Atkinson’s index 1   (F)  1 [  x 1  dF(x) ] 1/ [1  but welfare approach is indirect maybe introduces unnecessary assumptions 3: Alternative route: use distributional axioms directly see Cowell (2007)Cowell (2007) 30 January 2012Frank Cowell: EC426 4

Axioms: reinterpreted for inequality Anonymity permute individuals – inequality unchanged Population principle clone population – inequality unchanged Principle of Transfers poorer-to-richer transfer –inequality increases Scale Independence multiplying all incomes by (where > 0) leaves inequality unchanged relative inequality measures (Blackorby and Donaldson 1978)Blackorby and Donaldson 1978 (Alternative: Translation Independence) adding a constant  to all incomes leaves inequality unchanged absolute inequality measures (Blackorby and Donaldson 1980)Blackorby and Donaldson 1980 Decomposability independence: merging with an “irrelevant” income distribution does not affect welfare/inequality comparisons but here it is more instructive to look at decomposability interpretation 30 January 2012Frank Cowell: EC426 5

Structural axioms: illustration xixi xkxk xjxj 0  x *  Set of distributions, n=3  An income distribution  Perfect equality  Inequality contours  Anonymity  Scale independence  Translation independence  Irene, Janet, Karen  Inequality increases as you move away from centroid  What determines shape of contours?  Examine decomposition and independence properties 30 January 2012Frank Cowell: EC426 6

Inequality decomposition Relate inequality overall to inequality in parts of the population Incomplete information International comparisons Everyone belongs to one (and only one) group j: F (j) : income distribution in group j I j = I(F (j) ) : inequality in group j  j = #(F (j) ) / #(F) : population share of group j s j =  j  (F (j) )/  (F) : income share of group j Three types of decomposability, in decreasing order of generality: General consistency Additive decomposability Inequality accounting Which type is a matter of judgment Each type induces a class of inequality measures The “stronger” the decomposition requirement… …the “narrower” the class of inequality measures 30 January 2012Frank Cowell: EC426 7

Partition types and inequality measures General Partition any characteristic used for partition (age, gender, region, income…) Non-overlapping Partition weaker version: partition based on just income scale independence: GE indices + Gini translation independence:  indices + absolute Gini Can express Gini as a weighted sum   (x) x dF(x) where  (x) = [2F(x)  1] /  for absolute Gini just delete the symbol  from the above Note that the weights  are very special depend on rank or position in distribution May change as other members added/removed from distribution 30 January 2012Frank Cowell: EC426 8

Partitioning by income... Gini has a problem with decomposability Type of partition is crucial for the Gini coefficient Case 1: effect on Gini is proportional to [rank(x)  rank(x')] same in subgroup and population Case 2: effect on Gini is proportional to [rank(x)  rank(x')] differs in subgroup and population What if we require decomposability for general partitions? x*x* N1N1 N2N2 0 x ** N1N1 x x' x x  A transfer: Case 1  A transfer: Case 2  Non-overlapping groups  Overlapping groups 30 January 2012Frank Cowell: EC426 9

Three versions of decomposition General consistency I(F) =  (I 1, I 2,… ;  1,  2,… ; s 1, s 2,…) where  is increasing in each I j Additive decomposability specific form of  I(F) =  j  j I(F (j) ) + I(F between ) where  j is a weight depending on population and income shares  j = w(  j, s j ) ≥ 0 F between is distribution assuming no inequality in each group Inequality accounting as above plus  j  j = 1 30 January 2012Frank Cowell: EC426 10

A class of decomposable indices Given scale-independence and additive decomposability I takes the Generalised Entropy form: [  2   ]  1  [[ x/  (F)]   1] dF(x) Parameter  indicates sensitivity of each member of the class.  large and positive gives a “top -sensitive” measure  negative gives a “bottom-sensitive” measure Includes the two Theil (1967) indices and the coeff of variation:1967  = 0: –  log (x /  (F)) dF(x)  = 1:  [ x /  (F)] log (x /  (F)) dF(x)  = 2: ½  [[ x/  (F)]   1] dF(x) For  < 1 GE is ordinally equivalent to Atkinson (  = 1 –  ) Decomposition properties: the weight  j on inequality in group j is  j =  j 1−  s j  weights only sum to 1 if  = 0 or 1 (Theil indices) 30 January 2012Frank Cowell: EC426 11

Inequality contours Each  defines a set of contours in the Irene, Karen, Janet diagram each related to a concept of distance For example the Euclidian case other types  25   −   −   2 30 January 2012Frank Cowell: EC426 12

13 Example 1: Inequality measures and US experience Source: Source: DeNavas-Walt et al. (2005)DeNavas-Walt et al. (2005) 30 January 2012Frank Cowell: EC426 13

Example 2: International trends 14 Source: OECD (2011) Source: OECD (2011)OECD (2011)OECD (2011) 30 January 2012Frank Cowell: EC426 14

Application: International trends Break down overall inequality to analyse trends: I =  j  j I j + I between given scale independence I must take the GE form what weights should we use? Traditional approach takes each country as separate unit shows divergence – increase in inequality but, in effect, weights countries equally debatable that China gets the same weight as very small countries New conventional view (Sala-i-Martin 2006) (Sala-i-Martin 2006) within-country disparities have increased not enough to offset reduction in cross-country disparities. Components of change in distribution are important “correctly” compute world income distribution decomposition within/between countries is then crucial what drives cross-country reductions in inequality? Large growth rate of the incomes of the 1.2 billion Chinese 30 January 2012Frank Cowell: EC426 15

Inequality measures and World experience 30 January 2012Frank Cowell: EC Source: Sala-i-Martin (2006)Sala-i-Martin (2006)

Inequality measures and World experience: breakdown Source: Sala-i-Martin (2006)Sala-i-Martin (2006) 30 January 2012Frank Cowell: EC426 17

Another class of decomposable indices Given translation-independence and additive decomposability: Inequality takes the form   1  [e  [ x   (F)]  1] dF(x) (  ≠ 0) or  [ x    (F)  ] dF(x) (  = 0) Parameter  indicates sensitivity of each member of the class  positive gives a “top -sensitive” measure  negative gives a “bottom-sensitive” measure (Kolm 1976)Kolm 1976 Another class of additive measures These are absolute indices See Bosmans and Cowell (2010)Bosmans and Cowell (2010) 30 January 2012Frank Cowell: EC426 18

Absolute vs Relative measures Is inequality converging? ( Sala-i- Martin 2006 ) Sala-i- Martin 2006 Does it matter whether we use absolute or relative measures? In terms of inequality trends within countries, not much But worldwide get sharply contrasting picture Atkinson, and Brandolini (2010) World Bank (2005) 30 January 2012Frank Cowell: EC426 19

Frank Cowell: EC426 Overview... Inequality and structure Poverty Welfare and needs Inequality, Poverty Redistribution Poverty and its relation to inequality. Principles and trends January 2012Frank Cowell: EC426 20

Poverty measurement Sen (1979): Two main types of issues Sen (1979) identification problem aggregation problem Fundamental partition Depends on poverty line z Exogeneity of partition? Individual identification what kind of personal characteristics? Aggregation of information asymmetric treatment of information Jenkins and Lambert (1997): “3Is” Jenkins and Lambert (1997) Incidence Intensity Inequality 30 January 2012Frank Cowell: EC x z Non-Poor Poor

Counting the poor Use the concept of individual poverty evaluation applies only to the poor subgroup i.e. where x ≤ z Simplest version is (0,1) (non-poor, poor) headcount Perhaps make it depend on income poverty deficit Or on distribution among the poor? can capture the idea of deprivation major insight of Sen (1976)Sen (1976) Convenient to work with poverty gaps g (x, z) = max (0, z  x) Sometimes use cumulative poverty gaps:  (x, z) :=  x g (t, z) dF(t) x z 0 poverty evaluation xixi xjxj gigi gjgj 30 January 2012Frank Cowell: EC426 22

Poverty evaluation g 0 poverty evaluation poverty gap x = 0 Non-Poor Poor gigi A gjgj B  the “head-count”  the “poverty deficit”  sensitivity to inequality amongst the poor  Income equalisation amongst the poor 30 January 2012Frank Cowell: EC426 23

Brazil: How Much Poverty? Brasilia poverty line Brasilia poverty line compromise poverty line compromise poverty line  A highly skewed distribution  A “conservative” z  A “generous” z  An “intermediate” z  Censored income distribution $0$20$40$60$80$100$120$140$160$180$200$220$240$260$280$ January 2012Frank Cowell: EC $0$20$40$60 gaps  Distribution of poverty gaps Rural Belo Horizonte poverty line Rural Belo Horizonte poverty line

Additively Separable Poverty measures ASP approach simplifies poverty evaluation depends on income and poverty line: p(x, z) Poverty is an additively separable function P =  p(x, z) dF(x) Assumes decomposability amongst the poor ASP leads to several classes of measures Important special case (Foster et al 1984 )Foster et al 1984 make poverty evaluation depends on poverty gap. normalise by poverty line P =  [g(x, z)/z] a dF(x) a determines the sensitivity of the index 30 January 2012Frank Cowell: EC p(x,z)

Poverty rankings We use version of 2 nd -order dominance to get inequality orderings related to welfare orderings in some cases get unambiguous inequality rankings We could use the same approach with poverty get unambiguous poverty rankings for all povertylines? Concentrate on the FGT index’s particular functional form: P =  [g(x, z)/z] a dF(x) is  p(x, z) dF(x) ≥  p(x, z) dG(x) for all values of z  Z ? depends on sensitivity parameter (Foster and Shorrocks 1988a, 1988b)1988a1988b Theorem: Poverty orderings are equivalent to first-order welfare dominance for a = 0 second-degree welfare dominance for a = 1 (third-order welfare dominance for a = 2) 30 January 2012Frank Cowell: EC426 26

TIP / Poverty profile F(x)F(x)  (x,z) 0  Cumulative gaps versus population proportions  Proportion of poor  TIP curve (Jenkins and Lambert 1997)(Jenkins and Lambert 1997)  TIP curves have same interpretation as GLC  TIP dominance  implies unambiguously greater poverty for all poverty lines at z or lower 30 January 2012Frank Cowell: EC F(z)F(z)

Views on distributions Do people make distributional comparisons in the same way as economists? Summarised from Amiel-Cowell (1999)Amiel-Cowell (1999) examine proportion of responses in conformity with standard axioms in terms of inequality, social welfare and poverty InequalitySWFPoverty NumVerbalNumVerbalNumVerbal Anonymity83%72%66%54%82%53% Population58%66%66%53%49%57% Decomposability57%40%58%37%62%46% Monotonicity--54%55%64%44% Transfers35%31%47%33%26%22% Scale indep.51%47%--48%66% Transl indep.31%35%--17%62% 30 January 2012Frank Cowell: EC426 28

Empirical robustness Does it matter which poverty criterion you use? Look at two key measures from the ASP class Head-count ratio Poverty deficit (or average poverty gap) Use two standard poverty lines $1.08 per day at 1993 PPP $2.15 per day at 1993 PPP How do different regions of the world compare? What’s been happening over time? Use World-Bank analysis ( Ravallion and Chen 2006) Ravallion and Chen 2006) 30 January 2012Frank Cowell: EC426 29

Poverty rates by region 1981,2001 HeadcountPov Gap $1.08$2.15$1.08$2.15 China East Asia India South Asia Sub-Saharan Africa All Regions Latin-America, Caribbean M. East, N. Africa E. Europe, Central Asia Sub-Saharan Africa India South Asia All Regions China East Asia Latin-America, Caribbean E. Europe, Central Asia M. East, N. Africa January 2012Frank Cowell: EC426 30

Poverty: East Asia 30 January 2012Frank Cowell: EC426 31

Poverty: South Asia 30 January 2012Frank Cowell: EC426 32

Poverty: Latin America, Caribbean 30 January 2012Frank Cowell: EC426 33

Poverty: Middle East and N.Africa 30 January 2012Frank Cowell: EC426 34

Poverty: Sub-Saharan Africa 30 January 2012Frank Cowell: EC426 35

Poverty: E. Europe and Central Asia 30 January 2012Frank Cowell: EC426 36

Frank Cowell: EC426 Overview... Inequality and structure Poverty Welfare and needs Inequality, Poverty Redistribution Extensions of the ranking approach January 2012Frank Cowell: EC426 37

Social-welfare criteria: review Relations between classes of SWF and practical tools Additive SWFs W : W(F) =  u(x) dF(x) Important subclasses W 1  W : u() increasing W 2  W 1 : u() increasing and concave Basic tools : the quantile, Q(F; q) := inf {x | F(x)  q} = x q the income cumulant, C(F; q) := ∫ Q(F; q) x dF(x) give quantile- and cumulant-dominance Fundamental results: W(G) > W(F) for all W  W 1 iff G quantile-dominates F W(G) > W(F) for all W  W 2 iff G cumulant-dominates F 30 January 2012Frank Cowell: EC426 38

Applications to redistribution GLC rankings Straight application of welfare result Recall UK application from ONS Does “final income” 2-order dom “original income”? LC rankings Welfare result applied to distributions with same mean Does “final income” Lorenz dominate “original income”? For UK application – yes Tax progressivity Let two tax schedules T 1, T 2 have disposable income schedules c 1 and c 2 Then T 1 is more progressive than T 2 iff c 1 Lorenz-dominates c 2 (Jakobsson 1976)Jakobsson 1976 Requires a single-crossing condition on c 1 and c 2 (Hemming and Keen 1983)Hemming and Keen 1983 All these based on the assumption of homogeneous population no differences in needs 30 January 2012Frank Cowell: EC426 39

Income and needs reconsidered Standard approach using “equivalised income” assumes: Given, known welfare-relevant attributes a A known relationship  (a) Equivalised income given by x = y / is the "exchange-rate" between income types x, y Set aside the assumption that we have a single () Get a general result on joint distribution of (y, a) This makes distributional comparisons multidimensional Intrinsically difficult To make progress: simplify the structure of the problem again use results on ranking criteria see Atkinson and Bourguignon (1982, 1987), also Cowell (2000)1982Cowell (2000) 30 January 2012Frank Cowell: EC426 40

Alternative approach to needs Sort individuals be into needs groups N 1, N 2,… a proportion  j are in group N j. social welfare is W(F) =  j  j  a  Nj u(y) dF(a,y) To make this operational… Utility people get from income depends on needs: W(F) =  j  j  a  Nj u(j, y) dF(a,y) Consider MU of income in adjacent needs classes: ∂u(j, y) ∂u(j+1, y) ────  ────── ∂y ∂y “Need” reflected in high MU of income? If need falls with j then MU-difference should be positive Let W 3  W 2 be the subclass of welfare functions for which MU-diff is positive and decreasing in y 30 January 2012Frank Cowell: EC426 41

Main result Let F (  j) mean distribution for all needs groups up to and including j. Distinguish this from the marginal distribution F (j) Theorem (Atkinson and Bourguignon 1987) W(G) > W(F) for all W  W 3 if and only if G (  j) cumulant-dominates F (  j) for all j = 1,2,... To examine welfare ranking use a “sequential dominance” test check first the neediest group then the first two neediest groups then the first three, … etc Extended by Fleurbaey et al (2003) Fleurbaey et al (2003) Apply to household types in Economic and Labour Market Review… 30 January 2012Frank Cowell: EC426 42

Impact of Taxes and Benefits. UK Sequential GLCs 3+ adults with children 2+ adults with 3+children 2 adults with 2 children 2 adults with 1 child 1 adult with children 3+ adults 0 children 2+ adults 0 children 1 adult, 0 children 30 January 2012Frank Cowell: EC426 43

Conclusion Distributional analysis covers a number of related problems: Social welfare and needs Inequality Poverty Commonality of approach can yield important insights Ranking principles provide basis for broad judgments May be indecisive specific indices could be used But convenient axioms may not find a lot of intuitive support 30 January 2012Frank Cowell: EC426 44

References (1) Amiel, Y. and Cowell, F. A. (1999) Thinking about Inequality, Cambridge University Press, Cambridge, Chapter 7. Amiel, Y. and Cowell, F. A. (1999) Atkinson, A. B. and Bourguignon, F. (1982) “The comparison of multi-dimensional distributions of economic status,” Review of Economic Studies, 49, Atkinson, A. B. and Bourguignon, F. (1982) Atkinson, A. B. and Bourguignon, F. (1987) “Income distribution and differences in needs,” in Feiwel, G. R. (ed), Arrow and the Foundations of the Theory of Economic Policy, Macmillan, New York, chapter 12, pp Atkinson, A. B. and Brandolini. A. (2010) “On Analyzing the World Distribution of Income,” The World Bank Economic Review, 24 Atkinson, A. B. and Brandolini. A. (2010) Blackorby, C. and Donaldson, D. (1978) “Measures of relative equality and their meaning in terms of social welfare,” Journal of Economic Theory, 18, Blackorby, C. and Donaldson, D. (1978) Blackorby, C. and Donaldson, D. (1980) “A theoretical treatment of indices of absolute inequality,” International Economic Review, 21, Blackorby, C. and Donaldson, D. (1980) Bosmans, K. and Cowell, F. A. (2010) “The Class of Absolute Decomposable Inequality Measures,” Economics Letters, 109, Bosmans, K. and Cowell, F. A. (2010) Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Ch 2, Cowell, F. A. (2000) Cowell, F.A. (2007) “Inequality: measurement,” The New Palgrave, 2nd edn Cowell, F.A. (2007) *Fleurbaey, M., Hagneré, C. and Trannoy. A. (2003) “Welfare comparisons with bounded equivalence scales” Journal of Economic Theory, –336Fleurbaey, M., Hagneré, C. and Trannoy. A. (2003) DeNavas-Walt, C., Proctor, B. D. and Lee, C. H. (2005) “Income, poverty, and health insurance coverage in the United States: 2004.” Current Population Reports P60-229, U.S. Census Bureau, U.S. Government Printing Office, Washington, DC. DeNavas-Walt, C., Proctor, B. D. and Lee, C. H. (2005) Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, Foster, J. E., Greer, J. and Thorbecke, E. (1984) 30 January 2012Frank Cowell: EC426 45

References (2) Foster, J. E. and Shorrocks, A. F. (1988a) “Poverty orderings,” Econometrica, 56, Foster, J. E. and Shorrocks, A. F. (1988a) Foster, J. E. and Shorrocks, A. F. (1988b) “Poverty orderings and welfare dominance,” Social Choice and Welfare, 5, Foster, J. E. and Shorrocks, A. F. (1988b) Hemming, R. and Keen, M. J. (1983) “Single-crossing conditions in comparisons of tax progressivity,” Journal of Public Economics, 20, Hemming, R. and Keen, M. J. (1983) Jakobsson, U. (1976) “On the measurement of the degree of progression,” Journal of Public Economics, 5, Jakobsson, U. (1976) *Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis of UK poverty trends,” Oxford Economic Papers, 49, Jenkins, S. P. and Lambert, P. J. (1997) Kolm, S.-Ch. (1976) “Unequal Inequalities I,” Journal of Economic Theory, 12, Kolm, S.-Ch. (1976) OECD (2011) Divided We Stand: Why Inequality Keeps Rising OECD iLibrary. OECD (2011) Ravallion, M. and Chen, S. (2006) “How have the world’s poorest fared since the early 1980s?” World Bank Research Observer, 19, Ravallion, M. and Chen, S. (2006) *Sala-i-Martin, X. (2006) “The world distribution of income: Falling poverty and... convergence, period”, Quarterly Journal of Economics, 121Sala-i-Martin, X. (2006) Sen, A. K. (1976) “Poverty: An ordinal approach to measurement,” Econometrica, 44, Sen, A. K. (1976) Sen, A. K. (1979) “Issues in the measurement of poverty,” Scandinavian Journal of Economics, 91, Sen, A. K. (1979) Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4, Theil, H. (1967) The World Bank (2005) 2006 World Development Report: Equity and Development. Oxford University Press, New York The World Bank (2005) 30 January 2012Frank Cowell: EC426 46