Inequality, poverty and redistribution

Transcription

1 Inequality, poverty and redistribution EC February 2014

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 and welfare comparisons? also poverty? extend to comparisons when needs differ? 417 February Frank Cowell: EC426 2

3 Overview... Inequality, Poverty Redistribution Inequality and structure The composition of inequality. Is there convergence? Poverty Welfare and needs 17 February 2014 Frank Cowell: EC426 3

4 Income distributions n = 2 x j n 1 µ ( x) := xi n i= 1 Janet s income x µ1 x 0 45 µ(x) Irene s income x i 17 February 2014 Frank Cowell: EC426 4

5 Income distributions n = 3 x j A representation with 3 incomes Income distributions with given total Equal income distributions income distribution x 0 µ1 x x k Irene, Janet, Karen Inequality increases as you move away from centroid? What determines shape of contours? x i Axiomatic approach (Amiel-Cowell 1999 Appendix A, Cowell 2014 ) 17 February 2014 Frank Cowell: EC426 5

6 Inequality axioms (1) 1 Anonymity. Suppose x is a permutation of x. Then: I(x ) = I(x) 2 Population principle. I(x) I(y) I(x,x,,x) I(y,y,,y) 3 Transfer principle. (Dalton 1920) Suppose x i < x j then, for small δ: I(x 1,x 2..., x i + δ,..., x j δ,..., x n ) < I(x 1,x 2,..., x i,..., x n ) 17 February 2014 Frank Cowell: EC426 6

7 Income distributions n = 3 (close-up) [Janet] (ii) (iii) (i) x (iv) (vi) (v) [Irene] [Karen] 17 February 2014 Frank Cowell: EC426 7

8 x and x cannot be ranked [Janet] x x [Irene] [Karen] 17 February 2014 Frank Cowell: EC426 8

9 Two contour maps I CV var( ) ( ) := x x µ ( x). n n 1 IGini( x) := 2 xi xj 2 n µ ( x). i= 1 j= 1 17 February 2014 Frank Cowell: EC426 9

10 Scale invariance x j 0 x k x i 17 February 2014 Frank Cowell: EC426 10

11 Inequality axioms (2) 4 Decomposability. Suppose x' is formed by joining x with z and y' is formed by joining y with z. Then : I(x) I(y) I(x') I(y') 5 Scale invariance. For λ > 0: I(x) I(y) I(λx) I(λy) 6 Translation invariance. I(x) I(y) I(x+1δ) I(y+1δ) Axioms 1-5 yield the Generalised Entropy class of indices α 1 n α 1 xi IGE ( x) = 2 1 α α n i 1 µ ( ) = x Axioms yield the Kolm class + variance (Bosmans and Cowell 2010) I β K n 1 1 i ( x) := log e β n i= 1 [ x ( x) ] β µ 17 February 2014 Frank Cowell: EC426 11

12 Generalised Entropy measures Defines a class of inequality measures, given parameter α : α 1 n α 1 xi IGE ( x) = 1 2 α α n i= 1 µ ( ) x GE class is rich. Some important special cases for α < 1 it is ordinally equivalent to Atkinson (α = 1 ε ) 0 1 n α = 0: IGE ( x) := n i= 1 log( x i/ µ ( x)) (mean logarithmic deviation) 1 1 n α = 1: IGE ( x) = n i= 1[ xi/ µ ( x)]log( xi/ µ ( x)) (the Theil index) or α = 2 it is ordinally equivalent to (normalised) variance Parameter α can be assigned any positive or negative value indicates sensitivity of each member of the class α large and positive gives a top-sensitive measure α negative gives a bottom-sensitive measure each α gives a specific distance concept 17 February 2014 Frank Cowell: EC426 12

13 Generalised Entropy February 2014 Frank Cowell: EC426 13

14 Scale or translation independence? x j x j x * x * 0 µ1 x k 0 µ1 x k x i x i 417 February Frank Cowell: EC426 14

15 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: I j = I(x j ) : inequality in group j π j = n j / n: population share of group j s j = π j µ(x j )/µ(x) : income share of group j Three types of decomposability, in decreasing order of generality: General consistency I(x) = Φ(I 1, I 2, ; π 1, π 2, ; s 1, s 2, ) Additive decomposability I(x) = Σ j ω j I(x j ) + I between, ω j = w(π j, s j ) 0 Inequality accounting (as above, but also Σ j ω j = 1) 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 17 February 2014 Frank Cowell: EC426 15

16 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: β indices + absolute Gini Can express Gini as a weighted sum Σ i κ i x i where κ i := [2i/n 1] / µ for absolute Gini just delete the symbol µ from the above Note that the weights κ i are very special depend on rank or position in distribution May change as other members added/removed from distribution 17 February 2014 Frank Cowell: EC426 16

17 Partitioning by income... Non-overlapping groups Overlapping groups A transfer: Case 1 A transfer: Case 2 0 x N 1 N 2 x * x x' x ** N 1 x' x 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? 17 February 2014 Frank Cowell: EC426 17

18 Example 1: Inequality measures and US experience Gini GE0 GE1 A.25 A.50 A.75 0 Source: DeNavas-Walt et al. (2012) 17 February 2014 Frank Cowell: EC426 18

19 Example 2: International trends Source: OECD (2011) 17 February 2014 Frank Cowell: EC426 19

20 Application: International trends Break down overall inequality to analyse trends: I = Σ j ω 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) 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 17 February 2014 Frank Cowell: EC426 20

21 Inequality measures and World experience Gini GE0 GE1 A.50 A Source: Sala-i-Martin (2006) 17 February 2014 Frank Cowell: EC426 21

22 Inequality measures and World experience: breakdown GE0 GE0 betw GE0 within GE1 GE1 betw GE1 within Source: Sala-i-Martin (2006) 17 February 2014 Frank Cowell: EC426 22

23 Absolute vs Relative measures Is inequality converging? (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) Bosmans et al. (2013) World Bank (2005) 17 February 2014 Frank Cowell: EC426 23

24 Overview... Inequality, Poverty Redistribution Inequality and structure Poverty and its relation to inequality. Principles and trends Poverty Welfare and needs 17 February 2014 Frank Cowell: EC426 24

25 Poverty measurement Two main types of issues identification problem aggregation problem Fundamental partition Depends on poverty line z Exogeneity of partition? Aggregation of information Use the concept of individual poverty evaluation Simplest version is (0,1) the headcount approach Perhaps make it depend on income includes poverty deficit approach Perhaps make it depend on distribution among the poor deprivation approach (Sen 1976) Again use axiomatic approach Characterise P(x, z) Poor z Non-Poor x 17 February 2014 Frank Cowell: EC426 25

26 Poverty Axioms 7 Focus. If y i = x i for all x i, y i z then P(x, z) = P(y, z) 8 Monotonicity. If x i < y i for x i,y i z and x j = y j, j i then P(x, z) > P(y, z) 9 Independence. Suppose P(x, z) = P(y, z) and, for some i, x i = y i. Let x, y be formed by adding δ to both x i and y i. Then P(x, z) = P(y, z) Axioms yield the ASP class p is the poverty evaluation function These axioms + 5,6 yield the FGT class g i is the poverty gap max(0, z x i ) Foster et al (1984) 17 February 2014 Frank Cowell: EC426 26

27 Poverty contours, θ = 1 x j 0 x * µ1 x k x i 17 February 2014 Frank Cowell: EC426 27

28 Poverty evaluation, FGT p(x, z) p(x, z) 1 θ = 0 1 θ = 1 x x 0 z 0 z p(x, z) g θ = 2 0 x j x i x z 17 February 2014 Frank Cowell: EC426 28

29 Brazil: How Much Poverty? Rural Belo Horizonte poverty line A highly skewed distribution A conservative z A generous z An intermediate z Censored income distribution Distribution of poverty gaps compromise poverty line Brasilia poverty line $0 $20 $40 $60 $80 $100 $120 $140 $160 $180 $200 $220 $240 $260 $280 $300 $0 $20 $40 $60 gaps 17 February 2014 Frank Cowell: EC426 29

30 Poverty rankings We use 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 poverty lines? Theorem: (Foster and Shorrocks 1988) For FGT indices with parameter θ poverty orderings are equivalent to first-order welfare dominance for θ = 0 second-degree welfare dominance for θ = 1 (third-order welfare dominance for θ = 2) 17 February 2014 Frank Cowell: EC426 30

31 TIP / Poverty profile Cumulative gaps versus population proportions Proportion of poor Σg i TIP curve (Jenkins and Lambert 1997) TIP curves have same interpretation as GLC TIP dominance implies greater poverty for all poverty lines at z or lower F(x i ) = i/n 0 F(z) 17 February 2014 Frank Cowell: EC426 31

32 Views on distributions Do people make distributional comparisons in the same way as economists? Summarised from Amiel-Cowell (1999) examine proportion of responses in conformity with standard axioms in terms of inequality, social welfare and poverty Inequality SWF Poverty Num Verbal Num Verbal Num Verbal Anonymity 83% 72% 66% 54% 82% 53% Population 58% 66% 66% 53% 49% 57% Decomposability 57% 40% 58% 37% 62% 46% Monotonicity % 55% 64% 44% Transfers 35% 31% 47% 33% 26% 22% Scale indep. 51% 47% % 66% Transl indep. 31% 35% % 62% 17 February 2014 Frank Cowell: EC426 32

33 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) 17 February 2014 Frank Cowell: EC426 33

34 Poverty rates by region 1981,2001 Headcount Pov 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 February 2014 Frank Cowell: EC426 34

35 Poverty: East Asia 17 February 2014 Frank Cowell: EC426 35

36 Poverty: South Asia Headcount at $1.08 per day Headcount at $2.15 per day Poverty gap at $1.08 per day 100 Poverty gap at $2.15 per day February 2014 Frank Cowell: EC426 36

37 Poverty: Latin America, Caribbean 35 Headcount at $1.08 per day Headcount at $2.15 per day Poverty gap at $1.08 per day Poverty gap at $2.15 per day February 2014 Frank Cowell: EC426 37

38 Poverty: Middle East and N.Africa 35 Headcount at $1.08 per day Headcount at $2.15 per day Poverty gap at $1.08 per day Poverty gap at $2.15 per day February 2014 Frank Cowell: EC426 38

39 Poverty: Sub-Saharan Africa Headcount at $1.08 per day Headcount at $2.15 per day Poverty gap at $1.08 per day 90 Poverty gap at $2.15 per day February 2014 Frank Cowell: EC426 39

40 Poverty: E. Europe and Central Asia 25 Headcount at $1.08 per day Headcount at $2.15 per day Poverty gap at $1.08 per day Poverty gap at $2.15 per day February 2014 Frank Cowell: EC426 40

41 Overview... Inequality, Poverty Redistribution Inequality and structure Extensions of the ranking approach Poverty Welfare and needs 17 February 2014 Frank Cowell: EC426 41

42 Social-welfare criteria: review Additive SWFs W : W(F) = u(x) df(x) 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) 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 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 17 February 2014 Frank Cowell: EC426 42

43 Income and needs reconsidered All these based on the assumption of homogeneous population no differences in needs Standard approach using equivalised income assumes: given, known welfare-relevant attributes a a known relationship ν = ν(a) equivalised income given by x = y / ν ν is the "exchange-rate" between income types x, y Set aside the assumption that we have a single ν( ) Get a general result on joint distribution of (y, a) 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 (1987), Cowell (2000), Moyes (2012) 17 February 2014 Frank Cowell: EC426 43

44 Alternative approach to needs Sort individuals be into needs groups N 1, N 2, a proportion π j are in group N j. social welfare is W(F) = Σ j π j a Nj u(y) df(a,y) To make this operational utility people get from income depends on needs: W(F) = Σ j π 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 17 February 2014 Frank Cowell: EC426 44

45 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), Moyes (2012) Apply to household types in Economic and Labour Market Review 17 February 2014 Frank Cowell: EC426 45

46 Impact of Taxes and Benefits. UK Sequential GLCs All types orig Types 1-5 orig Type 1 orig 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 All types final Types 1-5 final Type 1 final 30,000 25,000 20,000 15,000 10,000 5, Proportion of population 17 February 2014 Frank Cowell: EC426 46

47 Conclusion Distributional analysis covers a number of related problems: inequality poverty social welfare and needs 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 17 February 2014 Frank Cowell: EC426 47

48 References Amiel, Y. and Cowell, F. A. (1999) Thinking about Inequality, Cambridge University Press, Cambridge, Chapter 7. 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 Bosmans, K. and Cowell, F. A. (2010) The Class of Absolute Decomposable Inequality Measures, Economics Letters, 109, Bosmans, K., Decancq, K. and Decoster, A. (2013) The Relativity of Decreasing Inequality Between Countries, Economica, forthcoming *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. (2014) Inequality and Poverty Measures, in Oxford Handbook of Well-Being And Public Policy, edited by Matthew D. Adler and Marc Fleurbaey Ebert, U. and P. Moyes (2002). A simple axiomatization of the Foster-Greer-Thorbecke poverty orderings. Journal of Public Economic Theory 4, *Fleurbaey, M., Hagneré, C. and Trannoy. A. (2003) Welfare comparisons with bounded equivalence scales Journal of Economic Theory, DeNavas-Walt, C., Proctor, B. D. and Smith, J. C. (2012) Income, poverty, and health insurance coverage in the United States: Current Population Reports P60-243, U.S. Census Bureau, Washington, DC. Foster, J. E., Greer, J. and Thorbecke, E. (1984) A class of decomposable poverty measures, Econometrica, 52, Foster, J. E. and Shorrocks, A. F. (1988) Poverty orderings and welfare dominance, Social Choice and Welfare, 5, 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, Moyes, P. (2012) Comparisons of heterogeneous distributions and dominance criteria, Journal of Economic Theory,147, OECD (2011) Divided We Stand: Why Inequality Keeps Rising OECD ilibrary. Ravallion, M. and Chen, S. (2006) How have the world s poorest fared since the early 1980s? World Bank Research Observer, 19, *Sala-i-Martin, X. (2006) The world distribution of income: Falling poverty and... convergence, period, Quarterly Journal of Economics, 121 Sen, A. K. (1976) Poverty: An ordinal approach to measurement, Econometrica, 44, World Bank (2005) 2006 World Development Report: Equity and Development. Oxford University Press, New York 17 February 2014 Frank Cowell: EC426 48