Inequality, poverty and redistribution

Frank Cowell: MSc Public Economics 2011/2http://darp.lse.ac.uk/ec426

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?

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Overview...

Inequality and structure

Poverty

Welfare and needs

Inequality, Poverty Redistribution

The composition of inequality. Is there convergence?

3Frank Cowell: EC426 330 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 m(F)1 [ x1 e dF(x) ] 1/ [1e]

• but welfare approach is indirect• maybe introduces unnecessary assumptions

• 3: Alternative route: use distributional axioms directly• see Cowell (2007)

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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 l (where l > 0) leaves inequality unchanged • relative inequality measures (Blackorby and Donaldson 1978)

• (Alternative: Translation Independence)• adding a constant d to all incomes leaves inequality unchanged• absolute inequality measures (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

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Structural axioms: illustration

xi

xk

xj

0 (m, m, m)•

x*

•

Set of distributions, n=3An income distributionPerfect equalityInequality contours Anonymity Scale independenceTranslation independence

Irene, Janet, Karen

Inequality increases as you move away from centroid

What determines shape of contours?

Examine decomposition and independence properties

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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 • Ij = I(F(j)) : inequality in group j • pj = #(F(j)) / #(F) : population share of group j• sj = pj m(F(j))/m(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

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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: b indices + absolute Gini

• Can express Gini as a weighted sum• k(x) x dF(x)• where k(x) = [2F(x) 1] / m• for absolute Gini just delete the symbol m from the above

• Note that the weights k are very special• depend on rank or position in distribution• May change as other members added/removed from distribution

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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*

N1 N2

0

x**

N1

x

x'x x'x

A transfer: Case 1 A transfer: Case 2

Non-overlapping groups Overlapping groups

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Three versions of decomposition• General consistency

• I(F) = F(I1, I2,… ; p1, p2,… ; s1, s2,…)• where F is increasing in each Ij

• Additive decomposability • specific form of F• I(F) = Sj wj I(F(j)) + I(Fbetween ) • where wj is a weight depending on population and income shares• wj = w(pj, sj) ≥ 0• Fbetween is distribution assuming no inequality in each group

• Inequality accounting• as above plus• Sj wj = 1

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A class of decomposable indices• Given scale-independence and additive decomposability I takes the

Generalised Entropy form:• [a2 a]1 [[ x/m(F)]a 1] dF(x)

• Parameter a indicates sensitivity of each member of the class.• a large and positive gives a “top -sensitive” measure• a negative gives a “bottom-sensitive” measure

• Includes the two Theil (1967) indices and the coeff of variation:• a = 0: – log (x / m(F)) dF(x)• a = 1: [ x / m(F)] log (x / m(F)) dF(x)• a = 2: ½ [[ x/m(F)]2 1] dF(x)

• For a < 1 GE is ordinally equivalent to Atkinson (a = 1 – e )• Decomposition properties:

• the weight wj on inequality in group j is wj = pj1−asj

a • weights only sum to 1 if a = 0 or 1 (Theil indices)

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Inequality contours

• Each a 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

a = 0.25 a = 0

a = −0.25 a = −1

a = 2

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13

Example 1: Inequality measures and US experience

• Source: DeNavas-Walt et al. (2005)

1967

1970

1973

1976

1979

1982

1985

1988

1991

1994

1997

2000

2003

0

0.1

0.2

0.3

0.4

0.5

0.6

GiniGE0GE1A.25A.50A.75

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Example 2: International trends

14

Source: OECD (2011)

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Application: International trends• Break down overall inequality to analyse trends:

• I = Sj wj Ij + Ibetween • 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

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Inequality measures and World experience

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

00.10.20.30.40.50.60.70.80.9

1

GiniGE0GE1A.50A1.0

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Source: Sala-i-Martin (2006)

Inequality measures and World experience: breakdown

00.10.20.30.40.50.60.70.80.9

1

GE0GE0 betwGE0 within GE1GE1 betwGE1 within

Source: Sala-i-Martin (2006)

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Another class of decomposable indices• Given translation-independence and additive decomposability:• Inequality takes the form

• b1 [eb[ x m(F)] 1] dF(x) (b ≠ 0)• or [ x2 m(F)2] dF(x) (b = 0)

• Parameter b indicates sensitivity of each member of the class• b positive gives a “top -sensitive” measure• b negative gives a “bottom-sensitive” measure (Kolm 1976)

• Another class of additive measures• These are absolute indices• See Bosmans and Cowell (2010)

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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)• World Bank (2005)

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Frank Cowell: EC426

Overview...

Inequality and structure

Poverty

Welfare and needs

Inequality, Poverty Redistribution

Poverty and its relation to inequality. Principles and trends

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Poverty measurement• Sen (1979): Two main types of issues

• 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”• Incidence• Intensity• Inequality

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xz

Non-PoorPoor

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)

• Convenient to work with poverty gaps • g(x, z) = max (0, z x)

• Sometimes use cumulative poverty gaps:G (x, z) := x g (t, z) dF(t)

xz

0

pove

rty e

valu

atio

n

xi xj

gi gj

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Poverty evaluation

g

0

poverty evaluation

poverty gap

x = 0

Non-Poor Poor

gi

A

gj

B

the “head-count” the “poverty deficit” sensitivity to inequality

amongst the poor Income equalisation

amongst the poor

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Brazil: How Much Poverty?

Brasilia 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 $300

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$0 $20 $40 $60

gaps

Distribution of poverty gaps 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 )

• 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

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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

a=0a=1a=1.5a=2a=2.5

p(x,z)

Poverty rankings• We use version of 2nd -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 zZ ? • depends on sensitivity parameter (Foster and Shorrocks 1988a, 1988b)

• 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)

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TIP / Poverty profile

F(x)

G(x,z)

0

Cumulative gaps versus population proportions

Proportion of poor TIP curve (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

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F(z)

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 VerbalAnonymity 83% 72% 66% 54% 82% 53%Population 58% 66% 66% 53% 49% 57%Decomposability 57% 40% 58% 37% 62% 46%Monotonicity - - 54% 55% 64% 44%Transfers 35% 31% 47% 33% 26% 22%Scale indep. 51% 47% - - 48% 66%Transl indep. 31% 35% - - 17% 62%

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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)

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Poverty rates by region 1981,2001Headcount Pov Gap

$1.08 $2.15 $1.08 $2.15China 63.80 1 88.10 3 23.41 1 50.82 1

East Asia 57.70 2 84.80 4 20.58 2 47.20 3India 54.40 3 89.60 1 17.27 3 47.22 2

South Asia 51.50 4 89.10 2 16.06 5 45.78 4Sub-Saharan Africa 41.60 5 73.30 5 17.03 4 38.54 5

All Regions 40.40 6 66.70 6 13.93 6 35.02 6Latin-America, Caribbean 9.70 7 26.90 8 2.75 7 10.66 7

M. East, N. Africa 5.10 8 28.90 7 1.00 8 8.81 8E. Europe, Central Asia 0.70 9 4.70 9 0.17 9 1.39 9

Sub-Saharan Africa 46.90 1 76.60 3 20.29 1 41.42 1India 34.70 2 79.90 1 7.08 2 34.43 2

South Asia 31.30 3 77.20 2 6.37 3 32.35 3All Regions 21.10 4 52.90 4 5.96 4 21.21 4

China 16.60 5 46.70 6 3.94 5 18.44 5East Asia 14.90 6 47.40 5 3.35 7 17.78 6

Latin-America, Caribbean 9.50 7 24.50 7 3.36 6 10.20 7E. Europe, Central Asia 3.70 8 19.70 9 0.79 8 5.94 9

M. East, N. Africa 2.40 9 23.20 8 0.45 9 6.14 8

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Poverty: East Asia

0

10

20

30

40

50

60

70

80

90

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

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Poverty: South Asia

0102030405060708090

100

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

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

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Poverty: Latin America, Caribbean

0

5

10

15

20

25

30

35

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

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

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Poverty: Middle East and N.Africa

0

5

10

15

20

25

30

35

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

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

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Poverty: Sub-Saharan Africa

0

10

20

30

40

50

60

70

80

90

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

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

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Poverty: E. Europe and Central Asia

0

5

10

15

20

25

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

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

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Frank Cowell: EC426

Overview...

Inequality and structure

Poverty

Welfare and needs

Inequality, Poverty Redistribution

Extensions of the ranking approach

373730 January 2012 Frank 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

• W1 Ì W : u(•) increasing• W2 Ì W1: u(•) increasing and concave

• Basic tools :• the quantile, Q(F; q) := inf {x | F(x) ³ q} = xq

• 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 WW1 iff G quantile-dominates F• W(G) > W(F) for all WW2 iff G cumulant-dominates F

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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 T1 , T2 have disposable income schedules c1 and c2 • Then T1 is more progressive than T2 iff c1 Lorenz-dominates c2 (Jakobsson 1976)• Requires a single-crossing condition on c1 and c2 (Hemming and Keen 1983)

• All these based on the assumption of homogeneous population• no differences in needs

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Income and needs reconsidered• Standard approach using “equivalised income” assumes:

• Given, known welfare-relevant attributes a• A known relationship n = n(a)• Equivalised income given by x = y / n• n is the "exchange-rate" between income types x, y

• Set aside the assumption that we have a single n(•)• 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)

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Alternative approach to needs• Sort individuals be into needs groups N1, N2 ,…

• a proportion pj are in group Nj .• social welfare is W(F) = Sj pj aNj u(y) dF(a,y)

• To make this operational… • Utility people get from income depends on needs:• W(F) = Sj pj aNj 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 W3 Ì W2 be the subclass of welfare functions for which MU-diff is positive and decreasing in y

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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 WW3 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) • Apply to household types in Economic and Labour Market Review…

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Impact of Taxes and Benefits. UK 2006-7. Sequential GLCs

£0

£5,000

£10,000

£15,000

£20,000

£25,000

£30,000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Proportion of population

All types orig All types finalTypes 1-5 orig Types 1-5 finalType 1 orig Type 1 final

3+ adults with children2+ adults with 3+children2 adults with 2 children2 adults with 1 child1 adult with children3+ adults 0 children2+ adults 0 children1 adult, 0 children

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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

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References (1)• Amiel, Y. and Cowell, F. A. (1999) Thinking about Inequality, Cambridge University Press, Cambridge,

Chapter 7. • Atkinson, A. B. and Bourguignon, F. (1982) “The comparison of multi-dimensional distributions of

economic status,” Review of Economic Studies, 49, 183-201• 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 350-370

• Atkinson, A. B. and Brandolini. A. (2010) “On Analyzing the World Distribution of Income,” The World Bank Economic Review, 24

• Blackorby, C. and Donaldson, D. (1978) “Measures of relative equality and their meaning in terms of social welfare,” Journal of Economic Theory, 18, 59-80

• Blackorby, C. and Donaldson, D. (1980) “A theoretical treatment of indices of absolute inequality,” International Economic Review, 21, 107-136

• Bosmans, K. and Cowell, F. A. (2010) “The Class of Absolute Decomposable Inequality Measures,” Economics Letters, 109,154-156

• Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Ch 2, 87-166

• Cowell, F.A. (2007) “Inequality: measurement,” The New Palgrave, 2nd edn• *Fleurbaey, M., Hagneré, C. and Trannoy. A. (2003) “Welfare comparisons with bounded equivalence

scales” Journal of Economic Theory, 110 309–336• 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.

• Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, 761-776

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References (2)• Foster , J. E. and Shorrocks, A. F. (1988a) “Poverty orderings,” Econometrica, 56, 173-177• Foster , J. E. and Shorrocks, A. F. (1988b) “Poverty orderings and welfare dominance,” Social

Choice and Welfare, 5,179-198• Hemming , R. and Keen , M. J. (1983) “Single-crossing conditions in comparisons of tax

progressivity,” Journal of Public Economics, 20, 373-380• Jakobsson, U. (1976) “On the measurement of the degree of progression,” Journal of Public

Economics, 5,161-168• *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, 317-327.• Kolm, S.-Ch. (1976) “Unequal Inequalities I,” Journal of Economic Theory, 12, 416-442• 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, 141-170• *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, 219-231• Sen, A. K. (1979) “Issues in the measurement of poverty,” Scandinavian Journal of Economics,

91, 285-307• Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4, 91-

134• The World Bank (2005) 2006 World Development Report: Equity and Development. Oxford

University Press, New York

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