Preliminary version - Freekumlai.free.fr/RESEARCH/THESE/TEXTE/INEQUALITY... · Poverty and Equity:...

Poverty and Equity:Measurement, Policy and Estimation with

DAD

Jean-Yves Duclos∗

Abdelkrim Araar†

25th February 2005

Preliminary version

∗ Centre interuniversitaire sur le risque, les politiqueseconomiques et l’emploi (CIRPEE)and Departement d’economique, Pavillon de Seve, Universite Laval, Quebec, Canada, G1K 7P4;Email: [email protected]; fax: 1-418-656-7798; phone: 1-418-656-7096

† CIRPEE and Poverty and Economic Policy (PEP) network, Pavillon de Seve, UniversiteLaval, Quebec, Canada, G1K 7P4; Email: [email protected]; fax: 1-418-656-7798; phone: 1-418-656-7507

CONTENTS 1

Contents

1 Preface 11

I Conceptual and methodological issues 13

2 Well-being and poverty 142.1 The welfarist approach. . . . . . . . . . . . . . . . . . . . . . . 142.2 Non-welfarist approaches. . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Basic needs and functionings. . . . . . . . . . . . . . . . 162.2.2 Capabilities. . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 A graphical illustration. . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Practical measurement difficulties for the non-welfarist approaches222.5 Poverty measurement and public policy. . . . . . . . . . . . . . 25

2.5.1 Poverty measurement matters. . . . . . . . . . . . . . . 252.5.2 Welfarist and non-welfarist policy implications. . . . . . 26

3 The empirical measurement of well-being 293.1 Survey issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Income versus consumption. . . . . . . . . . . . . . . . . . . . 313.3 Price variability . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Household heterogeneity. . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Estimating equivalence scales. . . . . . . . . . . . . . . 363.4.2 Sensitivity analysis. . . . . . . . . . . . . . . . . . . . . 383.4.3 Household decision-making and within-household inequal-

ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 References for chapters2 and3 . . . . . . . . . . . . . . . . . . . 41

II Measuring poverty and equity 44

4 Introduction 454.1 Continuous distributions. . . . . . . . . . . . . . . . . . . . . . 454.2 Discrete distributions. . . . . . . . . . . . . . . . . . . . . . . . 474.3 Poverty gaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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4.4 Cardinal versus ordinal comparisons. . . . . . . . . . . . . . . . 49

5 Measuring inequality and social welfare 515.1 Lorenz curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Gini indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2.1 Linear inequality indices and S-Gini indices. . . . . . . . 545.2.2 Interpreting Gini indices. . . . . . . . . . . . . . . . . . 575.2.3 Gini indices and relative deprivation. . . . . . . . . . . . 59

5.3 Social welfare and inequality. . . . . . . . . . . . . . . . . . . . 605.4 Social welfare. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4.1 Atkinson indices. . . . . . . . . . . . . . . . . . . . . . 635.4.2 S-Gini social welfare indices. . . . . . . . . . . . . . . . 655.4.3 Generalized Lorenz curves. . . . . . . . . . . . . . . . . 66

5.5 Decomposing inequality by population subgroups. . . . . . . . . 665.5.1 Generalized entropy indices of inequality. . . . . . . . . 675.5.2 A sub-group Shapley decomposition of inequality indices68

5.6 Statistical and descriptive indices of inequality. . . . . . . . . . . 695.7 Appendix: the Shapley value. . . . . . . . . . . . . . . . . . . . 715.8 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Measuring poverty 746.1 Poverty indices. . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.1.1 The EDE approach. . . . . . . . . . . . . . . . . . . . . 746.1.2 The poverty gap approach. . . . . . . . . . . . . . . . . 756.1.3 Interpreting FGT indices. . . . . . . . . . . . . . . . . . 766.1.4 Relative contribution to FGT indices. . . . . . . . . . . . 776.1.5 EDE poverty gaps for FGT indices. . . . . . . . . . . . . 78

6.2 Group-decomposable poverty indices. . . . . . . . . . . . . . . 796.3 Poverty and inequality. . . . . . . . . . . . . . . . . . . . . . . 806.4 Poverty curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.5 S-Gini poverty indices . . . . . . . . . . . . . . . . . . . . . . . 826.6 The normalization of poverty indices. . . . . . . . . . . . . . . . 836.7 Decomposing poverty. . . . . . . . . . . . . . . . . . . . . . . . 84

6.7.1 Growth-redistribution decompositions. . . . . . . . . . . 846.7.2 Demographic and sectoral decomposition of differences

in FGT indices . . . . . . . . . . . . . . . . . . . . . . . 876.7.3 The impact of demographic changes. . . . . . . . . . . . 886.7.4 Decomposing poverty by income components. . . . . . . 89

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6.8 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7 Estimating poverty lines 927.1 Absolute and relative poverty lines. . . . . . . . . . . . . . . . . 927.2 Social exclusion and relative deprivation. . . . . . . . . . . . . . 937.3 Estimating absolute poverty lines. . . . . . . . . . . . . . . . . . 95

7.3.1 Cost of basic needs. . . . . . . . . . . . . . . . . . . . . 957.3.2 Cost of food needs. . . . . . . . . . . . . . . . . . . . . 957.3.3 Non-food poverty lines. . . . . . . . . . . . . . . . . . . 987.3.4 Food energy intake. . . . . . . . . . . . . . . . . . . . .1007.3.5 Illustration for Cameroon. . . . . . . . . . . . . . . . . .102

7.4 Estimating relative and subjective poverty lines. . . . . . . . . . 1037.4.1 Relative poverty lines. . . . . . . . . . . . . . . . . . . .1037.4.2 Subjective poverty lines. . . . . . . . . . . . . . . . . .1057.4.3 Subjective poverty lines with discrete information. . . . . 106

7.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107

8 Measuring vertical equity and progressivity 1098.1 Taxes and transfers. . . . . . . . . . . . . . . . . . . . . . . . .1098.2 Concentration curves. . . . . . . . . . . . . . . . . . . . . . . .1108.3 Concentration indices. . . . . . . . . . . . . . . . . . . . . . . .1128.4 Decomposition of inequality into income components. . . . . . . 112

8.4.1 Using concentration curves and indices. . . . . . . . . . 1128.4.2 Using the Shapley value. . . . . . . . . . . . . . . . . .114

8.5 Progressivity comparisons. . . . . . . . . . . . . . . . . . . . .1148.5.1 Deterministic tax and benefit systems. . . . . . . . . . . 1148.5.2 General tax and benefit systems. . . . . . . . . . . . . .117

8.6 Tax and income redistribution. . . . . . . . . . . . . . . . . . .1188.7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

9 Horizontal inequity, reranking and redistribution 1239.1 Ethical and other foundations. . . . . . . . . . . . . . . . . . . .1239.2 Measuring reranking and redistribution. . . . . . . . . . . . . . .125

9.2.1 Reranking. . . . . . . . . . . . . . . . . . . . . . . . . .1269.2.2 S-indices of equity and redistribution. . . . . . . . . . . 1269.2.3 Redistribution and vertical and horizontal equity. . . . . 127

9.3 Measuring classical horizontal inequity and redistribution. . . . . 1299.3.1 Horizontally-equitable net incomes. . . . . . . . . . . . 129

CONTENTS 4

9.3.2 Change in inequality approach. . . . . . . . . . . . . . .1319.3.3 Cost of inequality approach. . . . . . . . . . . . . . . .132

9.4 Decomposition of classical HI. . . . . . . . . . . . . . . . . . .1339.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133

III Ordinal comparisons of poverty and equity 136

10 Distributive Dominance 13710.1 Ordering distributions. . . . . . . . . . . . . . . . . . . . . . . .13710.2 Sensitivity of poverty comparisons. . . . . . . . . . . . . . . . .13710.3 Ordinal comparisons. . . . . . . . . . . . . . . . . . . . . . . .13810.4 Ethical judgements. . . . . . . . . . . . . . . . . . . . . . . . .140

10.4.1 Paretian judgments. . . . . . . . . . . . . . . . . . . . .14010.4.2 First-order judgments. . . . . . . . . . . . . . . . . . . .14010.4.3 Higher-order judgments. . . . . . . . . . . . . . . . . .141

10.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143

11 Poverty dominance 14411.1 Primal approach. . . . . . . . . . . . . . . . . . . . . . . . . . .146

11.1.1 Dominance tests. . . . . . . . . . . . . . . . . . . . . .14611.1.2 Nesting of dominance tests. . . . . . . . . . . . . . . . .149

11.2 Dual approach. . . . . . . . . . . . . . . . . . . . . . . . . . . .15011.2.1 First-order poverty dominance. . . . . . . . . . . . . . .15011.2.2 Second-order poverty dominance. . . . . . . . . . . . . 15211.2.3 Higher-order dominance. . . . . . . . . . . . . . . . . .153

11.3 Assessing the limits to dominance. . . . . . . . . . . . . . . . .15311.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154

12 Welfare and inequality dominance 15612.1 Ethical welfare judgments. . . . . . . . . . . . . . . . . . . . .15612.2 Tests of welfare dominance. . . . . . . . . . . . . . . . . . . . .15712.3 Inequality judgments. . . . . . . . . . . . . . . . . . . . . . . .16012.4 Tests of inequality dominance. . . . . . . . . . . . . . . . . . .16112.5 Inequality and progressivity. . . . . . . . . . . . . . . . . . . . .16312.6 Social welfare and Lorenz curves. . . . . . . . . . . . . . . . . .16412.7 The distributive impact of benefits. . . . . . . . . . . . . . . . .16412.8 Pro-poor growth. . . . . . . . . . . . . . . . . . . . . . . . . . .166

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12.8.1 First-order pro-poor judgements. . . . . . . . . . . . . .16612.8.2 Second-order pro-poor judgements. . . . . . . . . . . . . 168

12.9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169

IV Poverty and equity: policy and growth 171

13 Poverty alleviation: policy and growth 17213.1 The impact of targeting. . . . . . . . . . . . . . . . . . . . . . .172

13.1.1 Group-targeting a constant amount. . . . . . . . . . . . . 17313.1.2 Inequality-neutral targeting. . . . . . . . . . . . . . . . .174

13.2 The impact of changes in the poverty line. . . . . . . . . . . . . 17613.3 Price changes. . . . . . . . . . . . . . . . . . . . . . . . . . . .17713.4 Tax and subsidy reforms. . . . . . . . . . . . . . . . . . . . . .17913.5 Income-component and sectoral growth. . . . . . . . . . . . . .182

13.5.1 Absolute poverty impact. . . . . . . . . . . . . . . . . .18213.5.2 Poverty elasticity. . . . . . . . . . . . . . . . . . . . . .183

13.6 Overall growth elasticity of poverty. . . . . . . . . . . . . . . .18413.7 The Gini elasticity of poverty. . . . . . . . . . . . . . . . . . . .186

13.7.1 Inequality and poverty. . . . . . . . . . . . . . . . . . .18613.7.2 Increasing bi-polarization and poverty. . . . . . . . . . . 187

13.8 The impact of policy and growth on inequality. . . . . . . . . . . 18813.8.1 Growth, fiscal policy, and price shocks. . . . . . . . . . . 18813.8.2 Tax and subsidy reform. . . . . . . . . . . . . . . . . . .190

13.9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191

14 Targeting in the presence of redistributive costs 19314.1 Poverty alleviation, redistributive costs and targeting. . . . . . . 19314.2 Costly targeting. . . . . . . . . . . . . . . . . . . . . . . . . . .196

14.2.1 Minimizing the headcount. . . . . . . . . . . . . . . . .19614.2.2 Minimizing the average poverty gap. . . . . . . . . . . . 19614.2.3 Minimizing a distribution-sensitive poverty index. . . . . 19814.2.4 Optimal redistribution. . . . . . . . . . . . . . . . . . .200

14.3 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201

V Estimation and inference for distributive analysis 203

CONTENTS 6

15 An introduction to DAD: A software for distributive analysis 20415.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .20415.2 Loading, editing and saving databases inDAD . . . . . . . . . . . 20615.3 Inputting the sampling design information. . . . . . . . . . . . . 20815.4 Applications inDAD: basic procedures. . . . . . . . . . . . . . .21015.5 Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21215.6 Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21415.7 Statistical inference: standard deviation, confidence intervals and

hypothesis testing. . . . . . . . . . . . . . . . . . . . . . . . . .21615.8 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218

16 Non parametric estimation inDAD 21916.1 Density estimation. . . . . . . . . . . . . . . . . . . . . . . . .219

16.1.1 Univariate density estimation. . . . . . . . . . . . . . . .21916.1.2 Statistical properties of kernel density estimation. . . . . 22116.1.3 Choosing a window width. . . . . . . . . . . . . . . . .22216.1.4 Multivariate density estimation. . . . . . . . . . . . . . .22416.1.5 Simulating from a nonparametric density estimate. . . . 224

16.2 Non-parametric regression. . . . . . . . . . . . . . . . . . . . .22616.3 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .228

17 Estimation and statistical inference 22917.1 Sampling design. . . . . . . . . . . . . . . . . . . . . . . . . . .22917.2 Sampling weights. . . . . . . . . . . . . . . . . . . . . . . . . .23117.3 Stratification. . . . . . . . . . . . . . . . . . . . . . . . . . . . .23217.4 Clustering (or multi-stage sampling). . . . . . . . . . . . . . . .23317.5 Impact of stratification, clustering, weighting and sampling with-

out replacement on sampling variability. . . . . . . . . . . . . .23617.5.1 Stratification . . . . . . . . . . . . . . . . . . . . . . . .23717.5.2 Clustering. . . . . . . . . . . . . . . . . . . . . . . . . .23917.5.3 Finite population corrections. . . . . . . . . . . . . . . .23917.5.4 Impact of weighting on sampling variance. . . . . . . . . 24117.5.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . .242

17.6 Formulae for computing standard errors of distributive estimatorswith complex sample design. . . . . . . . . . . . . . . . . . . .243

17.7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246

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18 Statistical inference in practice 24718.1 Asymptotic distributions. . . . . . . . . . . . . . . . . . . . . .24718.2 Hypothesis tests. . . . . . . . . . . . . . . . . . . . . . . . . . .24918.3 p-values and confidence intervals. . . . . . . . . . . . . . . . . .25118.4 Statistical inference using a non-pivotal bootstrap. . . . . . . . . 25218.5 Hypothesis tests and confidence intervals using pivotal bootstrap

statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25518.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257

19 Exercises. 26019.1 Household size and living standards. . . . . . . . . . . . . . . .260

19.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26019.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26019.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26019.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260

19.2 Choice of aggregating weights and poverty analysis.. . . . . . . . 26019.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26019.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26019.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260

19.3 Absolute and relative poverty. . . . . . . . . . . . . . . . . . . .26119.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26119.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26119.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261

19.4 Estimating poverty lines. . . . . . . . . . . . . . . . . . . . . .26119.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26119.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26119.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26119.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26219.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26219.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26219.4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26219.4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .262

19.5 Descriptive data analysis. . . . . . . . . . . . . . . . . . . . . .26219.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .262

19.6 Decomposing poverty . . . . . . . . . . . . . . . . . . . . . . .26219.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26319.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26319.6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263

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19.7 Poverty dominance. . . . . . . . . . . . . . . . . . . . . . . . .26319.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26319.7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26319.7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26319.7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26319.7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26319.7.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26319.7.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26419.7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26419.7.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .264

19.8 Fiscal incidence, growth, equity and poverty. . . . . . . . . . . . 26519.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26519.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26519.8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26519.8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26519.8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26519.8.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26619.8.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26619.8.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26619.8.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26619.8.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26619.8.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26619.8.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26719.8.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26719.8.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26719.8.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26719.8.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26719.8.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26719.8.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26819.8.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26819.8.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26819.8.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26819.8.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26819.8.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26919.8.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26919.8.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26919.8.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26919.8.27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .269

CONTENTS 9

19.8.28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26919.8.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26919.8.30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26919.8.31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27019.8.32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27019.8.33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27019.8.34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27019.8.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27019.8.36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27019.8.37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27019.8.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27019.8.39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27119.8.40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27119.8.41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27119.8.42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27119.8.43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27119.8.44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27119.8.45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27119.8.46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27119.8.47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27219.8.48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27219.8.49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27219.8.50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27219.8.51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27219.8.52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27219.8.53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27219.8.54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27319.8.55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27319.8.56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27319.8.57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273

19.9 Impact of a complex sample design on the estimated standard de-viation of various distributive statistics.. . . . . . . . . . . . . .27419.9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .274

19.10Impact of equivalence scales and choice of statistical units. . . . 27419.10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27419.10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27419.10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27519.10.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275

CONTENTS 10

19.10.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275

20 Graphs and tables 341

Symbols 388

Authors 395

1 PREFACE 11

1 Preface

This book is in large part due to the support of the International Develop-ment Research Center (IDRC) of the Government of Canada through two signifi-cant endeavors, the Micro Impacts of Macro Economic and Adjustment Program(MIMAP) and the Poverty and Economic Policy (PEP) international research net-work. The Secretariat d’appui institutionnela la rechercheeconomique en Afrique(SISERA), the World Bank Institute and the African Economic Research Consor-tium (AERC) also contributed to its development. The fundamental research wasfurther supported by grants from the Social Sciences and Humanities ResearchCouncil of Canada (SSHRC) and from the Fonds Quebecois de Recherche sur laSociete et la Culture (FQRSC) of the Province of Quebec.

This book is mostly targeted to senior undergraduate and graduate students ineconomics as well as to researchers and analytical policy makers. More generally,it is intended for social scientists and statisticians. Some of its content can also beinstructive to less specialized readers, such as those in the general public wishingto introduce themselves to the challenges posed by distributive analysis.

We certainly cannot pretend the book to be a comprehensive survey of themethods used to analyze poverty and equity. There is an obvious tendency forone’s exposition of a subject to be biased in favor of the work one knows best– and thus in favor of the work most closely related to our own. This book isa clear example of this bias. One advantage of such a bias, however, is that ittends to unify the exposition. Such a unification, we tried to enforce as much aswe could throughout the various parts of the book. This helped present in a singletext a unified treatment of distributive analysis from a conceptual, methodological,policy, statistical, empirical and practical point of view.

Most of the book’s footnotes refer to applications programmed inDAD. Thesefootnotes can thus guide the reader to where to go inDAD to test and implementmany of the measurement and statistical tools exposed in the book. In the marginsappear the exercise numbers which can be used to learn more about the book’stools. Most of these exercises involve the use ofDAD. The solutions to the exer-cises can be found onDAD’s web page.

To ease exposition within the main text, we also endeavored to limit as muchpossible references to the literature, except when such references were clearlyimproving readability. Instead, each chapter is followed by a reference section inwhich the chapter’s appropriate bibliographic references are mentioned and linkedto each other.

This book and the accompanying software are certainly perfectible. I suppose

1 PREFACE 12

it is the plight of all book writers to feel that their product is never really finished.We hope to correct this version’s shortcomings in future editions. For this, anycomments on this first edition will be gratefully received.

I wish to thank my co-authors and former students, Sami Bibi, Philippe Gregoire,Vincent Jalbert, Paul Makdissi and Martin Tabi for their insights and dedication.I am also very grateful to my co-authors on distributive analysis papers – Rus-sell Davidson, DamienEchevin, Carl Fortin, Peter Lambert, Magda Mercader,David Sahn, Steve Younger and Quentin Wodon – for their friendship and fruit-ful collaboration. Work at Universite Laval was made particularly productive andenjoyable by the encouragement of my colleagues – among whom Bernard De-caluwe and Bernard Fortin feature prominently as former heads of CREFA – andmore generally by the support of the Department of Economics and CIRPEE (for-merly CREFA). My thanks also extend to MIMAP and PEP co-workers, amongothers Touhami Abdelkhalek, Louis-Marie Asselin, Dorothee Boccanfuso, JohnCockburn, Anyck Dauphin, Samuel Kabore, Damien Mededji, Abena Oduro, LucSavard, Randy Spence, and to the teams of MIMAP and AERC researchers withwhich we have had the pleasure and privilege to work in the last decade. Theyprovided much of the motivation and inspiration for writing this book. I am alsograteful to my co-author, Abdelkrim Araar, for the trust and dedication he put intobuilding DAD and this book’s material over the last years, despite the uncertaintythat initially clouded the project.

Jean-Yves DuclosUniversite LavalQuebec

13

Part I

Conceptual and methodologicalissues

2 WELL-BEING AND POVERTY 14

2 Well-being and poverty

The assessment of well-being for poverty analysis is traditionally character-ized according to two main approaches, which, followingRavallion (1994), wewill term the welfarist and the non-welfarist approaches. The first approach tendsto concentrate in practice mainly on comparisons of ”economic well-being”, whichwe will also call ”standard of living” or ”income” (for short). As we will see, thisapproach has strong links with traditional economic theory, and it is also widelyused by economists in the operations and research work of organizations such asthe World Bank, the International Monetary Fund, and Ministries of Finance andPlanning of both developed and developing countries. The second approach hashistorically been advocated mainly by social scientists other than economists andpartly in reaction to the first approach. This second approach has nevertheless alsobeen recently and increasingly used by economists and non-economists alike as asound multidimensional complement to the classical standard of living approach.

2.1 The welfarist approach

The welfarist approach is strongly anchored in classical micro-economics,where, in the language of economics, ”welfare” or ”utility” are generally key inaccounting for the behavior and the well-being of individuals. Classical micro-economics usually postulates that individuals are rational and that they can bepresumed to be the best judges of the sort of life and activities which maximizetheir utility and happiness. Given their initial endowments (including time, landand physical and human capital), individuals make production and consumptionchoices using their set of preferences over bundles of consumption and produc-tion activities, and taking into account the available production technology andthe consumer and producer prices that prevail in the economy.

Under these assumptions and constraints, a process of individual and rationalfree choice will maximize the individuals’ utility; under additional assumptions(including that markets are competitive, that agents have perfect information, andthat there are no externalities – assumptions that are thus restrictive), a societyof individuals all acting independently under this freedom of choice process willalso lead to an outcome known as Pareto-efficient, in that no one’s utility could befurther improved by government intervention without decreasing someone else’sutility.

Underlying the welfarist approach to poverty, there is a premise that goodnote should be taken of the information revealed by individual behavior when


it comes to assessing poverty. This says, more particularly, that the assessmentof someone’s well-being should be consistent with the ordering of preferencesrevealed by that person’s free choices. For instance, a person could be observedto be poor by the total consumption or income standard of a poverty analyst. Thatsame person could nevertheless be able (i.e., have the working capacity) to be non-poor. This could be revealed by the observation of a deliberate and free choiceon the part of the individual to work and consume little, when the capability towork and consume more nevertheless exists. By choosing to spend little (possiblyfor the benefit of greater leisure), the person reveals that he is happier than if heworked and spent more. Although he would be considered poor by the standard ofa (non-welfarist) poverty analyst, a welfarist judgement could conclude that thisperson is not poor. As we will discuss later, this can have important implicationsfor the design and the assessment of public policy.

A pure welfarist approach faces important practical problems. To be opera-tional, pure welfarism requires the observation of sufficiently informative revealedpreferences. This is rarely the case, however. For instance, for someone to be de-clared poor or not poor, it is not enough to know that person’s current characteris-tics and income status: it must also be inferred from that person’s actions whetherhe judges his utility status to be above a certain utility poverty level.

Another – perhaps more fundamental – problem with the pure welfarist ap-proach is the need to assess levels of utility or ”psychic happiness”. How are weto measure the actual pleasure derived from experiencing economic well-being?Moreover, it is highly problematic to attempt to compare that level of utility acrossindividuals – it is well known that such a procedure poses serious ethical prob-lems. Preferences are heterogeneous, personal characteristics, needs and enjoy-ment abilities are diverse, households differ in size and composition, and pricesvary across time and space. More generally, because economic well-being (in par-ticular, utility) is typically seen as a subjective concept, most economists believethat interpersonal comparisons of economic well-being do not make sense.

Supposing that these criticisms are resolved, the welfarist approach wouldclassify as poor an individual who is materially well-off but not content, andasnot poor an individual not financially well-off but nevertheless content. It iscertainly not clear that we should accept as ethically significant these individualfeelings of utilities. Said differently, why should a difficult-to-satisfy rich per-son be judged less happy than an easily-contented poor person? Or, in the wordsof Sen (1983),p.160, why should a ”grumbling rich” be judged ”poorer” than a”contented peasant” ?

Hence, welfarist comparisons of poverty almost invariably use imperfect but


observable proxies for utilities, such as income or consumption. The ”working”definition of poverty for the welfarist approach is therefore alack of commandover commodities, measured by low income or consumption. These money-metricindicators are often adjusted for differences in needs, prices, and household sizesand compositions, but they clearly do remain far-from-perfect indicators of utilityand well-being. Indeed, economic theory tells us little about how to use con-sumption or income to make consistent interpersonal comparisons of well-being.Besides, the consumption and income proxies are rarely able to take full accountof the role for well-being of public goods and non-market commodities, such assafety, liberty, peace, health. In principle, such commodities can be valued usingreference or ”shadow” prices. In practice, this is very difficult to do accuratelyand consistently.

2.2 Non-welfarist approaches

2.2.1 Basic needs and functionings

There are two major non-welfarist approaches, the basic-needs approach andthe capability approach. The first focuses on the need to attain some basic multidi-mensional outcomes that can be observed and monitored relatively easily. Theseoutcomes are usually (explicitly or implicitly) linked with the concept of function-ings, a concept largely developed in Amartya Sen’s influential work:

Living may be seen as consisting of a set of interrelated ’function-ings’, consisting of beings and doings. A person’s achievement inthis respect can be seen as the vector of his or her functionings. Therelevant functionings can vary from such elementary things as beingadequately nourished, being in good health, avoiding escapable mor-bidity and premature mortality,etc., to more complex achievementssuch as being happy, having self-respect, taking part in the life of thecommunity, and so on (Sen (1992), p.39).

In this view, functionings can be understood to beconstitutiveelements of well-being. Onelives well if he enjoys a sufficiently large level of functionings. Thefunctioning approach would generally not attempt to compress these multidimen-sional elements into a single dimension such as utility or happiness. Utility orhappiness is viewed as a single and reductive aggregate of functionings, whichare multidimensional in nature. The functioning approach usually focuses in-stead on the attainment of multiple specific and separate outcomes, such as the


enjoyment of a particular type of commodity consumption, being healthy, literate,well-clothed, well-housed, not in shape,etc..

The functioning approach is closely linked with the well-known basic needsapproach, and the two are often difficult to distinguish in practice. Functionings,however, are not synonymous with basic needs. Basic needs can be understoodas the physical inputs that are usually required for individuals to achieve somefunctionings. Hence, basic needs are usually defined in terms of means ratherthan outcomes, for instance, as living in the proximity of providers of health careservices (but not necessarily being in good health), as the number of years ofachieved schooling (not necessarily as being literate), as living in a democracy(but not necessarily as participating in the life of the community), and so on. Inother words,

Basic needs may be interpreted in terms of minimum specified quan-tities of such things as food, shelter, water and sanitation that are nec-essary to prevent ill health, undernourishment and the like (Streetenand Others (1981)).

Unlike functionings, which can be commonly defined for all individuals, thespecification of basic needs depends on the characteristics of individuals and ofthe societies in which they live. For instance, the basic commodities required forsomeone to be in good health and not to be undernourished will depend on the cli-mate and on the physiological characteristics of individuals. Similarly, the clothesnecessary for one not to feel ashamed will depend on the norms of the society inwhich he lives, and the means necessary to travel, on whether he is handicappedor not. Hence, although the fulfillment of basic needs is an important elementin assessing whether someone has achieved some functionings, this assessmentmust also use information on one’s characteristics and socio-economic environ-ment. Human diversity is such that equality in the space of basic needs generallytranslates into inequality in the space in functionings.

Whether unidimensional or multidimensional in nature, most applications ofboth the welfarist and the non-welfarist approaches to poverty measurement dorecognize the role of heterogeneity in characteristics and in socio-economic en-vironments in achieving well-being.Streeten and Others (1981) and others havenevertheless argued that the basic needs approach is less abstract than the welfaristapproach in recognizing that role. As mentioned above, assessing the fulfillmentof basic needs can also be seen as a useful practical and operational step towardsappraising the achievement of the more abstract ”functionings”.


Clearly, however, there are important degrees in the multidimensional achieve-ments of basic needs and functionings. For instance, what does it mean preciselyto be ”adequately nourished”? Which degree of nutritional adequacy is relevantfor poverty assessment? Should the means needed for adequate nutritional func-tioning only allow for the simplest possible diet and for highest nutritional effi-ciency? These problems also crop up in the estimation of poverty lines in thewelfarist approach. A multidimensional approach extends them to several dimen-sions.

In addition, how ought we to understand such functionings as the functioningof self-respect? The appropriate width and depth of the concept of basic needs andfunctionings is admittedly ambiguous, as there are degrees of functionings whichmake life enjoyable in addition to being purely sustainable or satisfactory. Fur-thermore, could some of the dimensions be substitutes in the attainment of a givendegree of well-being? That is, could it be that one could do with lower needs andfunctionings in some dimensions if he has high achievements in the other dimen-sions? Such possibilities of substitutability are generally ignored (and are indeedhard to specify precisely) in the multidimensional non-welfarist approaches.

2.2.2 Capabilities

A second alternative to the welfarist approach is called the capability ap-proach, also pioneered and advocated in the last two decades by the work of Sen.The capability approach is defined by thecapacity to achieve functionings, asdefined above. InSen (1992)’s words,

the capability to function represents the various combinations of func-tionings (beings and doings) that the person can achieve. Capabilityis, thus, a set of vectors of functionings, reflecting the person’s free-dom to lead one type of life or another. (p.40)

What matters for the capability approach is the ability of an individual to functionwell in society; it isnot the functionings actually by the personper se. Havingthe capability to achieve ”basic” functionings is the source of freedom to livewell, and is thereby sufficient in the capability approach for one not to be poor ordeprived.

The capability approach thus distances itself from achievements of specificoutcomes or functionings. In this, it imparts considerable value to freedom ofchoice: a person will not be judged poor even if he chooses not to achieve some


functionings, so long as he would be able to attain them if he so chose. Thisdistinction between outcomes and the capability to achieve these outcomes alsorecognizes the importance of preference diversity and individuality in determiningfunctioning choices. It is, for instance, not everyone’s wish to be well-clothed orto participate in society, even if the capability is present.

An interesting example of the distinction between fulfilment of basic needs,functioning achievement and capability is given inTownsend (1979)’s (Table 6.3)deprivation index. This deprivation index is built from answers to questions suchas whether someone ”has not had an afternoon or evening out for entertainment inthe last two weeks”, or ”has not had a cooked breakfast most days of the week”.It may be, however, that one chooses deliberately not to go out for entertainment(he prefers to watch television), or that he chooses not to have a cooked breakfast(because he does want to spend the time to prepare it), although he does havethe capacity to do both. That person therefore achieves the functioning of beingentertained without meeting the basic need of going out once a fortnight, and doeshave the capacity to achieve the functioning of having a good breakfast, althoughhe chooses not to.

The difference between the capability and the functioning or basic needs ap-proach is in fact somewhat analogous to the difference between the use of incomeand consumption as indicators of living standards. Income shows the capability toconsume, and ”consumption functioning” can be understood as the outcome of theexercise of that capability. There is consumption only if a person chooses to enacthis capacity to consume out a given income. In the basic needs and functioningapproach, deprivation comes from a lack of direct consumption or functioning ex-perience; in the capability approach, poverty arises from the lack of incomes andcapabilities, which are imperfectly related to the actual functionings achieved.

Although the capability set is multidimensional, it thus exhibits a parallel withthe unidimensional income indicator, whose size determines the size of the ”bud-get set”:

Just as the so-called ’budget set’ in the commodity space representsa person’s freedom to buy commodity bundles, the ’capability set’ inthe functioning space reflects the person’s freedom to choose frompossible livings (Sen (1992), p. 40).

This shows further the fundamental distinction between the space of achieve-ments, the extents of freedoms and capabilities, and the resources required togenerate these freedoms and to attain these achievements.


2.3 A graphical illustration

To illustrate the relationships between the main approaches to assessing poverty,consider Figure12. Figure12 shows in four quadrants the links between income,consumption of two commodities – transportationT and clothingC goods – andthe functionings associated to the consumption of each of these two goods. Thenortheast quadrant shows a typical two-good budget set for the two goods andwith a budget constraintY 1. The curveU1 shows the utility indifference curvealong which the consumer chooses his preferred commodity bundle, which is herelocated at pointA.

The southeastern and the northwestern quadrants then transform the consump-tion of goodsT and C into associated functioningsFT and FC . This is donethrough the Functioning Transformation CurvesTCT andTCC , for transforma-tion of consumption ofT andC into transportation and clothing functionings, re-spectively. The curvesTCT andTCC appear respectively in the northwest and thesoutheast quadrants respectively. These curves thus bring us from the northeast-ern space of commodities,C, T, into the southwestern space of functionings,FC , FT. Using these transformation functions, we can draw a budget constraintS1 in the space of functionings using the traditional commodity budget constraint,Y 1. Since the consumer chooses pointA in the space of commodities, he enjoysB’s combination of functionings. But all of the functionings within the constraintS1 can also be attained by the consumer. The triangular area between the ori-gin and the lineS1 thus represents the individual’s capability set. It is the set offunctionings which he is able to attain.

Now assume that functioning thresholds ofzC andzT must be exceeded (ormust be potentially exceeded) for one not to be considered poor by non-welfaristanalysts. Given the transformation functionsTCT andTCF , a budget constraintY 1 makes the individualcapableof not being poor in the functioning space. Butthis does not guarantee that the individual will choose a combination of function-ings that will exceedzC andzT : this will also depend on the individual’s prefer-ences. At pointA, the functionings achieved are above the minimum functioningthreshold fixed in each dimension. Other points within the capability set wouldalso surpass the functioning thresholds: these points are shown in the shaded tri-angle to the northeast of pointB. Since part of the capability set allows the indi-vidual to be non-poor in the space of functionings, the capability approach wouldalso declare the individual not to be poor.

So would conclude, too, the functioning approach since the individualchoosesfunctionings abovezC andzT . Such a concordance between the two approaches


does not always have to prevail, however. To see this, consider Figure15. Thecommodity budget set and the Functioning Transformation Curves have not changed,so that the capability set has not changed either. But there has a been a shift ofpreferences fromU1 to U2, so that the individual now prefers pointD to pointA,and also prefers to consume less clothing than before. This makes his preferencesfor functionings to be located at pointE, thus failing to exceed the minimumclothing functioningzC required. Hence, the person would be considered nonpoor by the capability approach, but poor by the functioning approach. Whetheran individual with preferencesU2 is really poorer than one with preferencesU1 isdebatable, of course, since the two have exactly the same ”opportunity sets”, thatis, have access to exactly the same commodity and capability sets.

An important message of the capability approach is that two persons with thesame commodity budget set can face different capability sets. This is illustratedin Figure16, where the Functioning Transformation Curve for transportation hasshifted fromTCT to TC ′

T . This may due to the presence of a handicap, whichmakes it more costly in transportation expenses to generate a given level of trans-portation functioning (disabled persons would need to expend more to go fromone place to another). This shift of theTCT curve moves the capability constraintto S1′ and thus contracts the capability set. With the handicap, there is no pointwithin the new capability set that would surpass both functioning thresholdszC

andzT . Hence, the person is deemed poor by the capability approach and (nec-essarily so) by the functioning approach. Whether the welfarist approach wouldalso declare the person to be poor would depend on whether it takes into accountthe differences in needs implied by the difference between theTCT and theTC ′

T

curves.For the welfarist approach to be reasonably consistent with the functioning

and capability approaches, it is thus essential to consider the role of transforma-tion functions such as theTC curves. If this is done, we may (in our simpleillustration at least) assess a person’s capability status either in the commodity orin the functioning space.

To see this, consider Figure17. Figure17 is the same as Figure12 exceptfor the addition of the commodity budget constraintY 2 which shows the mini-mum consumption level needed for one not to be poor according to the capabilityapproach. According to the capability approach, the capability set must containat least one combination of functionings abovezC andzT , and this condition isjust met by the capability constraintS2 that is associated with the commoditybudgetY 2. Hence, to know whether someone is poor according to the capabilityapproach, we may simply check whether his commodity budget constraint lies


belowY 2.Even if the actual commodity budget constraint lies aboveY 2, the individual

may well choose a point outside the non-poor functioning set, as we discussedabove in the context of Figure15. Clearly then, the minimum total consumptionneeded for one to be non poor according to the functioning or basic needs ap-proach generally exceeds the minimum total consumption needed for one to benon poor according to the capability approach. More problematically, this mini-mum total consumption depends in principle on the preferences of the individu-als. On Figure15, for instance, we saw that the individual with preferenceU2 wasconsidered poor by the functioning approach, although another individual with thesame budget and capability sets was considered non-poor by the same approach.

2.3.1 Exercises

1. Show on a figure such as Figure12 the impact of an increase in the price ofthe transportation commodity on the commodity budget constraint and onthe capability constraint.

2. On a figure such as Figure17, show the minimal commodity budget set thatensures that the person

(a) is just able to attain one of the two minimum levels of functioningszC

or zT ;

(b) chooses a combination of functionings such that one of them exceedsthe corresponding minimum level of functioningszC or zT ;

(c) is just able to attain both minimum levels of functioningszC andzT ;

(d) chooses a combination of functionings such that both exceed the cor-responding minimum level of functioningszC andzT .

(e) How do these four minimal commodity budget constraints compareto each other? Which one corresponds to the different approaches toassessing poverty seen above?

2.4 Practical measurement difficulties for the non-welfarist ap-proaches

The measurement of capabilities raises various problems. Unless a personchooses to enact them in the form of functioning achievements, capabilities are


not easily inferred. Achievement of all basic functionings implies non-deprivationin the space of all capabilities; but a failure to achieve all basic functionings doesnot imply capability deprivation. This makes the monitoring of functioning andbasic needs an imperfect tool for the assessment of capability deprivation.

Besides, and as for basic needs, there are clearlydegreesof capabilities, somebasic and some wider. It is unlikely that true well-being is a dichotomic and dis-continuous function of achievement and capabilities. For most of the functioningsassessed empirically, there are indeed degrees of achievement, such as for beinghealthy, literate, living without shame,etc...It is important to take into account thevarying degrees of well-being in assessing and comparing the intensity of poverty,and not only to record dichotomic answers to multidimensional qualitative povertycriteria.

The multidimensional nature of the non-welfarist approaches also raises prob-lems of comparability across dimensions. How should we assess adequately thedegree of poverty of someone who has the capability to achieve two functioningsout of three, but not the third? Is that person necessarily ”better off” than some-one who can achieve only one, or even none of them? Are all capabilities of equalimportance when we assess well-being?

The multidimensionality of the non-welfarist criteria also translates into greaterimplementation difficulties than for the usual proxy indicators of the welfarist ap-proach. In the welfarist approach, the size of the multidimensional budget is ordi-narily summarized by income or total consumption, which can be thought of as aunidimensional indicator of freedom. Although there are many different combina-tions of consumption and functionings that are compatible with a unidimensionalmoney-metric poverty threshold, the welfarist approach will generally not imposemultidimensional thresholds. For instance, the welfarist approach will usuallynot require for one not to be poor that both food and non-food expenditures belarger than their respective food and non-food poverty lines. A similar transfor-mation into a unidimensional indicator is more difficult with the capability andbasic needs approaches.

One possible solution to this comparability problem is to use ”efficiency-income units reflecting command over capabilities rather than command overgoods and services” (??? Sen (1984), p.343), as we illustrated above when dis-cussing Figure17. This, however, is practically difficult to do, since commandover many capabilities is hard to translate in terms of a single indicator, andsince the ”budget units” are hardly comparable across functionings such as well-nourishment, literacy, feeling self-respect, and taking part in the life of the com-munity. On Figure17, anyone with an income belowY 2 would be judged capability-


poor. But by how much does poverty vary among these capability-poor? A naturalmeasure would be a function of the budget constraint. It is more difficult to makesuch measurements and comparisons in the capability set.


2.5 Poverty measurement and public policy

2.5.1 Poverty measurement matters

The measurement of well-being and poverty plays a central role in the dis-cussion of public policy. It is used, among other things, to identify the poor andthe non-poor, to design optimal poverty relief schemes, to estimate the errors ofexclusion and inclusion in the set of the poor (also known as Type I and Type II er-rors), and to assess the equity of poverty alleviation policy. Is growth ”pro-poor”?How do indirect taxes and relative price changes affect the poor? What should thetarget groups be for socially-efficient government interventions? What impact dotransfers have on poverty? How many of the poor are excluded from safety netprogrammes? Is it the poorest of the poor who benefit most from public policy?Would a different sort of poverty alleviation policy reduce deprivation further?

An important example of the central role of poverty measurement in the set-ting of public policy is the optimal selection of safety net targeting indicators. Thetheory of optimal targeting suggests that it will commonly be best to target indi-viduals on the basis of indicators that are as easily observable and as exogenousas possible, while being as correlated as possible with the true poverty status ofthe individuals. Indicators that are not readily observable by programme admin-istrators are of little practical value. Indicators that can be changed effortlesslyby individuals will be distorted by the presence of the programme, and will losetheir poverty-informative value. Whether available indicators are sufficiently cor-related with the deprivation of individuals in a population is given by a povertyprofile. The value of this profile will naturally be highly dependent on the partic-ular assumptions and the approach used to measure well-being and poverty.

Estimation of the errors of inclusion and exclusion of the poor is also a prod-uct of poverty profiling and measurement. These errors are central in the trade-offinvolved in choosing a wide coverage of the population – at relatively low ad-ministrative and efficiency costs – and a narrower coverage – with more generousforms of support for the fewer beneficiaries. However, asvan de Walle (1998a)puts it, a narrower coverage of the population, with presumably smaller errors ofinclusion of the non-poor, does not inevitably lead to a more equitable treatmentof the poor:

Concentrating solely on errors of leakage to the non-poor can lead topolicies which have weak coverage of the poor (van de Walle (1998a),p.366)).

The terms of this trade-off are again given by a poverty assessment exercise.


Another lesson of optimal redistribution theory is that it is ordinarily better totransfer resources from groups with a high level of average well-being to thosewith a lower one. What matters more, however, is the distribution of well-beingwithin each of the groups. For instance, equalising mean well-being across groupsdoes not usually eliminate poverty since there generally exist within-group in-equalities. Even within the richer group, for instance, there normally will befound some deprived individuals, whom a rich-to-poor cross-group redistributiveprocess would clearly not take out of poverty. The within- and between-groupdistribution of well-being that is required for devising an optimal redistributivescheme can be again revealed by a comprehensive poverty profile.

2.5.2 Welfarist and non-welfarist policy implications

The distinction between the welfarist and non-welfarist approaches to povertymeasurement often matters (implicitly or explicitly) for the assessment and thedesign of public policy. As described above, a welfarist approach holds that in-dividuals are the best judges of their own well-being. It would thus in principleavoid making appraisals of well-being that conflict with the poor’s views of theirown situation. A typical example of a welfarist public policy would be the provi-sion of adequate income-generating opportunities, letting individuals decide andreveal whether these opportunities are utility maximising, keeping in mind theother non-income-generating opportunities that are available to them.

A non-welfarist policy analyst would argue, however, that raising income op-portunities is not necessarily the best policy option. This is partly because in-dividuals are not necessarily best left with their own resolutions, at least in anintertemporal setting, for their educational and environmental choices for exam-ple. The poor’s short-run preoccupations may, for instance, harm their long-termself-interest. Individuals may choose not to attend skill-enhancing programmesbecause they deceivingly appear overly time costly in the short-run, and becausethey are not sufficiently convinced or aware of their long-term benefits. Besides,if left to themselves, the poor will not necessarily spend their income increase onfunctionings that basic-needs analysts would normally consider a priority, such asgood nutrition and health.

Thus, fulfilling ”basic needs” cannot be satisfied only by the generation ofprivate income, but may require significant amounts of targeted and in-kind pub-lic expenditures on areas such as education, public health and the environment.This would be so even if the poor did not presently believe that these areas weredeserving of public expenditures. Furthermore, social cohesion concerns are ar-


guably not well addressed by the maximization of private utility, and raisingincome opportunities will not fundamentally solve problems caused by adverseintra-household distributions of well-being, for instance.

An objection to the basic needs approach is that it is clearly paternalistic sinceit supposes that it is in the absolute interests of all to meet a set of often arbitrarilyspecified needs. Indeed, as emphasized above, non-welfarist approaches in gen-eral may use criteria for identifying and helping the poor that may conflict withthe poor’s views and utility maximizing choices. The welfarist school converselyemphasizes that individuals are generally better placed to judge what is good forthem. For instance,

To conclude that a person was not capable of living a long life wemust know more than just how long she lived: perhaps she preferreda short but merry life. (Lipton and Ravallion (1995))

To force that person to live a long but boring life might thus go against her prefer-ences.

For poverty alleviation purposes, the prescriptions of non-welfarist approachescould in principle go as far as, for instance, enforced enrolment in community de-velopment programmes, forced migration, or forced family planning. This maynot only conflict with the preferences of the poor, but would also clearly under-mine their freedom to choose. Freedom to choose is, however, arguably one of themost important basic capabilities which contribute fundamentally to well-being,and it should therefore be an important criterion for policy assessment.

A further example of the possible tension between the welfarist and non-welfarist approaches to public policy comes from optimal taxation theory, whichis linked to optimal poverty alleviation theory. In the tradition of classical microe-conomics, which values leisure in the production and labour market decisions ofindividuals, pure welfarists would incorporate the utility of leisure in the overallutility function of workers, poor and non-poor alike. In its support to the poor,the government would then take care of minimizing the distortion of their la-bor/leisure choices so as not to create overly high ”deadweight losses”. Classicaloptimal taxation theory then shows that giving a positive weight to such things aslabor/leisure distortions suggests a generally lower benefit reduction rates on theincome of the poor than otherwise. Taking into account such abstract things as”deadweight losses” is, however, less typical of the basic needs and functioningapproaches. Such approaches would, therefore, usually be less reluctant to targetprogramme benefits more sharply on the poor, and more likely to exact steeperbenefit reduction rates as income or well-being increases.


Relative to the pure welfarist approach, non-welfarist approaches are also typ-ically less reluctant to impose utility-decreasing (or ”workfare”) costs as side ef-fects of participation in poverty alleviation schemes. These side effects are in factoften observed in practice. For instance, it is well-known that income supportprogrammes frequently impose participation costs on benefit claimants. Theseare typically non-monetary costs. Such costs can be both physical and psycholog-ical: providing manual labor, spending energy, spending time away from home,sacrificing leisure and home production, finding information about application andeligibility conditions, corresponding and dealing with the benefit agency, queuing,keeping appointments, complying with application conditions, revealing personalinformation, feeling ”stigma” or a sense of guilt,etc...

Although non-monetary, these costs have a clear impact on participants’ netutility from participating in the programmes. When they are negatively corre-lated with unobserved (or difficult to observe) entitlement indicators, they canprovide self-selection mechanisms that enhance the efficiency of poverty allevia-tion programmes, for welfarists and non-welfarists alike. One unfortunate effectof these costs is, however, that many truly-entitled and truly deserving individualsmay shy away from the programs because of the costs they impose. Althoughprogram participation could raise their income and consumption above a money-metric poverty line, some individuals will prefer not to participate, revealing thatthey find apparent poverty utility dominant over program participation. Welfaristswould in principle take these costs into account when assessing the merits of theprogrammes. Non-welfarists would typically not do so, and would therefore judgethe programs more favorably.

Finally, the width of the definition of functionings is also important for theassessment and the design of public policy. For instance, public spending on ed-ucation is often promoted on the basis of its impact on productivity and growth.But education can also be seen as a means to attain the functioning of literacy andparticipation in the community. This provides an additional support for public ex-penditures on education. Analogous arguments also apply, for instance, to publicexpenditures on health, transportation, and the environment.

3 THE EMPIRICAL MEASUREMENT OF WELL-BEING 29

3 The empirical measurement of well-being

The empirical assessment of poverty and equity is customarily carried out us-ing data on households and individuals. These data can be administrative (i.e.,stored in government files and records), they can come from censuses of the entirepopulation, or (most commonly) they can be generated by probabilistic surveys onthe characteristics and living conditions of a population of households.

3.1 Survey issues

There are several aspects of the surveying process that are important for as-sessment of poverty and equity. First, there is the coverage of the survey: does itcontain representative information on the entire population of interest, or just onsome socio-economic subgroups? Whether the representativeness of the data isappropriate depends on the focus of the assessment. A survey containing obser-vations drawn exclusively from the cities of a particular country may be perfectlyfine if the aim is to design poverty alleviation schemes within these cities; its rep-resentativeness will, however, be clearly insufficient if the objective is to assessthe allocation of resources between the country’s urban and rural areas.

Then, there is the sample frame of the survey. Surveys are usually multi-staged, and built upon strata and clusters. Stratification ensures that a certainminimum amount of information is obtained from each of a given number of ar-eas within a population of interest. Population strata are often geographic andcan represent, for instance, the different regions or provinces of a country. Clus-tering facilitates the interviewing process by concentrating sample observationswithin particular population subgroups or geographic locations. Strata are thusoften divided into a number of different levels of clusters, representing, say, cities,villages, neighborhoods, or households. A complete listing of the clusters in eachstratum is used to select randomly within each stratum a given number of clusters.The selected clusters can then be subjected to further stratification or clustering,and the process continues until the last sampling units (usually households or indi-viduals) have been selected and interviewed. Thus, there is not only stratificationbut also multi-stage sampling.

Fundamental in the use of survey data is the role of the randomness of theinformation that is generated. Because households and individuals are not all sys-tematically interviewed (unlike in the case of censuses), the information generatedfrom survey data will depend on the particular selection of households and individ-uals made from a population. A poverty/equity assessment of a given population


will then vary across the various samples that can be selected from this same pop-ulation. For that reason, distributive assessments carried out using survey data willbe subject to so-called ”sampling errors”, that is, to sampling variability. Whengenerating distributive assessments from sample survey data, it is therefore im-portant to recognisee and assess the statistical imprecision of the sampling resultsobtained.

By ensuring that a minimum amount of information (typically, a certain num-ber of observations) is obtained from each of a number of strata, stratificationdecreases the extent of sampling errors. A similar effect is obtained by increas-ing the total size of the sample: the greater the number of households surveyed,the greater on average is the precision of the estimates obtained. Conversely, bybundling observations around common geographic or socio-economic indicators,clustering tends to reduce the informative content of the observations made andthus to increase the size of the sampling errors (for a given number of observa-tions). The sampling structure of a survey also impacts on its ability to provideaccurate information on certain population subgroups. For instance, if the clus-ters within a stratum represent regions, and between-region variability is large, itwould not be reasonable to try to use the information generated by the selectedregions to depict poverty in the other, non-selected, regions.

Survey data are also fraught with measurement and other ”non-sampling” er-rors. For instance, even though they may have been selected for appearance ina sample, some households will not be interviewed, either because they cannotbe reached or because they refuse to be interviewed. Such ”non-response” willraise difficulties for distributive assessments if it is correlated with observableand non-observable household characteristics. Even if interviewed, householdswill sometimes consistently misreport their characteristics and living conditions,either because of ignorance, mischief or self-interest. This tends to make distribu-tive assessments built from survey data diverge systematically from the true (andunobserved) population distributive assessment that would be carried out if therewere no non-sampling errors. Clearly, such a shortcoming can bias the under-standing of poverty and equity and the consequent design of public policy.

The empirical analysis of vulnerability and poverty dynamics is particularly”data demanding”. In general, it requires longitudinal (or panel) surveys, whichfollow each other in time and which interview the same final observational units.Because they link the same units across time, longitudinal data can contain moreinformation than transversal (or cross-sectional) surveys, and they are particularlyuseful for measuring vulnerability and for understanding poverty dynamics – inaddition to facilitating the assessment of the temporal effects of public policy on


well-being. It must be stressed, however, that measurement error problems renderthe analysis of vulnerability and mobility very difficult, and its results must beinterpreted with caution.

3.2 Income versus consumption

It is frequently argued that consumption is better suited than income as an in-dicator of living standard, at least in many developing countries. One reason isthat consumption is believed to vary more smoothly than income, both within anygiven year and across the life cycle. Income is notoriously subject to seasonalvariability, particularly in developing countries, whereas consumption tends to beless variable. Life-cycle theories also predict that individuals will try to smooththeir consumption across their low- and high-income years (in order to equalizetheir ”marginal utility of consumption” across time), through appropriate borrow-ing and saving. In practice, however, consumption smoothing is far from perfect,in part due to imperfect access to commodity and credit markets and to difficultiesin estimating precisely one’s ”permanent” or life-cycle income.

For the non-welfarist interested in outcomes and functionings, consumption isalso preferred over income because it is deemed to be a more ”direct” indicator ofachievements and fulfilment of basic needs. A caveat is, however, that consump-tion is indeed an outcome of individual free choice, an outcome which may differacross individuals of the same income and ability to consume, just like actualfunctionings vary across people of the same capability sets. At a given capabil-ity to spend, some individuals may choose to consume less (or little), preferringinstead to give to charity, to vow poverty, or to save in order to give importantbequests to their children.

Consumption is also held to be more readily observed, recalled and measuredthan income (at least in developing countries, although even then this is not alwaysthe case), to suffer less from underreporting problems, This is not to say that con-sumption is easy to measure accurately. Sources of income are typically far morelimited than types of expenditures, which may make it easier to collect incomeinformation. The periodicity of expenses on various goods varies, and differentrecall periods are therefore needed to ensure adequate expenditure coverage.

Moreover, consumption does not equal expenditures. The value of consump-tion equals the sum of the expenditures on the goods and services purchased andconsumed, plus the value of goods and services consumed but not purchased (suchas those received as gifts and produced by the household itself), plus the consump-tion or service value of assets and durable goods owned. Unlike expenditures,


therefore, consumption includes the value of own-produced goods. The valueof these goods is not easily assessed, since it has not been transacted in a mar-ket. Distinguishing consumption expenditures from investment expenditures isalso very difficult, but failure to do so properly can lead to double-counting in theconsumption measure. For instance, a $1 expenditure on education or machineryshould not be counted as current consumption if the returns and the utility of suchexpenditure will only accrue later in the form of higher future earnings.

Similarly, and as just mentioned, the value of the services provided by thosedurable goods owned by individuals ought also to enter into a complete consump-tion indicator, but the cost of these durable goods should not enter entirely into theconsumption aggregate of the time at which the good was purchased. An impor-tant example of this is owner-occupied housing. Further difficulties arise from theassessment of the value of various non-market goods and services – such as thoseprovided freely by the government – and the value of intangible benefits such asthe quality of the environment, the extent of security and peace, and so on.

3.3 Price variability

Whether it is income or consumption expenditures that are measured and com-pared, an important issue is how to account for the variability of prices acrossspace and time. Conceptually, this also includes variability in quality and in quan-tity constraints. Failure to account for such variability can distort comparisons ofwell-being across time and space. In Ecuador, for instance (Hentschel and Lan-jouw (1996)), and in many other countries, some households have free access towater, and tend to consume relatively large quantities of it with zero water ex-penditure. Others (often peri-urban dwellers) need to purchase water from privatevendors and consequently consume a lower quantity of it at necessarily highertotal expenditures. Ranking of households according to water expenditures couldwrongly suggest that those who need to buy water are better off and derive greaterutility from water consumption (since they spend more on it).

Microeconomic theory suggests that we may wish to account for price vari-ability by comparing real as opposed to nominal consumption expenditures (orincome). Several procedures can be followed to enable such comparisons. A firstprocedure estimates the parameters of the indirect utility function of the econ-omy’s consumers. These parameters identify the ordinal preferences of the con-sumer. Let these parameters be denoted byϑ and the indirect utility function bedefined byV (y, q, ϑ) , whereq is the price vector andy is total nominal expen-diture. Suppose that reference prices are given byqR. Equivalent consumption


expenditure is then given implicitly byyR:

V (yR, qR, ϑ) = V (y, q, ϑ). (1)

Inversion of the indirect utility function yields an equivalent expenditure func-tion e, which indicates how much expenditure at reference prices is needed tobe equivalent to (or to generate the same utility as) the expenditure observed atcurrent pricesq:

yR = e(qR, ϑ, V (y, q, ϑ)

). (2)

Distributive analysis would then proceed by comparing the real incomes definedin terms of the reference pricesqR.

An alternative procedure deflates by a cost-of-living index the level of totalnominal consumption expenditures. One way of defining such a cost-of-livingindex is to ask what expenditure is needed just to attain a poverty level of utilityvz at pricesq. This is given bye (q, ϑ, vz). A similar computation is carried outfor the expenditure needed to attainvz at pricesqR: this ise

(qR, ϑ, vz

). The ratio

e (q, ϑ, vz) /e(qR, ϑ, vz

)(3)

is then a cost-of-living index. Dividingy by (3) yields real consumption expendi-ture.

In practice, cost-of-living indices are often taken to be those aggregate con-sumer price indices routinely computed by national statistical agencies. Theseconsumer price indices vary usually across regions and time, but not across levelsof income (e.g., across the poor and the non-poor). In some circumstances (i.e.,for homothetic utility functions and when consumer preferences are identical), allof the above procedures are equivalent. In general, however, they are not the same.

The fact that utility functions are not generally homothetic, and that prefer-ences are highly heterogeneous, has important implications for distributive anal-ysis and public policy. First, the true cost-of-living index would normally be dif-ferent across the poor and the rich. Using the same price index for the two groupsmay distort comparisons of well-being. An example is the effect of an increase inthe price of food on economic well-being. Since the share of food in total con-sumption is habitually higher for the poor than for the rich, this increase shouldhurt disproportionately more the more deprived. Deflating nominal consumptionby the same index for the entire population will, however, suggest that the impactof the food price increase is shared proportionately by all.


Spatial disaggregation is also important if consumption preferences ad pricechanges vary systematically across regions. In few developing countries, however,are consumer price indices available or sufficiently disaggregated spatially. Thealternative is then to produce different poverty lines for different regions (based onthe same or different consumption baskets, but using different prices) or constructseparate food price indices. In both cases, the analyst would usually be usingregional price information derived from LSMS-style survey data. The resultingindices would then be interpreted as cost-of-living indices, and could help correctfor spatial price variation and regional heterogeneity in preferences.

To see why these adjustments are necessarily in part arbitrary, and to see whythey can matter in practice, consider the case of Figure29. It shows 3 indifferencecurvesU1, U2 andU3, for three consumers, 1, 2 and 3. Two of these consumershave relatively strong preferences for meat as opposed to fish, and the third (rep-resented byU3) has strong preferences for fish. Also shown are two budget con-straints, one using relative pricesqc (c for coastalarea), where the price of fish isrelatively low, and the other withqm (m for mountainousarea), where the price offish is high compared to the price of meat.

How is the standard of living for individuals 1,2 or 3 to be compared? Oneway to answer this question is to ”cost” the consumption of the three individuals.For this, we may use eitherqc or qm. If we use the mountains’ relative price,then the consumption bundles chosen by individuals 1 and 3 are equivalent interms of value: they lie on the same budget constraint of valueB in terms of meat(the numeraire). Individual 2 is clearly then the worst off of all three. If insteadwe use the coastal area’s relative price, then the consumption bundles chosen byindividuals 2 and 3 are equivalent, with a common value ofA in terms of meat –and individual 1 is the best off.

Hence, choosing reference prices to assess and compare living standards canmatter significantly. If we knewa priori that individuals 1 and 3 had equivalentliving standards, then reference pricesqm would be the right one (conversely:qc

would be the correct reference prices if 2 and 3 could be assumed equally welloff). But such information is generally not available. In some circumstances, suchas in comparing 1 and 2, we can be fairly certain that one individual is better offthan another, whatever the choice of reference prices, but even then, the extent ofthe quantitative difference in well-being can depend substantially on the choice ofreference prices.

The choice of reference prices and reference preferences will also matter forestimating the impact of price changes on well-being and equity. Consider againFigure29. Suppose that we wish to measure the impact on consumers’ well-being


of an increase in the price of fish. Assume for simplicity that this change in relativeprices is captured by a move fromqm to qc. If we were to choose as a referencebundle the bundle of meat and fish chosen by individuals 1 and 2 to capture theimpact of this change, then the price impact would be estimated to be fairly low.The reason is that both individuals consume little of fish. For instance, take meatas the numeraire and assess the real income value of being at U1. Underqm, thisis given byB and underqc by D. Using instead the preferences of individual 3 asreference tastes (and thusU3 as reference well-being), real consumption wouldmove fromA to B, a much greater change.

Furthermore, if 3 had been deemed better off than 1 before the increase in theprice of fish, it could well be that 3’s strong preferences for fish would make himless well off than 1 after the price change. Hence, when consumer preferencesare heterogeneous, price changes can reverse rankings in terms of well-being andpoverty. Indeed, in Figure29, the increase in the price of fish is visibly much morecostly for fish eaters than for meat ones. And, if 3 had been deemed poorer than1 initially, using 1’s preferences to capture the change of 3’s well-being wouldnot be appropriate since in this case the preferences of the richer are significantlydifferent from those of the poorer. This warns again against the use of a commonprice index across all regions, and as well as across all socio-economic groups –rich and poor.

3.3.1 Exercises

Suppose the following direct utility function over the two goodsx1 andx2,

U(x1, x2) = xν1x

1−ν2 ,

with ν = 1/3, and let pricesq1 andq2 be set to 1.

1. For a function%(q) independent ofy, prove that the indirect utility functionV (y, q) can be expressed as follows:

V (y, q) = %(q) · y.

2. If reference prices areqR1 = 1 andqR

2 = 1, show that equivalent consump-tion is given by:

yR =y

Υ(q)

for some functionΥ(q) independent ofy.


3. What is the expenditure needed to attain a poverty level of utility ofvz =159 at the reference prices? (Call thiszR.)

4. What are the quantities of goods 1 and 2 that are consumed at the povertylevel of utility?

5. Suppose that the price of good 2 is increased from 1 to 3. What is the newcost of the poverty level of utility? (Call thisz.)

6. Using definitions (1) and (2), prove the following :

yR/zR = y/z.

What does it imply?

7. Suppose now that a poverty analyst does not believe that consumption ofgoods 1 and 2 will adjust following good 2’s price increase. What is thepoverty linez that he would then obtain? (Hint: compute the cost of theinitial commodity basket using the new prices.)

8. Using indifference curves and budget constraints, show the difference thattaking account of changes in behavior can make for the computation of priceindices and the assessment of poverty.

3.4 Household heterogeneity

A fundamental problem arises when comparing the well-being of individualswho live in households of differing sizes and composition. Differences in house-hold size and composition can indeed be expected to create differences in house-hold ”needs”. It is essential to take these needs into account when comparing thewell-being of individuals living in differing households. This is typically doneusing equivalence scales. With these scales, the needs of a household of a partic-ular size and composition are said to be comparable to a household of a particularnumber of ”reference” or ”equivalent” adults.

3.4.1 Estimating equivalence scales

Strategies for the estimation of equivalence scales are all contingent on thechoice of comparable indicators of well-being. All such indicators are, however,intrinsically arbitrary. A popular example is food share in total consumption: at


equal household food shares, individuals of various household types are deemed tobe equally well-off. But, at equal well-being, one household type can well choosea food share that differs from that of the other household types. This would bethe case, for instance, for households of smaller sizes for which it would make”sense” to spend more on food than on those goods for which economies of scaleare arguably larger, such as housing.

Another difficulty arises when household size and composition are the resultof a deliberate free choice. It may be argued, for instance, that a couple whichelects freely to have a child cannot perceive this increase in household size to beutility-decreasing. This would be so even if the household’s total consumptionremained unchanged after the birth of the child (or even if it fell), despite the factthat most poverty analysts would judge this birth to increase household ”needs”.Another difficulty lies in the fact that the intra-household decision-making pro-cess can influence adversely the allocation of resources across household mem-bers, and thereby lead to wrong inferences of comparative needs. This is the case,for instance, when more is spent on boys than girls, not because of differentialneeds, but because of differential gender preferences on the part of the house-hold decision-maker. Such observations may lead analysts to overestimate thereal needs of boys relative to those of girls. Using these observed preferencesfor boys to estimate equivalence scales would then underestimate on average thelevel of deprivation experienced by girls and their households, since it would bewrongly assumed that girls are less ”needy”. An analogous analytical difficultyarises when the household decision-maker is a man, and the consumption of hisspouse is observed to be smaller than his own. Is this due to gender-biased house-hold decision-making, or to gender-differentiated needs?

To illustrate these issues, consider Figure30, which graphs consumption of areference goodxr(y, q) againsthouseholdincomey. The predicted consumptionof the reference good is plotted for two households, the first composed of onlyone man, and the other made of a couple (i.e., a man and a woman). A commonfeature of the literature on the estimation of equivalence scales is the estimationof the total household income at which a reference consumption of a referencegood is equal. The basic argument is that when the consumption of that referencegood is the same across households, the well-being of household members shouldalso be the same across households. Reference goods are often goods consumedexclusively by some members of the household, such as male or female clothing,perfume for women, a night out at the cinema for adults,etc..

For Figure30, take for instance the case of men’s clothing forxr(y, q). Sup-pose that the reference level of that good is given byx0. Leaving aside issues of


consumption heterogeneity within homogeneous households at a given incomelevel, one would estimate that the one-member household would need an in-comeyc in order to consumex0 (at pointc), and that the two-member householdwould require total household incomeyd to reach that same reference consump-tion. Hence, following this line of argument, the second household would needyd/yc as much income as the first one to be ”as well off” in terms of consumptionof men’s clothing. Said differently, the second household’s needs would beyd/ycthat of the single man household. The number of ”equivalent adults” in the secondhousehold would then be said to beyd/yc. This procedures applied to differenthousehold types provides a full equivalence scale, which expresses the needs andthe composition of households as a function of those of a reference (generally aone-adult) household.

This procedure faces many problems, most of which are very difficult to re-solve. First, there is the choice of the reference level ofxr(y, q). If a referencelevel ofx1 instead ofx0 were chosen in Figure30, the number of adult equivalentsin the second household would fall fromyd/yc to yf/ye. There is little that canbe done in general to determine which of these two scales is the right one. In suchcases, one cannot use a welfare-independent equivalence scale – the equivalencescale ratios depend on the levels of the households’ well-being.

Equivalence scale estimates also generally depend on the choice of the refer-ence good that is used to compare well-being across heterogeneous households.For instance, the choice of adult clothingversusthat of tobacco, alcohol or othercommodities consumed strictly by adults will generally matter in trying to com-pare the needs of households with and without children. This is in part becausepreferences for these goods are not independent of – and do not depend in the samemanner on – household composition. One additional problem is the issue of theprice dependence of equivalence scale estimates. Choosing a differentq in Figure30, for instance, would generally lead to the estimation of different equivalencescale ratios.

3.4.2 Sensitivity analysis

In view of these difficulties, recent work has emphasized that the choice of aparticular scale inevitably introduces important value judgements on how needs ofindividuals differing in non-income characteristics are assessed, and that it mighttherefore be appropriate to recognize the lack of agreement in this choice whenmeasuring and comparing inequality and poverty levels.

Allowing the assessment of needs to vary turns out to be especially relevant in


cross-country comparative analyses, particularly when countries compared differsignificantly in their socio-economic fabric. There is in this case the added issuethat not only can the appropriate scale rates be uncertain in a given country, butthey may also be different between countries. Testing the sensitivity of inequalityand poverty results to changes in the incorporation of needs is then a matter of cru-cial importance particularly for those international comparisons whose results caninfluence redistributive policies,e.g., through the transfer of resources from somecountries or regions to others, or in the assessment of transnational or alternativeanti-poverty policies.

To see how to carry out such sensitivity analysis, define an equivalence scaleE as an index of household needs. This index will typically depend on the char-acteristics of theM different household members, such as their sex and age, andon household characteristics, such as location and size. BecauseE is normalizedby the needs of a single adult, it can be interpreted as a number of ”equivalentadults”, viz, household needs as a proportion of the needs of a single adult. A”parametric” class of equivalence scales is usually defined as a function of one orof a few relevant household characteristics, with parameters indicating how needsare modified as these characteristics change.

A survey ofBuhmann, Rainwater, Schmaus, and Smeeding (1988) reported34 different scales from 10 countries, which they summarized as

E(M) = M s (4)

with s being a single parameter summarizing the sensitivity ofE to householdsizeM . This needs elasticity,s, can be expected to vary between 0 and 1. Fors = 0, no account is taken of household size. Fors = 1, adult-equivalent incomeis equal toper capitahousehold income. The larger the value ofs, the smaller arethe economies of scale in the production of well-being implicitly assumed by theequivalence scale, and the greater is the impact of household size upon householdneeds.

An obvious limitation of such a simple function such as (4) is its dependencesolely on household size and not on household composition or other relevant char-acteristics. Most equivalence scales do indeed distinguish strongly between thepresence of adults and that of children, and some like that ofMcclements (1977)even discriminate finely between children of different ages. An example of a classof equivalence scales that is more flexible than the above was suggested byCutlerand Katz (1992) – this class takes separately into account the importance of the


MA adults and theM −MC children:

E(M, MA) = (M + c (M −MA))s (5)

wherec is a constant reflecting the resource cost of a child relative to that of anadult, ands is now an indicator of the degree of overall economies of scale withinthe household. Whenc = 1, children count as adults (which is the assumptionmade in (4)); otherwise, adults and children have different needs.

3.4.3 Household decision-making and within-household inequality

Finally, and as elsewhere in distributive analysis, there is also the practicallyinsoluble difficulty of having to make interpersonal comparisons of well-beingacross individuals – compounded by the fact that individuals here are heteroge-neous in their household composition. On the basis of which observable variablecan we really make interpersonal comparisons of well-being? Again, note that theassumption that well-being for the man is the same as well-being for the couplewhenxr(y, q) is equalized in Figure30 is a very strong one. Furthermore, apartfrom influencing preferences and commodity consumption, household formationis as indicated above itself a matter of choice and is presumably the source ofutility in its own right. Preferences for household composition are themselvesheterogeneous, and so is the utility derived from a certain household status. Allof this makes comparisons of well-being across heterogeneous individuals andthe use of equivalence scales subject to arbitrariness and significant measurementerrors.

An additional problem in measuring individual living standards using surveydata comes from the presence of intrahousehold inequality. The final unit of ob-servation in surveys is customarily the household. Little information is typicallygenerated on the intrahousehold allocation of well-being (e.g., of the individualbenefits stemming from total household consumption). Because of this, the usualprocedure is to assume that the adult-equivalent consumption (once computed) isenjoyed identically by all household members.

This, however, is at best an approximation of the true distribution of economicwell-being in a household. If the nature of intrahousehold decision-making leadsto important disparities in well-being across individuals, assuming equal sharingwill significantly underestimate inequality and aggregate poverty. Not being ableto account for intra-household inequities will also have important implications forprofiling the poor, and also for the design of public policy. For instance, a povertyassessment that correctly showed the deprivation effects of unequal sharing within


households could indicate that it would be relatively inefficient to target supportat the level of the entire household – without taking into account how the targetedresources would subsequently be allocated within the household. Instead, it mightbe better to design public policy such as to self-select the least privileged individ-uals within the households, in the form of specific in-kind transfers or speciallydesigned incentive schemes.

A final and related difficulty concerns who we are counting in aggregatingpoverty: is it individuals or households? Although this distinction is fundamental,it is often surprisingly hidden in applied poverty profile and poverty measurementpapers. The distinction matters since there is habitually a strong positive corre-lation between household size and a household’s poverty status. Said differently,household poverty is usually found disproportionately among the larger house-holds. Because of this, counting households instead of individuals will typicallyunderestimate significantly the true proportion of individuals in poverty.

3.5 References for chapters2 and 3

The literature investigating the foundations and the impact of alternative ap-proaches to measuring well-being is large and (yet) rapidly increasing.

Influential discussions of the conceptual foundations can be found inDas-gupta (1993), Sen (1981), Sen (1983), Sen (1985), Streeten and Others (1981)andTownsend (1979).

Papers considering the impact of the accounting period (e.g., short-termvslong-term incomes) on the distribution of well-being includeAaberge and Al.(2002), Arkes (1998),Bjorklund (1993), Burkhauser, Frick, and Schwarze (1997),Burkhauser and Poupore (1997), Coronado, Fullerton, and Glass (2000), Creedy(1997), Creedy (1999a), Gibson, Huang, and Rozelle (2001), Harding (1993) andParker and Siddiq (1997).

The comprehensiveness of income concepts can also make a difference. Agood introduction to the general methodological issues isHentschel and Lanjouw(1996). The impact of the difference between cash and more comprehensive mea-sures of income is studiedinter alia in Formby, Kim, and Zheng (2001), Gustafs-son and Makonnen (1993), Gustafsson and Shi (1997), Harding (1995), Jenkinsand O’Leary (1996), Smeeding, Saunders, Coder, JENKINS, Fritzell, Hagenaars,Hauser, and Wolfson (1993), Smeeding and Weinberg (2001), Van den Bosch(1998), andYates (1994). The role of public services is also discussed inAnandand Ravallion (1993); see alsoPropper (1990) andDuclos (1995b) for adjustingthe value of public services for the costs of accessing them.


The sensitivity of the measurement of well-being to the choice between con-sumption and income measures is analyzed inBarrett, Crossley, and Worswick(2000b), Barrett, Crossley, and Worswick (2000a), Blacklow and Ray (2000),Blundell and Preston (1998), Cutler and Katz (1992), Jorgenson (1998) Mitrakosand Tsakloglou (1998), O’Neill and Sweetman (2001), Slesnick (1993) andZaidiand de Vos (2001).

Choosing the units of analysis, be they individuals, households or equivalentadults, also influences distributive analysis, as studied byBhorat (1999), Carlsonand Danziger (1999), Ebert (1999), andSutherland (1996). This is closely re-lated to the growing concerns expressed about the role of income pooling/sharingamong within families and households; see for instanceCantillon and Nolan(2001), Haddad and Kanbur (1990), Jenkins (1991), Kanbur and Haddad (1994),Lazear and Michael (1988), Lundberg, Pollak, and Wales (1997), Phipps and Bur-ton (1995), Quisumbing, Haddad, and Pena (2001), andWoolley and Marshall(1994). Ebert and Moyes (2003) explore the normative implications of a concernfor equality in living standards for the use of equivalence scales in applied studies.

Price adjustments can also be terribly important for making consistent com-parisons of well-being across time, space and socio-economic groups, and formeasuring equity and poverty properly. A good introduction to the methodolog-ical literature is given byDonaldson (1992). Empirical evidence can be foundin Araar (2002), Bodier and Cogneau (1998), Deaton (1988), Erbas and Sayers(1998), Finke, Chern, and Fox (1997), Idson and Miller (1999), Muller (2002),Pendakur (2002), Rao (2000), Ruiz Castillo (1998) andSlesnick (2002).

Justification and examples for the use by economists of non-money-metricmeasures of well-being can be foundinter alia in De Gregorio and Lee (2002)(for a link between education and income inequality),Haveman and Bershad-ker (2001) (self-reliance),Jensen and Richter (2001) (children’s health),Klasen(2000) andLayte and et al. (2001) (deprivation),Sahn and Stifel (2000) (welfareindex), Sefton (2002) (fuel poverty) andSkoufias (2001) (calorie). The wealthdistribution is also often of interest: see for instanceWolff (1998) for a reviewof the American evidence. Alternative measures of well-being are also exploredin Davies, Joshi, and Clarke (1997) (for a construction of a deprivation index),Desai and Shah (1988) (for estimates of relative deprivation),Hagenaars (1986)(for perceptions of poverty), andNarayan and Walton (2000) (for participatoryevidence on the living conditions and views of more than 20,000 poor people).

Survey measurement problems are numerous. See for instanceFields (1994)for a general discussion,Juster and Kuester (1991) for wealth measurement, andLanjouw and Lanjouw (2001) for the common procedure of estimating food and


non-food consumption expenditures.The sensitivity of distributive analysis to the ”equivalization” of incomes has

been the focus of much work in the last 15 years. This includesBanks and Johnson(1994), Bradbury (1997), Buhmann and Al. (1988), Burkhauser, Smeeding, andMerz (1996), Coulter, Cowell, and Jenkins (1992b), Coulter, Cowell, and Jenkins(1992a), de Vos and Zaidi (1997), Duclos and Mercader Prats (1999), Jenkinsand Cowell (1994), Lancaster and Ray (2002), Lanjouw and Ravallion (1995),Lyssiotou (1997), Meenakshi and Ray (2002), Phipps (1993), andRuiz Castillo(1998)

The econometric and theoretical difficulties involved in the estimation of equiv-alence scales are formidable, and these are discussedinter alia in Blundell andLewbel (1991), Blundell (1998), andPollak (1991). Estimation of equivalencescales is performed inBosch (1991), Nicol (1994), Pendakur (1999) (where theyare found to be ”base-independent”),Pendakur (2002) (where they are found to beprice-dependent),Phipps and Garner (1994) (where they are found to be differentacross Canada and the United States),Phipps (1998), andRadner (1997) (wherethey depend on the types of income considered).

An attempt to identify and estimate unconditional preferences for goods anddemographic characteristics isFerreira, Buse, and Chavas (1998). Whether equiv-alence scales should be income-dependent, and what happens of they are, is stud-ied among others inAaberge and Melby (1998), Blackorby and Donaldson (1993),andConniffe (1992).

The normative issues raised by the presence of heterogeneity in the popula-tion – heterogeneity other than in the dimension of income – are numerous, andsome of them are examined inEbert and Moyes (2003), Fleurbaey, Hagnere, andTrannoy (2003), Glewwe (1991) andLewbel (1989).

44

Part II

Measuring poverty and equity

4 INTRODUCTION 45

4 Introduction

In what follows in this book, we will denote living standards by the variabley.The indices we will use will sometimes require these living standards to be strictlypositive, and, for expositional simplicity, we may assume that this is always thecase. Strictly positive values ofy are required, for instance, for the Watts povertyindex and for many of the decomposable inequality indices. It is of course reason-able to expect indicators of living standards such as consumption or expendituresto be strictly positive. This assumption is less natural for other indicators, such asincome, for which capital losses or retrospective tax payments can generate nega-tive values. Also recall that, for expositional simplicity, we will also usually referto living standards as incomes.

Let p = F (y) be the proportion of individuals in the population who enjoy alevel of income that is less than or equal toy. F (y) is called the cumulative distri-bution function (cdf) of the distribution of income; it is non-decreasing iny, andvaries between 0 and 1, withF (0) = 0 andF (∞) = 1. For expositional simplic-ity, we assume thatF (y) is continuously differentiable and strictly increasing iny. These are reasonable approximations for large-population distributions of in-come. They are also reasonable assumptions from the point of view of describingthe data generating processes that generate the distributions of income observedin practice. The density function, which is the first-order derivative of thecdf,is denoted asf(y) = F ′(y) and is strictly positive sinceF (y) is assumed to bestrictly increasing iny.

4.1 Continuous distributions

A useful tool throughout the book will be ”quantiles functions”. The use ofquantiles will help simplify greatly the exposition and the computation of severaldistributive measures. Quantiles will also sometimes serve as direct tools to ana-lyze and compare distributions of living standards (to check first-order dominancein the dual approach for instance). The quantile functionQ(p) is defined implicitlyasF (Q(p)) = p, or using the inverse distribution function, asQ(p) = F (−1)(p).Q(p) is thus the living standard level below which we find a proportionp of thepopulation. Alternatively, it is the living standard of that individual whose rank –or percentile– in the distribution isp. A proportionp of the population is poorerthan he is; a proportion1− p is richer than him.

These tools are illustrated in Figure27. The horizontal axis shows percentilesp of the population. The quantilesQ(p) that correspond to differentp values are

4 INTRODUCTION 46

shown on the vertical axis. The larger the rankp, the higher the correspondingincomeQ(p). Alternatively, incomesy appear on the vertical axis of Figure27,and the proportions of individuals whose income is below or equal to thosey areshown on the horizontal axis. At the maximum income level,ymax, that proportionF (ymax) equals 1. The median is given byQ(0.5), which is the income valuewhich splits the distribution exactly in two halves.

Note that an important expositional advantage of working with quantiles isto normalize the population size to 1. This also means that everyone’s incomeand contribution to this book’s poverty and equity analysis can then appear onan interval of percentiles ranging from 0 to 1. In a sense, the population is thusscaled to a representative individual. Normalizing all population sizes to 1 alsomakes comparisons of poverty and equity accord with the population invarianceprinciple. This principle says that adding an exact replicate of a population to thatsame population should not change the value of its distributive indices. Puttingeveryone’s income within a common total population scale of 1 is one handy wayof comparing populations of different sizes. It also ensures that adding exactreplications to these populations will not change the distributive picture.

We will define most of the distributive measures (indices and curves) in termsof integrals over a range of percentiles. This is a familiar procedure in the con-text of continuous distributions. We will see below why this procedure is alsogenerally valid in the context of discrete distributions, even though the use ofsummation signs is more familiar in that context. Using integrals will make thedefinitions and the exposition simpler, and will help focus on what matters more,namely, the interpretation and the use of the various measures.

The most common summary index of a distribution is its mean. Using integralsand quantiles, it is defined simply as:

µ =

∫ 1

0

Q(p)dp. (6)

µ is therefore the area underneath the quantile curve. This corresponds to the greyarea shown on Figure27. Since the horizontal axis varies uniformly from 0 to1, µ is also the average height of the quantile curveQ(p), and this is given byµ on the vertical axis.µ is thus in some sense the income of the population’s”representative individual”.

The computation of the representative incomeµ gives here equal weight toall incomes in the population. We will see later in the book alternative weight-ing schemes for computing representative incomes. As for most distributions of

4 INTRODUCTION 47

income, the one shown on Figure27 is skewed to the left, which gives rise to ameanµ that exceeds the medianQ(p). Said differently, the proportion of individ-uals whose income falls underneath the mean,F (µ), exceeds one half.

4.2 Discrete distributions

To see how to rewrite the above definitions using familiar summation signs fordiscrete distributions, we need a little more notation. Say that we are interested ina distribution ofn incomes. We first order then observations ofyi in increasingvalues ofy, such thaty1 ≤ y2 ≤ y3 ≤ ... ≤ yn−1 ≤ yn. We then associatendiscrete quantiles over the interval ofp between 0 and 1. Forp such that(i −1)/n < p ≤ i/n, we then haveQ(p) = yi. Technically, this is equivalent todefining quantiles asQ(p) = miny|F (y) ≥ p. This is illustrated in Table11for n = 3 and where the three income values are 10, 20 and 30. Figure26 graphsthose quantiles as a function ofp.

The formulae for discrete distributions are then practically computed by re-placing the integral sign in the continuous case by a summation sign, by summingacross all quantiles, and by dividing that sum by the number of observationsn.Thus, the meanµ of a discrete distribution can be expressed as:

µ =1

n

n∑i=1

Q(pi). (7)

Whenever an expression like (6) arises, we can think of the integral sign as stand-ing for a summation sign and ofdp as standing for1/n.

Using (7), the mean of the discrete distribution of Table11, which is 20, is thensimply the integral of the quantile curve shown on Figure26. In other words, it isthe sum of the area of the three boxes each of length 1/3 that can be found under-neath the filled curve. For completeness, we will mention from time to time howindices and curves can be estimated using the more familiar summation signs. Formore information, you can also consultDAD’s User Guidewhere the estimationformulae shown use summation signs and thus apply to discrete distributions.

4.3 Poverty gaps

For poverty comparisons, we will also need the concept of quantiles censoredat a poverty linez. These are denoted byQ∗(p; z) and defined as:

4 INTRODUCTION 48

Q∗(p; z) = min(Q(p), z). (8)

Censored quantiles are therefore just the incomesQ(p) for those in poverty (belowz) andz for those whose income exceeds the poverty line. This is illustrated onFigure28, which is similar to Figure27. QuantilesQ(p) and censored quantilesQ∗(p; z) are identical up top = F (z), or up toQ(p) = z. After this point,censored quantiles equal a constantz and therefore diverge from the quantilesQ(p).

The mean of the censored quantiles is denoted asµ∗(z):

µ∗(z) =

∫ 1

0

Q∗(p; z)dp. (9)

This is the area underneath the curve of censored incomesQ∗(p; z). Censoringincome atz helps focus attention on poverty, since the precise value of those livingstandards that exceedz is irrelevant for poverty analysis and poverty comparisons(at least so long as we considerabsolutepoverty). The poverty gap at percentilep, g(p; z), is the difference between the poverty line and the censored quantile atp, or, equivalently, the shortfall (when applicable) of living standardQ(p) fromthe poverty line. Letf+ = max(f, 0). Poverty gaps can then be defined as1: E:19.7.6

g(p; z) = z −Q∗(p; z) = max(z −Q(p), 0) = (z −Q(p))+ . (10)

When income atp exceeds the poverty line, the poverty gap equals zero. A short-fall g(q; z) at rankq is shown on Figure28 by the distance betweenz andQ(q).The larger one’s rankp in the distribution – the higher up in the distribution ofincome – the lower the poverty gapg(p; z). The proportion of individuals with apositive poverty gap is given byF (z) (see the Figure). The average poverty gapthen equalsµg(z):

µg(z) =

∫ 1

0

g(p; z)dp. (11)

µg(z) is then the size of the area in grey shown in Figure28.

1DAD: Curves|Poverty gap

4 INTRODUCTION 49

4.4 Cardinal versus ordinal comparisons

There are two types of poverty and equity comparisons: cardinal and ordinalones. Cardinal comparisons simply involve comparing numerical estimates ofpoverty and equity indices. Ordinal comparisons rank poverty and equity acrossdistributions, without attempting to quantify the precise differences in poverty andequity that exist between these distributions. They can often say whether povertyand equity is larger or smaller, but not by how much.

Consider for instance the case of cardinal poverty comparisons. Numericalpoverty estimates attach a single number to the extent of aggregate poverty ina population,e.g., 40% or $200 per capita. But calculating cardinal poverty esti-mates requires making very specific and precise assumptions. These include,interalia, assumptions on the form of the poverty index, the definition of the indicatorof well-being, the choice of equivalence scales, the value of the poverty line, andhow that poverty line varies exactly across regions and time.

Once this information is provided, cardinal poverty estimates can tell, for in-stance, that the consumption expenditures of 30% of the individuals in a popula-tion used to lie underneath a poverty line, but that a recently-introduced govern-ment program has decreased that proportion to 25%. Cardinal poverty estimatescan also be used to carry out a money-metric cost-benefit analysis of the effectsof social programs. Thus, if the above government program involved yearly ex-penditures of $500 million, then we would know immediately that a 1% fall inthe proportion of the poor would seem to cost the government on average $100million. That amount could then be compared to the poverty alleviation cost ofother forms of government policy.

The main advantage of cardinal estimates of poverty and equity is their easeof communication, their ease of manipulation, and their (apparent) lack of ambi-guity. Government officials and the media often want the results of distributivecomparisons to be produced in straightforward and seemingly precise terms, andwill often feel annoyed when this is not possible. As hinted above, cardinal esti-mates of poverty and equity are, however, necessarily (and often highly) sensitiveto the choice of a number of arbitrary measurement assumptions.

It is clear, for example, that choosing a different poverty line will almost al-ways change the estimated numerical value of any index of poverty. The elasticityof the poverty headcount index to the poverty line is, for example, often signifi-cantly larger than 1 (see Section13.2). This implies that a variation of 10% in thepoverty line will then change by more than 10% the estimated proportion of thepoor in the population; this is a substantial impact for those interested in poverty

4 INTRODUCTION 50

alleviation, especially since poverty lines are rarely convincingly bounded withina narrow confidence interval.

Another source of cardinal variability comes from the choice of the form of adistributive index. Many procedures have been proposed for instance to aggregatenumerically the poverty of individuals. Depending on the chosen procedure, nu-merical estimates of poverty will appear larger or lower. As we will see later, forinstance, the identification of a ”socially representative poverty gap” will hingeparticularly on the relative weight given to the more deprived among the poor.There is little objective guidance in choosing that weight; the greater its value,however, the greater the socially representative poverty gap, and the greater thenumerical estimate of poverty.

Ordinal comparisons, on the other hand, do not attach a precise numericalvalue on the extent of poverty or equity, but only try to rank poverty and equityacross generally all indices that obey some generally-defined normative (or ethi-cal) principles. This can be highly useful when it suffices to know which of twopolicies will better alleviate poverty, or which of two distributions has the mostinequality but not precisely by how much. Because of this lower information re-quirement, ordinal rankings can also prove robust to the choice of a number ofmeasurement assumptions. For instance, ordinal poverty orderings can often rankpoverty over general classes of possible poverty indices and wide ranges of possi-ble poverty lines.

It is thus useful to consider in turn cardinal and ordinal comparisons of povertyand equity. We turn first to the construction of aggregate cardinal distributiveindices. Ordinal comparisons are considered in PartIII .

5 MEASURING INEQUALITY AND SOCIAL WELFARE 51

5 Measuring inequality and social welfare

5.1 Lorenz curves

The Lorenz curve has been for several decades the most popular graphicaltool for visualizing and comparing income inequality. As we will see, it providescomplete information on the whole distribution of income relative to the mean. Ittherefore gives a more comprehensive description of relative incomes than any oneof the traditional summary statistics of dispersion can give, and it is also a betterstarting point when looking at the inequality of income than the computation ofthe many inequality indices that have been proposed. As we will see, its popularityalso comes from its usefulness in establishing orderings of distributions in termsof inequality, orderings that are ”ethically robust”.

The Lorenz curve is defined as follows:

L(p) =

∫ p

0Q(q)dq∫ 1

0Q(q)dq

=1

µ

∫ p

0

Q(q)dq. (12)

The numerator∫ p

0Q(q)dq sums the incomes of the bottomp: proportion (the

poorest 100p%) of the population. The denominatorµ =∫ 1

0Q(q)dq sums the

incomes of all. Since population size is normalized to 1, the denominator givesaverage incomeµ. L(p) thus indicates the cumulative percentage of total incomeheld by a cumulative proportionp of the population, when individuals are orderedin increasing values of their income. For instance, ifL(0.5) = 0.3, then we knowthat the 50% poorest individuals hold 30% of the total income in the population2. E:19.8.2

A discrete formulation of the Lorenz curve is easily provided. Recall thatthe discrete income valuesyi are ordered such thaty1 ≤ y2 ≤ ... ≤ yn, withpercentilespi = i/n such thatQ(pi) = yi. For i = 1, ...n, the discrete Lorenzcurve is then defined as:

L(pi = i/n) =1

nµ

i∑j=1

Q (pj) . (13)

If needed, other values ofL(p) in (13) can be obtained by interpolation.

2DAD: Curves|Lorenz curve


The Lorenz curve has several interesting properties. As shown in Figure13,it ranges fromL(0) = 0 to L(1) = 1, since a proportionp = 0 of the populationnecessarily holds a proportion 0% of total income, and since a proportionp = 1of the population must hold 100% of aggregate income. It is increasing aspincreases, since more and more incomes are then added up. This is also seen bythe fact that the derivative ofL(p) equalsQ(p)/µ:

dL(p)

dp=

Q(p)

µ. (14)

This is positive if incomes are positive, as we are assuming throughout. Hence byobserving the slope of the Lorenz curve at a particular value ofp , we also knowthep-quantile relative to the mean, or in other words, the income of an individualat rankp as a proportion of mean income. An example of this can be seen onFigure13 for p = 0.5. The slope ofL(p) at that point isQ(0.5)/µ, the ratio ofthe median to the mean. The slope ofL(p) thus portrays the whole distribution ofmean-normalized incomes.

The Lorenz curve is also convex inp, since asp increases, the new incomesthat are being added up are greater than those that have already been counted. Thisis clear from equation (14) sinceQ(p) is increasing inp. Mathematically, a curveis convex when its second derivative is positive, and the more positive that secondderivative, the more convex is the curve. Formally, the second-order derivative ofthe Lorenz curve equals:

d2L(p)

dp2=

1

µ

dQ(p)

dp≥ 0. (15)

which is positive. Note that by definitionp ≡ F (Q(p)). Differentiating thisidentity with respect top, we have that1 ≡ f(Q(p)) d(Q(p))/dp. Thus,

dQ(p)

dp=

1

f(Q(p))(16)

and we therefore haved2L(p)

dp2=

1

µf(Q(p)). (17)

The larger the density of incomef(Q(p)) at a quantileQ(p), the more convexthe Lorenz curve atL(p). The convexity of the Lorenz curve is thus revealing of


the density of incomes at various percentiles. On Figure13, this density is thusvisibly larger for lower values ofp.

Some measures of central tendency can also be identified by a look at theLorenz curve. In particular, the median (as a proportion of the mean) is givenby Q(0.5)/µ, and thus, as mentioned above, by the slope of the Lorenz curve atp = 0.5. Since many distributions of incomes are skewed to the right, the meanexceeds the median andQ(p = 0.5)/µ will typically be less than one. The meanincome in the population is found at that percentile at which the slope ofL(p)equals 1, that is, whereQ(p) = µ and thus at percentileF (µ) (as shown on Figure13). Again, this percentile will often be larger than 0.5, namely, which is wherethe median income is located. The percentile of the mode (or modes) is whereL(p) is least convex, since by equation (15) this is where the densityf(Q(p)) ishighest.

Simple summary measures of inequality can readily be obtained from thegraph of a Lorenz curve. The share in total income of the bottomp proportionof the population is given byL(p); the greater that share, the more equal is thedistribution of income. Analogously, the share in total income of the richestpproportion of the population is given by1−L(p); the greater that share, the moreunequalis the distribution of income. These two simple indices of inequality areoften used in the literature.

An interesting but less well-known index of inequality is given by the mini-mum (hypothetical) proportion of total income that the government would need toreallocate across the population to achieve perfect equality in income. This pro-portion is given by the maximum value ofp − L(p), which is attained where theslope ofL(p) is 1 (i.e., atL(p = F (µ))). It is therefore equal toF (µ)−L(F (µ)).This index is usually called the Schutz coefficient.

Mean-preserving equalizing transfers of income are often called Pigou-Daltontransfers. In money-metric terms, they involve a marginal transfer of $1, say,from a richer person (of percentiler, say) to a poorer person (of percentileq¡r)and they keep mean income constant. All indices of inequality which do notincrease (and sometimes fall) following any such equalizing transfers are said toobey the Pigou-Dalton principle of transfers. These equalizing transfers also havethe consequence of moving the Lorenz curve unambiguously closer to the line ofperfect equality. This is because such transfers do not affect the value ofL(p)for all p up toq and for allp greater thanr, but they increase that value for allpbetweenq andr.

Hence, let the Lorenz curveLB(p) of a distributionB be everywhere abovethe Lorenz curveLA(p). We can think ofB as having been obtained fromA


through a series of equalizing Pigou-Dalton transfers applied to an initial distri-bution A. Hence, inequality indices which obey the principle of transfers willunambiguously indicate more inequality inA than inB. We will come back tothis important link in Chapter12.3when we discuss how to make ethically robustcomparisons of inequality.

5.2 Gini indices

If all had the same income, the cumulative% of total income held by any bot-tom proportionp of the population would also bep. The Lorenz curve would thenbeL(p) = p: population shares and shares of total income would be identical. Auseful informational content of a Lorenz curve is thus its distance,p−L(p), fromthe line of perfect equality in income. Compared to perfect equality, inequalityremoves a proportionp − L(p) of total income from the bottom100 · p% of thepopulation. The larger the deficit, the larger the inequality of income.

If we were then to aggregate that ”deficit” between population shares andshares in income across all values ofp between 0 and 1, we would get half thewell-known Gini index:

Gini index of inequality2

=

∫ 1

0

(p− L(p)) dp. (18)

The Gini index implicitly assumes that all “share deficits” acrossp are equallyimportant. It thus computes the average distance between cumulated populationshares and cumulated shares in income.

5.2.1 Linear inequality indices and S-Gini indices

One can, however, also think of other weights to aggregate the distancep −L(p). The class oflinear inequality measures is given by applying percentile-dependent weights to those distances. Let those weights be defined byκ(p)). Apopular one-parameter functional specification for such weights is given by

κ(p; ρ) = ρ(ρ− 1)(1− p)(ρ−2) (19)

and depends on the value of a single “ethical” parameterρ. That parameter mustbe greater than 1 for the weightsκ(p; ρ)) to be positive everywhere. The shape ofκ(p; ρ)) is shown on Figure48for values ofρ equal to 1.5, 2 and 3. The larger thevalue ofρ, the larger the value ofκ(p; ρ) for smallp.


Using (19) gives what is called the class of S-Gini (or “Single-Parameter”Gini) inequality indices,I(ρ):

I(ρ) =

∫ 1

0

(p− L(p))κ(p; ρ)dp. (20)

Note thatI(2) is the standard Gini index. This is becauseκ(p; ρ = 2) ≡ 2, whichthen gives equal weight to all distancesp−L(p). When1 < ρ < 2, relatively moreweight is given to the distances occurring at larger values ofp, as shown by Figure48. Conversely, whenρ > 2, relatively more weight is given to the distances foundat lower values ofp. Changingρ thus changes the “ethical” concern which is feltfor the “share deficits” at various cumulative proportions of the population.

Let ω(p; ρ) be defined as

ω(p; ρ) =

∫ 1

p

κ(q, ρ)dq = ρ(1− p)ρ−1. (21)

The shape ofω(p; ρ) is shown on Figure47 for ρ equal to 1.5, 2 and 3. Note thatω(p; ρ) > 0 and that∂ω(p; ρ)/∂p < 0 whenρ > 1. Since

∫ 1

0ω(p; ρ)dp = 1 for

any value ofρ, the area under each of the three curves on Figure47 equals 1 too.Using (21) and integrating by parts equation (20), we can then show that3: E:19.8.31

I(ρ) =1

µ

∫ 1

0

(µ−Q(p))ω(p; ρ)dp. (22)

This says that the deviation of income from the mean is weighted by weightswhich fall with the ranks of individuals in the population. Since, in equation (22),I(ρ) is a (piece-wise) linear function of the incomeQ(p), it is a member of theclass of linear inequality measures, a feature which will prove useful in measuringprogressivity and vertical equity later. The usual Gini index is then given simplyby:

I(ρ = 2) =2

µ

∫ 1

0

(µ−Q(p))(1− p)dp. (23)

Yaari (1988) defines “an indicator for the policy maker’s degree of equalitymindedness atp” as−ω(1)(p; ρ)/ω(p; ρ). This indicator thus captures the speed atwhich the weightsω(p; ρ) decrease with the ranksp. Forω(p; ρ), it gives:

−∂ω(p; ρ)/∂p

ω(p; ρ)= (ρ− 1)(1− p)−1. (24)

3DAD: Inequality|S-Gini index


Thus, the local degree of “equality mindedness” forω(p; ρ) is a proportional func-tion of the single parameterρ. As definition (24) makes clear, this degree of in-equality aversion is defined at a particular rankp in the distribution of income,independently of the precise value thatincometakes at that rank. The larger thevalue ofρ, the larger the local degree of equality mindedness, and the faster thefall of the weightsω(p; ρ) with an increase in the rankp. Therefore, the greaterthe value ofρ, the more sensitive is the social decision-maker to differences inranks when it comes to granting ethical weights to individuals.

The functionsκ(p; ρ) andω(p; ρ) can also be given an interpretation in termsof densities of the poor. Assume thatr individuals are randomly selected fromthe population. The probability that the income ofall of theser individuals willexceedQ(p) is given by[1− F (Q(p))]r. The probability of finding an incomebelowQ(p) in such samples is1 − [1− F (Q(p))]r = 1 − [1− p]r. 1 − [1− p]r

is thus the distribution function of the lowest income in samples ofr individuals.The density of the lowest rank of income in a sample ofr randomly selectedincome is the derivative of that distribution with respect top, which is

r (1− p)r−1 . (25)

This helps interpret the weightsκ(p; ρ) andω(p; ρ). By equation (19), κ(p; ρ)is ρ times the density of the lowest income in a sample ofρ− 1 randomly selectedindividuals; analogously, by equation (21), ω(p; ρ) is the density of the lowestincome in a sample ofρ randomly selected individuals.

We might be interested in determining the impact of some inequality-changingprocess on the inequality indices of type (22). One such process that can be han-dled nicely spreads income away from the mean by a proportional factorλ, andthus corresponds to some form of bi-polarization of incomes away from the mean(loosely speaking). This is equivalent to a process that adds(λ− 1) (Q(p) − µ)to Q(p), since

µ− (Q(p) + (λ− 1) (Q(p)− µ)) = λ (µ−Q(p)) (26)

does indeed spread income away from the mean by a proportional factorλ. Ascan be checked from equation (22), this changesI(ρ) proportionally byλ:

∂I(ρ)

∂λ= λI(ρ). (27)

Equation (27) also says that the elasticity ofI(ρ) with respect toλ, whenλ equals1 initially, is equal to 1 whatever the value of the parameterρ.


This bi-polarization away from the mean is also equivalent to a process thatincreases the distancep − L(p) by a factorλ. That this gives the same changein I(ρ) can be checked from equation (20). This bi-polarization process thusincreases the ”deficit”p − L(p) between population sharesp and income sharesL(p) by a constant factorλ across population shares. We will see later how thisdistance-increasing process leads to a nice illustration of the possible impact ofchanges in inequality on poverty.

As shown on Figure47, the larger the value ofρ, the greater the weight givento the deviation of low incomes from the mean. Whenρ becomes very large, theindexI(ρ) equals the proportional deviation from the mean of the lowest income.Whenρ = 1, the same weightω(p; ρ = 1) ≡ 1 is given to all deviations fromthe mean, which then makes the inequality indexI(ρ = 1) always equal to 0,regardless of the distribution of income under consideration. Thus, Gini indicesrange between 0 (when all incomes are equal to the mean or when the ethicalparameterρ is set to 1) and 1 (when incomes are concentrated in the hands of onlyone individual, or whenρ is large and the lowest income is close to 0). Since theLorenz curve moves towardsp when a Pigou-Dalton equalizing transfer is exerted,the value of the S-Gini indices also naturally decreases with such transfers.

Hence,ρ is a parameter of ”inequality aversion” that captures our ethical con-cern for the deviation of quantiles from the mean at various ranks in the pop-ulation. In this sense, it is analogous to the parameterε of relative inequalityaversion which we will discuss below in the context of the Atkinson indices.For the standard Gini index of inequality, we have thatρ = 2 and thus thatω(p; ρ = 2) = 2 · (1 − p); hence in assessing the standard Gini, the weight onthe deviation of one’s income from the mean decreases linearly with one’s rank inthe distribution of income. In a discrete formulation, the weightsω(p; ρ) take theform of:

ω(pi; ρ) =(n− i + 1)ρ − (n− i)ρ

nρ. (28)

5.2.2 Interpreting Gini indices

The S-Gini indices can be shown to be equal to the covariance formula

I(ρ) =− cov

(Q(p), ρ (1− p)(ρ−1)

)

µ, (29)

which can make their computation simple using common spreadsheet or statistical


softwares. The traditional Gini is then simply:

I(ρ = 2) =2 cov(Q(p), p)

µ(30)

and is just a proportion of the covariance between incomes and their ranks. Notehere the interesting analogy of (30) with the variance, given by

var(Q(p)) = cov(Q(p), Q(p)). (31)

A further useful interpretive property of the standard Gini index is that it equalshalf the mean-normalized average distance between all incomes:

I(ρ = 2) =

∫ 1

0

∫ 1

0|Q(p)−Q(q)|dpdq

2 µ. (32)

Thus, if we find that the Gini index of a distribution of income equals 0.4, then weknow that the average distance between the incomes of that distribution is of theorder of 80% of the mean. Again, note the interesting link of (32) with anotherdefinition of the variance,var(Q(p)) = 0.5

∫ 1

0

∫ 1

0|Q(p)−Q(q)|2dpdq.

The Gini index can also be computed as the integral of a simple transformationof the familiar distribution function. Recall thatF (y) and1 − F (y) are simplythe proportions of individuals with incomes below and abovey. If we integratethe product of these proportions across all possible values ofy, we obtain the Ginicoefficient:

I(ρ = 2) =

∫F (y) (1− F (y)) dy

µ=

∫F (y) (1− F (y)) dy∫

F (y)dy, (33)

where the last term is obtained by noting that∫

F (y)dy = µ. Note also thatF (y) (1− F (y)) is largest atF (y) = 0.5, which also explains why the Gini indexis most sensitive to changes in incomes occurring around the median income.

Now suppose that society is split into two classes, and that income is equallydivided within each class.

1. Assume that those in the first class hold no income. The Gini index of thetotal population is then given by the population share of that zero-incomeclass.

2. Assume that the population share of each group is 0.5. The Gini index of thetotal population is then given by0.5 − L(0.5). In other words, the incomeshare of the bottom class is 0.5 minus the Gini coefficient.


3. Assume that the population share of each group is again 0.5. Denote theincomes of the richer byyR and those of the poorer byyP . We then have:

yR

yP

=0.5 + G

0.5−G, (34)

or alternatively

G = 0.5

(yR − yP

yR + yP

). (35)

5.2.3 Gini indices and relative deprivation

A final interesting interpretation of the Gini index is in terms of average rel-ative deprivation, which has been linked in the sociological and psychologicalliterature to subjective well-being, social protest and political unrest. For this, it isfrequent to quote from the classic work ofRunciman (1966), who defines relativedeprivation as follows:

The magnitude of a relative deprivation is the extent of the differencebetween the desired situation and that of the person desiring it (as hesees it). (Runciman (1966), p.10)

Sen (1973), Yitzhaki (1979) andHey and Lambert (1980) follow Runciman’slead to propose for each individual an indicator of relative deprivation which mea-sures the distance between his income and the income of all those relative to whomhe feels deprived. For instance, let the relative deprivation of an individual withincomeQ(p), when comparing himself to another individual with incomeQ(q),be given by:

δ(p, q) =

0, if Q(p) ≥ Q(q)Q(q)−Q(p), if Q(p) < Q(q).

(36)

The expected relative deprivation of an individual at rankp is thenδ(p):

δ(p) =

∫ 1

0

δ(p, q)dq (37)

which, we can show, can be computed asδ(p) = µ(1− L(p))−Q(p)(1− p). Aswe did for the “shares deficits” above, we can aggregate the relative deprivation


at every percentilep by applying the weightsκ(p; ρ). We can show that this givesthe S-Gini index of inequality:

I(ρ) =1

ρ µ

∫ 1

0

δ(p)κ(p; ρ)dp. (38)

Hence, the S-Gini indices are also an indicator of the average relative deprivationfelt in a population. By equations (19), (25) and (38), they equal the expectedrelative deprivation of the poorest individual in a sample ofρ − 1 randomly se-lected individuals. The greater the value ofρ, the more important is the relativedeprivation of the poorer in computingI(ρ).

5.3 Social welfare and inequality

We now introduce the concept of a social welfare function. Unlike the conceptof relative inequality, which considers incomes relative to the mean, the concept ofsocial welfare aggregatesabsoluteincomes. We will see that under some popularconditions on the shape of social welfare functions, the measurement of inequalityand social welfare can often be nicely linked and integrated, and that the toolsused for the two concepts are then similar. This will explain why some inequalityindices are sometimes called ”normative”.

The social welfare functions we will consider will take the form of

W =

∫ 1

0

U(Q(p))ω(p)dp, (39)

where for expositional simplicity we will restrictω(p) to be of the special formω(p; ρ) defined by equation (21). U(Q(p)) is a “utility function” of incomeQ(p).Social welfare is then the expected utility of the poorest individual in a sample of(ρ− 1) individuals.

Another requirement that we wish to impose on the form ofW is that it beho-mothetic. Homotheticity ofW is analogous to the requirement on consumer utilityfunctions that the expenditure shares of the different consumption goods be con-stant as income increases, or the requirement on production functions that the ratioof the marginal products of inputs stays constant when output is increased. Forsocial welfare measurement, homotheticity implies that the ratio of the marginalsocial utilities (the marginal utility being given byU ′(Q(p))ω(p)) of any two indi-viduals in a population stays the same when all incomes are changed by the same


proportion. For (39) to be homothetic, we needU(Q(p)) to take the popular formof U(Q(p); ε), where

U(Q(p); ε) =

Q(p)1−ε

(1−ε), when ε 6= 1

ln Q(p), when ε = 1.(40)

Hence,W in equation (39) will depend on the parametersρ and onε, and we willdenote this asW (ρ, ε).

Homotheticity of a social welfare function has an important advantage: thesocial welfare function can then easily be used to measure relative inequality, themost common concept of inequality in the literature. To see how this can bedone, defineξ(ρ, ε) as the equally distributed income that is equivalent, in termsof social welfare, to the actual distribution of income (we will refer toξ as theEDE income).ξ(ρ, ε) is implicitly defined as:

∫ 1

0

U (ξ(ρ, ε); ε) ω(p; ρ) dp ≡∫ 1

0

U(Q(p); ε) ω(p; ρ) dp. (41)

Since∫ 1

0ω(p; ρ)dp = 1, ξ(ρ, ε) is also such that

U (ξ(ρ, ε); ε) =

∫ 1

0

U(Q(p); ε)ω(p; ρ)dp, (42)

or, alternatively,

ξ(ρ, ε) = U (−1)ε

(∫ 1

0

Uε(Q(p))ω(p; ρ)dp

)= U (−1)

ε (W (ρ, ε)) , (43)

whereU(−1)ε (·) is the inverse utility function:

U (−1)ε (x) =

(1− ε)x

11−ε , when ε 6= 1,

exp (x) , when ε = 1.(44)

The index of inequalityI corresponding to the social welfare functionW is thendefined as the distance between the EDE and the mean incomes, expressed as aproportion of mean income:

I =µ− ξ

µ= 1− ξ

µ. (45)

Usingξ(ρ, ε) in (45), this givesI(ρ, ε).


Clearly, then, the EDE income is a simple function of average income andinequality, with

ξ = µ · (1− I). (46)

Compared toW , ξ also has the advantage of being money metric and thus of beingeasily interpreted. It can, for instance, be compared to other economic indicatorsthat can also be expressed in money-metric terms. To increase social welfare, wecan either increaseµ or increase equality of income1− I by decreasing inequal-ity I. Two distributions of income can display the same social welfare even withdifferent average incomes if these differences are offset by differences in inequal-ity. This is shown in Figure37, starting initially with two different levels of meanincomeµ0 andµ1 and common zero inequality. We then have thatξ = µ0 andξ = µ1. To preserve the same level of social welfare in the presence of inequality,mean income must be higher: this is shown by the positive slope of the constantξfunctions. Furthermore, as inequality becomes large, further increases inI mustbe matched by higher and higher increases in mean income for social welfare notto fall.

Defined as in (45), inequality has an interesting interpretation: it measures thedifference between the mean level of actual income and the (lower) level neededinstead to achieve the same level of social welfare when income is distributedequally across the population. This difference being expressed as a proportion ofmean income,I thus shows theper capitaproportion of income that is wastedin social terms because of its unequal distribution. Society as a whole would bejust as well-off with an equal distribution of a proportion of just1− I of the totalactual income.I can thus be interpreted as a unit-free indicator of the social costof inequality.

Let a distributionB of income just be a proportional re-scaling of a distributionA. In other words, for a constantλ > 0, let QB(p) = λQA(p) for all p. If thesocial welfare function used for the computation ofI is homothetic, it must bethat IA = IB. This is illustrated in Figure34 for the case of two incomesyA

1

andyA2 for the case of an initial distributionA, and two incomesyB

1 andyB2 for

a ”scaled-up” distributionB (sinceλ > 1). Social welfare inA is given byWA.The social indifference curveWA shown in Figure34also depicts the many othercombinations of incomes that would yield the same level of social welfare. Thecombinations at pointF correspond to a situation of equality of income whereboth individuals enjoyξA. ξA is therefore the equally distributed income that issocially equivalent to the distribution(yA

1 , yA2 ).

The average income inA is given byµA, which leads to pointG = (µA, µB)


in Figure34. Hence two distributions of income, one made of the vector(yA1 , yA

2 )and the other of the vector(ξA, ξA), generate the same level of social welfare,the first with an unequally distributed average incomeµA and the other with anequally distributed average incomeξA. Hence, the vertical (or horizontal) distancebetween pointF and pointG in Figure 34 can be understood as the ”cost ofinequality” in the distributionA of income. Taking that distance as a proportionof µA (see equation (45)) gives the index of inequalityIA.

ThatyB1 = λyA

1 andyB2 = λyA

2 for the sameλ can be seen from the fact thatthe two vectors of income lie along the same ray from the origin. If the functionW is homothetic, then inequality inA must be the same as inequality inB. Inother words, the distance between pointsD andE as a proportion of the distanceOE must be the same as the distance between pointsF andG as a proportion ofthe distanceOG.

5.4 Social welfare

5.4.1 Atkinson indices

Two special cases ofW (ρ, ε) are of particular interest in assessing social wel-fare and relative inequality. The first is when income ranks are not importantperse in computing social welfare: this is obtained whenρ = 1, and it yields thewell-known Atkinson additive social welfare function,W (ε):

W (ε) = W (ρ = 1, ε) =

∫ 1

0

U (Q(p); ε)) dp. (47)

This Atkinson social welfare function has had two major interpretations: 1) first,as a utilitarian social welfare function, whereU(Q(p); ε) is an individual utilityfunction displaying decreasing marginal utilities of income, and 2) second, as aconcave social evaluation of a concave individual utility of income.

It can be argued, however, that “it is fairly restrictive to think of social welfareas a sum of individual welfare components”, and that one might feel that “thesocial value of the welfare of individuals should depend crucially on the levelsof welfare (or incomes) of others” (Sen 1973, p.30 and 41). The unrestrictedform W (ρ, ε) allows for such interdependence and may therefore be thought moreflexible than the Atkinson additive formulation. In the light of the above, wecan also interpretW (ρ, ε) as the expected utility of the poorest individual in agroup ofρ randomly selected individuals, or the expected social valuation of theutility of such individuals. This interpretation of the social evaluation function


W (ρ, ε) confirms why it is not additive or separable in individual welfare: thesocial welfare weight onU(Q(p); ε) depends on the rankp of the individual in thewhole distribution of income.

Figure35 shows the shape of the utility functionsU (y; ε) for different valuesof ε.4 Incomes are shown on the horizontal axis as a proportion of their mean, andutility U (y; ε) can be read on the vertical axis. The normalizationU (µ; ε) = 1has been applied for graphical convenience. Although for all values ofε, theslope ofU (y; ε) is positive, that slope is not constant. This is made more expliciton Figure36 which shows the marginal social utility of incomeU (1) (y; ε)) fordifferent values ofε. Again, a normalization ofU (1) (µ; ε) = 1 is applied. Forε = 0, the marginal social utility is constant: increasing by a given amount apoor person’s income has the same social welfare impact as increasing by thesame amount a richer person’s income. Forε > 0, however, increasing the poor’sincome is socially more desirable than increasing the rich’s. The larger the valueof ε, the faster the marginal social utility falls withy.

By (43) and (45), the Atkinson inequality index is then given by:

I(ε) = I(ρ = 1, ε) =

1− (∫ 10 Q(p)(1−ε)dp)

11−ε

µ, when ε 6= 1,

1− exp(∫ 10 ln(Q(p))dp)

µ, when ε = 1.

(48)

The Atkinson indices are said to exhibit constant relative inequality aversion, sincethe elasticity ofU (1)(Q(p); ε) with respect toQ(p) is constant and equal toε:

Q(p) U (2)(Q(p); ε)

U (1)(Q(p); ε)= ε. (49)

The parameterε is usually referred to as the Atkinson parameter of relative in-equality aversion.

Figure31 illustrates graphically the link between the Atkinson social evalua-tion functionsW (ε) and their associated inequality indices. For this, suppose apopulation of only two individuals, with incomesy1 andy2 as shown on the hori-zontal axis. Mean income is given byµ = (y1 + y2) /2 (the middle point betweeny1 and y2). The ”utility function” U(y; ε) has a positive but decreasing slope.W (ε) is then given by(U (y1) + U (y2)) /2, the middle point betweenU (y1) andU (y2).

If equally distributed, an average mean income ofξ would be sufficient togenerate that same level of social welfare, since on Figure31we have thatW (ε) =

4This paragraph draws fromCowell (1995), pp.40-41.


U(ξ; ε). The cost of inequality is thus given by the distance betweenµ and ξ,shown asC on Figure31. Inequality is the ratioC/µ.

Graphically, the more ”concave” the functionU(y; ε), the greater the cost ofinequality and the greater the inequality indicesI(ε). This can be seen on Figure32where two functionsU(y; ε) have been drawn, with different relative inequalityaversion parametersε0 < ε1. We have thatW (ξ0) = U (ξ0; ε0) andW (ξ1) =U (ξ1; ε1) This difference in relative inequality aversion parameters leads toξ0 >ξ1, and therefore toI(ε0) < I(ε1). A specification with greater inequality aversionleads to a greater inequality index, and to the judgement that inequality costssocially a greater proportion of average income.

5.4.2 S-Gini social welfare indices

The second special case is obtained when the utility functionsU(Q(p); ε) arelinear in the levels of income, and thus whenε = 0. This yields the class of S-Ginisocial welfare functions,W (ρ):

W (ρ) = W (ρ, ε = 0) =

∫ 1

0

Q(p)ω(p; ρ) dp. (50)

Social welfare is thus the expected income of the poorest individual in a group ofρ randomly selected individuals. By (43), this is also the EDE income. Hence, theassociated inequality indices are given by:

I(ρ, ε = 0) = 1−∫ 1

0Q(p)ω(p; ρ)dp

µ(51)

=

∫ 1

0(µ−Q(p))ω(p; ρ)dp

µ(52)

which is seen by (22) to be the same as the S-Gini inequality indicesI(ρ). Hence,social welfare and the EDE income equal theper capitaincome corrected by theextent of relative deprivation in those incomes:

W (ρ) = µ− 1

ρ

∫ 1

0

δ(p)κ(p; ρ)dp. (53)


5.4.3 Generalized Lorenz curves

A useful curve for the analysis of the distribution of absolute incomes is theGeneralized Lorenz curve. It is defined asGL(p):

GL(p) = µ · L(p) =

∫ p

0

Q(q)dq, (54)

and is illustrated on Figure14. The Generalized Lorenz curve has all of the at-tributes of the Lorenz curve, except for the fact that it does not normalize incomesby their mean.GL(p) gives the absolute contribution toper capitaincome of thebottomp proportion (the100p% poorest) of the population.GL(p) is thus alsothe per capitaincome that would be available if society could rely only on theincome of the bottomp proportion of the population. Assume for instance thatµ = $20000 and thatGL(0.5) = $5000. Then,per capitaincome would be only$5000 if we assumed that the richest 50% of the population were suddenly to re-tire and earn no income... Note also thatGL(p)/p gives the average income of thebottomp proportion of the population.

Combining (20), (45) and (50) further shows that the Generalized Lorenzcurve has a nice graphical link with the S-Gini index of social welfare:

W (ρ) =

∫ 1

0

GL(p)κ(p; ρ)dp. (55)

5.5 Decomposing inequality by population subgroups

A frequent goal is to explain the total amount of inequality in a distribution bythe extent of inequality found among socio-economic groups (“intra” or “within”group inequality) and across them (“inter” or “between” group inequality). Thereare several ways to do this. One method relies on the class of inequality indicesthat are exactly decomposable into terms that account for within- and between-groups inequality. Although that class can be given a justification in terms ofsocial welfare functions, this exercise is less transparent and intuitive than for theclasses of relative inequality indices considered hitherto. Another method appliesthe Shapley decomposition to any type of inequality indices. We discuss eachmethod in turn.


5.5.1 Generalized entropy indices of inequality

For most practical purposes, we can express these decomposable inequalityindices as Generalized indices of entropy. We can define them asI(θ):

I(θ) =

1θ(θ−1)

(∫ 1

0

(Q(p)

µ

)θ

dp− 1

), if θ 6= 0, 1

∫ 1

0ln

(µ

Q(p)

)dp, if θ = 0

∫ 1

0Q(p)

µln

(Q(p)

µ

)dp, if θ = 1.

(56)

Some special cases of (56) are worth noting. First, if we constrainθ to be nogreater than 1 and letθ = 1 − ε, I(θ) becomes ordinally equivalent to the familyof Atkinson indices. This simply means that if an Atkinson indexI(ε) indicatesthat there is more inequality in a distributionA than in a distributionB, then theindexI(θ) with θ = 1− ε will also necessarily indicate more inequality inA thanin B. Second, the special caseI(θ = 0) gives the Mean Logarithmic Deviation,sinceI(θ = 0) can also be expressed as

∫ 1

0

(ln µ− ln Q(p)) dp, (57)

that is, as the average deviation between the logarithm of the mean and the log-arithms of incomes.I(θ = 1) gives the well-known Theil index of inequality.I(θ = 2) is half the square of the coefficient of variation sinceI(θ = 2) can berewritten as

0.5

∫ 1

0

(Q(p)− µ

µ

)2

dp. (58)

Now assume that we can split the population intoK mutually exclusive pop-ulation subgroups,k = 1, ..., K. The indices in (56) can then be decomposed asfollows:

I(θ) =K∑

k=1

φ(k)

(µ(k)

µ

)θ

I(k; θ)

︸︷︷︸within group

inequality

+ I(θ),︸︷︷︸between group

inequality

(59)

whereφ(k) is the proportion of the total population that belongs to subgroupkandµ(k) is the mean income of subgroupk.


• I(k; θ) is inequality within subgroupk, defined in exactly the same way asin (56) for the total population. The first term in (59) can thus be interpretedas a weighted sum of the within-group inequalities in the distribution ofincome.

• I(θ) is total population inequality when each individual in subgroupk isgiven the mean incomeµ(k) of his subgroup (namely, when within sub-group inequality has been eliminated): it can thus be interpreted as the con-tribution of between-group inequality to total inequality.

Note, however, that only whenθ = 0 is it the case that the within-groupinequality contributions do not depend on mean income in the groups; the termsI(k; θ = 0) are then strictly population-weighted. Otherwise, the within-groupinequalities are weighted by weights which depend on the mean income in thesubgroupsk. Depending on the context, this can makeI(θ = 0) a more attractivedecomposable index than for other values ofθ.

5.5.2 A sub-group Shapley decomposition of inequality indices

This decomposition has two steps. The first one is to decompose total inequal-ity into global between-group and within-group contributions. The second step isto express total within-group contribution as a sum of within-group contributionsof each of the groups.

In the first step, we suppose that the two Shapley factors are between-groupand within-group inequality. The rules followed to compute the contribution ofeach of these factors are:

1. first, to eliminate within-group inequality and to calculate between-groupinequality, we use a vector of incomes in which each observation is assignedthe average incomeµ(k) of the observation’s groupk;

2. to eliminate between-group inequality and to calculate within-group in-equality, we use a vector of incomes where each observation has its incomemultiplied by the ratioµ(k)/µ of its groupk.

To be more precise, let an inequality indexI depend on the incomes of indi-viduals ink = 1, ..., K groups, each group withn(k) individuals. Lety(k) be then(k)-vector of incomes in groupk. We want to express total inequality as a sumof between and within group inequality:

I(y(1), ...,y(K)) = Ibetween+ Iwithin (60)


To compute the contribution of between-group inequality, we compute thefall of inequality observed when the mean incomes of the groups are equalized.This can be done either before or after within-group inequality has been removed.Hence, the Shapley contribution of between-group inequality is given by:

Ibetween= 0.5 [I(y(1), ...y(K))− I(µ/µ(1) · y(1), ..., µ/µ(K) · y(K))+ I(µ(1) · 1(1), ..., µ(K) · 1(K))− 0] ,

(61)where1(k) is a unit vector of sizenk. Within-group contribution is then given as

Iwithin = 0.5 [I(y(1), ...y(K))− I(µ(1) · 1(1), ..., µ(K) · 1(K))+ I(µ/µ(1) · y(1), ..., µ/µ(K) · y(K))− 0] ,

(62)

The second step consists in decomposing total within-group inequality as asum of within-group inequality across groups. To do this, we proceed by replac-ing the incomes of those in a groupk by µ(k) in order to eliminate groupk’scontribution to total within-group inequality. The fall in inequality induced bythis equalization of incomes is the contribution of groupk to total within-groupinequality. We compute this for each group. Given that this computation dependson the ordering of the groups in the sequence of elimination of within-group in-equality, we compute the average contribution of a groupk over all possible or-derings of groups. This gives the Shapley value of groupk’s contribution to totalwithin-group inequality.

To formalize this, suppose that there are only two groups,k = 1, 2. The firstgroup’s contribution to total within-group inequality is given as

0.25 [I(y(1),y(2))− I(µ(1)1(1),y(2))+ I(y(1), µ(2)1(2))− I(µ(1)1(1), µ(2)1(2))+ I(µ/µ(1) · y(1), µ/µ(2) · y(2))− I(µ1(1), µ/µ(2) · y(2))+ I(µ/µ(1) · y(1), µ1(2))− I(µ1(1), µ1(2))] ,

(63)

and symmetrically for the second group.

5.6 Statistical and descriptive indices of inequality

A popular descriptive index of inequality is the quantile ratio. This is simplythe ratio of two quantiles,Q(p2)/Q(p1) using percentilesp1 andp2. Popular val-ues ofp1 andp2 includep1 = 0.25 andp2 = 0.75 (the quartile ratio), as well as


p1 = 0.10 andp2 = 0.90 (the decile ratio). Note that these values ofp1 andp2

are often reversed. Median income is also a popular choice forQ(p1). Observealso that these ratios are by definition insensitive to changes that affect quantilesother thanQ(p1) andQ(p2). Moreover, none of them is normatively consistentwith Lorenz inequality orderings: it can be that the Lorenz curve for a distributionA is always above that of distributionB, but that quantile ratios suggest thatBhas less inequality thanA. For inequality analysis, an arguably better choice fornormalizingQ(p2) is mean income – an index such asQ(p2)/µ can indeed beshown to be consistent with first-order (restricted) inequality dominance.

The coefficient of variation is the ratio of the standard deviation to the meanof income. It is therefore given by

√∫ 1

0

(Q(p)/µ− 1)2dp. (64)

Recall that it is also given as a simple transformation of the entropy indexI(θ =2).

Two other popular measures of inequality use distances in logarithms of in-come. The first one, which we can call the logarithmic variance, is defined as

∫ 1

0

(ln Q(p)− ln µ)2dp (65)

and the second, the variance of logarithms, as

∫ 1

0

(ln Q(p)−

∫ 1

0

ln Q(q)dq

)2

dp. (66)

These two last measures do not, however, always obey the Pigou-Dalton principleof transfers – that is, they will sometimes increase following a spread-reducingtransfer of income between two individuals.

Finally, the relative mean deviation is the mean of the absolute deviation frommean income, normalized by mean income:

∫ 1

0

|Q(p)− µ|µ

dp. (67)

Note that this measure is insensitive to transfers made between individuals whoseincome lies on the same side of the mean.


5.7 Appendix: the Shapley value

The Shapley value is a solution concept often employed in the theory of co-operative games. Consider a setS of s players who must divide some surplusamong themselves. The question to resolve is the following one: how can wedivide the surplus between thes players? To see this, note that thes players canform coalitionsΣ (these coalitions are subsets ofS) to extract a part of the surplusand redistribute it between theirσ members. Suppose that the functionV deter-mines the extracting force of the coalition,viz, that amount of the surplus that itcan extract without resorting to an agreement with those players that are outsideof the coalition. The value of an additional playeri in a coalitionΣ is given by

MV (Σ, i) = V (Σ ∪ i)− V (Σ) (68)

The termMV (Σ, i) equals the marginal value added by playeri after his adhesionto the coalitionΣ. What will then be the expected marginal contribution of playeri over the different possible coalitions that can be formed and which he can join?Note first that the number of possible permutations of thes players equalss!.Note also that the size of coalitionsΣ is limited toσ ∈ 0, 1, ...s− 1. Out ofs!possible permutations of players, the number of times that the same firstσ playersare located in a same coalitionΣ is given by the number of possible permutationsof theσ players in coalitionΣ, that is, byσ!. For every permutation in the coalitionΣ, we find(s−σ−1)! permutations for the players that complement the coalitionΣ (excluding playeri). The Shapley value gives the expected marginal value thatplayeri generates after his adhesion to a coalitionΣ of any possible sizeσ. It isthus given by:

Ci =∑

Σ⊂S\iσ∈0,s−1

σ!(s− σ − 1)!

s!MV (Σ, i) (69)

This decomposition procedure has two useful properties. The first is symmetry,ensuring that the contribution of each factor is independent of the order in whichit appears in the list or sequence of factors. The second property is exactness andadditivity, from which the total surplus is given by

∑si=1 Ci.

For decompositions of inequality or poverty indices, say, applying a Shapleyapproach consists in computing the marginal effect on such indices of removingeach contributing factor (between or within group inequality, inequality in incomecomponent, differences in mean income, etc.) in a given sequence of elimination.Repeating the computation for all possible elimination sequences, we estimate


the mean of the marginal effects for each factor. This mean provides the contribu-tion of each such factor, yielding an exact, additive decomposition of distributiveindices and variations in them intos contributions.

introduced by Shapley ??? in 1953, (Owen, 1977; Moulin, 1968; Shorrocks,1999; For example, Chantreuil and Trannoy (1999) use it to decompose inequalityby income source.)

5.8 References

The literature on the measurement of inequality and social welfare is verylarge. General references includeAtkinson (1983), Atkinson and Bourguignon(2000), Atkinson and Micklewright (1992), Bishop, Formby, and Smith (1993),Chakravarty (1990), Champernowne and Cowell (1998), Cowell (1995), Cow-ell (2000), Essama Nssah (2000), FOSTER and Sen (1997), Johnson and Shipp(1997), Lambert (2001), Sen (1973), Sen (1992), Sen (1992), andSaunders (1994).

Applications to real data are evidently very numerous too – among the mostinfluential recent ones featureBourguignon and Morrisson (2002), Danziger andGottschalk (1995), Gottschalk and Smeeding (1997), Gottschalk and Smeeding(2000), Jantti (1997) andMilanovic (2002).

Seminal work on inequality measurement and Lorenz curves includeAtkin-son (1970), Blackorby and Donaldson (1978), Dalton (1920), Dasgupta, Sen, andStarret (1973), ?) Hainsworth (1964), Kakwani (1977a), Kolm (1969), andRoth-schild and Stiglitz (1973). Aaberge (2000) rationalizes the use of ”moments ofLorenz curves” as measures of inequality, andAaberge (2001a) presents axiomaticbases for the use of Lorenz curve orderings.Foster and Ok (1999) analyze theconcordance of the variance of logarithms with Lorenz dominance.

Discussion and interpretation of linear (or rank-dependent) indices of inequal-ity can be found inAaberge (1997), Aaberge (2000), ?) Barrett and Salles (1995),Ben Porath and Gilboa (1994), Blackburn (1989), Blackorby, Bossert, and Don-aldson (1994), Bossert (1990), Chakravarty (1988), Chew and Epstein (1989),Donaldson and Weymark (1980) and Donaldson and Weymark (1983) (for S-Ginis), Duclos (1997a), Weymark (1981), Yaari (1988), Yitzhaki (1983) (for ex-tended Ginis, equivalent to S-Ginis – see alsoKakwani (1980)), andWang andTsui (2000). The most popular member of the class of linear inequality indices isthe Gini index: it is discussed in detail inDeutsch and Silber (1997), Milanovic(1994b), Milanovic (1997), Subramanian (2002) andYitzhaki (1998).

The theory and the economic measurement of relative deprivation is exploredinter alia in Berrebi and Silber (1985), Chakravarty and Chakraborty (1984),


Clark and Oswald (1996), Davis (1959), Duclos (2000), Ebert and Moyes (2000),Festinger (1954), Hey and Lambert (1980), Merton and Rossi (1957), Paul (1991),Podder (1996), Runciman (1966), Silver (1994), Wang and Tsui (2000), Yitzhaki(1979), Yitzhaki (1982a) andNolan and Whelan (1996).

Discussion and use of the Theil index appearsinter alia in Beblo and Knaus(2001), Duro and Esteban (1998) andGoerlich Gisbert (2001).

Other inequality indices are discussed inAraar and Duclos (1999) andBerrebiand Silber (1981) (a combination of Atkinson and Gini inequality indices),Chakravarty(2001) (a defense of the use of the variance),del Rio and Ruiz Castillo (2001) (for”intermediate inequality measures”), andFoster and Shneyerov (2000) (for ”path-independent decomposable measures”).

Decomposition of inequality across population subgroups has also been thefocus of a large literature. This has mostly involved using additive and general-ized entropy indices – see, for instance,Bourguignon (1979), Cowell (1980), Fos-ter and Shneyerov (1999), Mookherjee and Shorrocks (1982), Shorrocks (1980),Shorrocks (1984), Schwarze (1996) andZandvakili (1999). Decompositions ofthe Gini and rank-dependent inequality indices are investigated inDagum (1997),Deutsch and Silber (1999a), Deutsch and Silber (1999b), Milanovic and Yitzhaki(2002), Sastry and Kelkar (1994), Tsui (1998) andYitzhaki and Lerman (1991). Amoney-metric cost-of-inequality approach to decomposing inequality across sub-populations is derived inBlackorby, Donaldson, and Auersperg (1981), Duclosand Lambert (2000b) andEbert (1999). Alternative decomposition approachesare also explored inCowell and Jenkins (1995), Fields and Yoo (2000), Fournier(2001), Hyslop (2001), Jenkins (1995), Parker (1999), andSchultz (1998).

6 MEASURING POVERTY 74

6 Measuring poverty

6.1 Poverty indices

Two approaches have been used to devise cardinal indices of poverty. Thefirst uses the concept of the equally distributed equivalent (EDE) income of asociety where incomes have been censored at the poverty line, and compares it tothe poverty line. The second combines income and the poverty line into povertygaps, and aggregates these gaps in social-welfare like functions to assess overallpoverty. We look at these two approaches in turn.

6.1.1 The EDE approach

For the EDE approach to building poverty indices, we simply use the distribu-tion of incomeQ(p). Since, for poverty comparisons, we want to focus on thoseincomes that fall below the poverty line (the “focus axiom”), the incomesQ(p)are censored at the poverty linez, to giveQ∗(p; z). The censored incomes arethen aggregated using one of the many social welfare functions that have beenproposed in the literature. A poverty index is obtained by taking the differencebetween the poverty line and the EDE income. For instance, for the social wel-fare functions proposed in section5.3, this leads to the following class of povertyindices:

P (z; ρ, ε) = z − ξ∗(z; ρ, ε) (70)

whereξ∗(z; ρ, ε) is the EDE income of the distribution of censored incomeQ∗(p; z)and where we needρ ≥ 1 andε ≥ 0 for the Pigou-Dalton transfer principle not tobe violated.P (z; ρ, ε) can then be interpreted as the “socially representative” orEDE poverty gap.

Examples of such poverty indices include a transformation of the Clark, Hem-ming and Ulph’s (CHU) second class of poverty indices, given by ,P (z; ε) =P (z; ρ = 1, ε):

P (z; ε) =

z −(∫ 1

0Q∗(p; z)(1−ε)dp

) 11−ε

, when ε 6= 1,

z − exp(∫ 1

0ln(Q∗(p; z))dp

), when ε = 1.

(71)

The CHU indices are then obviously closely related to the Atkinson social welfarefunctions and inequality indices. Whenε = 1, the CHU poverty index is also the


EDE poverty gap corresponding to the Watts poverty index, an index which isdefined as :

PW (z) =

∫ 1

0

ln

(z

Q∗(p; z)

)dp. (72)

For 0 ≤ ε ≤ 1, the CHU indices also correspond to the EDE poverty gap of theclass of poverty indices proposed by Chakravarty,PC(z; ε):

PC(z; ε) = 1−∫ 1

0

(Q∗(p; z)

z

)1−ε

dp, 0 ≤ ε ≤ 1. (73)

Moreover, if we chooseε = 0 for the class of indices defined in (70), we obtainthe class of S-Gini indices of poverty,P (z; ρ) :

P (z; ρ) ≡ P (z; ρ, ε = 0) = z −∫ 1

0

Q∗(p; z)ω(p; ρ)dp. (74)

P (z; ρ = 2) is then a ”Gini-like” index of poverty.

6.1.2 The poverty gap approach

The second approach to constructing poverty indices uses the distribution ofpoverty gaps,g(p; z) = z − Q∗(p; z). Once this distribution is known, no otheruse of the poverty line is allowed or needed in the aggregation of poverty. Becauseof this, the poverty gap approach to constructing poverty indices is slightly morerestrictive and also puts more structure on the shape of the allowable poverty in-dices than the previous EDE approach. After the distribution of poverty gapshas been computed, we may use aggregating functions analogous to those usedin section5.3 for the analysis of social welfare. Unlike social welfare functions,however, where we normally want an increase in someone’s income to increasesocial welfare, we would normally wish the poverty indices to bedecreasinginpoverty gaps. Further, whereas an equalizing Pigou-Dalton transfer would oftenincrease the value of a social welfare function, we would typically wish a povertyindex to decrease when such an equalizing transfer of income takes place amongthe poor.

A popular class of poverty gap indices that can obey these axioms is knownas the Foster-Greer-Thorbecke (FGT) class. It differentiates its members using an


ethical parameterα ≥ 0 and is generally defined as

P (z; α) =

∫ 1

0

(g(p; z)

z

)α

dp (75)

for the normalized FGT poverty indices and as

P (z; α) =

∫ 1

0

g(p; z)αdp (76)

for the un-normalized version (which can sometimes be more useful than the moreusual normalized form). Note that poverty gap indices other than the FGT onescan also be easily proposed, simply by using some other aggregating functionsof poverty gaps that obey some of the desirable axioms (such as that of beingincreasing and convex in poverty gaps) discussed in the literature5. E:19.7.4

6.1.3 Interpreting FGT indices

Whenα = 0, the FGT index gives the simplest and most common example ofa poverty index. This is called the poverty headcount, and is simply the proportionof a population that is in poverty (those with a positive poverty gap),F (z) 6. The E 19.1.1next simplest and most commonly used index,µg(z), is given by the averagepoverty gap,P (z; α = 1), and is the average shortfall of income from the povertyline:

µg(z) = P (z; α = 1) =

∫ 1

0

g(p; z)dp. (77)

To see how to interpret the form of the FGT indices for general values ofα, con-sider Figure41. It shows the (absolute) contributions to total povertyP (z; α) ofindividuals at different ranksp. These contributions are given by(g(p; z)/z)α. Forα = 0, the contribution is a constant 1 for the poor and 0 for the rich (those whoserank exceedsF (z) on the Figure, or equivalently those whose incomeQ(p) ex-ceedsz). The headcount is then the area of the dotted rectangle on Figure41. Forα = 1, the contribution of someone atp equals precisely his normalized povertygap,g(p; z)/z. The normalized average poverty is then the area underneath theg(p; z)/z drawn on Figure41. The same reasoning is valid for higher values ofα.For instance, the absolute contribution toP (z; α = 3) of individuals at rankp is

5DAD: Poverty|FGT Index6DAD:Poverty | FGT index


given by(g(p; z)/z)3 on Figure41, andP (z; α = 3) equals the area underneaththe(g(p; z)/z)3 curve.

Notwithstanding the above, interpreting the numerical value of FGT indicesfor α different from 0 and 1 can be problematic. We can easily understand whatis meant by a proportion of the population in poverty or by an average povertygap, but what, for instance, can a squared-poverty-gap index actually signify?And how to explain it to a government Minister?... A further difficulty with suchindices emerges from another look at Figure41, which suggests that the absolutecontribution of the poor (including the poorest) to total povertydecreaseswith α– the contribution curves(g(p)/z)α move down asα rises. This also implies thatthe normalized FGT indices necessarily fall asα increases. This is paradoxicalsince it is usually argued that the higher the value ofα, the greater the focus onthose who suffer most ”severely” from poverty. It would thus be more natural ifan increase inα also increasedP (z; α).

6.1.4 Relative contribution to FGT indices

One partial solution to these interpretive problems is to switch one’s focusfrom theabsoluteto therelativecontribution to an FGT index of individuals withdifferent poverty gaps. Such a relative contribution is depicted on Figure42 forα = 0, 1 and 2. It shows the ratio of the absolute contributionsg(p)α to totalpovertyP (z; α) (these ratios are the same for normalized and un-normalized FGTindices). Since this graph shows relative contributions to total poverty, the areaunderneath each of the three curves must in all cases equal 1.

For α = 0, each poor contributes relatively the same constant1/F (z) to thepoverty headcount. The poor’s relative contribution to the average poverty gapincreases with their own poverty gap, as shown by the curveg(p)/P (z; α = 1).That relative contribution equals 1 for those individuals whose own poverty gapis precisely equal to the average poverty gap. The rank of such individuals isgiven byF (µg(z)), as is also shown on Figure42. Thus, those located atp =F (µg(z)) have a poverty gap that is representative of the average poverty gap inthe population. Increasingα from 1 to 2 decreases the relative contribution of thenot-so-poor, but increases reciprocally the contribution of those with the highestpoverty gaps. This then becomes consistent with the general view that, in theaggregation of poverty, higher values ofα put more emphasis on those who suffermost severely from poverty – those with lower values ofp and higher values ofg(p; z).


6.1.5 EDE poverty gaps for FGT indices

Figure42 does not, however, solve the main interpretation problems associ-ated with the FGT indices. As mentioned above, explaining to non-technicians orpolicymakers the practical meaning of FGT indices for general values ofα canprove hazardous since these indices are averages ofpowersof poverty gaps. Theyare also neither unit-free nor money-metric (except forα = 0 and 1). An anotheralready-mentioned difficulty is that the usual FGT indices will generally fall withan increase in the value of their poverty-severity parameter,α.

A valuable and intuitive solution to these two problems is to transform theFGT indices into EDE poverty gaps. An EDE poverty gap is that poverty gapwhich – if it were assigned equally to all individuals – would yield the sameaggregate poverty index as that which is currently observed. An EDE povertygap can then usefully be interpreted as a socially-representative poverty gap. Thistransformation provides a money-metric measure of poverty which can be usefullycompared across different poverty indices and/or across different values ofα. Aswe will see later, it also allows the analyst to determine the impact of poverty-gapinequality upon the level of poverty. For the un-normalized FGT indices, the EDEpoverty gap is given simply by (forα > 0)

ξg(z; α) = [P (z; α)]1/α . (78)

For the normalized FGT indices, it is justξg(z; α) = ξg(z; α)/z.

Figure43 shows such socially-representative poverty gapsξg(z; α) for differ-ent values ofα. In each case, we obtain a socially-weightedmoney-metricindica-tor of the distribution of deprivation in the population. This summary aggregateindicator can also be compared to the individual distribution of poverty, given bytheg(p; z) curve. Those whoseg(p; z) exceedsξg(z; α) experience more povertythan the socially representative average. Those exactly atξg(z; α) are located ex-actly at the socially representative poverty gap. Those representative individualsare thus found at the ranks given byF (ξg(z; α)), which are also shown on Figure43 for different values ofα.

An important point to note is that an increase inα moves the socially-representativepoverty gap closer to that experienced by the poorest individuals. This is sinceξg(z; α + 1) ≥ ξg(z; α) for anyα > 0. (This is unlike the usual definition of theFGT indices, for which we haveP (z; α + 1) ≤ P (z; α) for anyα > 0.) Hence,we can readily interpret increases inα as increases in the socially-representativepoverty gap, and thus in the relative weight given to the poorer of the poor. The


larger the value ofα, the more important are the most severe cases of deprivationin computing a socially-representative aggregate level of poverty.

Note finally that, besides being already in an EDE poverty gap form, the S-Gini index of poverty has the property of being a poverty gap index. Indeed, by(74), we have that

P (z; ρ) =

∫ 1

0

g(p; z)ω(p; ρ)dp, (79)

6.2 Group-decomposable poverty indices

Much of the literature on the construction of poverty indices has focussed onwhether indices are decomposable across population subgroups. This has led tothe identification of a subgroup of poverty indices known as the “class of decom-posable poverty indices”. These indices have the property of being expressibleas a weighted sum (more generally, as a separable function) of the same povertyindices assessed across population subgroups. They most commonly include theFGT and the Chakravarty classes of indices as well as the Watts index.

Let the population be divided intoK mutually exclusive population subgroups,whereφ(k) is the share of the population found in subgroupk. For the FGTindices, we then have that:

P (z; α) =K∑

k

φ(k)P (k; z; α) (80)

whereP (k; z; α) is the FGT poverty index of subgroupk 7. The Watts and E: 19.6Chakravarty indices are expressible as a sum of the poverty indices of each sub-group in exactly the same way as for the FGT indices in (80).

To illustrate the practical contents of this property, consider the following two-group (K = 2) example. Let the first group contain 40% of the total population,and let the poverty in group 1 be 0.8 and in group 2 be 0.4. Poverty in the totalpopulation is then a simple weighted mean of the group poverty, and is immedi-ately computable as0.4 · 0.8 + 0.6 · 0.4 = 0.56. Estimates of total poverty in apopulation can then be constructed in a decentralized manner, first by estimatingpoverty within communities or regions, and then by averaging over these aggre-gate estimates, without there being a need for all of the micro data to be regroupedin one single register.

7DAD:Decomposition|FGT: Decomposition by groups


Subgroup decomposability also implies that an improvement in well-being inone of the subgroups will necessarily improve aggregate poverty if the incomesin the other groups have not changed. It will further also mean that the optimaldesign of social safety nets and benefit targeting within any given group can becomputed independently of the income distribution in the other groups. This en-ables targeting also to be done in a decentralized manner: only the distributivecharacteristics of the relevant group matter for the exercise. If targeting succeedsin decreasing poverty at a local level, then it must also succeed at the aggregatelevel.

Subgroup decomposability is therefore useful, although it is certainly not im-perative for poverty analysis. In particular, it is certainly the case that it is notbecause an index facilitates poverty profiling and targeting analysis that this indexis ethically fine. Ease of computation and ethical soundness are two different cri-teria. Among other things, imposing the decomposability and additivity propertycan mean sacrificing some important ethical features in the aggregation of poverty.In that context,Ravallion (1994) notes that when measuring poverty ”one possi-ble objection to additivity is that it attaches no weight to one aspect of a povertyprofile: the inequality between sub-groups in the extent of poverty”. This canbe an important flaw if between-group relative deprivation is considered ethicallysignificant.

6.3 Poverty and inequality

Expressing poverty indices in the form of EDE poverty gaps enables the de-composition of poverty as a sum of average poverty and inequality in povertygaps. Letξg(z) be the EDE poverty gap andΞg(z) be the cost of inequality inpoverty gaps. We then have:

Ξg(z) ≡ ξg(z)− µg(z), (81)

or, alternatively,ξg(z) ≡ Ξg(z) + µg(z). (82)

For instance, for the popular FGT indices, we have that:

Ξg(z; α) = ξg(z; α)− µg(z). (83)

Whenα = 1, we have that the socially representative poverty gapξg is just theaverage poverty gapµg(z) and inequality in poverty gaps is thus not taken into


account in assessing poverty. The cost of inequality on poverty is then nil. Sinceµg(z) is insensitive toα, and sinceξg(z; α) is increasing inα, it follows thatΞg(z; α) is also increasing inα; the larger the value ofα, the larger the impactof inequality on the level of aggregate poverty. We can thus interpretα as a pa-rameter of inequality aversion in measuring poverty. For0 < α < 1, we havethatξg(z; α) < µg(z), and inequality in poverty is then deemed toreducepoverty:Ξg(z, α) < 0. Ceteris paribus, we then have that the greater the level of inequal-ity, the lower the socially representative level of poverty. Forα > 1, Ξg(z; α) > 0and inequality has a positive poverty cost.

A similar decomposition can be done using (70) and the EDE level of censoredincome. The EDE poverty gap corresponding to that approach is defined as

z − ξ∗(z; ρ, ε) = z − µ∗(z)(1− I∗(z; ρ, ε))= µg(z) + Ξ∗(z; ρ, ε),

(84)

whereΞ∗(z; ρ, ε) = µ∗(z) · I∗(z; ρ, ε) is the cost of inequality in censored incomeand whereI∗(z; ρ.ε) is the index of inequality in censored income.

6.4 Poverty curves

It is often informative to portray the whole distribution of poverty gaps on asimple graph, in a way which shows both the incidence and the inequality of thedeprivation in income. Particularly useful are the poverty gap curves, which plotg(p; z) as a function ofp. The poverty gap curve shows the ”intensity of poverty”felt at each rank in the population. The curve naturally decreases with the rankp inthe population, and reaches zero at the value ofp equal to the headcount ratio. Theintegral under the curve gives the average poverty gap, and its steepness indicatesthe degree of inequality in the distribution of poverty gaps.

Another quantile-based curve that is graphically informative and that is usefulfor the measurement and comparison of poverty is called the Cumulative PovertyGap (CPG) curve (also sometimes referred to as the inverse generalized Lorenzcurve, the “TIP” curve, or the poverty profile curve). The CPG curve cumulatesthe poverty gaps of the bottomp proportion of the population. It is defined as:

G(p; z) =

∫ p

0

g(q; z)dq. (85)

A CPG curve is drawn on Figure18. The slope ofG(p; z) at a given valueof p shows the poverty gapg(p; z). Sinceg(p; z) is non-negative,G(p; z) is


non-decreasing.G(p = 1; z) equals the average poverty gapµg(z). The per-centile at whichG(p; z) becomes horizontal (whereg(p; z) becomes zero) yieldsthe poverty headcount. Furthermore, the higher his rankp in the population, thericher is an individual, and therefore the lower is his poverty gap.G(p; z) is there-fore concave8. Because of this, the CPG curve exhibits for poverty analysis theE:19.7.8same descriptive interest as the Lorenz and Generalized Lorenz curves for theanalysis of inequality and social welfare. The distance ofG(p; z) from the lineof perfect equality of poverty gaps (namely, the line 0B in Figure18) shows theinequality of poverty gapsamong the total population. The distance ofG(p; z)from the line of perfect equality of poverty gapsamong the poor(namely, the line0A in Figure18) displays the inequality of poverty gaps among the poor. Finally,as for Lorenz curves, the concavity ofG(p; z) is inversely related to the density ofpoverty gaps atp.

6.5 S-Gini poverty indices

When weighted byκ(p; ρ), the area underneath the CPG curve generates theclass of S-Gini poverty indices:

P (z; ρ) =

∫ 1

0

G(p; z)κ(p; ρ)dp. (86)

Recall thatκ(p; ρ) = ρ(ρ − 1) (1− p)ρ−2 . P (z; ρ = 1) thus equals the averagepoverty gap,µg(z), P (z; ρ = 2) is the poverty index that is analogous to thestandard Gini index of inequality, and the well-known Sen index of poverty isgiven by:

P (z; ρ = 2)

F (z) · z . (87)

An interesting feature of theP (z; ρ) indices is their link with absolute andrelative deprivation. Let absolute deprivation,AD(z), be given by the averageshortfall from the poverty line, that is, byµg(z). Recalling (36) and (37), we candefine relative deprivation in censored income at percentilep as:

δ∗(p; z) =

∫ 1

p

(Q∗(q; z)−Q∗(p; z))dq. (88)

8DAD: Curves|CPG curve


Average relative deprivation across the whole population is then:

RD(z; ρ) =

∫ 1

0

δ∗(p; z)κ(p; ρ)dp. (89)

It is then possible to show that:

P (z; ρ) = AD(z) + RD(z, ρ). (90)

The larger the value ofρ, the larger is relative deprivation,RD(z, ρ), and the largeris P (z; ρ) and the contribution of relative deprivation and inequality to poverty.This provides an alternative link between inequality and poverty.

6.6 The normalization of poverty indices

Most of the poverty indices discussed above have initially been introducedin the literature in a normalized form, that is, by dividing censored income andpoverty gaps by the poverty line. The FGT indices, for instance, are generallyexpressed as:

P (z; α) = z−α

∫ 1

0

(g(p; z))α dp (91)

(see (75)). Normalizing poverty indices will make no substantial difference andlittle expositional difference for poverty analysis when the distributions of incomebeing compared have identical poverty lines. This will typically be the case, forinstance, when incomes are expressed in real (or constant) values, and when thefocus is on absolute poverty with constant real poverty lines. Normalizing povertyindices by the poverty line will

• make the EDE poverty gap lie between 0 and 1,

• make poverty indices insensitive to and independent of the monetary units(e.g., dollars or cents) used in assessing income, and

• make the indices invariant to an equi-proportionate change in all incomesand in the poverty line.

This is particularly useful if the poverty lines are to be interpreted essentially asprice indices, and thus used to enable comparisons of nominal income across timeand space (recall that price indices are used to convert nominal incomes into base-year real incomes).


Normalized poverty indices are usually referred to as ”relative poverty in-dices”. Changing all incomes and the poverty line by the same proportion willnot affect the value of relative poverty indices. FGT and other poverty gap in-dices that are not normalized are often called “absolute” poverty indices; equalabsolute additions to all incomes and to the poverty line will not affect their value.Increasing all incomes and the poverty line by the same proportion will, however,increase the value of such absolute poverty indices.

When poverty lines are different across distributions, and when their ratioacross time or space cannot be interpreted simply as a ratio of price indices, thenormalization of poverty indices by these poverty lines can, however, be prob-lematic, and is surely open to criticism. This is the case for instance when weare interested in comparing the absolute shortfalls of “real” income from a “real”poverty line, when these real poverty lines vary across populations or populationsubgroups. Examples can arise,inter alia, in comparing the poverty measure offamilies of different sizes and composition, or in comparing poverty across distri-butions with different social or cultural bases for the definition of a poverty line.

To see this more clearly, consider the following example in which all incomesand poverty lines are expressed in real terms (namely, they have been adjusted fordifferences in the cost of living, and they are therefore comparable). In countryA,the poverty line is $1,000, and a poor personi has an income of $500. Because,say, of cultural and/or sociological differences (these differences may exist acrosstime or space), the poverty line in countryB is larger and equal to $2,000, and apoor personj in it has an income equal to $1,100. Which ofi andj is poorer?If we adopt the relative view to building poverty indices,i will be considered thepoorer since as a proportion of the respective poverty lines he is farther away fromit thanj. If, instead, absolute poverty indices are used,j will be deemed the poorersince his absolute poverty gap ($900) is by far larger than that ofi ($500). Whichof these two views should prevail is open to debate.

6.7 Decomposing poverty

6.7.1 Growth-redistribution decompositions

It is often useful to determine whether it is mean-income growth or changes inthe relative income shares accruing to different parts of the population that are re-sponsible for the evolution of poverty across time. Investigating this can also helpassess whether these two factors, mean-income changes and inequality changes,work in the same or in opposite directions when it comes to the behavior of ag-


gregate poverty. Similarly, we may wish to assess whether differences in povertyacross countries or regions are due to differences in inequality or to differences inmean levels of income.

There are several ways to do this. To illustrate them, assume that we wish tocompare distributionsA andB to determine if it is the difference in their mean in-come or the difference in their income inequality that accounts for their differencein poverty. The common feature of all existing growth-redistribution decomposi-tion procedures is

1. first, to scale the two distributionsA andB such that they have the samemean, and interpret the difference in poverty across these two scaled distri-butions as the impact on poverty of their difference in inequality

2. and second, to interpret the difference in poverty between one of the distri-butions (say,A) and that same distribution scaled to the mean income of theother distribution (B) as the impact on poverty of their difference in meanincome.

Starting from this, the precise growth-redistribution decomposition proceduresthat are chosen differ by the solution they apply to a basic problem known gener-ally in the national-accounts literature as the ”index problem”. Specifically here,should we scaleA to the mean ofB, or B to the mean ofA, to assess the im-pact of differences in inequality? And, in estimating the impact of differences inmean incomes, should we compareA with A-scaled-to-the-mean-of-B, orB withB-scaled-to-the-mean-of-A?

The first paper that implemented a growth-redistribution decomposition ofpoverty differences (Datt and Ravallion (1992)) used the initial distribution asa reference ”anchor point”. To see how, it is easiest to use the normalized FGTindicesP (z; α) defined in (75), although the growth-redistribution decomposi-tion methodologies can be used with any relative poverty indices, additive or not.The change in poverty betweenA andB is expressed as a sum of a ”growth”(difference in mean income) effect and of a ”redistributive” (difference in relativeincome shares) effect, plus an error term that originates from the above-mentionedindex problem. This gives:

PB (z; α)− PA (z; α)

=

(PA

(zµA

µB

; α

)− PA (z; α)

)

︸︷︷︸growth effect

+

(PB

(zµB

µA

; α

)− PA (z; α)

)

︸︷︷︸redistribution effect

+ error term.

(92)


The first expression in the first term on the left of (92), PA

(zµA

µB; α

), is poverty

in A after A’s incomes have been scaled byµB/µA to yield a distribution with

meanµB and inequality unchanged.(PA

(zµA

µB; α

)− PA (z; α)

)is thus the dif-

ference between two distributions with the same relative income shares but with(possibly) different mean incomes. WhenµB > µA, this growth term is nega-

tive. The first expression in the second term,PB

(zµB

µA; α

)is poverty inB after

B’s incomes have been scaled byµA/µB to yield a distribution with meanµA.(PB

(zµB

µA; α

)− PA (z; α)

)is thus the difference between two distributions with

identical mean incomes but with (possibly) different inequality. When the Lorenzcurve forB is everywhere above the Lorenz curve forA, this redistribution termis necessarily negative whenα > 1, but it can also be positive whenα < 1.

The error term in (92) can be expressed as:

PB (z; α) + PA (z; α)− PB

(zµB

µA

; α

)− PA

(zµA

µB

; α

). (93)

This error term can be shown to be either the difference between the growth effectmeasured usingB as a reference distribution and that usingA as the referencedistribution,

PB (z; α)− PB

(zµB

µA

; α

)−

(PA

(zµA

µB

; α

)− PA (z; α)

), (94)

or the difference between the redistribution effect measured usingB as the refer-ence distribution and the redistribution effect usingA as the reference distribution,

PB (z; α)− PA

(zµA

µB

; α

)−

(PB

(zµB

µA

; α

)− PA (z; α)

). (95)

An alternative decomposition uses the posterior distributionB as the referencedistribution for assessing the growth and redistribution effects. This yields:


=

(PB (z; α)− PB

(zµB

µA

; α

))

︸︷︷︸alternative growth effect

+

(PB (z; α)− PA

(zµA

µB

; α

))

︸︷︷︸alternative redistribution effect

+ error term.

(96)Clearly, a middle way between these two alternative decomposition methodolo-gies is to measure the growth effect as the average of the two growth effects, and


likewise to measure the redistribution effect as the average of the two redistribu-tion effects. Proceeding this way has the advantage of eliminating the error term inthe poverty decomposition, since the error terms of each of the two alternative de-compositions sum to zero. This middle way is in fact what would be given by theuse of the Shapley value to perform a growth-redistribution decomposition – seeAppendix5.7 for more details on the Shapley value. This leads to the followinggrowth-redistribution decomposition:


=

(PA

(zµA

µB

; α

)− PA (z; α)

)+

(PB (z; α)− PB

(zµB

µA

; α

))

︸︷︷︸Shapley growth effect

+

(PB

(zµB

µA

; α

)− PA (z; α)

)+

(PB (z; α)− PA

(zµA

µB

; α

))

︸︷︷︸Shapley redistribution effect

.

(97)

6.7.2 Demographic and sectoral decomposition of differences in FGT in-dices

Equation (80) shows how poverty can be expressed as a sum of the povertycontributions of the various subgroups that make a population. Each subgroupcontributes in proportion to its share in the population and to the level of povertyfound in that subgroup. Hence, we may wish to express changes in poverty acrosstime, or differences in poverty across entities, as a function of differences in thesefactors. More precisely, differences in poverty across distributions can be at-tributed to differences in demographic or sectoral composition across these distri-butions, or to differences in poverty across these demographic or sectoral groups.


We may express this as follows:


=∑K

k φA(k)(PB(k; z; α)− PA(k; z; α)

)︸︷︷︸

within-group poverty effects+

∑Kk PA(k; z; α) (φB(k)− φA(k))︸︷︷︸

demographic or sectoral effects

+K∑

k

(PB(k; z; α)− PA(k; z; α)(φB(k)− φA(k))

)

︸︷︷︸interaction term

.

(98)

Note that the decomposition in (98) suffers from the same index number problemas the earlier one in (92). For example, one could prefer to useφB(k) instead ofφA(k) to compute the within-group poverty effects. It may be more convenient toweight the within-group poverty effects by the average population shares, and toweight the demographic and sectoral effects by the average poverty index. Thisyields:


=∑K

k φ(k)(PB(k; z; α)− PA(k; z; α)

)︸︷︷︸

within-group poverty effects+

∑Kk P (k; z; α) (φB(k)− φA(k))︸︷︷︸

demographic or sectoral effects

,

(99)

whereφ(k)= 0.5 (φA(k) + φB(k)) andP (k; z; α) = 0.5(PA(k; z; α) + PB(k; z; α)

).

Note from (99) that this decomposition procedure removes the error term. De-pending on the context, the decomposition in (99) could serve to show, for in-stance, how variations in the size and in the poverty of various sectors of the econ-omy account for variations of total poverty across economies, how differencesin the size and in the poverty of various demographic groups explain differencesin total poverty across societies, how migration and differential poverty acrossregions account for changes in poverty across time,etc..

6.7.3 The impact of demographic changes

An alternative use of the decomposition in (80) computes the impact of achange in the proportion of the population that is found in a groupk, this change


being accompanied by an exactly offsetting change in the proportion of the othergroups. This may be useful, for instance, if one wishes to predict the impact of mi-gration or demographic changes on national poverty. Let the population share of agroupt, φ(t), increase by a proportionpc to φ(t)(1 + pc), with a proportional fallin the other groups’ population share fromφ(k) to φ(k) (1− φ(t)pc/ (1− φ(t))).Note that the new population shares will add up to 1 since

φ(t)(1 + pc) +∑

k 6=t

φ(k)

(1− φ(t)pc

1− φ(t)

)=

∑

k

φ(k) = 1. (100)

The net impact of this on poverty is then

∆P (z; α) =

(P (t; z; α)−

∑

k 6=t

φ(k)

1− φ(t)P (k; z; α)

)φ(t)pc. (101)

We may indeed wish to predict the impact of an absolute increase in the pop-ulation share of a groupt. Let this changes be frompc to φ(t) + pc, with a cor-responding fall in the other groups’ population share that is proportional to theirinitial share (a fall fromφ(k) to φ(k) (1− pc/ (1− φ(t)))). The resulting changein poverty is analogously given as

∆P (z; α) =

(P (t; z; α)−

∑

k 6=t

φ(k)

1− φ(t)P (k; z; α)

)pc. (102)

Note that the only difference between (100) and (101) comes from the size in theincrease inφ(t), which isφ(t)pc in (100) andpc in (101).

6.7.4 Decomposing poverty by income components

LetC income components add up to total incomeX(p), with X(p) =∑C

c=1 X(c)(p)andX(c)(p) being the expected value of income componentc at rankp in the distri-bution of total income.X(c)(p) can be, for instance, agricultural or capital income,or income of those living in some geographic area, or some type of expenditurethat enters total expenditureX.

We may wish to know by what amount total poverty is reduced by the presenceof an income component. Clearly, we would expect those components with a largemeanµX(c)

to be more effective in helping to alleviate total poverty. But we mustalso take into account the distribution ofX(p). Suppose for instance that urban


capital income is larger than rural capital income, but that poverty is low in urbanareas because urban labour income is large there. Then, it is unclear whether highcapital income in urban areas is more effective at alleviating poverty than lowercapital income in rural areas, where poverty is more concentrated.

The contribution of an income componentc to poverty alleviation can be givenby the fall in poverty afterX(c)(p) is added to initial income. But this fall dependson what this initial income is. This path-dependency difficulty can again be cir-cumvented by the use of the Shapley value. We start by assuming maximumpoverty, that is, poverty when total income is nil for everyone. We then estimatethe contribution of componentc to poverty alleviation as the expected value of itsmarginal contribution when it is added randomly to anyone of the various subsetsof income components that one can choose from the set of all components. Whena component is missing from that set for an individual, we assume that its value is0.

6.8 References

Rowntree (1901) predated by far the modern quantitative approach to povertymeasurement. General and recent references includeGlewwe (2001) (for a veryextensive coverage of the nature, evolution, and causes of poverty),Chen andRavallion (2001) (for wide empirical evidence on poverty),Deaton (2001) (for theempirical difficulties associated with ”counting the poor”),Jantti and Danzinger(2000) (for poverty in more advanced countries),Lipton and Ravallion (1995) (forpoverty and policy),Constance and Michael (1995) (for the US debate on povertymeasurement),Ravallion (1994) andRavallion (1996) (for a non-technical overviewand discussion of poverty measurement issues),Smeeding, Rainwater, and O’Higgins(1990) (for early results using Luxembourg Income Study data) andZheng (1997)(for a review of poverty indices).

The papers byWatts (1968), Sen (1976) and Foster, Greer, and Thorbecke(1984) influenced greatly much of the subsequent and large literature on povertyindices. Relatively early contributions on poverty measurement are found inAnand(1977), Blackorby and Donaldson (1980), Chakravarty (1983b), Chakravarty (1983a),Clark, Hamming, and Ulph (1981), Donaldson and Weymark (1986), Foster (1984),Hagenaars (1987), Kakwani (1980), Kundu and Smith (1983), Takayama (1979),and Thon (1979). Recent works includeChakravarty (1997), Myles and Picot(2000), Osberg and Xu (2000) andShorrocks (1995) on a revisited and improvedform of theSen (1976) poverty index,Duclos and Gregoire (2002) on the link be-tween linear poverty indices and relative deprivation,Morduch (1998) andZheng


(1993) on the Watts index,Pattanaik and Sengupta (1995) on the original Senindex, andShorrocks (1998) on ”deprivation profiles”.

Applied poverty studies using these developments have been almost innumer-able. A small subset of the studies that have been published includesCoulombeand McKay (1998) (Mauritania),Coulombe and McKay (1998) (Ghana),David-son and Duclos (2000b) (using LIS data)?) (Northern countries),Grootart andKanbur (1995) (Cote d’Ivoire), Gustafsson and Shi (2002) )(China),Hagenaarsand De Vos (1988) (the Netherlands),Hill and Michael (2001) (US), Iceland andet al. (2001) (US), Milanovic (1992) (Poland),Osberg and Xu (1999) (Canada),Osberg (2000) (Canada and the US),Pendakur (2001) (Canada),Rady (2000)(Egypt),Ravallion and Bidani (1994) (Indonesia),Ravallion and Chen (1997) (67less developed countries),Rodgers and Rodgers (2000) (Australia), andSzulc(1995) (Poland).

The empirical links between growth, poverty and inequality have also oftenbeen analyzed in recent years. Studies on whether growth is beneficial to the poor,both absolutely and relatively speaking, includeBigsten and Shimeles (2003) (forEthiopian evidence),Datt and Ravallion (2002) (for a survey of the Indian ev-idence),Dollar and Kraay (2002) (for an influential study of the experience of42 countries over 4 decades),Essama Nssah (1997) (for Madagascar evidence),Ravallion and Chen (1997) (where growth is found not to increase inequality moreoften than it decreases it),Ravallion (2001) (where a warning against the use ofcross-country regressions is made), andRavallion and Datt (2002) (for differentialevidence across Indian states).Ravallion (1998a), De Janvry and Sadoulet (2000)andDeininger and Squire (1998) also apply causal tests to determine whether in-equality favors or impedes growth. See alsoTsui (1996) andRavallion and Chen(2003) for the use of the poverty gap and the Watts index as indices of whethergrowth is beneficial to the poor.

7 ESTIMATING POVERTY LINES 92

7 Estimating poverty lines

Three major issues arise in the discussion of poverty lines. First, we must de-fine the space in which well-being is to be measured. As discussed in Chapter2,this can be the space of utility, incomes, ”basic needs”, functionings, or capabili-ties. Second, we must determine whether we are interested in an absolute or in arelative poverty line in the space considered. Third, we must choose whether it isby someone’s ”capacity to function” or by someone’s ”actual functioning” that wewill judge if that person is poor. We consider first the issue of the choice betweenan absolute and a relative poverty line.

7.1 Absolute and relative poverty lines

An absolute poverty line can be interpreted as fixed in any one of the spacesin which we wish to assess well-being. Conversely, a relative poverty line de-pends on the distribution of well-being (including the utilities, living standards,functionings or capabilities) found in a society. Considerable controversy existson whether absoluteness or relativity is a better property for a poverty threshold.Most analysts would probably agree that a poverty threshold defined in the spaceof functionings and capabilities should be absolute (but even on this there is nounanimity). An absolute threshold in these spaces would, however, imply rela-tivity of the corresponding thresholds in the space of the commodities and in thelevel of basic needs required to achieve these functionings.

There are two main reasons for this. First, the relative prices and the avail-ability of commodities depend on the distribution of incomes. For instance, as asociety initially develops, the affordability and accessibility of public transporta-tion usually first increases as rising numbers of people need to travel to work andto trade, without first being able to afford the costs of private transportation. Associeties become richer on average, however, their citizens make increasing useof private forms of transportation, which causes a fall in the supply and avail-ability of public transportation, and leads to an increase in its relative price. Thismakes the capacity to travel (arguably an important capacity) more or less costly,depending on the state of economic development.

Second, not to be deprived of some capability may require the absence ofrelative deprivation in the space of some commodities. In support of this, thereis Adam Smith’s famous statement that the commodities needed to go withoutshame (an oft-mentioned basic functioning) can be to some extent relative to thedistribution of incomes in a society:


By necessities, I understand not only the commodities which are in-dispensably necessary for the support of life but whatever the customof the country renders it indecent for creditable people, even of thelowest order, to be without. (?))

?) reinforces this by distinguishing clearly the two dimensions of capabilities andcommodities:

I would like to say that poverty is an absolute notion in the space ofcapabilities but very often it will take a relative form in the space ofcommodities and characteristics (?, ?).

This has led some writers (particularly in developed countries) to conclude thatattempts to preserve some degree of absoluteness in the space of commodities areuntenable:

In summary, it does not seem possible to develop an approach topoverty measurement which is linked to absolute standards. Whilesome analysts are uneasy with relativist concepts of poverty on thegrounds that they are difficult to comprehend and can be seen assomewhat arbitrary and open to manipulation, no real practical al-ternative to relativist concepts exists. (Saunders (1994), p. 227)

7.2 Social exclusion and relative deprivation

Complete relativity of the poverty line in the space of commodities would nev-ertheless draw poverty analysis very close to the analysis of social exclusion (asexemplified by?) at the International Labor Organization) and relative deprivation(as propounded for instance byTownsend (1979). Social exclusion entails ”thedrawing of inappropriate group distinctions between free and equal individualswhich deny access to or participation in exchange or interaction” [Silver (1994),p.557]. This includes participation in property, earnings, public goods, and in theprevailing consumption level [Silver (1994), p.541]. Relative deprivation focuseson the inability to enjoy living standards and activities that are ordinarily observedin a society.Townsend (1979), p. 30, defines it as a situation in which

Individuals, families and groups in the population (. . . ) lack the re-sources to obtain the types of diet, participate in the activities and


have the living conditions and amenities which are customary or atleast widely encouraged or approved, in the society to which theybelong.

Equating absolute deprivation in the space of capabilities with relative depri-vation in the space of commodities can, however, be a source of confusion inpoverty comparisons. First, it tends to blur the operational and conceptual dis-tinction between poverty and inequality. Second, it can hinder the identificationof ”core” or absolute poverty in any of the spaces. Identifying core poverty is,however, probably the most relevant task in the design of public policy in devel-oping countries. Third, although the ethical appeal of Sen’s capability approachhas variously been invoked to justify the use of an entirely relative poverty line inthe space of commodities, Sen himself does not accept this:

Indeed, there is an irreducible core of absolute deprivation in our ideaof poverty, which translates reports of starvation, malnutrition andvisible hardship into a diagnosis of poverty without having to ascer-tain first the relative picture. Thus the approach of relative deprivationsupplements rather than supplants the analysis of poverty in terms ofabsolute dispossession (?), p.17)).

Furthermore,

(. . . ) considerations of relative deprivation are relevant in specifyingthe ’basic’ needs, but attempts to make relative deprivation the solebasis of such specification is doomed to failure since there is an ir-reducible core of absolute deprivation in the concept of poverty (?),p.17).

Given the measurement difficulties involved in estimating relative povertylines that correspond to absolute poverty lines in the space of functionings and ca-pabilities, analysts often find most transparent to use the space of living standardsas the space in which to define an absolute threshold. If this is done, however, itmust subsequently be admitted that the procedure will imply a set of thresholds inthe space of functionings and capabilities that depend at least partly on the condi-tions of the society in which an individual lives. Indeed, for a given absolute levelof living standard in the space of commodities, an individual’s capabilities are rel-ative, that is, they depend on his social environment, at least for functionings suchas shamelessness and participation in the life of the community.


7.3 Estimating absolute poverty lines

Methodologies for the estimation of poverty lines have been most developedin the context of the need to fulfil basic physiological functionings. Although suchmethodologies have often been set in a welfarist framework, they also matter forthe basic needs, functioning or capability approaches since these approaches arealso concerned with basic physiological achievements. These methodologies haverecently been most often applied to developing countries.

7.3.1 Cost of basic needs

The estimation of the ”cost of basic needs” usually involves two steps. First,an estimation is made of the minimal food expenditures that are necessary forliving in good health; we will denote this byzF . Second, an analogous estimateof the required non-food expenditures,zNF , is computed and added tozF to yielda total poverty line,zT . We consider now in some detail each of these two steps.

7.3.2 Cost of food needs

The first step in the computation of a global poverty line is usually to estimatea food poverty line. The determination of a food poverty line generally proceedsby asking what amount of food expenditures is required to achieve some minimalrequired level of food-energy intake (or nutrient intake, such as proteins, vitamins,fat, minerals,etc...). Early examples of the application of this approach includeRowntree (1901) and?). A basket of food commodities is designed or estimatedby ”food specialists” such as to provide those minimally required levels of food-energy intake. The cost of that basket yields the food poverty linezF .

To illustrate how this exercise can be carried out in practice, consider Figure19, which plots consumptionx1(p) andx2(p) of two goods, goods 1 and 2, over arange of percentilesp. For simplicity, Figure19supposes that good 1 is ”income-inelastic” (x1(p) is constant) but that the consumption of good 2 increases with therank in the distribution of income (it is income elastic). The idea then is to selecta combination ofx1(p) andx2(p) that provides a given level of minimal calorieintake. For the purposes of this illustration, assume that this minimum energyintake is 3000 calories per day, and that 1 unit of good 1 and 2 provides 2000 and1000 calories each respectively. Also assume that each unit of good 1 and 2 costsq$.

The cheapest way to achieve the minimum calorie intake would be to con-sume only of good 1, since good 1 is the most calorie-efficient (we can think of


good 1 as ”cereals” and good 2 as ”meat”). Indeed, each calorie provided by theconsumption of good 1 costsq$/2000, whereas each calorie provided by the con-sumption of good 2 costs twice as much, that is,q$/1000. 1.5 units of good 1 (1.5units *2000 calories/unit =3000 calories) would then be required for the minimalenergy intake to be met, andzF would then equal1.5q$.

This, however, would suppose a food commodity basket that no individual inFigure19would be observed to consume. Even at the very bottom of the distribu-tion of income, individuals consume indeed at least some of good 2 at the expenseof a diminished consumption of the more calorie-efficient good 1. We should pre-sumably take this information into account if we wished to respect somewhat thecultural and culinary preferences of those whose well-being we aim to evaluate.This raises the obvious question of which preferences we should consider. Notethat the preferred ratio of good 2 over good 1 increases continuously withp inFigure19. For convenience, denote that ratio byρ(p) = x2(p)/x1(p). Simplealgebra then shows that the cost of attaining the minimum calorie intake is givenby zF (p) = 3q$(1 + ρ(p))/(2 + ρ(p)), wherezF (p) indicates thatzF depends onthe rankp of those whose preferences we use to build the commodity basket andto compute the food poverty line.

Figure19plotszF (p) and shows that it is not neutral to the choice ofp. Usingthe preferences of the poorest, we obtainzF (p = 0) = 1.8q$, but if we usethe preferences of the median population, we getzF (p = 0.5) = 2.1q$. Thisis in fact just a special example of a more general standard observation in theliterature on poverty lines that the choice of reference parameters matters for theestimation of poverty lines. In Figure19, the farther are the preferencesρ(p) fromthe most calorie-efficient choice, the more costly is the estimated food povertyline zF (p). Arguably, the preferencesρ(p) should be those of individuals aroundthe total poverty line, but this is a (partly) circular argument sinceρ(p) is itselfa determinant of that total poverty line. In practice, an arbitrary value ofp isoften chosen, reflecting somea priori belief on the position of those at the edgeof the total poverty line. A more common (though arguably less commendable)procedure is to compute an average value ofx2(p)/x1(p) over a range ofp, suchas the bottom 25% or 50% individuals of a population.

Even if we were to agree on the positionp at which we wish to observe pref-erences such asρ(p), there still remains the awkward fact that preferences willoften vary significantly even at this given value ofp. Said differently, there are inpractice many different actual consumption patterns for a group of ”typical poor”.One solution is simply to ignore these differences and estimate the typical poor’saverage consumption patterns. Following this line of argument, consumption ex-


penditures on various food items are regressed against income and the estimatedparameters of these regressions are then used to predict the consumption patternsof the ”typical poor”. These regressions have often been parametric – assumingfor instance that expenditures on cereals and meat are globally quadratic or log-linear in total expenditures. It is unlikely, however, that such parametric forms fitappropriately at all income levels, low and high alike. A better statistical proce-dure would probably be to regress consumption expenditures non parametricallyon total expenditures, which would allow for a better fit of the preferences of thosearound the ”typical poor”.

An additionally important issue then is whether variations in culinary tastesand food habits across socio-economic characteristics should be taken into ac-count. If no account of such variations are taken, then we could choose amongour reference group the observed diet that minimizes food cost while providingthe minimum required level of food-energy intake. This would typically generatean unreasonably low level of expenditures for many of our reference individu-als, with an implied dietary basket of food commodities that could again be verydifferent from those they typically consume.

If, however, full account of diversity in culinary tastes were to be taken, a se-rious risk would exist of overestimating the poverty lines of those individuals andgroups of individuals with a greater taste for expensive foods (e.g., of higher qual-ity or better taste). This is commonly the case, for instance, for urban households,who customarily have more sophisticated culinary tastes than rural dwellers (forthe same overall living standards), and have also greater access to a larger vari-ety of imported and expensive foods. This procedure would then assign greaterpoverty lines to the urban versus the rural individuals. It would also mean thatthe equivalents of individual food poverty lines in terms of reference living stan-dards and ”utilities” would depend on the peculiarities of the individuals’ foodpreferences. This would generally lead to inconsistent comparisons of well-beingacross urban and rural inhabitants, and would exaggerate the degree of poverty inthe urban as compared to the rural areas.

We can illustrate this using Figure20. Figure20 shows baskets of two foodcommodities,x1 and x2, with three food budget constraints of total food con-sumption equal toY0 , Y1 , andY2 (these total budgets are expressed in units ofx1). Figure20 also shows a ”minimum calorie constraint”, along which the totalcalories provided by the consumption ofx1 andx2 would equal the required min-imum level of calorie intake. If no account whatsoever were taken of preferences,Y0 would yield the food poverty line. But along the food budget constraintY0 ,there is only one point which meets the minimum calorie constraint, and it is of


course unlikely that individuals willchoosea food basket to be precisely at thatcorner. An individual with preferencesU0 , for instance, would not locate him-self on the minimum calorie constraint. It is only with the more generous budgetconstraintY1 that this individual will consume the minimally required level ofcalorie intake, as shown on Figure20.

But not all individuals will necessarily choose to be ”calorie-sufficient” evenwith a total food budget ofY1 . Individuals with greater preferences – as in thecase ofU2 – for the less-calorie efficient goodx2 will not choose a food basketon or above the minimum calorie constraint. Individual with preferencesU2 willinstead needY2 to be calorie-sufficient. Yet, whether individuals with preferencesU1 and budgetY1 are just as well off as individuals with preferencesU2 andbudgetY2 is highly debatable. Such would be the assumption, however, if weused two distinct poverty linesY1 andY2 for the two different tastes.

As mentioned above, such comparability assumptions are often implicitly madein practice when individuals living in different regions, rural or urban for instance,are assigned different poverty lines for reasons independent of differences in needsor prices. As illustrated in Figure20, this supposes that an individual with ”sophis-ticated” preferences (an urban dweller who has been accustomed to food variety)needs a higher budget to be as ”well off” as an individual with less expensive pref-erences (a rural dweller who is content with eating basic food types). Probablymore convincing, however, would be the view thatU2 with Y 2 in Figure20 pro-vides greater utility and well-being thanU1 with Y 1. Assigning different povertylinesY 1 andY 2 would then lead to inconsistent or biased poverty estimates.

Minimally required food expenditures can also be (and are often) adjusted fordifferences in climate, sex, or age, when such differences impact on needs ratherthan on tastes (as we discussed above). These expenditures can also be adjustedfor variations in activity levels, although activity levels depend on the level ofone’s well-being, and thus on one’s poverty status. Activity-level adjustmentswould thus involve a poverty line that evolves endogenously with the standard ofliving of individuals, a slightly awkward feature for comparing well-being acrossindividuals and across time.

7.3.3 Non-food poverty lines

The subsequent step is usually to estimate the non-food component of the totalpoverty line. The most popular method for doing this is simply to go straight to anestimate of the total poverty line by dividing the food poverty line by the share offood in total expenditures. The intuition for this is as follows. The larger the food


share in total expenditures, the closer the food poverty line should be to the totalpoverty line. Therefore, the smallest should be the necessary adjustment to thefood poverty line (the closer to 1 should be the denominator that divides the foodpoverty line). The problem of which food share to use is of course an importantissue. It is a problem analogous to the one discussed above on what the foodbasket should be for computing a food poverty line. Popular practices vary, butoften make use of:

A- the average food share of those whose total expenditures equal the foodpoverty line9; E: 19.4.5

B- the average food share of those whose food expenditures equal the foodpoverty line10; E: 19.4.3

C- the average food share of a bottom proportion of the population (e.g., the25% or 50% poorest).

In addition to this, another popular method

D- adds tozF the non-food expenditures of those whose total expendituresequalzF

11. E: 19.4.7

To see how methods A, B and D work and differ from each other, considerFigure21. Figure21 shows (predicted) total expenditures against various levelsof food expenditures. The regression can be done parametrically, but a generallybetter approach would be to predict total expenditures using a non-parametricregression on food expenditures. On each of the two axes is shown the level ofthe (previously estimated) food poverty linezF . These two levels meet at the 45degree line.

As indicated above, method A makes use of the average food share of thosewhose total expenditures equal the food poverty line. Total expenditures equal thefood poverty line,zF , at point E on Figure21. The food share at point E is givenby the inverse of the slope of the line OE that goes from the origin to point E. Thetotal poverty line according to method A is therefore given by the height of a lineOE that extends to just above a level of food expenditures ofzF . This gives thevertical height of point A as the total poverty line according to method A.

9DAD:Density|Non parametric regression10DAD:Density|Non parametric regression11DAD:Density|Non parametric regression


Method B makes use of the average food share of those whose food expen-ditures equal the food poverty line. Those who consumezF in food are locatedat point B on Figure21. Their food share is given by the inverse of the slope ofthe straight line that would extend from point O to point B. Hence, dividingzF bythat food share brings us back to point B, which is therefore the total poverty lineaccording to method B.

The total poverty line according to method B is more generous than that ac-cording to method A since the food share used for B is lower than that used forA. Indeed, method A focusses on the food share of a rather deprived population:those who,in total, only spend thefoodpoverty line. Method B focusses on thefood share of a less deprived population: those who,on food only, spend the foodpoverty line. Since food shares tend to decline with standards of living, methodB’s food share is lower than method A’s.

Finally, method D considers the non-food expenditures of those whose totalexpenditures equalzF . As for method A, these individuals are found at point Eon Figure21. Their non-food expenditures are given by the length of line EG onthe Figure. Adding these non-food expenditures tozF yields a total poverty linegiven by the height of point D.

The choice of methods and food shares and the estimation of the non-foodpoverty lines is rather arbitrary, and the resulting estimate of the total poverty linewill also be somewhat arbitrary. Moreover, and perhaps more worryingly, someof the estimates will also vary with the distribution of living standards, as in thecase of method C where the food share is an average over a range of individuals.To avoid inconsistencies in the comparisons of poverty, it would therefore seempreferable to use the same food share across the distributions being compared,and to use methods that do not make estimates overly dependent on a particulardistribution of living standards.

7.3.4 Food energy intake

A slightly different method for estimating poverty lines that is popular in theliterature is the so-called Food-Energy-Intake (FEI) method. Estimates of the ob-served calorie intake of persons are first computed and then graphed against theirobserved (total or food) expenditures. The analyst then estimates the expendituresof those whose calorie intake is just at the minimum required for healthy subsis-tence. When these expenditures are on food, this provides a food poverty line,which can then be used as described above in Section7.3.3 to provide an esti-mate of a global poverty line. When the expenditures are total expenditures, the


FEI method provides a direct link between a minimum calorie intake and a totalpoverty line12. E:19.4.1

Figure22 illustrates how this method works. The curve shows the level of ex-penditure (measured on the vertical axis) that is observed (on average) at a givenlevel of calorie intake (shown on the horizontal axis). The curve is increasing andconvex, since calorie intake is usually expected to increase at a diminishing ratewith food or total expenditures. Abovezk, the minimum calorie intake recom-mended for a healthy life, we readz, the food or total poverty line according tothe FEI method.

As just exposed, the FEI method may appear straightforward and simple toimplement. A number of conceptual and measurement problems are, however,hidden behind this apparent simplicity. Note for instance that the line traced onFigure22 is theexpectedlink between expenditure and calorie intake; there isin real life a significant amount of variability around this line. How are we tointerpret this variability? If it is due to measurement errors, then we may perhapsignore it. If it is due to variability in preferences, then we may wish to modelthe calorie-intake-expenditure relationship separately for different groups of thepopulation, as is often done in practice, for urban and rural areas for instance. Asin the cost-of-basic-needs method, however, we then run the risk of estimatinghigher poverty lines for those groups that have more expensive or sophisticatedtastes for food. This would lead to inconsistent comparisons of well-being andpoverty, as discussed in Section7.3.2.

To compute expected expenditure (given the variability of actual observedspending) at a given calorie intake, we can estimate the parameters of a paramet-ric regression linking expenditures to calorie intake. Again, the regression is oftenpostulated to be log-linear or quadratic. This parametric specification supposes,however, that the functional relationship between expenditures and calorie intakeis known by the analyst, up to some unknown parameter values. This is unlikelyto be true everywhere, especially for those far from the level of calorie intake ofinterest (e.g., those at the lower and upper tails of the distribution of spending andcalorie intake). In such cases, the parametric procedure will make the estimatedexpenditure poverty line affected by the presence of ”outliers” that are relativelyfar from the minimum level of calorie intake. This procedure will then generatea biased estimator of the ”true” poverty line. A more flexible and arguably betterapproach would be to estimate the link between expenditures and calorie intakenon parametrically.

12DAD: Density|Non parametric regression


7.3.5 Illustration for Cameroon

To see whether differences in some of the methodologies described above mat-ter, consider the case of Cameroon. Table7 shows the result of estimating food,non-food and total poverty lines for the whole of Cameroon and for each of 6regions separately. Note that the figures are in Francs CFA adjusted for price dif-ferences, with Yaounde being the reference region. The food poverty line wasestimated using the FEI method at 2400 calories per day per adult equivalent. Anon parametric regression using DAD was performed for the whole of Cameroonand separately for each of the 6 regions. The lower non-food poverty line was ob-tained (non parametrically) using method D in section7.3.3, and the upper non-food poverty line using method B. Again, the relevant regressions were carriedout for the whole of Cameroon and separately for each of the 6 regions.

As can be seen, the link between calorie intake and food expenditures variessystematically across regions. Expected food expenditure at 2400 calories per dayare significantly higher in urban areas (Yaounde, Douala and Other Cities) than inthe rural ones. In Douala, for instance, a household would need 408 Francs CFAper day per adult equivalent to reach an intake of 2400 calories per day. In theHighlands, no more than 170 Francs CFA would on average be needed. The linkbetween food and total expenditures also varies across the regions in Cameroon.Combined with the different estimates for the food poverty lines, this leads to verysignificant variations across regions in the total poverty lines. Using method D, alower total poverty line of 589 Francs CFA is obtained for Douala, but that samepoverty line is only 235 Francs CFA for the Highlands. Note also that the choiceof method Bvsmethod D has a very significant impact on the estimate of the totalpoverty line. For the whole of Cameroon, the lower and the upper total povertylines are respectively 373 and 534 Francs CFA, a difference of 43%.

Unsurprisingly, these large differences across regions and across methods havea large impact on poverty estimates and on regional poverty comparisons. This isillustrated in Table8, which shows the proportion of individuals underneath var-ious poverty lines for various indicators of well-being. ”Calorie poverty” (firstcolumn) is fairly constant across Cameroon. In the whole of Cameroon, 68.1% ofthe population was observed to consume less than 2400 calories per day per adultequivalent. This proportion varies between 59.9% (for Other cities) and 86.5%(for Forest) across regions. Roughly the same limited variability and the samepoverty rankings appear when food poverty is estimated using for each region itsown food poverty line (third column). However, when acommonfood poverty lineis used to assess food poverty in each region (second column), national poverty


stays roughly unchanged at around 69% but urban regions now appear signifi-cantly less poor than the rural ones. For instance, the poverty headcount in Douala(42.0 %) is now only half that of the Highlands (82.5 %).

The rest of Table8 confirms these lessons. When a common poverty line isused to compare the regions, rural areas are significantly poorer than urban ones.When region-specific poverty lines are used, these differences are much reduced,and the regional rankings are often even reversed. For example, using a com-mon lower total poverty line (fourth column), the Highlands have a headcountratio more than three times that of the urban regions. When regional lower to-tal poverty lines are used instead, the Highlands become prominently the leastpoor of all regions. Setting common as opposed to regional poverty lines canthus have a crucial impact on poverty rankings and the determination of subse-quent poverty alleviation policies. The choice of a lower as against an upper totalpoverty line also makes a difference. For the whole of Cameroon, the proportionof the Cameroonian population in poverty increases from 43.9% to 68.0% whenwe move from a common lower total poverty line (fourth column) to a commonupper total poverty line (sixth column). Clearly, this changes significantly one’sunderstanding of the incidence of poverty in Cameroon.

These results also implicitly caution that the choice of well-being indicatorsis not neutral to the identification of the poor. In our context, this is because thecorrelation between calorie intake, food expenditure and total expenditure is im-perfect. Table9 indicates, for example, that in bidimensional poverty analysesusing any two of these three indicators of well-being, around 20% to 25% of thepopulation is characterized as poor in one dimension but non poor in the other.In the first part of9, we note for instance that 11.2% of the population would bejudged poor in terms of calorie intake but not poor in terms of food expenditure.Conversely, 9.6% of the population would be deemed non poor in terms of calo-rie intake but poor in terms of food expenditure. These proportions are slightlyhigher for the other bidimensional poverty analyses, which compare food withtotal expenditure poverty, and calorie with total expenditure poverty, respectively.

7.4 Estimating relative and subjective poverty lines

7.4.1 Relative poverty lines

There are two other popular methodologies for the estimation of poverty lines.The first deals with purely relative poverty lines, which, as we saw above, can beuseful to determine the commodities needed for ”living without shame” and for


participating in the ”prevailing consumption level”. A relative poverty line is typi-cally set as an arbitrary proportion of the mean or some quantile (often the median)of living standards. Clearly, such a poverty line will vary with the central tendencyof the distribution of living standards, and will not be the same across regions andtime. One awkward feature of the use of a relative poverty line approach is that apolicy which raises the living standards of all, but proportionately more those ofthe rich, will increase poverty, although the absolute living standards of the poorhave risen. Conversely, a natural catastrophe which hurts absolutely everyone willdecrease poverty if the rich are proportionately the most hurt13. E:19.3

Another awkward feature of the use of relative poverty lines is that anim-provementin the absolute living standards of some of the poor, with no changein the living standards of the others, may in factincreasepoverty. To see why,let η andς be small positive values and let an income distribution be defined asQ(p) + η(p), with

η(p) =

η, if p ∈ [p0 − ς, p0 + ς]0, otherwise,

(103)

and withη set initially to 0. Choosez = λµ. The un-normalized FGT index isthen given by

P (z; α) =

∫ 1

0

(λµ−Q(p) + η(p))+α dp. (104)

Note that∂µ/∂η|η=0 = 2ς > 0, which simply says that the relative poverty lineincreases with an increase inς. We may then check how increases inη affectoverall poverty, for a smallς. For the headcount index, we find

limς→0

(∂P (λµ; α = 0)

∂η

∣∣∣∣η=0

/2ς

)= λ · f(λµ) > 0, (105)

which says that the headcount necessarily increases whenever someone’s incomeincreases, regardless of whether that person is poor or rich. Whenα > 0,

limς→0

(∂P (λµ; α > 0)

∂η

∣∣∣∣η=0

/2ς

)= α

λP (λµ; α− 1)︸︷︷︸

A

− (λµ−Q(p0))α−1+︸︷︷︸

B

.

(106)

13DAD: Poverty|FGT Index


The termA on the right-hand side of (106) is positive: an increase in incomesincreases the relative poverty line and thus tends to increase poverty. Whenp0 > F (λµ), the increase in income is beneficial to the rich: the termB is thennil, and poverty then necessarily increases. Whenp0 < F (λµ), the increase inincome benefits some of those below the poverty line, and this increase in theirabsolute living standards explains why the termB is then negative. Whether it issufficiently negative to offset the positive termA depends 1) on how far below thepoverty line these poor are, and 2) on the value of the ethical parameterα. Hence,even withα > 0, relative poverty may increase when growth is beneficial to thepoor14. E:19.3

When used alone, relative poverty lines thus drift the analysis towards theconcept of relative inequality. Because of this, they are probably best used inconjunction with absolute living standard thresholds, at least when the aim is tocapture both absolute deprivation in basic physiological capabilities and socialexclusion and relative deprivation in more social capabilities.

7.4.2 Subjective poverty lines

An alternative poverty line approach relies on the use of subjective informationon the link between living standards and well-being. One source of informationcomes from interviews on what is perceived to be a sound poverty line, usinga query found for instance inGoedhart, Halberstadt, Kapteyn, and Van Praag(1977):

We would like to know which net family income would, inyour cir-cumstances, be the absolute minimum foryou. That is to say, that youwould not be able to make both ends meet if you earned less. (p.510)

The answers are subsequently regressed on the living standards of the respondents.The subjective poverty line is given by the point at which the predicted answer tothe minimum income question equals the living standard of the respondents. Thebasic idea for this is that unless someone earns that poverty line, he will not trulyknow that it is indeed the appropriate minimum income needed to ”make bothends meet”.

This method is illustrated in some detail on Figure23. Each point representsa separate answer to the above query, namely, the minimum income judged to

14DAD:Poverty|FGT Index


be needed to make both ends meet as a function of the actual income of the re-spondents. The filled line shows the predicted response of individuals at a givenlevel of income. For low income levels, this predicted minimum subjective in-come is well above the respondents’ income. The predicted minimum subjectiveincome increases with actual income, but not as fast as income itself. Those withincome abovez∗ answer that they need on average less than their own income.At z∗, which is also where the 45-degree line crosses the line of predicted mini-mum subjective income, that predicted minimum subjective income equals actualincome.

One difficulty with the subjective approach is the sensitivity of poverty lineestimates to the formulation of interview questions. Another problem comes fromthe considerable variability in the answers provided, even within groups of rela-tively socio-economically homogeneous respondents. The presence of this vari-ability is apparent on Figure23 with points sometimes quite far away from thepredicted response line. This variability has some awkward consequences. OnFigure23, for instance, an individual at pointa is a point at which someone wouldbe judged poor according to the subjective income method since his income fallsbelow z∗. An individual ata feels, however, that his income exceeds the mini-mum income he feels to be needed (pointa is to the right of the 45-degree line).He would therefore feel that he is not poor. Conversely, someone at pointb feelsthat he is poor, since his reported minimum income exceeds his actual income,but he would be judged not to be poor by the subjective poverty line method.

How, therefore, ought we to interpret this variability? Is it due to measure-ment errors? If so, then we may probably best ignore it. Is it rather that the linkbetween living standards and real well-being varies systematically within homo-geneous groups of people? If so, then we should perhaps not attempt to use livingstandards or other direct or indirect indicators of well-being to classify the poorand the non poor. Instead, we should take individuals at their word on whetherthey declare themselves to be poor or not. But then, this would clearly raise im-portant practical problems for the assessment and the implementation of publicpolicy. For instance, can the implementation of public policy really depend on theprovision of subjective information on the part of individuals?

7.4.3 Subjective poverty lines with discrete information

An alternative approach to estimating subjective poverty lines is to ask re-spondents whether they feel that their living standards are below the poverty line,without direct indications of what the value of that poverty line may well be. An-


swers are coded 0 or 1 – according to whether respondents feel that they are pooror not – alongside the respondents’ living standards. The estimate of the povertyline is that which best reconciles the distribution of those answers with that of therespondents’ incomes.

This is illustrated in Figure24. Each ”dot” is an observation of whether a re-spondent of a certain income level felt poor (1) or not (0). The implicit assumptionis that respondents compare their income to acommonsubjective poverty linez∗.z∗ is unobserved and must be estimated. Not everyone with an income belowz∗

says that he is poor; conversely, not everyone abovez∗ says that he is not poor.These ”classification errors” would be explained by measurement and/or misre-porting errors. Hence, on Figure24, there are ”false poor” and ”false rich”, asshown within the ellipses at the bottom left and at the top right of the Figure.

One natural estimation procedure forz∗ would seem to be to maximize thelikelihood that the respondents’ declarations of poverty status correspond to thatwhich would be inferred by comparingz∗ to their incomes. Said differently, theestimator ofz∗ would minimize the likelihood of observing observations withinthe ellipses in Figure24.

7.5 References

The literature on the estimation of poverty lines is both significant and varied.Note that there is often a sharp distinction in tone and in content between thoseworks which focus on poverty in less developed countries and those which addresspoverty in more developed economies.

Early reviews of the literature includeGoedhart, Halberstadt, Kapteyn, andVan Praag (1977) andHagenaars and Van Praag (1985). An excellent and com-prehensive recent review can be found inRavallion (1998b) – this chapter hasbeen much influenced by it.Greer and Thorbecke (1986) has been influential inguiding the FEI method of estimating a poverty line. A method based on ”basicneeds budget” is described inRenwick and Bergmann (1993). The differentialeffects for poverty measurement of choosing FEIvsCBN methods for estimatingpoverty lines can be foundinter alia in Ravallion and Bidani (1994) andWodon(1997a).

The consequences and the issues that accompany the choice between absoluteand relative poverty lines are discussed inBlackburn (1998) (on the empiricalsensitivity of poverty comparisons to that choice),de Vos and Zaidi (1998) (onwhether poverty lines should be country specific),Foster (1998) andZheng (1994)(on the consequences for the choice of poverty indices), andFisher (1995) and


Madden (2000) (on the empirical income elasticity of poverty lines).”Subjective” methods for setting poverty lines are discussed and explored

in de Vos and Garner (1991) (for comparisons of results between the US andthe Netherlands),Pradhan and Ravallion (2000) (on perceived consumption ade-quacy),Stanovnik (1992) (for an application to Slovenia),Van den Bosch, Callan,Estivill, Hausman, Jeandidier, Muffels, and Yfantopoulos (1993) (for a compari-son across 7 European countries),Blanchflower and Oswald (2000) (for reportedlevels of happiness in Great Britain and in the US), andRavallion and Lokshin(2002) (for perceptions of well-being in Russia)

Barrington (1997), Fisher (1992), Glennerster (2000) andOrshansky (1988)provide critical reviews of the literature on the setting of the official poverty linein the United States.

8 MEASURING VERTICAL EQUITY AND PROGRESSIVITY 109

8 Measuring vertical equity and progressivity

As is well-known, the assessment of tax and transfer systems draws on twofundamental principles: efficiency and equity. The former relates to the pres-ence of distortions in the economic behavior of agents, while the latter focuses ondistributive justice. Vertical equity as a principle of distributive justice is rarelyquestioned as such, although the extent to which it must be precisely weightedagainst efficiency is of course a matter of intense disagreement among policy an-alysts. A principle of redistributive justice which gathers even greater support isthat of horizontal equity, the equal treatment of equals. TheHE principle is oftenseen as a consequence of the fundamental moral principle of the equal worth ofhuman beings, and as a corollary of the equal sacrifice theories of taxation. Thischapter and the next cover in turn the measurement of each of these principles.

8.1 Taxes and transfers

Let X andN represent respectively gross and net incomes, and letT be taxesnet of transfers – the net tax for short. Gross income is pre-tax and/or pre-transferincome, and net income is post-tax and/or post-transfer income, that is,N =X − T . For expositional simplicity, we assume in this chapter that gross incomesare exogenous. This is a common assumption in the literature on the measurementof the impact of taxes and transfers, but it can certainly fail to capture the trueimpact of tax and transfer policies on well-being when these taxes and transfersare non-marginal.

We can expect a part of the net tax to be a function of the value of gross incomeX. Otherwise, taxes would be lump sum and orthogonal to gross income. Wedenote this deterministic part byT (X). For several reasons, we also expectT tobe stochastically linked toX. In real life, taxes and transfers depend on a numberof variables other than gross incomes, such as family size and composition, age,sex, area of residence, sources of income, consumption and savings behavior, andthe ability to avoid taxes or claim transfers. Thus, we can think ofT as being astochastic function ofX, with

T = T (X) + ν, (107)

whereν is a stochastic tax determinant.We denote byFX,N(·, ·) the joint cumulative distribution function (cdf) of

gross and net incomes. LetQX(p), QN(p) andQT (p) be thep-quantile func-tions for gross incomes, net incomes and net taxes, respectively. LetFN |x(·) be


thecdf of N conditional on gross income being equal tox. Theq-quantile func-tion for net incomes conditional on ap-quantile value for gross incomes is thentechnically defined asQN(q|p) = infs ≥ 0|FN |QX(p)(s) ≥ q for q ∈ [0, 1],assuming that net incomes are non-negative.QN(q|p) thus gives the net incomeof the individual whose net income rank isq among all those with gross incomeequal toQX(p).

The expected net income of those withQX(p) is then given by

N(p) =

∫ 1

0

QN(q|p)dq, (108)

and the expected net tax of those withQX(p) is obtained as

T (p) = QX(p)−N(p). (109)

8.2 Concentration curves

An important descriptive and normative tool for capturing the impact of taxand transfer policies is the concentration curve. Concentration curves can helpcapture the horizontal and vertical equity of existing tax and transfer systems.They can also serve to predict the impact of reforms to these systems.

The concentration curve forT is15: E:19.8.11

CT (p) =

∫ p

0T (q)dq

µT

(110)

whereµT =∫ 1

0QT (p)dp = µX − µN is average taxes across the population.

CT (p) shows the proportion of total taxes paid by thep bottom proportion of thepopulation.

In practice, concentration curves are usually estimated by ordering a finitenumbern of sample observations(X1, N1), ..., (Xn, Nn) in increasing values ofgross incomes, such thatX1 ≤ X2 ≤ ... ≤ Xn, with percentilespi = i/n, i =1, ..., n. For i = 1, ...n, the sample (or “empirical”) concentration curve for taxes(Ti = Xi −Ni) is then defined as

CT (p = i/n) =1

nµT

i∑j=1

Tj. (111)

15DAD: Curves|Concentration curve


As for the empirical Lorenz curves, other values ofCT (p) can be estimated byinterpolation.

The concentration curveCN(p) for net incomes is analogously defined as

CN(p) =

∫ p

0N(q)dq

µN

(112)

and typically estimated as

CN(p = i/n) =1

nµN

i∑j=1

Nj, (113)

where theNj have been ordered in increasing values of the associatedgross in-comesNj (see (111)). Note thatCN(p) is different from the Lorenz curve of netincomes,LN(p), defined as:

LN(p) =

∫ p

0QN(q)dq

µN

. (114)

Empirically, the Lorenz curve for net income is indeed typically estimated as

LN(p = i/n) =1

nµN

i∑j=1

Nj, (115)

where the observations have been re-ordered in increasing values ofnet incomes,with N1 ≤ N2 ≤ ... ≤ Nn. Thus,CN(p) sums up the expected value of netincomes up to gross income percentilep. LN(p), however, sums up net incomesup to a net income percentilep.

Denote ast the average tax as a proportion of average gross income, witht = µT /µX . Whent 6= 0, we can show that

CN(p)− LX =t

1− t[LX − CT (p)] . (116)

For a positivet, this indicates that the more concentrated are the taxes among thepoor (the smaller the differenceLX − CT (p)), the less concentrated among thepoor will net incomes be. The reverse is true for transfers (negativet): the moreconcentrated they are among the poor, the more concentrated net income is amongthe poor. This link will prove useful later in defining indices of tax progressivity.


8.3 Concentration indices

As for the Lorenz curves and the S-Gini indices of inequality introduced ear-lier, we can aggregate the distance betweenp and the concentration curvesC(p)to obtain summary indices of concentration. These indices of concentration areuseful to compute aggregate indices of progressivity and vertical equity. Moregenerally, they can also serve to decompose the inequality in total income or to-tal consumption into a sum of the concentration of the components of that totalincome or consumption, such as different sources of income (different types ofearnings, interests, dividends, capital gains, taxes, transfers,etc.) or differenttypes of consumption (of food, clothing, housing,etc.).

To define indices of concentration, we can simply weight the distancep−C(p)by an ethical weightκ(p), of which a popular form is again given byκ(p; ρ) inequation (19). This gives the following class of S-Gini indices of concentration,IC (ρ):

IC (ρ) =

∫ 1

0

(p− C(p))κ(p; ρ)dp. (117)

8.4 Decomposition of inequality into income components

8.4.1 Using concentration curves and indices

An S-Gini inequality index for a variable can easily be decomposed as a sum ofthe concentration indices of the component variables that add up to that variable.This can be useful, for instance, for decomposing total income inequality as a sumof concentration indices for the different sources of income (employment, capital,transfers,etc.), or total expenditure inequality as a sum of concentration indicesfor food and non-food expenditures. For example, letX(1) andX(2) be two typesof expenditures, and letX = X(1) + X(2) be total consumption. LetCX(1)

(p)andCX(2)

(p) be the concentration curves of each of the two types of consumption(usingX as the ordering variable). The concentration indices forX(c),ICX(c)

(ρ)),c = 1, 2, are as follows:

ICX(c)(ρ) =

∫ 1

0

(p− CX(c)(p))κ(p; ρ)dp. (118)


Inequality inX can then be decomposed as a sum of the inequality inX(1) and inX(2). The Lorenz curve for total consumption is given by:

LX(p) =

(µX(1)

CX(1)(p) + µX(2)

CX(2)(p)

)

µX

, (119)

which is a simple weighted sum of the concentration curves for each of the twotypes of consumption. The index of inequality in total consumption is similarlya simple weighted sum of the concentration indices of each of the two types ofconsumption:

IX(ρ) =µX(1)

ICX(1)(ρ) + µX(2)

ICX(2)(ρ)

µX

. (120)

For givenµX(1)and µX(2)

, the higher the concentration indicesICX(1)(ρ) and

ICX(2)(ρ), the larger the S-Gini index of inequality in total consumption. More-

over, the higher the shareµX(c)/µX of the more highly concentrated expenditure,

the higher the inequality in total expenditures16. E:19.8.32One possible difficulty with the above is that a component which has the same

value for all will be judged by the decompositions in (119) and (120) to havea zero contribution to total inequality. This is becauseCX(c)

(p) = p for all pandICX(c)

= 0 if componentc is equally distributed across all individuals. Itmay be argued, however, that in such a case contributionc should be seen ascontributingnegativelyto total inequality. Being the same for all, componentc indeed decreases the inequality introduced by other components. One way tocapture this is to rewrite the decompositions (119) and (120) in reference toLX(p)andIX(ρ). This gives:

µX(1)

(CX(1)

(p)− LX(p))

µX

+µX(2)

(CX(2)

(p)− LX(p))

µX

= 0 (121)

and

µX(1)

(ICX(1)

(ρ)− IX(ρ))

µX

+µX(2)

(ICX(2)

(ρ)− IX(ρ))

µX

= 0. (122)

The two terms on the left of each of these last two expressions give respectivelythe contributions of components 1 and 2 to the Lorenz curve and the inequalityindex of total expenditureX.

16DAD: Decomposition|S-Gini: by sources


8.4.2 Using the Shapley value

An alternative approach uses the Shapley value to express inequality in totalincome as a sum of the contributions of inequality in individual income compo-nents. For expositional simplicity, assume that there are only two components.Total inequality is then given byI

(X(1)(p), X(2)(p)

). Suppose that we replace

the two income componentsX(1)(p) andX(2)(p) by their mean valueµX(1)and

µX(2), to yield I

(µX(1)

, µX(2)

). Clearly, inequality would be zero after such a

substitution. Total inequality is then simply given by

I(X(1)(p), X(2)(p)

)=

I(X(1)(p), X(2)(p)

)− I(µX(1)

, X(2)(p))

+I(µX(1)

, X(2)(p))

.

(123)

The second line would show the contribution of component 1 and the third wouldindicate the contribution of component 2. These estimated contributions are ingeneral dependent upon the order in which they are replaced by their mean value.The contribution of component 1 could for instance be estimated alternatively as

I(X(1)(p), µX(2)

). The use of the Shapley value thus defines the contribution of a

componentc to total inequality as its expected contribution to inequality reductionwhen it is added randomly to anyone of the various subsets of components thatone can choose from the set of all components. With two components, this gives:

I(X(1)(p), X(2)(p)

)=

0.5

I(X(1)(p), X(2)(p)

)− I(µX(1)

, X(2)(p))

+ I(X(1)(p), µX(2)

)

︸︷︷︸Shapley contribution of component 1

+ 0.5

I(X(1)(p), X(2)(p)

)− I(X(1)(p), µX(2)

)+ I

(µX(1)

, X(2)(p))

︸︷︷︸Shapley contribution of component 2

.

(124)

8.5 Progressivity comparisons

8.5.1 Deterministic tax and benefit systems

Let us for a moment assume that the tax system is non-stochastic (or determin-istic), namely, thatν equals a constant zero. Suppose also for now that the deter-ministic tax system does not rerank individuals, or equivalently thatT (1)(X) ≤ 1.


Furthermore, denote the average rate of taxation at gross incomeX by t(X)(t(X) = T (X)/X)17. Assuming no reranking, a net tax (possibly including aE:19.8.9transfer or subsidy)T (X) is said to be

• locally progressive atX = x if the average rate of taxation increases withX, that is, ift(1)(x) > 0

• locally proportional atX = x if the average rate of taxation stays constantwith X, that is, ift(1)(x) = 0,

• and locally regressive atX = x if the average rate of taxation decreaseswith X, that is, ift(1)(x) < 018. E:19.8.10

There are two popular ”local” measures to capture the change in taxes and netincome as gross income increases. One is the elasticity of taxes with respect toX, also called Liability Progression,LP(X):

LP(X) =X

T (X)T (1)(X) =

T (1)(X)

T (X). (125)

LP(X) is simply the ratio of the marginal tax rate over the average tax rate atX. It is possible to show that a tax system is everywhere progressive (namely,t(1)(X) > 0 everywhere) ifLP(X) > 1 everywhere. The larger this measure ateveryX, the more concentrated among the richer are the taxes.

One problem withLP(X) is that it is not defined whenT (X) = 0, and thatit is awkward to interpret when a net tax is sometimes negative and sometimespositive across gross income. Another problem is that it is linked to the rela-tive distribution of taxes, not with the relative distribution of the associated netincomes.

These problems are avoided by a second local measure of progression, calledResidual Progression (RP(X)), which is the elasticity of net income with respectto gross income:

RP(X) =∂(X − T (X))

∂X· X

N=

1− T (1)(X)

1− T (X). (126)

Unlike LP(X), RP(X) is well defined and easily interpretable even when taxesare sometimes negative, positive or zero, so long as gross and net incomes are

17DAD: Distribution|Non parametric regression18DAD: Distribution|Non parametric regression


strictly positive. It is then possible to show that a tax system is everywhere pro-gressive (again, this means thatt(1)(X) > 0 everywhere) ifRP(X) < 1 every-where.

There is a nice link between these measures of progressivity and the redis-tributive impact of taxes.

Progressivity and inequality reductionAssuming no reranking, the following conditions are equivalent:

1. t(1)(X) > 0 for all X;

2. LP(X) > 1 for all X (assumingT (X) > 0);

3. RP(X) < 1 for all X ;

4. LX(p) > CT (p) for all p and for any distributionFX of gross income (as-sumingµT > 0);

5. LN(p) > LX(p) for all p and for any distributionFX of gross income.

Progressive taxation will thus make the distribution of net incomes unambigu-ously more equal than the distribution of gross incomes, regardless of that actualdistribution of gross incomes. Moreover, if the residual progression for a tax sys-temA is always lower than that of a tax systemB, whatever the value ofX, thenthe tax systemA is said to be everywhere more residual-progressive than the taxsystemB, and the distribution of net incomes will always be more equal underAthan underB, again regardless of the distribution of gross incomes.

Hence, an important distributive consequence of progressive taxation is tomake the inequality of net incomes lower than that of gross income. Analo-gously, proportional taxation will not change inequality, and regressive taxationwill increase inequality. The more progressive the tax system, the more inequality-reducing it is. To check whether a deterministic tax system is progressive, pro-portional or regressive, we may thus simply plot the average tax rate as a functionof X and observe its slope. Alternatively, we may estimate and graph its liabilityprogression or its residual progression at various values ofX. To check whethera tax system is more residual-progressive (and thus more redistributive) than an-other one, we simply plot and compare the elasticity of net incomes with respectto gross incomes. All of this can be done using non-parametric regressions ofT (X) andN againstX.


Another informative descriptive approach is to compare the share in taxes andbenefits to the share in the population of individuals at various ranks in the distri-bution of gross income. This is most easily done by plotting on a graph the ratiosT (X)/µT or T (p)/µT for various values ofX or p. If these ratios exceed 1, thenthose individuals with those incomes or ranks pay a greater share of total taxesthan their population share. A similar intuition applies whenT (·) is a benefit: aratioT (X)/µT or T (p)/µT that exceeds 1 indicates that the benefit share exceedsthe population share. IfT (X) or T (p) increases proportionately faster thanX orQX(p), then the tax system is everywhere locally progressive.

A competing descriptive tool is to plot the ratio of taxes over gross income, thatis, T (X)/X, perhaps assessed at some rankp to giveT (p)/QX(p). Such a graphshows how the average tax rate evolves with gross income or ranks. When theseratios increase everywhere withX, the tax is everywhere locally progressive.

8.5.2 General tax and benefit systems

Although graphically informative, the above simple descriptive approachespresent some problems. First, ifT (1)(X) > 1, the tax system will induce rerank-ing, even if it is a deterministic function ofX. As we will see below, reranking(and, more generally, horizontal inequity) decreases the redistributive effect oftaxation, besides being of significant ethical concern in its own right.

Second, and more importantly in empirical applications, taxes are typicallynot a deterministic function of gross income, and randomness in taxes will intro-duce greater variability and inequality in net incomes than the above deterministicapproach would predict.X − T (X) may then be an unreliable guide to the distri-bution of net incomes, and the above theorems relating local progression measuresto global redistributive impact lose a great part of their practical usefulness. Ran-domness in taxes will also introduce further reranking. These two features willreduce the redistributive effect of the tax, and may even in the most extreme casesincrease inequality even when the “deterministic trend” of the tax is progressive,or such thatt(1)(X) > 0.

Finally, the actual redistribution effected by taxes depends on the distributionof gross incomes, and not only on the shape of the tax functionT . Said differently,the actual redistributive effect of liability or residual progression will depend onthe actual distribution of gross incomes. Arguably, the actual redistribution oper-ated by a tax system is probably of greater interest than its potential impact. A taxmay be very locally progressive over some ranges of gross income, but the actualredistributive impact will depend on the interaction of this local progression with


the distribution of gross incomes.

8.6 Tax and income redistribution

To deal with these difficulties, we can use theactualdistribution of taxesT andnet incomesN (instead of their predicted valuesT (X) andX − T (X)) to deter-mine whether the actual tax system isreally progressive and inequality-reducing.This amounts to combining the local measures of progressivity with the distribu-tion of gross incomes to generateglobalmeasures of progressivity.

There are two leading approaches for this exercise. The first is the Tax-Redistribution (TR) approach, and the second is the Income-Redistribution (IR)approach. The global definitions of tax progressivity associated to each of theseapproaches are as follows.

1. ForTR progressivity19: E:19.8.2

(a) A tax T is TR-progressive if

CT (p) < LX(p) for all p ∈]0, 1[. (127)

(b) A benefitB is TR-progressive if

CB(p) > LX(p) for all p ∈]0, 1[. (128)

(c) A tax T(1) is moreTR-progressive than a taxT(2) if

CT(1)(p) < CT(2)

(p) for all p ∈]0, 1[. (129)

(d) A benefitB(1) is moreTR-progressive than a benefitB(2) if

CB(1)(p) > CB(2)

(p) for all p ∈]0, 1[. (130)

(e) A tax T is moreTR-progressive than a benefitB if

LX(p)− CT (p) > CB(p)− LX(p) for all p ∈]0, 1[. (131)

2. For IR progressivity20: E:19.8.3

19DAD: Redistribution|Tax or Transfer20DAD: Redistribution|Tax or Transfer


(a) A net taxT is IR-progressive if

CN(p) > LX(p) for all p ∈]0, 1[. (132)

(b) A net taxT(1) is moreIR-progressive than a tax (and/or a transfer)T(2)

ifCN(1)

(p) > CN(2)(p) for all p ∈]0, 1[. (133)

These twoTR and IR approaches are consistent with the use above of li-ability and residual progression in a deterministic tax system. Ift(1)(X) > 0and T (1)(X) ≤ 1 (namely, no reranking), then, whatever the actual distribu-tion of gross incomes,T (X) is bothTR- and IR-progressive. Furthermore, ifLP (1) > LP (2) at all values ofX, then the tax system 1 is necessarily moreTRprogressive than the tax system 2. And ifRP (1) < RP (2) at all values ofX, thenthe tax system 1 is necessarily moreIR progressive than the tax system 2.

Note that these progressivity comparisons have as a reference point the initialLorenz curve. In other words, a tax is progressive if the poorest individuals bear ashare of the total tax burden that is less than their share in total gross income. Asmentioned above, an alternative reference point would be the cumulative shares inthe population. This is often argued in the context of state support – the referencepoint to assess the equity of public expenditures is population share. The ana-lytical framework above can easily allow for this alternative view – for instance,simply by replacingLX(p) by p in the above definitions ofTR progressivity. Thiswill make more stringent the conditions to declare a benefit to be progressive, butit will also make it easier for a tax to be declared progressive – to see this, compare(127) and (128).

8.7 References

Many of the classical texts on the concept, the role and the measurement oftax progressivity date from the 1950’s but they are still very relevant today – theyinclude Blum and Kahen Jr. (1963), Musgrave and Thin (1948), Slitor (1948)and Vickrey (1972). See alsoOkun (1975) for an influential discussion of theinteraction between efficiency and equity issues, as well asPechman (1985) onincidence analysis.

The measurement of progressivity and vertical equity moved forward signif-icantly in the middle of the 1970’s following the slightly earlier advances on themeasurement of inequality – see, for instance,Fellman (1976), Jakobsson (1976)


andKakwani (1977a) for the link between progressivity and inequality reduction,andKakwani (1977b), Suits (1977) andReynolds and Smolensky (1977) for in-fluential indices of tax progressivity and vertical equity. Reviews of the literaturecan be found inLambert (1993) andLambert (2001).

For papers that address general links between progressivity and inequality, seeDavies and Hoy (2002) (for the inequality-reducing properties of ”flat taxes”),Latham (1993) (for how to assess whether one tax is more progressive than an-other), Liu (1985) (for tax progressivity and Lorenz dominance),Moyes andShorrocks (1998) (on the difficulties that arise for the measurement of progressiv-ity when households differ in needs), andThistle (1988) (for residual progressionand progressivity).

Numerous versions of other specific tax progressivity indices have been dis-cussed and presented over the years. These include, for example,Baum (1987),for ”relative share adjustment” indices;Blackorby and Donaldson (1984) andKiefer (1984), for normatively-based indices of progressivity);Duclos (1995a),Duclos and Tabi (1996) and Duclos (1997b), for indices of the ”social perfor-mance” of tax progressivity);Duclos (1998), for normative foundations for theSuits progressivity index;Hayes, Slottje, and Lambert (1992), for effective taxprogression across percentiles); andZandvakili (1994), Zandvakili (1995) andZandvakili and Mills (2001) for the use of progressivity indices derived from Gen-eralized entropy and Atkinson inequality indices.

Linear indices of progressivity derive from the class of linear inequality in-dices introduced inMehran (1976). They are discussedinter alia Duclos (2000),Kakwani (1987), Pfahler (1983) andPfahler (1987).

Some of the literature has also tended to focus on the tension and on the linksbetween local and global progressivity. See, for instance,Baum (1998), Cassady,Ruggeri, and Van Wart (1996), Formby, Seaks, and Smith (1984), Formby, Smith,and Thistle (1987), Formby, Smith, and Thistle (1990) andFormby, Smith, andSykes (1986). See alsoDuclos (1995a) for a method for estimating the averageresidual progression of unevenly progressive tax and benefit systems, andKeen,Papapanagos, and Shorrocks (2000) andLe Breton, Moyes, and Trannoy (1996)for the impact of changes in tax components (such as sizes of allowances) on theprogressivity of the overall tax system.

The influence of the ”initial distribution” (that of gross incomes) on progres-sivity measurement is studied inDardanoni and Lambert (2002) (for a ”transplant-and-compare” procedure) and inLambert and Pfahler (1992) – see also the com-ment byMilanovic (1994a). Yardsticks for assessing the effectiveness of tax andbenefit policies in reducing initial inequality are proposed inFellman, Jantti, and


Lambert (1999) andFellman (2001).How income is measured is also of importance for the measurement of pro-

gressivity and redistribution. See in particularAltshuler and Schwartz (1996)(for the annualvs a ”time-exposure” incidence of the US child care tax credit),Caspersen and Metcalf (1994) (for the annualvslifetime incidence of value-addedtaxes),Creedy and van de Ven (2001) (for the annualvs lifetime incidence of theAustralian tax and benefit system),Lyon and Schwab (1995) (for the annualvslifetime incidence of taxes on cigarettes and alcohol),Metcalf (1994) (for thelifetime incidence of US state and local taxes),Nelissen (1998) (for the lifetimeincidence of Dutch social security),

Empirical studies of progressivity and redistribution have been very numerousover the last three decades. They includeBishop, Chow, and Formby (1995a)(redistribution in six LIS countries),Borg, Mason, and Shapiro (1991) (regressiv-ity of taxes on casino gambling),Davidson and Duclos (1997) (progressivity inCanada),Decoster and Van Camp (2001) (the redistributive effect of a shift fromdirect to indirect taxation in Belgium),Dilnot, Kay, and Norris (1984) (progres-sivity in the UK between 1948 and 1982),Duclos and Tabi (1999) (redistributionin Canada),Giles and Johnson (1994) (redistribution in the UK),Gravelle (1992)(the redistributive effect of the 1986 US tax reform),Hanratty and Blank (1992)(the comparative poverty effect of redistributive policies in the US and in Canada),Heady, Mitrakos, and Tsakloglou (2001) (the redistributive effect of social trans-fers in the European Union),Hills (1991) (the redistributive effect of British hous-ing subsidies),Howard, Ruggeri, and Van Wart (1994) (the redistributive effect oftaxes in Canada),Khetan and Poddar (1976) (redistribution in Canada),Loomisand Revier (1988) (the redistributive effect of excise taxes),Mercader Prats (1997)(redistribution in Spain, 1980-1994),Milanovic (1995) (the redistributive effect oftransfers in Eastern Europe and in Russia),Morris and Preston (1986) (redistri-bution in the UK),Norregaard (1990) (tax progressivity in the OECD countries),O’higgins and Ruggles (1981) (redistribution in the UK),O’higgins, Schmaus,and Stephenson (1989) (comparative redistribution of taxes and transfers in sevencountries),Persson and Wissen (1984) (the impact of tax evasion on redistribu-tion), Price and Novak (1999) (the regressivity of implicit taxes on lottery games),Ruggeri, Van Wart, and Howard (1994) (the redistributive impact of governmentspending in Canada),Ruggles and O’higgins (1981) (the redistributive impact ofgovernment spending in the US),Schwarz and Gustafsson (1991) (redistributionin Sweden),Smeeding and Coder (1995) (redistribution in 6 LIS countries),vanDoorslaer and et al. (1999) (the redistributive impact of health care financingin 12 OECD countries),Vermaeten, Gillespie, and Vermaeten (1995) (the redis-


tributive impact of taxes in Canada, 1951-1988),Wagstaff and Doorslaer (1997)(the redistributive impact of health care financing in the Netherlands),Wagstaff,van Doorslaer, Hattem, Calonge, Christiansen, Citoni, Gerdtham, Gerfin, Gross,and Hakinnen (1999) (the redistributive impact of personal income taxation in 12OECD countries),Younger, Sahn, Haggblade, and Dorosh (1999) (tax incidencein Madagascar).

Benefit incidence analysis is also regularly carried out in les developed economies– see, for instance,Lanjouw and Ravallion (1999) for the role of differentiated”program capture” in explaining the evolution of the incidence of benefits,Sahn,Younger, and Simler (2000) for a dominance analysis of benefit incidence in Ro-mania,van de Walle (1998a) for a discussion of general issues, andWodon andYitzhaki (2002) for the role of program allocation rules in the study of benefitincidence.

There have been numerous papers decomposing the Gini indices into sums ofcontributions of income sources. These includeAaberge and Al. (2000), Achdut(1996), Cancian and Reed (1998), Gustafsson and Shi (2001), Keeney (2000),Leibbrandt, Woolard, and Woolard (2000), Lerman (1999), Lerman and Yitzhaki(1985), Morduch and Sicular (2002), Podder (1993), Podder and Mukhopad-haya (2001), Podder and Chatterjee (2002), Reed and Cancian (2001), Shorrocks(1982), Silber (1989), Silber (1993), Silber (1989), Sotomayor (1996), Wodon(1999), andYao (1997).

9 HORIZONTAL INEQUITY, RERANKING AND REDISTRIBUTION 123

9 Horizontal inequity, reranking and redistribution

In this section, we examine in more detail a more neglected aspect of thenotion of redistributive equity: horizontal equity (HE ) in taxation (including neg-ative taxation).21 Two main approaches to the measurement ofHE are found inthe literature, which has evolved substantially in the last twenty years. The classi-cal formulation of theHE principle prescribes the equal treatment of individualswho share the same level of welfare before government intervention.HE mayalso be viewed as implying the absence of reranking: for a tax to be horizontallyequitable, the ranking of individuals on the basis of pre-tax welfare should not bealtered by a fiscal system. Most of the analysis below will involveethical indices.We will see that, depending on the choice of the underlying social welfare func-tion or inequality index, horizontal inequity will be captured either by a “classical”horizontal inequity index or by a “reranking” one.

9.1 Ethical and other foundations

Why should concerns for horizontal equity influence the design of an opti-mal tax and transfer system? Several answers have been provided, using either oftwo approaches. The traditional or “classical” approach definesHE as the equaltreatment of equals (seeMusgrave (1959)). While this principle is generally wellaccepted, different rationales are advanced to support it. First, a tax which dis-criminates between comparable individuals is liable to create resentment and asense of insecurity, possibly also leading to social unrest.

Second, the principles of progressivity and income redistribution, which arekey elements of most tax and transfer systems, are generally undermined byHI(as we shall see in our own treatment below). This has indeed been one of themain themes in the development of the reranking approach in the last decades.Hence, a desire forHE may simply derive from a general aversion to inequality,without any further appeal to other normative criteria.HI may moreover suggestthe presence of imperfections in the operation of the tax and transfer system, suchas an imperfect delivery of social welfare benefits, attributable to poor targetingor to incomplete take-up . It can also signal tax evasion, which caninter alia costthe government significant losses of tax revenue.

Third,HE can be argued to be an ethically more robust principle thanVE . VErelates to the reduction of welfare gaps between unequal individuals. Depending

21This section draws heavily on Duclos, Jalbert and Araar, where more details can be found.


on the retained specification of distributive fairness, the requirements of verticaljustice can vary considerably, while the principle of horizontal equity remainsessentially invariant. This has led several authors to advocate thatHE be treatedas a separate principle fromVE , and thus to form one of the objectives betweenwhich an optimal trade-off must be sought in the setting of tax policy.

The theory of relative deprivation also suggests that people often specificallycompare their relative individual fortune with that of others in similar or closecircumstances. The first to formalize the theory of relative deprivation,Davis(1959), expressly allowed for this by suggesting how comparisons with similarvsdissimilar others lead to different kinds of emotional reactions; he used the expres-sion “relative deprivation” for “in-group” comparisons (i.e., for HI ), and “relativesubordination” for “out-group” comparisons (i.e., for VE ) (Davis (1959), p.283).Moreover, in the words ofRunciman (1966), another important contributor to thattheory, “people often choose reference groups closer to their actual circumstancesthan those which might be forced on them if their opportunities were better thanthey are” (p.29).

In a discussion of the post-war British welfare state, Runciman also notes that“the reference groups of the recipients of welfare were virtually bound to remainwithin the broadly delimited area of potential fellow-beneficiaries. It was anoma-lies within this area which were the focus of successive grievances, not the relativeprosperity of people not obviously comparable” (p.71). Finally, in his theory ofsocial comparison processes,Festinger (1954) also argues that “given a range ofpossible persons for comparison, someone close to one’s own ability or opinionwill be chosen for comparison” (p.121). In an income redistribution context, itis thus plausible to assume that comparative reference groups are established onthe basis of similar gross incomes and proximate pre-tax ranks, and that individ-uals subsequently make comparisons of post-tax outcomes across these groups.Individuals would then assess their relative redistributive ill-fortune in referencegroups of comparables by monitoringinter alia how they fare compared to similarothers, and by assessing whether they are overtaken by or overtake these compa-rables in income status, thus providing a plausible “micro-foundation” for the useof HE as a normative criterion.

This suggests that comparisons with close individuals (but not necessarily ex-act equals) would be at least as important in terms of social and psychologicalreactions as comparisons with dissimilar individuals, and thus that analysis ofHIand reranking in that context should be at least as important as considerations ofVE . It also says that, although classicalHI and reranking are both necessary andsufficient signs ofHI , they are (and will be perceived as) different manifestations


of violations of theHE principle.The value of studying classicalHI has nonetheless been questioned by a few

authors, rejecting the premise that the initial distribution is necessarily just orpointing out that utilitarianism and the Pareto principle may justify the unequaltreatment of equals (as discussed above). A number of authors have also expresseddissatisfaction with the classical approach toHE because of the implementationdifficulties it was seen to present. Indeed, since no two individuals are ever exactlyalike in a finite sample, it was argued that analysis of equals had to proceed on thebasis of groupings of unequals which were ultimately arbitrary. The proposedalternative was then to linkHI and reranking and to note that the absence ofrerankingimpliesthe classical requirement ofHE . For instance,Feldstein (1976),p.94, argues that

the tax system should preserve the utility order, implying that if twoindividuals would have the same utility level in the absence of taxa-tion, they should also have the same utility level if there is a tax

Various other ethical justifications have also been suggested for the requirement ofno-reranking. For instance,King (1983) argues in favor of adding (for normativeconsistency) the qualification ”and treating unequals accordingly” to the classicaldefinition ofHE . It then becomes clear that classicalHE alsoimpliesthe absenceof reranking. Indeed, if two unequals are reranked by some redistribution, thenit could be argued at a conceptual level that at a particular point in that processof redistribution, these two unequals became equals and were then made unequal(and reranked), thus violating classicalHE . Hence, from the above, it wouldseem that (quoting again from (King (1983), p. 102) ”a necessary and sufficientcondition for the existence of horizontal inequity is a change in ranking betweenthe ex ante and the ex post distributions”. We thus follow each of the approachesin turn, starting with reranking.

9.2 Measuring reranking and redistribution

We first show how to decompose the net redistributive effect of taxes andtransfers into vertical equity (VE ) and reranking (RR) components. TheVEeffect measures the tendency of a tax system to “compress” the distribution of netincomes, which is linked to the progressivity of the tax system. TheRR termcontributes negatively to the net redistributive effect of the tax system. The use ofLorenz and concentration curves and of the associated S-Gini indices of inequalityand redistribution will enable this integration of reranking and horizontal inequity.


9.2.1 Reranking

Recall the definition of a concentration curve for net income in (112). We canshow thatCN(p) will never be lower than the Lorenz curveLN(p), and will bestrictly greater thanLN(p) for at least one value ofp if there is “reranking” inthe redistribution of incomes22. Intuitively, CN(p) cumulates some net incomeswhose percentiles in the net income distribution exceedp. These are net incomesthat exceedN(p) andQN(p). Such high incomes are nevertheless possible, how-ever, due to the stochastic termν in (107). LN(p) only cumulates the net incomeswhich equalQN(p) or less. Hence,CN(p) ≥ LN(p). This can also be seen bycomparing the estimators in equations (113) and (115). In (115), observations ofNj are cumulated in increasing values ofNj, but in (113), observations ofNj arecumulated in increasing values ofXj, which means that some higher values ofNj

may be cumulated before some lower ones.It is therefore straightforward to conclude that a net taxT will cause reranking

(and hence horizontal inequity) if and only ifCN(p) > LN(p) for at least onevalue of p ∈]0, 1[. The distanceCN(p) − LN(p) can therefore be used as anindicator of reranking23. A natural S-Gini index of reranking is then obtained as aE:19.8.5weighted distance between the two curves:

RR(ρ) =

∫ 1

0

(CN(p)− LN(p)) κ(p; ρ)dp. (134)

DenotingICN(ρ) as the index of concentration of net incomes (recall (117)), thisindex of reranking can also be obtained as

RR(ρ) = IN(ρ)− ICN(ρ). (135)

9.2.2 S-indices of equity and redistribution

As for comparisons of inequality and concentration, it is often useful to sum-marize the progressivity, vertical equity, horizontal inequity as well as the redis-tributive effect of taxes and transfers into summary indices. We can do this byweighting the differences expressed above by the weightsκ(p; ρ) of the S-Gini in-dices to obtain S-Gini indices ofTR-progressivity (IT (ρ)), IR-progressivity andvertical equity (IV (ρ)), reranking (RR(ρ)), and redistribution (IR(ρ)):

22In a continuous distribution, a sufficient condition for reranking is thatν in (107) is not de-generate, namely, that it is not a constant.

23DAD: Curves|Lorenz curve & Curves|Concentration curve


IT (ρ) =

∫ 1

0

(LX(p)− CT (p))κ(p; ρ)dp, (136)

IV (ρ) =

∫ 1

0

(CN(p)− LX(p))κ(p; ρ)dp, (137)

RR(ρ) =

∫ 1

0

(CN(p)− LN(p))κ(p; ρ)dp, (138)

IR(ρ) =

∫ 1

0

(LN(p)− LX(p))κ(p; ρ)dp. (139)

These indices can also be computed as differences between S-Gini indices of in-equality and concentration:

IT (ρ) = ICT (ρ)− IX(ρ) (140)

IV (ρ) = IX(ρ)− ICN(ρ) (141)

RR(ρ) = IN(ρ)− ICN(ρ) (142)

IR(ρ) = IX(ρ)− IN(ρ). (143)

Many of these indices have first been proposed withρ = 2, which correspondsto the case of the standard Gini index.IT (ρ = 2) is known as the Kakwani indexof TR progressivity24, IV (ρ = 2) is known as the Reynolds-Smolensky index ofE:19.8.4IR progressivity and vertical equity, andRR(ρ = 2) is known as the Atkinson-Plotnick index of reranking.

9.2.3 Redistribution and vertical and horizontal equity

The difference between the Lorenz curve of net and gross incomes is given by:

LN(p)− LX(p) = CN(p)− LX(p)︸︷︷︸VE

− (CN(p)− LN(p))︸︷︷︸RR

. (144)

Alternatively, the net redistribution can be expressed in terms of S-indicesCurves|Lorenzcurve25: E:19.8.6

IX(ρ)− IN(ρ) = IX(ρ)− ICN(ρ)︸︷︷︸VE

− (IN(ρ)− ICN(ρ))︸︷︷︸RR

. (145)

24DAD: Inequality|S-Gini Index & Redistribution|Concentration index25DAD: Inequality|S-Gini Index & Redistribution!Concentration Index


The first termVE in each of the above two expressions is clearly linked to thedefinition of IR-progressivity in equation (132). As shown in equation (116), itcan also be expressed in terms ofTR-progressivity whent 6= 0:

CN(p)− LX(p) =t

1− t[LX(p)− CT (p)] , (146)

and, using S-indices,

IX(ρ)− ICN(ρ) =t

1− t[ICT (ρ)− IX(ρ)] . (147)

Furthermore, if there is more than one tax and/or benefit that make upT , wecan decompose totalVE as a sum of theIR andTR progressivity of each tax andtransfer. Lett(j) be the overall average rate of the taxT(j), with j = 1, ..., J , suchthat

∑Jj=1 t(j) = t, and letCT(j)

(p) andCN(j)(p) be the concentration curves of

net income and taxes corresponding to taxT(j), with N(j) = X − T(j) .Then, wehave

CN(p)− LX(p) =J∑

j=1

(1− t(j))(CN(j)(p)− LX(p))

(1− t)(148)

and

IX(ρ)− ICN(ρ) =J∑

j=1

(1− t(j))

(1− t)(IX(ρ)− ICN(j)

(ρ)). (149)

CN(j)(p)−LX(p) andIX(j)

(ρ)−ICN(ρ) capture the vertical equity of tax or trans-fer j at percentilep, and again can be easily seen to be an element of the definitionof IR-progressivity. Each of theseVE contributions can also be expressed as afunction ofTR progressivity atp (whent(j) 6= 0):

CN(j)(p)− LX(p) =

t(j)1− t(j)

[LX(p)− CT(j)

(p)]. (150)

and

IX(ρ)− ICN(j)(ρ) =

t(j)1− t(j)

[ICT(j)

(ρ)− IX(ρ)]. (151)

The second term on the right-hand side of (144) and (145) is the redistribution-reducing reranking effect. As is well known from the literature on reranking (seeAtkinson, 1979, and Plotnick, 1981, for instance), taking into account rerankingwhen using rank-dependent inequality indices increases measured inequality and


decreases the redistributive effect of taxation, and this explains whyIN gener-ally exceedsIN , and also why the difference can be interpreted as the impact ofreranking on the net redistributive effect of taxation.

To interpret that second term, we may also think of individuals resenting beingoutranked by others, but enjoying outranking others, and then assessing their netfeeling of resentment by the amount by which the net income of the richer (thanthemselves) actually exceeds what the net income of the richer class would havebeen had no ”new rich” displaced ”old rich” in the distribution of net incomes. Wecan then show thatµN (IN(ε)− ICN(ε)) is the expected net income resentment ofthe poorest person in samples ofv−1 randomly selected individuals, and thus thatRR is an ethically-weighted indicator of such net resentment in the population.

9.3 Measuring classical horizontal inequity and redistribution

We now turn to the measurement of classical horizontal equity.

9.3.1 Horizontally-equitable net incomes

One natural avenue for measuring whether equals are treated equally is to es-timate the variability of taxes and net incomes conditional on some initial value ofgross income. We may, for instance, wish to estimate the conditional variabilityof T at a given value ofX. Alternatively, and perhaps better for expositional pur-poses, we may want to show that conditional variability over a range of percentilesp of gross incomeX, and thus estimate for example the conditional variance ofTat gross incomeX(p), σ2

T (p)26: E:19.8.7

σ2T (p) =

∫ 1

0

(T (q|p)− T (p)

)2dq. (152)

Recent work has, however, attempted to make the measurement of classicalHI flow from ethical (as opposed to descriptive or statistical) foundations. Weshow how this can be done using the popular Atkinson social welfare functionW (ε) introduced in(47). For the distribution of net incomes, this social welfarefunction equals:

WN(ε) =

∫ 1

0

U (QN(p); ε) dp. (153)

26DAD: Distribution|Conditional Standard deviation


Recall that the expected net income of those at rankp in the distribution of grossincome is given byN(p). Hence, if the tax system were horizontally equitableand if all individuals at rankp in the distribution of gross income were all grantedN(p) in net incomes, the local level of utility would beU

(N(p); ε

)and net-

income social welfare would equal

WN(ε) =

∫ 1

0

U(N(p); ε

)dp. (154)

The expected net income utility of those at rankp in the distribution of grossincome is, however, equal to

U(p; ε) =

∫ 1

0

U (QN(q|p); ε) dq. (155)

If, instead ofU(N(p); ε), we assigned individuals at rankp their expected netincome utilityU(p; ε), social welfare would equal

WU(ε) =

∫ 1

0

U(p; ε)dp. (156)

WN(ε) is social welfare usingex anteexpected net income;WU(ε) is social wel-fare usingex anteexpected net income utility. By the concavity of the utilityfunction, we have thatU(N(p); ε) − U(p; ε) ≥ 0, and this difference capturesthe local utility cost of net income uncertainty atp. Hence, we also have thatWN(ε) = WU(ε) ≤ WN(ε), a feature which we are about to use to capture theglobal social welfare cost ofHI and its impact on redistribution.

To show the social welfare cost ofHI and its impact on redistribution, we canfollow either of two approaches. Recall that we have just providedtwo locallyhorizontally-equitable tax systems:

• one in which each individual at rankp in the distribution of gross incomesreceivesN(p) and utilityU(N(p); ε),

• and one in which each of these individuals receivesU(p; ε).

In the first case,WN(ε) ≤ WN(ε) but mean income is the same under all thetwo distributionsN(p) andN(p) sinceµN =

∫ 1

0

∫ 1

0N(q|p)dq dp =

∫ 1

0N(p)dp ≡

µN . Hence, a consequence ofHI is to increase inequality and to decrease theredistribution fall in inequality brought about by tax and benefit systems. This isfurther developed in Section9.3.2.


The second case imposes a horizontally-equitable local distribution of utilityU(p) that equals theex anteexpected local utility. Compared to the actual dis-tribution of net incomes, this reduces inequality but maintain the overall level ofsocial welfare. Hence, it must be that average income underU(p) is lower thanunderN(p). It also implies that the cost of inequality is lower with withU(p).This is further developed in Section9.3.3.

9.3.2 Change in inequality approach

Let the equally distributed equivalent (EDE) incomes forWN(ε), WN(ε) andWU(ε) beξN(ε), ξN(ε) andξU(ε), respectively. As before, inequality can be mea-sured by the differences between thoseξ and the correspondingµ, as a proportionof µ. Now observe that

IN(ε) = 1− ξN(ε)

µN

≤ 1− ξN(ε)

µN

= IN(ε) (157)

sinceµN = µN and WN(ε) ≥ WU(ε). Hence,HI increases inequality. Theoverall redistributive change in inequality that results from the effect of taxes andtransfers can then be expressed as

∆I(ε) = IX(ε)− IN(ε). (158)

Note also that, by (45), (158) is equivalent toξN(ε)− ξX(ε) when the means ofXandN are the same.

Hence, using(158) we obtain the following decomposition of the net redis-tributive change in inequality:

∆I(ε) = IX(ε)− IN(ε)︸︷︷︸VE

− (IN(ε)− IN(ε))︸︷︷︸HI≥0

. (159)

VE represents the decrease in inequality yielded by a tax which treats equalsequally. Thus,VE can be interpreted as a measure of the underlying verticalequity of horizontally-equitable net taxesX(p) − N(p). HI measures the fall inredistribution attributable to the unequal post-tax treatment of pre-tax equals. Theexcess ofIN(ε) overIN(ε) is due to the appearance of post-tax income inequalitywithin groups of pre-tax equals.


9.3.3 Cost of inequality approach

In the above change-in-inequality approach, average income is kept the samewhile comparing distributions of actual and horizontally equitable net incomes.Social welfare and inequality do, however, vary across the distributions ofN(p)andN(p). In the second approach, the cost-of-inequality approach, social welfareis kept the same across the distributions being compared but the mean incomerequired to attain this level of welfare varies. Each element of the decompositionin this section thus corresponds to a difference in means at equal social welfareWN(ε).

The cost of inequality in the distribution of net income can be expressed as:

CN(ε) = µN − ξN(ε) = µNIN(ε). (160)

Recall thatCN(ε) represents the level ofper capitanet income that society coulduse for the elimination of inequality with no loss of social welfare.

Let CF (ε) represent the cost of inequality subsequent to a flat (or proportional,and thus inequality neutral) tax on gross incomes that generates the same level ofsocial welfare as the distribution of net incomes. Denote the average income underthis welfare-neutral flat tax byµF . The net effect of redistribution on the cost ofinequality then becomes:

∆C(ε) = CF (ε)− CN(ε). (161)

SinceξN(ε) = µN − CN(ε) = µF − CF (ε) and sinceξN(ε) = µN(1 − IN(ε)) =µF (1− IX(ε)), we also have

∆C(ε) =(IX(ε)− IN(ε))

(1− IX(ε))µN (162)

which is positive ifIX(ε) > IN(ε). The more progressive the net tax system, thegreaterthe value of∆C. If the net tax system is progressive, the greater the valueof ε, the greater the redistributive fall in the cost of inequality.

We then write the decomposition of the total variation in the cost of inequalityas:

∆C(ε) = CF (ε)− CU(ε)︸︷︷︸V E

− (CN(ε)− CU(ε))︸︷︷︸HI≥0

. (163)

The redistributive fall in the cost of inequality then decomposes into two effects.First, CU is the cost of inequality under a (horizontally-equitable) certainty-

equivalent level of net income at all ranksp. This certainty-equivalent net income


is given byξU(p; ε) = U (−1)(U(p; ε)

). Hence, for constant social welfare, an

horizontally-equitable tax system corresponds to a distribution ofξU (p; ε) to eachindividual at pre-tax percentilep. Since social welfare is kept constant, it followsthat the measure of classical horizontal inequity in (163) equals

CN(ε)− CU(ε) = µN −∫ 1

0

γξU(p; ε) dp. (164)

CN(ε)−CU(ε) is then clearly a money-metric measure of the welfare cost causedby classical HI.

Second,CF (ε)−CU(ε) in (163) measures the difference in the cost of inequal-ity of two horizontally equitable tax systems, the first being a flat tax system, andthe second granting everyone his certainty equivalent level of net income, withboth systems yielding the same level of social welfareWN . CF (ε) − CU(ε) ispositive if the tax system is progressive in anex ante, certainty-equivalent, sense.In such a case, the distribution across percentiles of the certainty-equivalent netincomes is less inequality costly than the distribution of gross incomes.

9.4 Decomposition of classical HI

We may also wish to know at which percentile or for which population groupHI is more pronounced, and by how much it contributes to total classicalHI . Forthis, define the local cost of classical violations ofHE atp as:

N (p)− ξN (p; ε) . (165)

This is the ”risk-premium” of net income uncertainty at percentilep, and it isthus a money-metric cost of local classical HI atp. It is then possible to showthat aggregating (165) using population weights yields the global index of totalclassicalHI in (163):

CN(ε)− CU(ε) =

∫ 1

0

N (p)− ξN (p; ε) dp. (166)

9.5 References

The literature on horizontal inequity has evolved very significantly over thelast 25 years. Recent literature surveys can be found inJenkins and Lambert(1999), Lambert and Ramos (1997a) and Lambert (2001) (see also the com-ment byPlotnick (1999) and the earlier reviews ofMusgrave (1990) andPlotnick


(1985)). See also?), Feldstein (1976), ?), ?) and?) for a treatment of horizontalequity as a separate principle from vertical equity, andKaplow (1989), Kaplow(1995) andKaplow (2000) for a critique of the principle horizontal inequity.

The early reranking approach was much influenced byAtkinson (1979), Plot-nick (1981) andPlotnick (1982) (for the RR(2) index), andKing (1983) (for anormative link between inequality, mobility and reranking). See alsoChakravarty(1985) for normative links between inequality and reranking,Dardanoni and Lam-bert (2001) for a statistically-based look at the association between gross and netincomes,Duclos (1993) for the general form of theIR(ρ) indices,Jenkins (1988a)for a ”within-group” horizontal equity focus,Kakwani and Lambert (1999) for aHI-related analysis of tax discrimination,Kakwani and Lambert (1998) for an ax-iomatic construction of equity measures,Rosen (1978) for a (rare) utility-basedevaluation of horizontal inequity, andLerman and Yitzhaki (1995) for reasons forwhich reranking maydecreaseinequality.

Classical horizontal equity has seen extensive developments particularly in thelast 10 years: see, for instance,Aronson, Johnson, and Lambert (1994), Aronsonand Lambert (1994), Aronson, Lambert, and Trippeer (1999) and van de Ven,Creedy, and Lambert (2001), for the use of the Gini for calculating both rerank-ing and classical horizontal inequity;Duclos and Lambert (2000a), for a cost-of-inequality approach; andAuerbach (2002) andLambert and Ramos (1997b), fora change-in-inequality approach.

Empirical enquiries into the extent of horizontal inequity have also been rela-tively numerous. They includeinter aliaAnkrom (1993) for comparative Swedish,British and American evidence,Berliant and Strauss (1985) for the US federal in-come tax system,Bishop, Formby, and Lambert (2000) for the effects of noncom-pliance and tax evasion,Creedy (2001) andCreedy (2002) for the impact of non-uniform indirect taxes on horizontal inequity in Australia,Creedy and van de Ven(2001) for the impact on measured horizontal inequity of using different equiv-alence scales and of using annualvs lifetime income,Decoster, Schokkaert, andVan Camp (1997) for indirect taxation and horizontal inequity in Belgium,Duc-los (1995c) for the role of imperfections in poverty alleviation programs,Jenk-ins (1988b) andNolan (1987) for the extent of reranking in the UK,Sa Aadu,Shilling, and Sirmans (1991) for whether the treatment of capital gains on owner-occupied housing matters for horizontal inequity, andStranahan and Borg (1998)for whether an implicit ”lottery tax” is a source of horizontal inequity.

The advances in the measurement of horizontal inequity have also led to adesire to decompose the overall measurement of redistribution as a function ofprogressivity, vertical equity, reranking and classical horizontal inequity. This is


doneinter alia in Duclos (1993) (with the S-Gini),Duclos (1995c) (with redis-tributive imperfections),Kakwani (1984) andKakwani (1986) by using the Giniindex but not attempting to measure classical horizontal inequity; and inAronson,Johnson, and Lambert (1994), Aronson and Lambert (1994), van Doorslaer andet al. (1999) (for health financing in 12 OECD countries),Wagstaff and Doorslaer(1997) (for health financing in the Netherlands),Wagstaff, van Doorslaer, Hattem,Calonge, Christiansen, Citoni, Gerdtham, Gerfin, Gross, and Hakinnen (1999)(for personal income taxes in 12 OECD countries), all using the Gini index andincorporating both reranking and classical horizontal inequity. See alsoWagstaffand van Doorslaer (2001) for a decomposition of total tax progressivity in compo-nents such as the progressivity of tax credits, marginal tax rates, allowances anddeductions.

136

Part III

Ordinal comparisons of poverty andequity

10 DISTRIBUTIVE DOMINANCE 137

10 Distributive Dominance

10.1 Ordering distributions

We have, up to now, focussed mostly on measuring and comparing cardinalindices of poverty and equity. As discussed in Section4.4, this has several expo-sitional advantages. The greatest of these advantages is probably that of focussingon only one (or a few) numerical assessments of poverty and equity. It is then rel-atively straightforward to compare poverty and equity across distributions just bycomparing the values of these cardinal indices. The conclusions are then (seem-ingly) ”clear-cut”.

There are, however, important reasons to consider instead ordinal comparisonsof poverty and equity. The most important one is that comparisons of cardinalpoverty and equity indices (comparisons across time, regions, socio-demographicgroups or fiscal regimes, for instance) may be disturbingly sensitive to the choiceof indices and poverty lines. For instance, we can find for some poverty lines andindices that poverty is greater in a regionA than in a regionB, but we can alsofind the opposite for other lines and indices. We could support the introductionof a particular fiscal policy or macroeconomic adjustment programme for somesocial welfare indices, but could be in doubt as to whether the same support wouldbe warranted with other indices. Since there is rarely unanimity as to the rightchoice of poverty lines and distributive indices, it is clear that such sensitivity canseriously undermine one’s confidence in comparing distributions of making policyrecommendations.

10.2 Sensitivity of poverty comparisons

To see this better in the context of poverty comparisons, consider the hypo-thetical example of Table10. The second, third and fourth lines in the table showthe incomes of three individuals in two hypothetical distributions,A andB. Thus,distributionA contains three standards of living of 4, 11 and 20 respectively. Thebottom 3 lines of the table show the value of the two most popular indices ofpoverty, the headcount and the average poverty gap indices, at two alternativepoverty lines, 5 and 10. Recall from Section6.1.2 that the poverty headcountgives the proportion of individuals in a population whose standard of living fallsunderneath a poverty line. At a poverty line of 5, there is only one such personin poverty in distributionA, and the headcount is thus equal to 0.33. The aver-age poverty gap index is the sum of the distances of the poor’s standards of living


from the poverty line, divided by the total number of people in the population. Forinstance, at a poverty line of 10, there are 2 people in poverty inB, and the sum oftheir distances from the poverty line is (10-6)+(10-9)=5. Divided by 3, this gives1.66 as the average poverty gap inB for a poverty line of 10.

The last column of Table10 gives the poverty ranking of the two distribu-tions according to the different choices of poverty lines and poverty indices. At apoverty line of 5, the headcount inA is clearly greater than inB , but this rank-ing is spectacularly reversed if we consider instead the same headcount but at apoverty line of 10. The ranking changes again if we use the same poverty line of10 but now focus on the average poverty gapµg(z): µg

A(10) = 2 < 1.66 = µgB(z).

Clearly, the poverty rankingA andB can be quite sensitive to the precise choiceof measurement assumptions.

10.3 Ordinal comparisons

The alternative to comparing the value of one or a few cardinal indices is tocheck whether distributive orderings of poverty and equity are valid for a class ofethical judgments. These classes are defined over classes of indices, as well asover ranges of poverty lines (for poverty comparisons). In other words, we do notwish to quantify poverty or equity. We only try to determine whether poverty andequity is higher or lower in one distribution than in another, for a class of ethicaljudgments. When inferred, an ordinal ranking of poverty and equity across twodistributions establishes the sign of the differences across these two distributionsof anyone of the cardinal poverty and equity indices of that class. Note that it cansay only whether poverty and equity is higher in one distribution than in another,but not by how much.

Ordinal comparisons of poverty do not, therefore, provide precise numericalvalues to compare with numerical indicators of other aspects or effects of govern-ment policy, such as its administrative or efficiency cost. This is seemingly theirmain defect. It is arguably also their greatest advantage. As seen above in thecontext of Table10, simple poverty differences can be deceptive when it comesto ranking distributions. They can also quantify deceptively differences acrossdistributions. To illustrate this, consider Table12 with distributionsA andB anda poverty linez = 1. The three FGT poverty indices agree that poverty has notincreased in moving fromA to B. But the quantitative change in poverty variessignificantly with the value ofα. With the poverty headcount, poverty remains thesame, but the average poverty falls by 33% and theP (z; α = 2) falls by 56%.

In short, ordinal poverty comparisons can be highly robust to the choice of


measurement assumptions, since they will sometimes be valid for wide classesand ranges of such assumptions. When the problem is simply of resolving whichof two policies will better alleviate poverty, or determining which of two distri-butions displays the greatest level of social welfare, or assessing which of twodistributions is the most unequal, ordinal comparisons can be sufficiently infor-mative, and cardinal estimates will then not be needed.

A focus on ordinal comparisons can save most of the considerable energy andtime often spent on choosing poverty lines and poverty indices. It can avoidinteralia the difficult debate on the choice of appropriate theoretical and econometricmodels for estimating poverty lines. It can also escape arguments on the relativemerits and properties of the many distributive indices that have been proposed inthe scientific literature, and of which the previous chapters introduced only a few.Again, this is because ordinal distributive comparisons simply order distributions,and for this, differences in numerical indices do not need to be estimated. Forinstance, we will see later in Section11.1that we can order robustly distributionsA andB in Table10 for all ”distribution-sensitive” poverty indices and for anychoice of poverty line. If such ordering is considered sufficiently strong and infor-mative, then, in comparingA andB, we can effectively stop quibbling on whetherwe should use the Watts index or the average poverty gap as a poverty index, andon whether the poverty line should be 5 or 10.

As we will see in detail below, ordinal comparisons of poverty and equityinvolve using classes of distributive indices. It is useful to define these classes byreferring to ”orders of normative (or ethical) judgements”, an order being denotedass = 0, 1, 2, .... An ethical judgement of orders thus serves to define a class ofindices also of orders. Whether an ordering of poverty and equity is valid for allof the indices that are members of a class of orders is empirically tested throughdominance tests, which happen to be convenient variants of well-known stochasticdominance tests also of orders. When two dominance curves of a given order donot intersect, all indices that obey the ethical principles associated to this order ofdominance then rank identically the two distributions. Hence, a dominance testof orders serves to test whether some distributive ranking is valid for all of theindices of a class of orders, and that class of orders can be interpreted throughthe use of ethical judgements of the same orders.


10.4 Ethical judgements

10.4.1 Paretian judgments

A first natural property of normative judgements is that a society should bejudged improved whenever the income of one of its members increases and noone else’s income decreases. For poverty, this would mean that indices of povertyshould (weakly) fall whenever someone’s income increases, everything else beingthe same. (”Weakly fall” means that the index should at the very least not increasefollowing the change, and conversely for ”weakly increase”. Thiscaveatappliesto all of the ethical statements considered in this chapter.) For social welfare com-parisons, this would imply that social welfare indices should increase followingthis improvement in someone’s income. Because the ethical condition imposedfor membership in that class is very weak, we can consider that class to be ofethical order 0. Note that a focus on relative poverty might seem to provide an ex-ception to this, since an increase in someone’s income could increase the relativepoverty line and possibly also increase the poverty index. To deal with this possi-ble exception, it is best to think of the poverty line as constant in the discussion ofthe ethical principles.

All of the indices which obey the above ethical condition then belong to(poverty or social welfare) classes of orders = 0. Such indices are in effect”Paretian”: the only property that they must possess is that of responding favor-ably to Pareto-improving changes in the distribution of income. Such indices thisobey theParetoprinciple . It has, however, long been recognized that searchesfor Pareto improvements in distributions of incomes are generally doomed to fail-ure, because of inherent randomness in economic status and because of variabilityin preferences, endowments and markets. For a distributive change to be Paretoimproving, it must indeed not decrease anyone’s income, whatever one’s pecu-liar circumstances. This is unlikely to be possible, even if we were to focus onlyon those with incomes below some poverty line. Besides, checking for Pareto-improving temporal changes would always require the use of panel data in orderto observe individual-specific changes in incomes. Even with these data, it wouldnot be possible to infer Pareto-improvement in the population as opposed to sim-ply within the available sample.

10.4.2 First-order judgments

It is thus natural to consider ethical principles of order higher than that of thePareto principle. In the light of the above, a plausible higher-order ethical judge-


ment would require that the distributive indices beanonymousin the incomes ofthe individuals. That is,ceteris paribus, whether it is an individual nameda or bthat enjoys some level of income should not affect the value of a distributive index.It also follows from this property that interchanging two income levels should notaffect distributive indices: these indices obey thesymmetryor anonymityprin-ciple. Clearly, this principle would not be acceptable for an index of horizontalequity, but it would seem relatively uncontroversial for comparing inequality, so-cial welfare or poverty across distributions.

First-order classes of distributive indices then regroup all indices that showa social improvement when the income at some percentile in the population in-creases. These indices have properties that are analogous to those of Paretianindices:ceteris paribus, the larger the individual incomes, the better off is society.They are in addition symmetric in income since they obey the anonymity princi-ple. Indices of poverty and welfare that obey this principle are therefore such thatpoverty decreases and social welfare increases when incomes at some percentileincrease.

10.4.3 Higher-order judgments

Even with this higher-order ethical constraint, it is likely that some of thefirst-order distributive indices will clash in their distributive ranking. Some of thefirst-order poverty indices could declare a policy reform to worsen poverty, whileothers might indicate that the reform improves poverty. To resolve this ambiguity,we may move to a second-order class of distributive indices. As before, this isdone by constraining such indices to obey additional ethical principles.

To do this, assume that distributive indices must show a social improvementwhen a mean-preserving redistributive transfer from a richer to a poorer individualoccurs. This corresponds to imposing the well-knownPigou-Daltonprinciple onthem. The second-order classes of distributive indices thus contain those first-order indices that have a greater ethical preference for the poorer than for thericher. They display a preference for equality of income and are therefore said tobedistribution-sensitive. For instance, all other things the same, the more equalthe distribution of income among the poor, the lower the level of poverty.Ceterisparibus, if a transfer from a richer to a poorer person takes place, all second-ordersocial welfare indices will increase and all second-order inequality and povertyindices will fall. Note again that all indices that belong to a second-order class ofpoverty and welfare indices also belong to the relevant first-order class.

There are often sound ethical reasons to be socially more sensitive to what


happens towards the bottom of the distribution of income than higher up in it.We may thus be less concerned about a ”bad” disequalizing transfer higher up inthe distribution of income than we are pleased about a ”good” equalizing transferlower down. To make this more precise, imagine four levels of income, for indi-viduals 1, 2, 3, and 4, such thaty2−y1 = y4−y3 > 0 andy1 < y3 . Let a marginaltransfer of $1 of income be made from individual 2 to individual 1 (an equalizingtransfer) at the same time as an identical $1 is transferred from individual 3 toindividual 4 (a disequalizing transfer). This is called in the literature a ”favorablecomposite transfer”.

Note that the equalizing transfer is made lower down in the distribution ofincome than the disequalizing transfer. This can be seen by the fact the recipientof the first transfer, individual 1, has a lower standard of living than the donor ofthe second transfer, individual 3, sincey3 > y1. For a given distance betweenrecipients and donors, the social improvement effect of equalizing transfers isdecreasing in the income of the recipient. Said differently, Pigou-Dalton transferslose their social improvement effects as recipients become more affluent.

Second-order indices which respond favorably to such a ”favorable compositetransfer” obey thetransfer-sensitivityprinciple and therefore belong to the third-order class of indices. Again, such a favorable composite transfer is made of abeneficial Pigou-Dalton transfer within the lower part of the distribution, coupledwith a reverse Pigou-Dalton transfer within the upper part of the distribution, withno change in the variance of the distribution. Third-order welfare indices willincrease following this change, and third-order poverty and inequality indices willfall.

We can, if we wish, define subsequent classes of indices in an analogous man-ner. To define fourth-order indices, for instance, we consider a combination of twoexactly opposite and symmetric composite transfers, the first one being favorableand occurring within the lower part of the distribution, and the second one beingunfavorable and occurring within the higher part of the distribution. The third-order indices that respond favorably to this combination of composite transfersare also fourth-order indices.

As can be seen, higher-order transfer principles essentially postulate that, asthe order increases, the ethical weight assigned to the effect of income changesoccurring at the bottom of the distribution also increases. Thus, as the orders ofthe class of poverty indices increases, the indices become more and more sensi-tive to the distribution of income among the poorest. At the limit, ass becomesvery large, only the income of the poorest individual matters in comparing povertyacross two distributions. In that sense, these indices become more and more Rawl-


sian.

10.5 References

Much of normative welfare economics has been influenced by the philosoph-ical work of Nozick (1974), Rawls (1971) (seeRawls (1974) for a very shortsynthesis addressed to economists) andSen (1982). The combined work ofKolm(1970) andKolm (1976) was the first to introduce the transfer-sensitivity condi-tion into the inequality literature, andKakwani (1980) subsequently adapted it topoverty measurement. See alsoDavies and Hoy (1994) (who describe that con-dition as a Rawlsian extension of the Lorenz criterion),Shorrocks (1987) for acomplete characterization of this transfer principle, andZheng (1997) for an in-formative discussion of it. Higher-order principles can be interpreted using thegeneralized transfer principles ofFishburn and Willig (1984) – see alsoBlacko-rby and Donaldson (1978) for a description of these principles as becoming ”moreRawlsian”. Surveys of the normative and axiomatic foundations of modern in-equality measurement can be found inBlackorby, Bossert, and Donaldson (1999)andChakravarty (1999).

Other papers which explore variations to the normative principles typicallyused in distributive analysis areMosler and Muliere (1996) (for an alternativeprinciple of transfers),Ok (1995) (for a ”fuzzy” measurement of inequality),Ok(1997) (for ranking over opportunity sets),Salas (1998) (for marginal populationinvariance),Shorrocks (1987) (for ”transfer sensitivity”),Zoli (1999) (for a posi-tional transfer principle when Lorenz curves intersect), andTam and Zhang (1996)(for an alternative Pareto principle defined in terms of growth over the poor).

Experimental evidence on the normative attitudes of individuals and societiestowards the measurement of inequality and poverty has also grown fast in thelast decades. Methods and results can be found inAmiel and Cowell (1992) (onattitudes to inequality – which question the acceptability of transfer and decom-posability principles),Amiel and Cowell (1999) (on attitudes to poverty, socialwelfare and inequality),Amiel and Cowell (1997) (on attitudes towards povertymeasurement), and inAmiel, Creedy, and Hurn (1999) (on quantifying inequal-ity aversion usingOkun (1975)’s ”leaky bucket experiment”). A survey of suchattitudes can be found inCorneo and Gruner (2002).

Fong (2001) tests whether such attitudes can be explained by self-interest orby values about distributive justice.Dolan and Robinson (2001) further explorewhether there is a ”reference point” problem in such studies, andRavallion andLokshin (2002) reports that expectations about future levels of well-being can

11 POVERTY DOMINANCE 144

influence individuals’ desire for redistributive policies.See alsoStodder (1991) for empirical evidence as to why inequality aversion

can matter for ranking distributions, andChristiansen and Jansen (1978) for anexample of the estimation of social preferences using the revealed structure of anexisting tax system (the Norwegian one).

A number of studies have recently also attempted to distinguish between atti-tudes towards inequality and towards risk aversion: seeinter alia Amiel, Cowell,and Polovin (2001), Beck (1994), Cowell and Schokkaert (2001), andKroll andDavidovitz (2003).

11 Poverty dominance

To see what the material of Chapter10 means analytically for poverty domi-nance, we focus for simplicity on classes of additive poverty indices denoted asΠs(z). The additive poverty indicesP (z) that are members of that class can thenbe expressed as

P (z) =

∫ 1

0

π(Q(p); z) dp (167)

wherez is a poverty line. For expositional simplicity, assume thatπ(Q(p); z)is continuously differentiable inQ(p) between 0 andz up to the appropriateorder. Denote theith-order derivative ofπ(Q(p); z) with respect toQ(p) asπ(i)(Q(p); z). We can think of the functionπ(Q(p); z) as the contribution of anindividual with incomeQ(p) to overall povertyP (z). Hence, we can also assumethat π(Q(p); z) = 0 if Q(p) > z. This ensures that the poverty indices fulfillthe well-knownpoverty focusprinciple, which simply states that changes in theincomes of the rich should not affect the poverty measure. We will also denote byz+ the upper bound of the range of poverty lines that can be reasonably envisaged.

The first class of poverty indices (denoted byΠ1(z+)) then regroups all ofthe poverty indices that decrease when the income of someone in the populationincreases and whose poverty line does not exceedz+. Formally, indices withinΠ1(z+) are such that:

Π1(z+) =

P (z)

∣∣∣∣π(1)(Q(p); z) ≤ 0 whenQ(p) ≤ zz ≤ z+,

(168)

where the first-order ethical judgment appears through the form of a non-positivefirst-order derivativeπ(1)(Q(p); z).


The second class of poverty indices,Π2(z), contains those first-order indicesthat have a greater ethical preference for the poorer among the poor. Increasingthe income of a poorer individual is better for poverty reduction that increasingby the same amount the income of a richer person. The absolute value of the first-order derivative is therefore decreasing withQ(p), and the indices are thus convexin income. This classΠ2(z+) is then:

Π2(z+) =

P (z)

∣∣∣∣∣∣

P (z) ∈ Π1(z+),π(2)(Q(p); z) ≥ 0 whenQ(p) ≤ z,π(z; z) = 0,

(169)

We will discuss further below the role of the continuity conditionπ(z, z) = 0.Clearly,Π2(z+) ⊂ Π1(z+), but not necessarily the reverse.

Technically, obeying the ”transfer-sensitivity” principle requires for theP (z)indices that the second derivativeπ(2)(Q(p); z) be decreasing inQ(p). Poverty in-dices belonging to the third-order class of poverty indicesΠ3(z+) are then definedas:

Π3(z+) =

P (z)

∣∣∣∣∣∣

P (z) ∈ Π2(z+),π(3)(Q(p); z) ≤ 0 whenQ(p) ≤ z,π(z, z) = 0andπ(1)(z, z) = 0

. (170)

As before,Π3(z+) ⊂ Π2(z+).Subsequent classes of poverty indices are defined in an analogous manner.

Generally speaking, poverty indicesP (z) will be members of classΠs(z+) if(−1)s π(s)(Q(p); z) ≤ 0 and ifπ(i)(z, z) = 0 for i = 0, 1, 2..., s− 2. As the orders of the class of poverty indices increases, the indices become more and moresensitive to the distribution of income among the poorest. At the limit, and asmentioned above, only the income of the poorest individual matters in comparingpoverty across two distributions. Increasing the orders makes us focus on smallersubsets of poverty indices, in the sense thatΠs(z) ⊂ Πs−1(z).

All poverty indices seen in Chapter6 fit into some of the classes defined above.The poverty headcount clearly belongs toΠ1(z+) (wheneverz ≤ z+). As wewill see, it also plays a crucial role in tests of first-order dominance. But it doesnot belong to the higher-order classes since it is not continuous at the povertyline. The average poverty gap belongs toΠ1(z+) and toΠ2(z+), but not to thehigher-order classes. The square of the poverty gaps index belongs toΠ1(z+),Π2(z+) andΠ3(z+), but not toΠ4(z+). More generally, the FGT indices, forwhich π(Q(p); z) = g(p; z)α, belong toΠs(z+) whenα ≥ s − 1 andz ≤ z+.


The Watts index belongs toΠ1(z+) and toΠ2(z+), but not toΠ3(z+) since itdoes not obey theπ(1)(z, z) = 0 restriction. A transformation of the Watts index,by which π(Q(p); z) = g(p; z) [ln(z)− ln (Q∗(p))], would, however, belong toΠ3(z+). The Chakravarty and Clarket al. indices belong toΠ1(z+) andΠ2(z+),and so do as well the S-Gini indices of poverty.

We can now ask how to determine poverty inA is greater than inB for allindices that are members of any one of these classes. For this, there exist twoapproaches: a primal and a dual one. We consider them in turn.

11.1 Primal approach

11.1.1 Dominance tests

We are interested in whether we may assert confidently that poverty in a distri-butionA, as measured byPA(z), is larger than poverty in a distributionB, PB(z),for all of the poverty indicesP (z) belonging to one of the classes of povertyindices defined above. We are therefore interested in checking whether the fol-lowing difference in poverty indices∆P (z) = PA(z)− PB(z) is positive:

∆P(z) =∫ 1

0π(QA(p); z)− π(QB(p); z)dp

=∫ z

0π(y; z)∆f(y)dy,

(171)

where on the second line a change of variable has been effected and where∆f(y)is the difference in the densities of income. To demonstrate the dominance con-ditions, we will make repetitive use of integration by parts of (171). This processwill involve the use of stochastic dominance curvesDs(z), for orders of domi-nances = 1, 2, 3, .... D1(z) is simply thecdf, F (z), namely, the proportion ofindividuals underneath the poverty linez. The higher order curves are iterativelydefined as

Ds(z) =

∫ z

0

Ds−1(y)dy, (172)

Thus,D2(z) is simply the area underneath thecdf curve for a range of incomesbetween 0 andz. This is illustrated in Figure39. The curve shows thecdf F (y)at different values ofy. The grey-shaded area underneath that curve (up toz) thusgivesD2(z).

Defined as in (172), dominance curves may seem complicated to calculate.Fortunately, there is a very useful link between the dominance curves and thepopular FGT indices, a link that greatly facilitates the computation ofDs(z). We


can indeed show that

Ds(z) = c · ∫ z

0(z − y)s−1 dy

= c · P (z; α = s− 1),(173)

wherec = 1/(s− 1)! is a constant that can be ignored. Therefore, to compute thedominance curve of orders, we need only compute the FGT index atα = s − 1,which is P (z; α = s − 1) (see (76)). Recall thatP (z; α = 1): is the averagepoverty gap. Hence, the dominance curve of order 1 is simply the average povertygap for different poverty lines. This can also be seen on Figure39. The distancebetweenz andy gives (when it is positive) the poverty gap at a given value ofincomey. Fory = y′, for instance, Figure39 shows that distancez − y′. dF (y′)– as measured on the vertical axis – gives the density of individuals at that levelof income. The rectangular area given by the product of(z − y′) anddF (y′) thenshows the contribution of those with incomey′ to the population average povertygap. Integrating all such positive distances betweeny andz across the populationthus amounts to calculating the average poverty gap – again, this is the sum ofindividual rectangles of lengths(z − y) and heightsdF (y), or simply the grey-shaded area of Figure39.

Let us now integrate by parts equation (171). This gives:

∆P (z) = π(z; z) ∆D1(z)−∫ z

0

π(1)(y; z)∆D1(y)dy, (174)

where∆Ds(y) is defined asDsA(y)−Ds

B(y). If we wish to ensure that∆P (z) ispositive for all of the indices that belong toΠs(z), we need to ensure that (174) ispositive for all of the poverty indices that satisfy the conditions in (168), whateverthe values of their first-order derivativeπ(1)(y; z), so long as that derivative iseverywhere non-positive between 0 andz+. For this to hold, we simply need that(recall thatD1(y) = F (y)):

FA(y) ≥ FB(y), for all y ∈ [0, z+]. (175)

We refer to this as first-order poverty dominance ofB overA. The result can besummarized as follows:

First-order poverty dominance (primal):

PA(z)− PB(z) ≥ 0 for all P (z) ∈ Π1(z+)iff D1

A(y) > D1B(y) for all y ∈ [0, z+].

(176)


The dominance condition in (176) is relatively stringent: it requires the head-count index inA never to be lower than the headcount inB, for all possible povertylines between 0 andz+. If, however, the condition is found to hold in practice,a very robust poverty ordering is obtained: we can then unambiguously say thatpoverty is higher inA than inB for all of the poverty indices inΠ1(z+) (includingthe headcount index). Since (almost) all of the poverty indices that have been pro-posed obey this restriction, this is a very powerful conclusion indeed. Note againthat this ordering is valid for any choice of poverty line up toz+.

Moving to second-order poverty dominance, we integrate once more by partsequation (174) and find that:

∆P (z) = π(z; z) ∆D1(z)−π(1)(z; z) ∆D2(z)+

∫ z

0

π(2)(y; z)∆D2(y)dy. (177)

Recall that the indices that are members ofΠ2(z+) are such thatπ(2)(Q(p); z) ≥ 0Q(p) whenQ(p) ≤ z and withπ(z, z) = 0. Hence, if we wish∆P (z) to bepositive for all of the indices that belong toΠ2(z+), we must have:

∆D2(y) ≥ 0 for all y ∈ [0, z+]. (178)

This is second-order poverty dominance ofB overA; it can be summarized as:

Second-order poverty dominance (primal):

PA(z)− PB(z) ≥ 0 for all P (z) ∈ Π2(z+)iff D2

A(y) ≥ D2B(y) for all y ∈ [0, z+].

(179)

Recall from173thatD2(z) = P (z; α = 1). Second-order poverty dominancethus requires that the average poverty gap inA be always larger than the averagepoverty gap inB, for all of the poverty lines between 0 andz. If the conditionin (179) is found to hold in practice, then we can say that poverty is higher inA than inB for all of the poverty indices that are continuous at the poverty lineand that are equality preferring (their second-order derivative is positive). That, ofcourse, also includes the average poverty gap itself. Most of the indices found inthe literature fall into that category, a major exception being the headcount and theSen index. And that ordering is again valid for any choice of poverty line between0 andz+.

We can repeat this process for any arbitrarily higher order of dominance, bysuccessive integration by parts and by determining the conditions under which allof the poverty indicesP (z) that are members of a classΠs(z+) will indicate more


poverty inA than inB, and this for all of the poverty linesz between 0 andz+.This gives the general formulation ofsth order poverty dominance:

sth-order poverty dominance (primal):

PA(z)− PB(z) ≥ 0 for all P (z) ∈ Πs(z+)iff Ds

A(y) ≥ DsB(y) for all y ∈ [0, z+].

(180)

This condition is illustrated in Figure25for generals-order dominance, wheredominance holds untilz+, but would not hold ifz+ exceededzs. Checking povertydominance is thus conceptually straightforward. For first-order dominance, weuse what has been termed “the poverty incidence curve”, which is the headcountindex as a function of the range of poverty lines[0, z+]. For second-order domi-nance, we use the “poverty deficit curve”, which is the area underneath the povertyincidence curve or more simply the average poverty gap, again as a function of therange of poverty lines[0, z+]. Third-order dominance makes use of the area un-derneath the poverty deficit curve, or the square of poverty gaps index (also calledthe poverty severity curve) for poverty lines between 0 andz+. Dominance curvesfor greater orders of dominance simply aggregate greater powers of poverty gaps,graphed against the same range of poverty lines[0, z+].

11.1.2 Nesting of dominance tests

The condition (179) for second-order dominance is less stringent than (176)for first-order poverty dominance. To see why, consider (172) again. When first-order dominance over[0, z+] holds, then second-order dominance over[0, z+]must also hold. Hence, when we find that the poverty indices inΠ2(z+) showmore poverty inA, we also know that the poverty indices inΠ1(z+) will do thesame. That is of course consistent with the fact thatΠ2(z+) ⊂ Π1(z+).

Suppose, however, that we have that∆D2(y) > 0 for all y ∈ [0, z+], but notthat∆D1(y) > 0 for all y ∈ [0, z+]. Hence, we have first-order, but not second-order, dominance. Poverty is larger inA for all of the indices inΠ2(z+) but notfor all those inΠ1(z+). This is possible sinceΠ1(z+) is larger thanΠ2(z+).

These relationships are in fact sequentially valid for higher orders as well.This is illustrated in Figures49 and50. Figure49 shows that a class of indicesΠs+1(z+) is a subset of the lower class of indicesΠs(z+). Whenever an orderingis made overΠs(z+), it is also necessarily valid over the subsetΠs+1(z+). Figure50 analogously illustrates the sizes of the set of distributions(A,B) that can beordered by the dominance condition in (180). The greater the value ofs, the


more likely can a couple(A,B) fall into those sets, and therefore the more likelycan they be compared unambiguously by that dominance condition. Taken jointly,Figures49and50show the trade-off that exists between wishing to assert whetherA really has more poverty thanB, and wishing to assert this for as large a class ofpoverty indices and poverty lines as possible.

For a simple illustration of these relationships, consider a comparison of dis-tributionsA andB in Table10. The first-order dominance condition (176) onlyholds ifz+ is lower than 9. Hence, we can conclude thatA has more poverty thanB for any choice of first-order indices so long as the poverty line is less than 9.Indeed, it is not hard to find some first-order indices that will show more povertyin B when z exceeds 9: the headcount between 9 and 11 clearly shows morepoverty inB. We can, however, verify that the second-order condition is obeyedfor any choice ofz+. This then implies that all second-order indices (those thatare members ofΠ2(z+)) will show more poverty inA, regardless of the choice ofpoverty line. This is quite a robust statement, since it is valid for all distribution-sensitive poverty indices (the headcount is not distribution-sensitive, hence it doesnot always indicate more poverty inA) and again for any choice of poverty line.Again, as mentioned above, second-order poverty dominance is a less stringentcriterion than first-order dominance to check in practice. The price of this, how-ever, is that the set of indices over which poverty dominance is checked is smallerfor second-order dominance than for first-order dominance.

11.2 Dual approach

There exists a dual approach to testing first-order and second-order povertydominance, which is sometimes called ap, percentile, or quantile approach. Whereasthe primal approach makes use of curves that censor the population’s income atvarying poverty lines, the dual approach makes use of curves that truncate the pop-ulation at varying percentile values. The dual approach has interesting graphicalproperties, which makes it useful and informative in checking poverty dominance.

11.2.1 First-order poverty dominance

To illustrate this second approach, we focus on indices that aggregate povertygaps using weights that are functions ofp:

Γ(z) =

∫ 1

0

g(p; z)ω(p)dp (181)


Note that using aggregates of poverty gaps as in (181) is more restrictive than us-ing functionsπ(Q(p); z) defined separately overQ(p) andz, as is done in (167).When the poverty lines are the same across distributions (as was implicitly as-sumed above for the primal approach, and as is almost always assumed to be thecase in practice), the dominance rankings are, however, the same, as we will seebelow.

Membership in the (dual) first-order classΠ1(z+) of poverty indices only re-quires that the weightsω(p) be non-negative functions ofp:

Π1(z+) =

Γ(z)

∣∣∣∣ω(p) ≥ 0z ≤ z+.

(182)

If we want to check whether∆Γ(z) = ΓA(z) − ΓB(z) is positive for all ofthe indices that belong toΠ1(z+), it is clear that we need only assess whethergA(p; z+) ≥ gB(p; z+) for all p ∈ [0, 1]. This yields the following dual first-orderpoverty dominance :

First-order poverty dominance (dual):

ΓA(z)− ΓB(z) ≥ 0 for all Γ(z) ∈ Π1(z+)iff gA(p; z+) ≥ gB(p; z+) for all p.

(183)

Condition (183) requires poverty gaps to be nowhere lower inA than inB,whatever the percentilesp considered. It thus amounts to ordering the povertygap curves. It is not difficult to show that this is also equivalent to checking theprimal first-order poverty dominance condition in (176). In other words, if we canorder poverty overΠ1(z+), then we can also do so overΠ1(z+), andvice versa. Infact, first-order poverty dominance (primal or dual) implies ordering all povertyindices (additive or otherwise) that are (weakly) decreasing in income. To checksuch wide degree of ethical robustness, we can use either the primal or the dualfirst-order poverty dominance condition.

First-order poverty dominanceThe following conditions are equivalent:

1. Poverty is higher inA than inB for any of the poverty indices that obey thefocus (see p.144), the anonymity (p.141) and the Pareto (p.140) principlesand for any choice of poverty line between 0 andz+;

2. PA(z; α = 0) ≥ PB(z; α = 0) for all z between 0 andz+;

3. gA(p; z+) ≥ gB(p; z+) for all p between 0 and 1.


11.2.2 Second-order poverty dominance

Membership in the dual second-order classΠ2(z) of poverty indices requiresthat the weightsω(p) be positive and decreasing functions of the ranksp:

Π2(z+) =

Γ(z)

∣∣∣∣Γ(z) ∈ Π1(z+)ω(1)(p) ≤ 0.

(184)

To show what dominance condition applies to (184), recall thatG(p; z) is theCumulative Poverty Gap (CPG) curve, and integrate by parts (181):

Γ(z) = G(p; z)ω(p)|10 −∫ 1

0

G(p; z)ω(1)(p)dp. (185)

For∆Γ(z) to be positive for all of the indices that belong toΠ2(z+) (and thereforealso for all poverty linesz ≤ z+), we need to order the the CPG curves. The resultis summarized as:

Second-order poverty dominance (dual):

ΓA(z)− ΓB(z) ≥ 0 for all Γ(z) ∈ Π2(z+)iff GA(p; z+) ≥ GB(p; z+) for all p ∈ [0, 1].

(186)

Again, we can show that the condition in (186) is equivalent to the primalsecond-order poverty dominance condition in (179). In other words, if and onlyif ΓA(z) − ΓB(z) ≥ 0 for all Γ(z) ∈ Π2(z+), thenPA(z) − PB(z) ≥ 0 forall P (z) ∈ Π2(z+). Thus, to check robustness of poverty ordering over alldistribution-sensitive poverty indices, we can use either the primal or the dualsecond-order poverty dominance condition. This is summarized as follows:

Second-order poverty dominanceThe following conditions are equivalent:

1. Poverty is higher inA than inB for any of the poverty indices that obeythe focus (see p.144), the anonymity (p.141), the Pareto (p.140) and thePigou-Dalton (p.141principles and for any choice of poverty line between0 andz+;


3. GA(p; z+) ≥ GB(p; z+) for all p between 0 and 1.


11.2.3 Higher-order dominance

Dual conditions for higher-order poverty dominance are not as convenient andsimple as those just stated for first- and second-order dominance. It is there-fore usual to check higher-order dominance using the primal conditions of (180).Stated in terms of ethical principles, third-order dominance reads for instance as:

Third-order poverty dominanceThe following conditions are equivalent:

1. Poverty is higher inA than inB for any of the poverty indices that obey thefocus (see p.144), the anonymity (p.141), the Pareto (p.140), the Pigou-Dalton (p. 141 and the transfer-sensitivity principles (p.142 and for anychoice of poverty line between 0 andz+;


11.3 Assessing the limits to dominance

Whether we use the primal or the dual approach, testing for poverty dominanceinvolves specifying an upper boundz+ for the ordering of the dominance curves.This bound can presumably be obtained from empirical or ethical work on whatreasonable range of poverty lines should be used to compare poverty. It can ofcourse also be specified arbitrarily by the researcher. An alternative strategy is touse the available sample information and estimate directly from that informationthe upper bound up to which a distributive comparison can be inferred to be robust.We can then interpret this statistics as a ”critical” bound. In the light of the resultsabove, this critical value will limit the range of poverty lines over which we willbe able to order poverty acrossA andB.

Assume for instance that a primal poverty dominance curveDsA(y) for A is

initially higher than that forB for low values ofy, but that this ranking is reversedfor higher values ofy. Let ζ+(s) be the first crossing point of the curves, such thatDs

A(ζ+(s)) = DsB(ζ+(s)). DistributionB then has less poverty than distribution

A for all of the poverty indices inΠs(z+), so long asz+ ≤ ζ+(s). As the notationimplies, this calculation can be done for any desired orders of poverty dominance.

It may be, however, that we feel (for some orders) that ζ+(s) is too low.Said differently, being able to order poverty only over a narrow range[0, ζ+(s)]may seem unsatisfactory. We may change this by moving to a higher order ofdominance. Indeed, we can show thatζ+(s) is increasing ins, with ζ+(s + 1) >


ζ+(s), wheneverDsA(z) > Ds

B(z) for somez < ζ+(s). We may thus increase therange of poverty lines over which a poverty ranking is robust by moving up to ahigher class of indices.

This is illustrated in Figure51, wherez+ < z++. For the sake of illustration,suppose that the first-order dominance curves ofA andB cross somewhere be-tween 0 andz++. It is then impossible to order poverty over all of the indices thatbelong toΠ1(z++). Assume, however, that decreasing the upper bound fromz++

to z+ does rank the distributions overΠ1(z+), and that increasing the order ofdominance from 1 to 2 while maintaining the upper bound atz++ also ranks thedistributions in terms of poverty. In either case, poverty is now ordered, but overdifferent sets. The alternative is then to choose between an ordering on indicesthat are ethically more restrictive (such asΠ2(z++)), and an ordering on indiceswith a more restrictive range of poverty lines (such asΠ1(z+)).

11.4 References

Methods for testing poverty dominance are relatively recent, and postdatemuch of the literature on inequality and social welfare dominance. One of theearly influential papers isAtkinson (1987) – that paper also introduced the idea of”restricted” dominance. The theoretical poverty dominance conditions have beenfurther and rigorously explored inFoster and Shorrocks (1988b) andFoster andShorrocks (1988c). Bounds to poverty dominance are discussed inDavidson andDuclos (2000a). Zheng (2000a) provides a different approach based on ”minimumdistribution-sensitivity” poverty indices.

The Pigou-Dalton principle has been framed alternatively as a strong and as aweak axiom for the study of poverty indices (seeDonaldson and Weymark (1986)andZheng (1999a)). In the weak version, the axiom says that the poverty indexmust increase following a transfer from one individual to another wealthier indi-vidual, providing that both are initially below the poverty line and that the transferdoes not lift the wealthier person above this threshold. The strong axiom postu-lates that the index must increase even if this transfer pushes the higher-incomerecipient above the poverty line. The strong formulation of the axiom is usuallypreferred.

Del Rio and Ruiz Castillo (2001), Jenkins and Lambert (1998a), Jenkins andLambert (1998a), Jenkins and Lambert (1998b) andSpencer and Fisher (1992)discuss the use of CPG (or ”TIP”) curves (initially proposed byJenkins andLambert 1997) for second-order poverty dominance. Surveys and integrative re-views of the literature can be found inZheng (1999a), Zheng (2000b) andZheng


(2001a). US applications includeBishop, Formby, and Zeager (1996) (for themarginal impact of food stamps on US poverty) andZheng, Cushing, and Chow(1995) (for another US application).

12 WELFARE AND INEQUALITY DOMINANCE 156

12 Welfare and inequality dominance

12.1 Ethical welfare judgments

As for poverty, we may wish to determine if the ranking of two distributionsof income in terms of social welfare is robust to the choice of social welfare in-dices. Of course, one way to check such robustness would be to verify the welfareranking of the two distributions for a large number of the many social welfareindices that have been proposed in the literature. This, however, would certainlybe a tedious task. Besides, new social welfare indices can always and easily bedesigned.

A simpler and potentially more powerful alternative is to apply tests of wel-fare dominance. Unlike for poverty, welfare dominance tests take into accountthe whole distributions of income, as opposed to just the censored or truncateddistributions used for poverty comparisons.

As for poverty dominance, there are two testing approaches, a primal (income-censoring) and a dual (percentile-truncating) one. The primal approach has theadvantage of being applicable to any desired (however large) order of dominance,and uses curves of the well-known FGT indices for an infinite range of ”povertylines” or income censoring points. The dual approach is practically convenientonly for first and second order dominance, but it uses curves that are graphicallyinstructive and that have been documented extensively in the literature. As forpoverty dominance, if, for first and second order dominance, a welfare rankingis obtained using one of these two testing approaches, the same ranking will beobtained using the other approach. In other words, the two approaches are equiva-lent in terms of their ability to rank distributions robustly over classes of first- andsecond-order social welfare indices.

As for poverty dominance, for both of these approaches we will make useof classes of social welfare indices defined by the reactions of their indices tochanges in or reallocations of income. These social welfare indices do not need tobe additive, but for expositional convenience assume that they are defined in thesimple rank-dependent utilitarian format ofW in (39):

W =

∫ 1

0

U (Q(p)) ω(p) dp. (187)

The first-order class of social welfare indices regroups all of the symmetric (oranonymous) social welfare indices that are increasing in income. Int terms of


(187), this can be formulated as the classΩ1 with

Ω1 =

W

∣∣∣∣U (1)(Q(p)) ≥ 0ω(p) ≥ 0.

(188)

The second-order class of social welfare indices regroups all of the first-order in-dices that are increasing in mean-preserving equalizing transfers. Recall that suchtransfers redistribute one dollar of income from a richer to a poorer person. Theseindices thus obey the Pigou-Dalton principle of transfers. Using (187) again, thissuggests the class ofΩ2 indices:

Ω2 =

W

∣∣∣∣∣∣

W ∈ Ω1,U (2)(Q(p)) ≤ 0,ω(1)(p) ≤ 0.

(189)

The third-order class of social welfare indices includes all second-order indicesthat further obey the transfer-sensitivity principle – requiring that equalizing trans-fers have a greater impact on social welfare when they occur lower down in thedistribution of income. Expressed in terms of (187), this requirement forcesω(p)to be a constant and requires the concavity of individual utility functions to bedecreasing in income. This suggestsΩ3:

Ω3 =

W

∣∣∣∣∣∣

W ∈ Ω2

ω(1)(p) = 0U (3)(Q(p)) ≥ 0.

(190)

As hinted above on page142, higher orders of classes can be defined analogously.Generally speaking, membership in a higher-order class of welfare indices re-quires these indices to be more sensitive to the income of the very poor.Ωs im-plies membership inΩs−1, and fors-order additive welfare indices, we also needthat(−1)(i)U (i)(Q(p)) ≤ 0 for i = 1, ..., s.

12.2 Tests of welfare dominance

As for poverty dominance, both primal and dual conditions can be used fortesting first- and second-order welfare dominance. The two types of tests ordersocial welfare on exactly the same class of indices.

First-order welfare dominanceThe following conditions are equivalent:


1. Social welfare is larger inB than inA for any of the social welfare indicesthat obey the anonymity (p.141) and the Pareto (p.140) principles;

2. WB −WA ≥ 0 for all W ∈ Ω1;

3. PA(z; α = 0) ≥ PB(z; α = 0) for all z between 0 and infinity;

4. D1A(z) ≥ D1

B(z) for all z between 0 and infinity;

5. QA(p) ≤ QB(p) for all p between 0 and 1.

First-order welfare dominance can thus be checked by verifying whether the head-count index is higher forA than forB for all poverty linesz. There is thereforea useful analogue between tests of poverty and welfare dominance. Ordering twodistributions of incomes over the first-order class of social welfare indices can alsobe done by comparing the incomes of the two distributions over the entire rangeof percentiles. More graphically, it requires checking that ”Pen’s parade of dwarfsand giants” be everywhere higher inB than inA, whatever the percentiles beingcompared. The two distributions ”parade” simultaneously alongside each other,and the distributive analyst ”observes” if one parade dominates everywhere theother.

A similar result can be stated for second-order welfare dominance. To see this,first recall the definition of the Generalized Lorenz curveGL(p) (see (54)):

GL(p) =

∫ p

0

Q(q)dq. (191)

The generalized Lorenz curve sums all incomes up to percentilep, and is thereforethe cumulative Pen’s parade. We then obtain:

Second-order welfare dominanceThe following conditions are equivalent:

1. Social welfare is larger inB than inA for any of the social welfare indicesthat obey the anonymity (p.141), the Pareto (p.140) and the Pigou-Dalton(p. 141) principles ;

2. WB −WA ≥ 0 for all W ∈ Ω2;

3. PA(z; α = 1) ≥ PB(z; α = 1) for all z between 0 and infinity;

4. D2A(z) ≥ D2



5. GLA(p) ≤ GLB(p) for all p between 0 and 1.

An exactly similar result applies for higher-order welfare dominance. As forpoverty dominance, the dual conditions are less convenient and are omitted here.

Higher-order welfare dominanceThe following conditions are equivalent:

1. WB −WA ≥ 0 for all W ∈ Ωs;

2. PA(z; α = s− 1) ≥ PB(z; α = s− 1) for all z between 0 and infinity;

3. DsA(z) ≥ Ds


Checking fors-order welfare dominance thus simply requires comparing the FGTindices forα = s− 1 over all possible poverty lines.


12.3 Inequality judgments

As for poverty and welfare dominance, we can define classes of relative in-equality indices over which to check the robustness of the inequality orderings oftwo distributions of income. As we will see, these classes of inequality indiceshave properties which are analogous to those of the classes of social welfare in-dices. They react to income changes or income reallocations in a manner thatdepends on the order of the classes to which they belong. Unlike social welfarefunctions, however, relative inequality indices also need to be homogeneous of de-gree 0 in all income. This means that an equi-proportionate change in all incomeswill not affect the value of these relative inequality indices.

Consider first the classΥ1(l+) of inequality indices of the first-order. Recallthat income shares (or normalized quantiles) are given byQ(p) = Q(p)/µ. Υ1(l+)is a class of inequality indices that is not usually considered in the literature be-cause it censors the effects of changes affecting income shares beyondl+. Indeed,besides being homogeneous of degree 0 in income, the indices that are membersof Υ1(l+) are such that, for a given mean, inequality decreases when an individ-ual’s income increases, so long as that individual’s income share does not exceedl+. Said differently, the inequality indices inΥ1(l+) are decreasing in the incomeshares of those withQ(p) ≤ l+. If the income of an individual with income sharegreater thanl+ changes, then an index that is a member ofΥ1(l+) cannot change.We can think of keeping mean income constant, following these changes, througha decrease in the income of the richest individual, since that will not by definitionaffect the first-order inequality indices. In addition to being symmetric in income,these indices are therefore in some loose sense of the Pareto type.

The Pareto principle underlyingΥ1(l+) is thus an alternative ethical principleto the well-known Pigou-Dalton principle of transfers, which has been at the heartof inequality analysis for many decades. But the scope of this Pareto principle iscensored: it only applies to income shares belowl+. This makes the first-orderclass of inequality indices a poverty-like class. For this reason, we do not havethatΥ2 ⊂ Υ1(l+).

Unlike the censored Pareto principle, the Pigou-Dalton principle will postu-late that a mean-preserving transfer of income from a higher-income person to alower-income person decreases inequality, whatever the income shares of thoseaffected by this income reallocation. All inequality indices that belong to theclassΥ2 of second-order inequality indices obey this principle and decrease aftera mean-preserving equalizing transfer. These inequality indices are also said to beSchur-convex. Almost all of the frequently used inequality indices (including the


Atkinson, S-Gini and generalized entropy indices, (with the notable exception ofthe variance of logarithms) are members ofΥ2.

Inequality indices that belong to the classΥ3 of third-order inequality indicesalso belong toΥ2, and weakly decrease after a favorable composite transfer. Thisincludes the Atkinson indices and some of the generalized entropy indices, butnot the S-Gini indices. ClassesΥs of higher order inequality indices can be sim-ilarly defined. For instance, to be members of the class of fourth-order inequalityindices, inequality indices must be members ofΥ3 and must be more sensitive tofavorable composite transfers when they take place lower down in the distributionof income. Again, all of the Atkinson indices belong toΥ4. The higher the valueof s, the more Rawlsian are the indices since the more sensitive they are to theincome shares of the poorest.

Comparing the definitions of the classesΩs andΥs, note that when the meansof the distributions are equal, the social welfare ranking is the same as the inequal-ity ranking, in the sense that ifIA ≥ IB for all I in Υs, thenWA ≤ WB for allW in Ωs, andvice versa. In such cases, checking for inequality dominance can bedone by checking for welfare dominance. When the means are not equal, we cannormalize all incomes by their mean (this does not affect relative inequality), andthen use the welfare dominance results described in section12 for Ωs to checkfor dominance over a classΥs of relative inequality indices. Hence, to check forinequality dominance, we can simply test for welfare dominance once incomeshave been normalized by their mean. WhenB has more welfare thanA at orders, we can say thatIB is lower thanIA for all of the inequality indices that belongto Υs.

12.4 Tests of inequality dominance

As indicated above, checking for inequality dominance can be done most eas-ily by using the welfare dominance conditions of Section12.2and normalizingincomes by their mean. For the primal dominance curves, we will thus need thenormalized stochastic dominance curveD

s(lµ),

Ds(lµ) =

Ds(l · µ)

(lµ)(s−1). (192)


Ds(lµ) has a nice equivalence in terms of the normalized FGT indicesP (z; α):

Ds(lµ) = cP (z = lµ; α = s− 1)

= c

∫ 1

0

(g(p; lµ)

lµ

)s−1

dp, (193)

wherec is as before a constant that we can ignore. Thus, estimating the normalizeddominance curve atlµ and orders is equivalent to computing the normalized FGTindex for a poverty line equal tolµ and forα equal tos− 1.

Similarly, for dual dominance conditions, we may use the poverty gaps nor-malized by mean income

g(p; z) = g(p; z)/z. (194)

This leads to:

First-order restricted inequality dominanceThe following conditions are equivalent:

1. IA − IB ≥ 0 for all I in Υ1(l+);

2. D1

A(λµA) ≥ D1

B(λµB) for all λ between 0 andl+;

3. gA(p; l+µA) ≥ gB(p; l+µB) for all p between 0 and 1.

Note that the conditionD1

A(λµ) ≥ D1

B(λµ) is easily interpreted. It simply com-pares the proportion of those with income less thanl times the mean inA and inB. If there are fewer such individuals inB than inA, for all l ≤ l+, inequality isgreater inA for all of the indices inΥ1(l+).

Second-order inequality dominanceThe following conditions are equivalent:

1. Relative inequality is larger inB than inA for any of the inequality indicesthat obey the anonymity (p.141) and the Pigou-Dalton (p.141principles ;

2. IA − IB ≥ 0 for all I ∈ Υ2;

3. PA(λµA; α = 1) ≥ PB(λµB; α = 1) for all λ between 0 and infinity;

4. D2

A(λµA) ≥ D2

B(λµB) for all λ between 0 and infinity;

5. LA(p) ≤ LB(p) for all p between 0 and 1.


Testing for second-order inequality dominance can thus be done simply by com-paring the usual normalized average poverty gap forA and forB for all possibleproportions of the mean as poverty lines. An alternative equivalent test is that ofcomparing the Lorenz curves forA andB. This is the well-known and classicalLorenz test, which has long been considered the golden rule of relative inequalityrankings. Dual conditions for higher-order (i.e., third-order) inequality dominancehave also been proposed in the literature, but they are again less convenient thanthe primal conditions.

A generals-order inequality dominance condition is then simply stated as:

s-order inequality dominanceThe following conditions are equivalent:

1. IA − IB ≥ 0 for all I ∈ Υs;

2. PA(λµA; α = s−1) ≥ PB(λµB; α = s−1) for all λ between 0 and infinity;

3. Ds

A(λµA) ≥ Ds

B(λµB) for all λ between 0 and infinity.

12.5 Inequality and progressivity

We can combine some of the results derived above to what we saw in Chap-ter 8 on the measurement of vertical equity to link progressivity and inequalitydominance.

First, in the absence of reranking, it is clear that a tax and/or a transfer that isTR- or IR-progressive, will decrease all of the inequality indices that are membersof Υ2. This is most easily seen by considering equations (144) and (146) and bynoting thatCN(p) = LN(p) when there is no reranking. ForIR progressivity,this follows from the fact that a concentration curve for net income that lies abovethe Lorenz curve of gross income pushes the Lorenz curve of net income above,which decreases inequality for all second-order indices of inequality.

Further, again in the absence of reranking, if a tax and/or transferT1 is moreIR-progressive than a tax and/or transferT2, thenT1 necessarily reduces inequal-ity by more thanT2 when inequality is measured by any of the inequality indicesthat belong toΥ2. This can be seen by the sum of theIR-progressivity terms in(148) (see also equation (144)) and by noting again thatCN(p) = LN(p) in theabsence of reranking.

We may also be concerned about the impact of a tax and benefit system on theclass of first-order inequality indices,viz, on indices that are monotonic in some


lower income shares, but not always in terms of cumulative income shares. Tocheck whether this impact reduces first-order inequality indices, we must checkwhetherT (X)/X is always lower thanµT /µX for all of theX that are below somecensoring pointl+µ. This supposes again, however, that the tax does not inducereranking. When it does, one way to account for the reranking effect is to compute”income growth curves”, which are given by(N(p)−X(p))/X(p) . When thesecurves exceed the growth in average income – given by(µN − µX)/µX – for allp ≤ FX(l+µX), then all of the first-order inequality indices inΥ1(l+) will fall.

12.6 Social welfare and Lorenz curves

It often occurs that two income distributionsA and B are compared usingestimates of average income and inequality separately. Using second-order dualconditions, it is possible to combine these estimates to assess whether social wel-fare is greater inA than inB by noting from54 thatGL(p) = µL(p).

Say that we dispose of the entire Lorenz curves of each of the two distribu-tions. Figure38shows four cases of comparisons of average income and inequal-ity across these two distributions. In Case 1,A Lorenz-dominatesB, and it alsohas a higher average income. Hence, there is generalized-Lorenz-dominance ofAoverB, and we are therefore assured thatWA−WB ≥ 0 for all W ∈ Ω2. In Case2, A also dominatesB according to the Lorenz criterion, butµA < µB; becauseof this, GLA(p) crossesGLB(p) and there can be no unambiguous second-ordersocial welfare ranking. Comparing the slopes of each of these two curves gives,however, the quantiles at various percentilesp. Since these quantiles are visiblylarger inA than inB for a large lower range ofp, A has less poverty thanB for alarge range of possible poverty lines and for many poverty indices. Case 3 depictsan ambiguous ranking of inequality acrossA andB. However, becauseµA is wellaboveµB, the generalized Lorenz curve forA is above that forB. Finally, Case4 shows a circumstance in which inequality and social welfare rankings clash.Ahas unambiguously less inequality thanB according to the Lorenz criterion, butµA being significantly belowµB, A has unambiguously less social welfare thanBaccording to the generalized-Lorenz criterion.

12.7 The distributive impact of benefits

The impact of government benefits and transfers on the distribution of incomescan also be visualized using curves that are linked to the poverty, social welfareand inequality dominance curves.


Say, for instance, that the expected benefit at rankp of some government pro-gram – or some economic change – is given byB(p). (This could be estimatednon-parametrically.) An impact indicator of the cumulative effect of that benefitup to rankp is given by:

GC B(p) =

∫ p

0

B(q)dq. (195)

with µB = GC B(1). In analogy to the Generalized Lorenz curve, we may callGC B(p) a Generalized concentration curve.GC B(p) shows approximately theabsolute contribution of the bottom proportionp of the population to theper capitabenefits. The impactGC B(p) is only approximate since it ignores the possiblereranking of individuals by the program. The concentration curve of the benefitup to rankp can then be defined as:

CB(p) =GC B(p)

µB

. (196)

Recall that the concentration curveCB(p) at p gives the percentage of the totalbenefits that accrue to those with initial rankp or lower. UsingCB(p) andGC B(p)can help assess the distributive effect of the program. For instance:

1. For understanding the approximate impact of the benefit on social welfare,we may wish to test whetherB(p) is always positive, regardless ofp. If so,then the benefit will tend to increase social welfare for all first-order welfareindices. If not, we can test ifGC B(p) is always positive regardless ofp. Ifso, then the approximate impact of the benefit is to increase social welfarefor all second-order welfare indices.

2. For understanding the approximate impact of the benefit on poverty, weproceed basically as in point1 just above, with the only difference that weassess the curvesB(p) andGC B(p) only for all p ∈ [0, F (z+)]. If B(p) isalways positive over that range ofp, then the benefit will tend to decreasepoverty for all first-order poverty indicesΠ1(z+), and ifGC B(p) is alwayspositive for allp ∈ [0, F (z+)], then the approximate impact of the benefit isto decrease poverty for all poverty indices inΠ2(z+).

3. For assessing the impact of the benefit on inequality and relative poverty, wemay compareB(p)/µB with X(p)/µX , andCB(p) with LX(p). ComparingB(p)/µB with X(p)/µX sheds light on the approximate impact of the ben-efit on first-order inequality indices, whereas comparingCB(p) with L(p)


shows the approximate impact of the benefit on second-order inequality in-dices. We compare these curves for allp ∈ [0, 1] if we are concerned aboutthe whole population, for allp ∈ [0, F (z+)] if we are only concerned aboutthe poor, or for allp ∈ [0, F (l+µ)] if we are concerned about first-orderinequality indices.

12.8 Pro-poor growth

Assessing whether distributional changes are ”pro-poor” has become increas-ingly widespread in academic and policy circles. We will see that is relativelystraightforward to use the tools developed above to make such an assessment.There are, however, two important issues that we must first discuss.

The first issue is whether our pro-poor standard should be absolute or rela-tive. This is equivalent to asking whether we should be interested in the impact ofgrowth on absolute poverty or on relative inequality. It is indeed important to dis-tinguish between expectations that growth should change the incomes of the poorby the same absolute or by the same proportional amount – these expectations areconceptually not the same, and their empirical fulfillment also varies significantly.

The second issue is whether pro-poor judgements should put relatively moreemphasis on the impact of growth upon the poorer of the poor. This is equivalentto deciding whether our pro-poor judgements should obey higher-order ethicalprinciples such as the Pigou-Dalton principle. We will consider two orders ofpro-poor judgements; the first will obey the focus, the anonymity and the Paretoprinciples, and the second will also obey the Pigou-Dalton principle.

12.8.1 First-order pro-poor judgements

Let a distributive change entail a movement from a distributionX(p) to adistributionN(p). Let ”income growth curves” be defined as the proportionalchange in income observed at various percentiles:

g(p) =N(p)−X(p)

X(p). (197)

If the income-growth curve is positive everywhere overp ∈ [0, 1], then it is clearfrom the first-order welfare dominance results of page157 that the change in-creases social welfare for all of the welfare indices that belong toΩ1. It is alsoclear from the first-order poverty dominance results of page151 that the changedecreases poverty for all of the poverty indices that belong toΠ1(∞) (and thus


for all those that obey the first-order — focus, Pareto and Pigou-Dalton — ethicalprinciples). This result is valid for any choice of poverty lines.

A test with a greater chance to succeed is to check whether the income-growthcurve is positive everywhere overp ∈ [0, FX(z+)]. If so, then the distributivechange decreases poverty for all poverty indicesP (z) that belong toΠ1(z+). Insuch circumstances, the change can be called ”absolutely pro-poor”, in the sensethat the poor benefit in absolute terms from the distributive change. We then have:

First-order absolute pro-poor judgementsThe following statements are equivalent:

1. A movement fromX to N is first-order absolutely pro-poor for all choicesof poverty lines between 0 andz+;

2. Poverty is higher inX than inN for all of the poverty indices that obey thefocus (p.144), the anonymity (p.141) and the Pareto (p.140) principles andfor any choice of poverty line between 0 andz+;

3. PX(z; α = 0) ≥ PN(z; α = 0) for all z between 0 andz+;

4. g(p) ≥ 0 for all p between 0 andFX(z+).

Income growth curves can also be used to test whether a distributive changeis ”relatively pro-poor”, in the sense that the change increases the incomes of thepoor at a faster rate than that of the incomes of the rest of the population. Forthat purpose, we only need to compare the income growth curveg(p) at variouspercentiles of the poor to the growth in mean income. If the income growth curveat all p ∈ [0, F (z+)] is higher than the growth in mean income, then the changecan be said to be first-order relatively pro-poor. An exactly equivalent test can bedone by comparing the normalized quantiles for the initial and posterior incomes –recall that normalized quantilesQ(p) = Q(p)/µ are just incomes as a proportionof mean income. If the normalized quantiles of the poor are increased by thechange, then the change is first-order relatively pro-poor. We thus have:

First-order relative pro-poor judgementsThe following statements are equivalent:

1. A movement fromX to N is first-order relatively pro-poor for all choicesof poverty lines between 0 andz+;

2. g(p) ≥ µN−µX

µXfor all p between 0 andFX(z+);


3. QX(p)/µX ≤ QN(p)/µN for all p between 0 andFX(z+);

4. IX − IN ≥ 0 for all I in Υ1(z+/µX);

5. FX(λµX) ≥ FN(λµN) for all λ between 0 andz+/µX

12.8.2 Second-order pro-poor judgements

Testing for first-order pro-poor judgements can be demanding. It requires allquantiles of the poor to undergo a rate of growth that is either positive (for absolutejudgements) or at least as large as the growth rate in average income (for relativejudgements). We may want to relax this on the basis that a large rate of growthfor the poorer among the poor may sometimes be deemed ethically sufficient tooffset a low rate of growth for some percentiles of the not-so-poor. This thereforesays that pro-poor judgements could give greater weight to the growth experienceof the poorer among the poor. Implementing this is done by forcing pro-poorjudgements to obey the Pigou-Dalton principle.

Second-order absolute pro-poor judgementsThe following statements are equivalent:

1. A movement fromX toN is second-order absolutely pro-poor for all choicesof poverty lines between 0 andz+;

2. Poverty is higher inX than inN for all of the poverty indices that obey thefocus (p.144), the anonymity (p.141), the Pareto (p.140) and the Pigou-Dalton principles (p.141) and for any choice of poverty line between 0 andz+;

3. PX(z; α = 1) ≥ PN(z; α = 1) for all z between 0 andz+;

4. GN(p; z+) ≤ GX(p; z+) for all p ∈ [0, 1].

Recall that the cumulative income up to rankp is given by the GeneralizedLorenz curve. Denote its proportional change by

G(p) =GLN(p)−GLX(p)

GLX(p). (198)

A sufficient condition for a second-order absolute pro-poor change is then that thegrowth in cumulative incomes be positive:

G(p) ≥ 0 for all p ∈ [0, FX(z+)]. (199)


As for first-order pro-poor judgements, we may wish second-order judgementsto require that the incomes of the poor at least keep up with those of the rest of thepopulation. This yields:

Second-order relative pro-poor judgementsThe following statements are equivalent:

1. A movement fromX toN is second-order relatively pro-poor for all choicesof poverty lines between 0 andz+;

2. PX(λµX ; α = 1) ≥ PN(λµN ; α = 1) for all λ between 0 andz+/µX .

If the above conditions hold forz+ = ∞, then the change also reduces all ofthe inequality indices that are members ofΥ2. From the Theorem on second-orderinequality dominance on page162, this is therefore also equivalent to checkingwhether the Lorenz curve is pushed up by the distributive change.

A sufficient condition for second-order relative pro-poorness can also be im-plemented by comparing the growth in the cumulative incomes of the poor to thegrowth in average income. If, for allp lower thanF (z+), the percentage growthin the cumulative incomes of a bottom proportionp of the population is largerthan the percentage growth in mean income, then the change can be said to besecond-order relatively pro-poor:

G(p) ≥ µN − µX

µX

for all p ∈ [0, FX(z+)]. (200)

Income growth curves and cumulative income growth curves may also be usedto assess the impact of a distributive change on relative poverty. The procedureis similar to that of checking whether the change is pro-poor – we compare in-come growth for the poor to the growth of some central tendency of the incomedistribution. One difference with the measurement of pro-poor growth is that thecentral tendency of interest may be some quantile (such as the median income) ifthe relative poverty line is set as a proportion of that quantile.

12.9 References

Methods for establishing inequality dominance surprisingly predate those forestablishing welfare dominance in welfare economics. The seminal works arethose byAtkinson (1970), Dasgupta, Sen, and Starret (1973) andKolm (1969)for inequality dominance andShorrocks (1983) andFoster and Shorrocks (1988c)


for welfare dominance.Foster and Shorrocks (1988a) explore the links betweenrelative poverty and relative inequality dominance (see alsoDavidson and Duclos(2000b) andFormby, Smith, and Zheng (1999)). Welfare economists have madeextensive use of the literature on the ranking of distributions under risk aversion– see among many othersFishburn and Vickson (1978), Pratt (1964), Whitmore(1970) andYitzhaki (1982b).

Descriptions and theoretical foundations of dual stochastic dominance toolscan be foundinter alia in Pen’s parade of ”dwarfs and giants” (Pen 1971, chap-ter 3), in Yaari (1987), in Moyes (1999) (for links with Lorenz curves), and inDavies and Hoy (1995) andMuliere and Scarsini (1989) (for when Lorenz curvesintersect).

Empirical tests for inequality and welfare dominance are numerous; they in-clude inter alia Bishop, Formby, and Smith (1991d) (Lorenz dominance in theUS), Bishop, Chow, and Formby (1991b) (first-order and truncated dominance),Bishop, Formby, and Thistle (1991e) (Pen or ”rank” dominance),Bishop, Formby,and Smith (1991c) (Lorenz dominance across 9 countries),Bishop, Formby, andThistle (1992) (convergence of US regional distributions),Bishop, Formby, andSmith (1993) (welfare and inequality dominance using LIS data),Chen, Datt,and Ravallion (1994) (comparisons of 44 less developed countries),Gouveia andTavares (1995) (Portuguese distributions),Makdissi and Groleau (2002) (Cana-dian distributions),Ravallion (1992) (Indonesia),Sahn and Stifel (2002) (appliedto nutritional data), andWang, Shi, and Zheng (2002) (comparing inequality andsocial welfare in China).

171

Part IV

Poverty and equity: policy andgrowth

13 POVERTY ALLEVIATION: POLICY AND GROWTH 172

13 Poverty alleviation: policy and growth

13.1 The impact of targeting

For policy purposes, it is often as useful to assess the impact ofreformstoa benefit or public expenditure program as it is to evaluate the effect ofexistingprograms. For administrative or political reasons, it may indeed be impossibleto eliminate or to amend dramatically the structure of existing programs. Hence,comparing a current tax or benefit program with one in which it is supposed notto exist may not be very useful for practical purposes. Marginal reforms to suchprograms are nevertheless often feasible, and we therefore focus on them in thischapter. As we will see, focusing on marginal reforms also has the advantage ofmaking the measurement of the welfare impact of such reforms independent ofthe behavioral adjustment that individuals may make in reaction to these reforms.

We consider five such marginal reforms in this chapter. The first one channelspublic expenditure benefits to members of specific and easily observable socio-economic groups. The main issue then is: In which socio-economic group isadditional public money best spent to reduce aggregate poverty? The second typeof reform consists in an increase in public expenditures that raises all incomes insome socio-economic groups by some proportional amount. Again, an importantquestion is: For which socio-economic group would this increase in public expen-ditures reduce aggregate poverty the fastest? This second type of reform can alsobe thought as (for instance) a process that increases the quality of infrastructureand the quantity of economic activity in a particular group or region – in a waythat affects proportionally all incomes and that is thus distributionally neutral inthe sense of not affecting inequality within the groups affected.

The third type of reform considers a change in the price of some commodities,either through some macroeconomic or external shocks, or through a change incommodity taxes or subsidies. How is the distribution of well-being, and povertyin particular, affected by such a price change? The fourth question we ask is: whattype of reform to a system of commodity taxes and subsidies could we implement,with no change in overall government revenues, but with a fall in poverty? Thatis, which commodities should be prime targets for a reduction in their tax rate orfor an increase in their rate of subsidy? The fifth and last type of reform affectsproportionally all incomes of a certain type – such as some type of farm income,the labor income of some type of workers,etc... Which sort of income sourcesshould the government attempt to bolster if the aim is to alleviate poverty?

For all such reforms, we measure their poverty impact by the change in the


FGT poverty indices that they cause. Recall that the use of the FGT indices isclosely connected to checks for stochastic dominance and ethical robustness ofpoverty changes. Hence, we can use the methods below to determine how thereforms affect poverty as measured not only by the FGT poverty indices, but byall poverty indices which obey some ethical conditions. For instance, if we findthat some form of targeting decreases a FGT index of someα value for a range[0, z+] of poverty lines, then we know that the reform will also decrease all povertyindices of ethical orderα + 1, whatever the choice of poverty line within[0, z+].

13.1.1 Group-targeting a constant amount

We consider first the effect of a transfer of a constant amount of income toevery one in a groupk. For this, recall that the FGT index can be decomposed as:

P (z; α) =K∑

k=1

φ(k)P (k; z; α). (201)

Theper capitacost to the government of granting anequalamountη(k) to eachmember of a groupk is equal to:

R =K∑

k=1

φ(k)η(k). (202)

Aggregate poverty after such transfers equals,P (k; z; α):

P (k; z; α) =

∫ 1

0

[z −Q(k; p)− η(k)]α+ dp. (203)

To determine which groupk should be of greatest priority for the targeting ofgovernment expenditures, we need to determine for which groupk targeted gov-ernment expenditures (in the form ofηk) reduce aggregate poverty the most perdollar spent. In other words, we need to compare acrossk the aggregate povertyreduction benefits of targeting $1 to a groupk.

Whenα 6= 0, we can show that the marginal reduction of aggregate povertyper dollar ofper capitagovernment expenditures is given by:

∂P (z; α)

∂η(k)

/∂R

∂η(k)= −αP (k; z; α− 1) ≤ 0, (204)


and, for the normalized FGT, by:

∂P (z; α)

∂η(k)

/∂R

∂η(k)= −αz−1P (k; z; α− 1) ≤ 0. (205)

To reduceP (z; α) the most, we must therefore target those groups for whichP (k; z; α − 1) is the greatest. It is thus simply the FGT withα − 1 that guidespolicy based on reducing FGT withα27. The greater the value ofα, the greater E:19.8.45the chance that we will favor those groups where extreme poverty is highest28. E:19.8.34

Whenα = 0, the per dollar reduction of aggregate poverty is given byf(k; z),the groupk’s density of income at the poverty line:

∂P (z; α = 0)

∂η(k)

/∂R

∂η(k)= −f(k; z) ≤ 0. (206)

We must then target those groups with the greatest proportion of people justaround the poverty line, regardless of how much poverty there is below that povertyline – another consequence of the insensitivity of the headcount index to the dis-tribution of incomes belowz.

13.1.2 Inequality-neutral targeting

Consider now a transfer that increases by a proportionλ(k) − 1 the incomeQ(k; p) of each member of a groupk. The increase in income is thus(λ(k)− 1) Q(k; p).The FGT index for groupk after such a transfer is then:

P (k; z; α) =

∫ 1

0

[z −Q(k; p) · λ(k)]α+ dp. (207)

Forα 6= 0, the marginal impact of a change inλ(k) is given by

∂P (z; α)

∂λ(k)

∣∣∣∣λ(k)=1

= αφ(k) [P (k; z; α)− zP (k; z; α− 1)] ≤ 0. (208)

and by

∂P (z; α)

∂λ(k)

∣∣∣∣λ(k)=1

= αφ(k)[P (k; z; α)− P (k; z; α− 1)

] ≤ 0. (209)

for the normalized FGT index29. E:19.8.3727DAD: Poverty|Lump-sum targeting28DAD: Poverty|Lump-sum targeting


How (208) (and (209)) varies across values ofk depends on two factors. First,there is the factor[P (k; z; α)− zP (k; z; α− 1)]. Groups in which there is a sig-nificant presence of extreme poverty will tend to see theirP (k; z; α) poverty in-dices fall significantly withα, thus leading to a large value of[P (k; z; α)− zP (k; z; α− 1)].We may thus expect that these groups should be a priority for government target-ing. However, those groups with considerable incidence of extreme poverty arealso those for which a proportional increase in income has the least impact on theaverage income of the poor – since there is then little income on which growthmay take effect. Hence, whether those groups with higher incidence of extremepoverty will exhibit a higher value of[P (k; z; α)− zP (k; z; α− 1)] is ambiguous.

The second factor that enters into (208) is population shareφ(k). Ceterisparibus, targeting government expenditures (in the form of an increase inλ(k)) togroups with a high population share will naturally tend to decrease overall povertyfastest. But this fails to take into account that a given increase inλ(k) will gener-ally be more costly for the government to attain for groups with a large share ofthe population. Because of this, we may instead wish to compare across groupsthe ratio of the benefit in poverty reduction to the groupper capita increase inincome. Assume that the cost of this groupper capitaincome increase is entirelyborne by the government. Theper capitarevenue impact of such a transfer on thegovernment budget equals∂R/∂λ(k), where:

R = φ(k)µ(k) (λ(k)− 1) . (210)

Whenα 6= 0, the reduction of aggregate poverty per dollarper capitaspent isthen

∂P (z; α)

∂λ(k)

/∂R

∂λ(k)=

α [P (k; z; α)− zP (k; z; α− 1)]

µ(k)≤ 0. (211)

and

∂P (z; α)

∂λ(k)

/∂R

∂λ(k)=

α[P (k; z; α)− P (k; z; α− 1)

]

µ(k)≤ 0. (212)

for the normalized form. To reduceP (z; α) the fastest, the government shouldtherefore target those groups for which the term on the right is the greatest inabsolute value. Compared to (208), (211) does not feature population shares asa factor, since it is cancelled by the revenue impact of the government transfer.

29DAD: Poverty|Inequality Neutral targeting


There now appears, however, the termµ(k) in the denominator. Indeed, if it mustbear the entire cost of the income increase, the government will have to pay moreto achieve a given increase inλ(k) for those groups with a high average incomethan for those with a lower average income level. Finally, and for the same reasonsas above, whether those groups with higher incidence of extreme poverty willexhibit a higher value of[P (k; z; α)− zP (k; z; α− 1)] is ambiguous.

Whenα = 0, the per dollar reduction of aggregate poverty from a proportional-to-income transfer is given by

∂P (z; α = 0)

∂λ(k)

/∂R

∂λ(k)= −z · f(k; z)

µ(k)≤ 0. (213)

Those groups with a high density of income at the poverty line, and whose aver-age income is small, are then a prime target for poverty-efficient proportional-to-income transfer scheme.

13.2 The impact of changes in the poverty line

Variability of poverty line estimates across time, regions, or poverty analy-ses and institutions can occur for several reasons. There may be methodologicaluncertainty and divergences as to how poverty lines should be estimated (recallChapter7). Estimation (sampling and non-sampling) errors also occur for purelystatistical and survey reasons. Poverty lines may also be updated with time due tonew data becoming available, or in line with the evolution of some form of repre-sentative income. Whatever the reason, it may be useful given this uncertainty toknow how responsive poverty measurement will be to such variability in povertyline estimates.

To do this, consider first the case of the un-normalized FGT indices. We findthat

∂P (z; α)

∂z=

f(z) if α = 0αP (z; α− 1) if α > 0.

(214)

For the headcount index, what matters is thus the income density at the povertyline. For higher-α indices, the sensitivity to the poverty line is given simply byP (z; α− 1). The elasticity of FGT indices to the poverty line then follows as

∂P (z; α)/∂z

P (z; α)/z=

zf(z)F (z)

if α = 0αzP (z;α−1)

P (z;α)if α > 0.

(215)


Note that the elasticity of the headcount index has a useful graphical interpreta-tion. Consider Figure44 which shows the income densityf(y) at different val-ues ofy. The area underneath thef(y) curve up toy = z gives the headcountP (α = 0; z) = F (z). The value ofzf(k; z) is given by the size of the rectanglewith width z and heightf(z) in Figure44. Hence, the elasticity of the headcountwith respect to the poverty line is simply the ratio of the rectangular areazf(k; z)over the shaded areaF (z). This elasticity is larger than 1 whenever the povertyline z is lower than the (first) mode of the distribution, and will in fact be above 1in Figure44for any poverty line up to approximativelyz′. For poverty lines largerthanz′, the poverty elasticity falls below 1. Thus, it is only for societies in whichthe headcount is initially high that we can expect the elasticity of the headcountwith respect to the poverty line to be lower than 1. Otherwise, a change of 1% inthe poverty line will cause a change of more than 1% in the headcount index.

For normalized FGT indices, we obtain:

∂P (z; α)

∂z=

f(z) if α = 0αz−1

(P (z; α− 1)− P (z; α)

)if α > 0,

(216)

and∂P (z; α)/∂z

P (α; z)/z=

zf(z)F (z)

if α = 0

α(

P (z;α−1)−P (z;α)

P (z;α)

)if α > 0

(217)

for the corresponding elasticities.

13.3 Price changes

The level of prices is an important determinant of the distribution of incomes,and can therefore matter significantly for poverty analysis. Governments can af-fect their levels directly or indirectly, through the use of sales and indirect taxes,competition policy, export taxes and import duties, subsidies on food, education,energy or transportation, etc..

To see how changes in prices (and therefore how price-changing reforms) canimpact poverty, lety be a household-specific level of exogenous income, andexpress consumers’ preferences asϑ . The indirect utility function is given byV (y, q; ϑ), whereq is a vector of consumer and producer prices. We define a vec-tor of reference prices asqR – this is necessary to assess consumers’ well-beingat constant prices. Denote the real income in the post-reform situation byyR,whereyR is measured on the basis of the reference pricesqR. yR is implicitly


defined byv(yR, qR; ϑ

)= v (y, q; ϑ), and explicitly by the real income function

yR = R(y, q, qR; ϑ

), where

V(R

(y, q, qR; ϑ

), qR; ϑ

) ≡ V (y, q; ϑ) . (218)

By definition,yR gives the level of income that provides underqR the same realincome asy yields underq.

We then wish to determine how real incomes are affected by a marginal changein prices. Letxc (y, q; ϑ) be the net consumption of goodc (which can be negativeif the individual or household is a net producer of goodc) of a consumer/producerwith incomey, preferencesϑ and facing pricesq. Let qc be the price of goodc.Differentiating (218), we find:

∂R(y, q, qR; ϑ)

∂qc

∂V(yR, qR; ϑ

)

∂yR=

∂V (y, q; ϑ)

∂qc

. (219)

Using Roy’s identity and setting reference prices to pre-reform prices, this leadsto:

∂R(y, q, qR; ϑ)

∂qc

∣∣∣∣q=qR

=∂V (y, q; ϑ) /∂qc

∂V (yR, qR; ϑ) /∂yR

∣∣∣∣q=qR

= − ∂V(y, qR; ϑ

)/∂y

∂V (yR, qR; ϑ) /∂yR· xc

(y, qR; ϑ

)

= −xc

(y, qR; ϑ

). (220)

Equation (220) says that observed pre-reform net consumption of goodc isa sufficient statistic to know the impact on the real income of a marginal changein the price of goodc . This simple relationship is also valid for rationed goods.Equation (220) gives a ”first-order approximation” to the true change in real in-come that occurs from a change in the price of goodc . The approximation isexactwhen the price change is marginal. It becomes less exact if the price changeis non-marginal and if the compensated demand for goodc varies significantlywith qc.

Assume that preferencesϑ and exogenous incomey are jointly distributedaccording to the distribution functionF (y, ϑ). The conditional distribution ofϑgiveny is denoted byF (ϑ |y ), and the marginal distribution of incomey is givenby F (y). Let preferences belong to the setΘ, and assume income to be distributedover[0, a]. Expected consumption of goodc at incomey is given byxc(y, q), suchthat

xc(y, q) = Eϑ [xc (y, q; ϑ)] =

∫

Θ

xc (y, q; ϑ) dF (ϑ|y). (221)


By (220), −xc(y, q) is also proportional to the expected fall in real incomes ofthose with incomey following an increase inqc.

Let xc(q) then be theper capitaconsumption of goodc, defined asxc(q) =∫ a

0xc (y, q) dF (y) . By (220), xc(q) is also the average welfare cost of an increase

in the price of goodc. As a proportion ofper capitaconsumption, consumptionof goodc at incomey is expressed asxc(y, q) = xc(y, q)/xc(q).

It is now at last useful to see how the FGT indicesP (z; α) are affected by achange in the price of goodc30. Using (220), we find that:

∂P (z; α)

∂qc

∣∣∣∣q=qR

=

xc

(z, qR

)f(z), if α = 0

αz−α∫ z

0xc

(y, qR

)(z − y)α−1 dF (y) if α 6= 0,

(222)

wheref(z) is again the density of income atz. When graphed over a rangeof poverty linesz, this effect generates the so-called ”consumption dominance”CD c(z; α) curve of a goodc31: E:19.8.39

CD c(z; α) =∂P (z; α)

∂qc

. (223)

Note that the impact on poverty depends onα andz32. By (222), CD c(z; α = E:19.8.400) only takes into account the consumption pattern of those precisely atz. Theimpact of an increase in the price of goodc on the headcount index will be largeif there are whose income borders the poverty line (f(z) is then large) and/or ifthey consume much of goodc – xc

(z, qR

)is then large. TheCD c(z; α = 1) curve

gives the absolute contribution to total consumption ofc of those individuals withincome less thanz. It is therefore an informative statistics on the distributionof consumption expenditures, similar in content to the generalized concentrationcurveGCxc(p) for goodc – which gives the absolute contribution to totalxc con-sumption of those below a certain rankp. Forα = 2, 3, ..., progressively greaterweight to the shares of those with higher poverty gaps.

13.4 Tax and subsidy reforms

The above section gave us the tools needed to assess the impact of marginalprice changes on poverty. We may also use these tools to assess whether a revenue-neutral tax and subsidy reform could be implemented that would reduce aggregatepoverty.

30For un-normalized FGT indices, we simply multiply the results byzα.31DAD: Curves|CD curve32DAD: Curves|CD curve


For this, we need to take into account the government budget constraint, andmore particularly the net revenues that the government raises from a policy ofcommodity taxes and subsidies. Lett be the vector of tax rates on theC goods.Setting producer prices to1 and assuming them to be constant (for simplicity) andinvariant to changes int, we then haveq = 1 + t anddqc = dtc, wheretc denotesthe tax rate on goodc. Let per capita net commodity tax revenues be denoted asR(q). They are equal toR(q) =

∑Cc=1 tcxc(q). Without loss of generality, assume

that the government’s tax reform increases the tax rate on thejth commodity anduses the extra revenue raised to decrease the tax rate (or to increase the subsidy)on thelth commodity. Revenue neutrality of the tax reform requires that

dR(q) =

[xj(q) +

C∑c=1

tc∂xc(q)

∂qj

]dqj +

[xl(q) +

C∑c=1

tc∂xc(q)

∂ql

]dql = 0. (224)

Now defineγ as

γ =xl(q) +

∑Cc=1 tc

∂xc(q)∂ql

xl(q)

/xj(q) +

∑Cc=1 tc

∂xc(q)∂qj

xj(q). (225)

The numerator in (225) gives the marginal tax revenue of a marginal increase inthe price of goodl, per unit of the average welfare cost that this price increase im-poses on consumers. Equivalently, this is1 minus the deadweight loss of taxinggoodl, or the inverse of the marginal economic efficiency cost of funds (MECF)from taxingl (seeWildasin (1984)). The denominator gives exactly the same mea-sures for an increase in the price of goodj. γ is thus the economic (or “average”)efficiency of taxing goodl relative to taxing goodj. We may thus interpretγ asthe efficiency cost of taxingj relative to that of taxingl (the MECF forj over thatfor l). The higher the value ofγ, the less economically efficient is taxing goodj.

By simple algebraic manipulation, we can then rewrite equation (224) as

dqj = −γ

(xl(q)

xj(q)

)dql, (226)

which fixesdqj as a revenue-neutral proportion ofdql. This last relationship givesus a nice synthetic expression for the impact on a FGT indexP (z; α) of a revenue-neutral tax reform that increases the tax on a goodl for the benefit of a fall in thetax on a goodj:

∂P (z; α)

∂tl

∣∣∣∣revenue neutrality

= CD l(z; α)− γxl(q)CD j(z; α)/xj(q). (227)


We then wish to check whether such a tax reform would lead to a fall inpoverty. For the fall to ”ethically robust”, we would want to check that it oc-curs for any one of the poverty indices of some ethical order and for a range ofpoverty lines. To test this, it is useful to define and use normalizedCD curves, de-noted asCD c(z; α). NormalizedCD curves are just the above-definedCD curvesfor goodc normalized by the average consumption of that good,xc(q):

CD c(z; α) =CD c(z; α)

xc(q). (228)

CD curves are thus the ethically weighted (or social) cost of taxingc as a pro-portion of the average welfare cost. Comparing normalizedCD c(z; α) curvesthus allows comparing the distributive benefits of decreasing tax rates (or in-creasing subsidies) across commodities, per dollar of average welfare benefit. IfCD l(z; α) > CD j(z; α) ≥ 0, then poverty falls faster per dollar of welfare benefitif taxes onl are decreased (instead of taxes onj)33. E:19.8.30

For overall social efficiency, we must also take into the parameter of economicefficiency,γ. This parameter translates tax revenue into average welfare changes.Suppose that we were to envisage a revenue neutral tax reform that decreasestlbut increasestj. It is clear from (227) that this tax reform is poverty reducing isand only if

CD l(z; α)− CD j(z; α)

CD j(z; α)≥ γ − 1. (229)

Recall from (225) that whenγ exceeds 1, the economic efficiency cost of taxingj exceeds that of taxingl. Considering economic efficiency alone then suggestsincreasingtl and decreasingtj.

The left-hand-side of (229) shows the distributive benefit of the reform. Itcompares the fall in poverty following a decrease intl versus that following ofa fall in tj, in each case per dollar of average welfare gain. Ignoring economicefficiency considerations, decreasingtl and increasingtj is then poverty reducingif that difference is positive.

Condition (229) therefore says that decreasingtl but increasingtj reducespoverty if the distributive benefit of such a reform is larger than its economicefficiency cost. We may then check whether a tax reform is ”poverty efficient” orethically robust by verifying whether the following condition holds34: E:19.8.52

CD l(z; α)− γCD j(z; α) ≥ 0,∀z ∈ [0, z+

]. (230)

33DAD: Curves|CD curve34DAD: Dominance|Inderect tax dominance


To interpret the implication of (230), it is useful to recall the general povertydominance results of (180). Using this, it follows that if condition (230) holds,then all of the poverty indices that are members of the classΠα+1(z+) (of ethicalorderα + 1) will decrease following a fall intl and a rise intj. This can besummarized as:

sth-order poverty dominant tax reform: A revenue-neutral marginaltax reform that decreasestl and increasestj will decrease all poverty in-dices that are members ofΠs(z+) if and only if35 E:19.8.41

CD l(z; α)− γCD j(z; α) ≥ 0,∀z ∈ [0, z+

]. (231)

Considering the relationship between poverty and welfare dominance (seepage159), a similar result holds for welfare dominance:

sth-order welfare dominant tax reform: A revenue-neutral marginal taxreform that decreasestl and increasestj will increase all social welfareindices that are members ofΩs if and only if

CD l(z; α)− γCD j(z; α) ≥ 0,∀z ∈ [0,∞[ . (232)

13.5 Income-component and sectoral growth

It is just a matter of notational change to use the tools developed above toassess the poverty impact of growth in some income component, in some sectorof economic activity, or for some socio-economic group. We may assess, forinstance, by how much aggregate poverty would fall per percentage of growthrate in the industrialized sector (a sectoral change), or per dollar of growth inagricultural income (an income component that enters into aggregate income), orin some region.

13.5.1 Absolute poverty impact

Assume that total incomeX is the sum ofC income components, with quantileX(p) =

∑Cc=1 λcX(c)(p), whereλc is a factor that multiplies income component

X(c) and whereX(c)(p) is the expected value of income componentc at rankp inthe distribution of total income. Again,X(c)(p) can be, for instance, agriculturalor capital income, or income of those living in some geographic area.

The derivative of the normalized FGT index with respect toλc is then givenby36 E:19.8.42

35DAD: Dominance|Indirect tax dominance


∂PX(z; α)

∂λc

∣∣∣∣λc=1

= −CD c(z; α), (233)

where thisCD c(z; α) curve can now be interpreted as a ”component dominance”curve for income componentX(c). It can be defined formally as:

CD c(z; α) =

X(c)(FX(z))f(z) if α = 0,

αz−α∫ 1

0X(c)(p) [z −X(p)]α−1

+ dp if α 6= 0.(234)

Multiplied by a proportional changedλc, CD c(z; α) gives the marginal change inthe FGT indices that we can expect from growth in componentc. Note that thederivative of the un-normalized indexP (z; α) is simplyzαCD c(z; α).

We can intuitively expect, however, that a given percentage change will have alarger poverty impact when it applies to a larger sector or income component. Totake this element into account and to normalize by the importance of the compo-nent, we may wish instead to compute the change in the FGT indicesper dollarof per capitagrowth in the overall economy, when that growth comes exclusivelyfrom growth in componentc. This is given by the normalizedCD curves forcomponentc:

∂P (z; α)/∂λc

∂µX/∂λc

∣∣∣∣λc=1

= −CD c(z; α)

µX(c)

= −CD c(z; α), (235)

or by−zαCD c(z; α) for the un-normalized FGT index.Note that the richer the society, the lower will the fall in poverty tend to be per

dollar of per capitagrowth. This is so for two reasons. First, a richer society willtend to have a lower level of poverty and fewer poor, and hence there is less scopein such an environment for poverty to decrease significantly in absolute terms.This is captured in (234) by a lower value off(z) and [z −X(p)]+. Second, ina richer society, a 1% increase in some component will generate a larger level ofper capitagrowth in dollar terms. This is captured by a largerµX(c)

. Both factorswill thus tend to push (235) downwards. Thus, growth willarithmeticallytend tohave a smaller absolute poverty impact in richer societies.

13.5.2 Poverty elasticity

An alternative indicator of the poverty impact of growth is theelasticityofpoverty with respect to overall growth, where again that overall growth comes

36DAD: Poverty|Income component proportional growth


strictly from growth in a componentX(c). From (235), this is given by

−CD c(z; α)

P (z; α)· µX = −CD c(z; α)/P (z; α)

µX(c)/µX

, (236)

for both normalized and un-normalized FGT indices. Expressed as elasticities, theimpact of income component and sectoral growth will tend to revert to comparablemagnitudes between rich and poor countries. As shown by the right-hand-side of(236), that magnitude will mostly depend on the importance of componentX(c)

among the poor (the termCD c(z; α)/P (z; α)) as a proportion of the importanceof componentX(c) in total income (the termµX(c)

/µX).Note, therefore, that the use of poverty elasticities as opposed to poverty

changes will often give a different picture of where growth is (or has been) mosteffective in reducing poverty. Using absolute poverty changes (235) will usuallysuggest that growth reduces poverty most in poorer countries; using elasticities(236) may instead imply that growth reduces poverty most in richer countries.

13.6 Overall growth elasticity of poverty

How fast can inequality-neutral growth in the economy be expected to reducepoverty? From which group can inequality-neutral growth be expected to reduceaggregate poverty the fastest? And in which group would poverty fall the fastestdue to such growth?

Using (208) above, it can be shown that the elasticity of total FGT povertywith respect to total income – when growth in total income comes exclusivelyfrom inequality-neutral growth in groupk – equalsεy(k; z; α)37: E:19.8.47

εy(k; z; α) =α [P (k; z; α)− zP (k; z; α− 1)]

P (z; α)· µ

µ(k)(237)

for α 6= 0. Whenα = 0, (237) becomes:

εy(k; z; α = 0) =−zf(k; z)

F (z)· µ

µ(k). (238)

Equations (237) and (238) can be used and interpreted in a number of interest-ing ways.

37DAD: Poverty|FGT-Elasticity


1. ReplacingP (k; z; α) by P (z; α), P (k; z; α − 1) by P (z; α − 1) andµ(k)by µ in (237) and (238) gives as a special case the elasticity of total povertywith respect to inequality-neutral growth in the overall economy,εy(z; α).

2. ReplacingP (z; α) by P (k; z; α), F (z) by F (k; z) andµ by µ(k) in (237)and (238) yields the elasticity of poverty in groupk with respect to inequality-neutral growth in the income of that same group.

3. As discussed above, the most beneficial source of growth (for overall povertyreduction) may not come from those groups with greatest poverty. Groupsin which poverty is highest will tend to have a large[P (k; z; α)− zP (k; z; α− 1)],but we also need to consider the ratio ofµ(k) to µ: high poverty in a groupcan in principle be associated to a high level ofaverageincome.

4. The growth elasticity of the headcount,−zf(z)F (z)

, has a nice graphical inter-pretation. To see this, consider Figure45where the densityf(y) of incomeat differenty is shown. Recall that the area underneath thef(y) curve up toy = z gives the headcountF (z). The termz · f(z) in (238) is the area inFigure45 of the rectangle with widthz and heightf(z). Hence, the elas-ticity (in absolute value) of the headcount with respect to inequality-neutralgrowth is given in Figure45by the ratio of the rectangular areaz ·f(z) overthe shaded areaF (z).

It is clear, then, that this elasticity is larger than one whenever the povertyline z is lower than the (first) mode of the distribution. In fact, it will beabove one in Figure45 for any poverty line up to approximativelyz′. Forpoverty lines larger thanz′, the growth elasticity will in absolute value fallbelow 1.

This can have important policy consequences. For societies in which thepoverty line is deemed to be lower than the mode (which is usually not farfrom the median), then the headcount in these societies will fall at a pro-portional rate that is faster than the growth rate in average incomes. But forsocieties in which the headcount is initially high (larger than 0.5, say), wecan expect the growth elasticity of the headcount to be lower than 1. Thisimplies that inequality-neutral growth can be expected to have aproportion-atelysmaller impact on the number of the poor in poorer societies than inricher ones.


13.7 The Gini elasticity of poverty

13.7.1 Inequality and poverty

It may also be of interest to predict how changes in inequality will affectpoverty. The immediate difficulty here is that, unlike the case of growth in meanincome, it is not immediately obvious which pattern of changing inequality weshould consider. Indeed, as discussed above, a natural reference case for analyz-ing the impact of growth is the case of inequality-neutral growth – all incomes varyproportionately by the same growth rate in mean income. For inequality changes,which inequality index should we use to measure inequality? And, supposing thatwe were to agree on the choice of such a summary inequality index, which of themany different ways in which that index can change by a given amount should wechoose? Each of these different ways can have a dramatically different impact onpoverty.

To make this difficulty slightly more concrete, suppose that we wish to under-stand the impact of an increase in the Gini index on the poverty headcount (thisis often done in macroeconomic ”inequality-poverty-growth regressions”). Alsosuppose that this increase in the Gini comes from a mean-neutral increase by someconstant in the gap between two quantilesQ(p1) andQ(p2), with p2−p1 = η > 0.From (23), note that the impact of this on the Gini is the same, whatever the valueof p1. There are, however, several possible reactions of the headcount followingthis increase in the Gini:

1. If p1 is well aboveF (z), the headcount will not change.

2. If p1 is just aboveF (z), the headcount will increase.

3. If p1 is belowF (z) andp2 is aboveF (z), the headcount will not change.

4. If p2 is just belowF (z), the headcount will fall.

5. If p2 is well belowF (z), the headcount will not change.

Clearly, even in this special setting, the relationship between poverty and inequal-ity is far from being monotonic.

So trying to predict the effect on poverty of a process of changing inequality,through the use of a single indicator of inequality, is really to ask too much of sum-mary indices of inequality. There cannot exist any stable structural relationshipbetween inequality indices and poverty, even assuming mean income to be con-stant. This in fact casts serious doubt on the structural validity of the many studies


that regress past changes in poverty indices upon past changes in inequality in-dices, and which then try to explain or predict the impact of changing inequalityon poverty.

13.7.2 Increasing bi-polarization and poverty

What can be done, however, is toillustrate how some peculiar and simplisticpattern of changing inequality can affect poverty. Such an illustration can be madeusing the single-parameter (λ) process of bi-polarization shown by equation (26).How does poverty change when inequality changes due to this bi-polarization?For this, we use the most popular indices of poverty and inequality, the FGT andthe Gini indices (the result is exactly the same if we use the broader class of S-Gini indices). Assume that the change in inequality comes from aλ that movesmarginally away from 1. The impact on the normalized FGT index is given by

∂P (z; α)/∂λ∣∣λ=1

= α(P (z; α) +

(µ

z− 1

)P (z; α− 1)

). (239)

Thus, the elasticity of the (normalized and un-normalized) FGT poverty indiceswith respect to the Gini index is obtained as (whenα > 0) εG(z; α):

εG(z; α) = α

(1 +

P (z; α− 1)

P (z; α)

(µ

z− 1

)). (240)

When the headcount is used, we have

∂F (z)

∂λ

∣∣∣∣λ=1

= f(z)(µ− z) (241)

and thus

εG(z; α = 0) =f(z) (µ− z)

F (z). (242)

Note that even with this highly simplified process of changing inequality, the im-pact on poverty is ambiguous. It depends in part on the sign of(µ− z). Whenmean income is below the poverty line, an increase in the Gini index can – and,for the headcount index, will – imply afall in poverty.


13.8 The impact of policy and growth on inequality

13.8.1 Growth, fiscal policy, and price shocks

We may now turn to the impact of policy and growth on inequality. The ap-proach we use enables us to consider the impact on inequality of several ways inwhich income changes may occur. One is growth that takes place within a particu-lar socio-economic group. Another is growth that affects the value of some incomesources – such as agricultural income or informal urban labor income. Anotheris the impact of price changes, which affect real income and its distribution. Onemore is the impact of changes in some tax or benefit policies, such as changingthe subsidy rates on some production or consumption activity, or increasing theamount of monetary transfers made to some socio-economic groups.

For each such income-changing phenomenon, we may be interested in theabsolute amount by which inequality will change, or in the absolute amount bywhich inequality will change for each percentage change in mean real income, orin the elasticity of inequality with respect to mean income.

Assume that we have as above that total standard of livingX is the sumof C components,X(c), to which we apply again a factorλc to yield X(p) =∑C

c=1 λcX(c)(p). We then have thatµX =∑C

c=1 λcµX(c). If we are interested in

total consumption, then we may think of theX(c) as different types of consump-tion expenditures. If we are thinking of tax and benefit policy, then some of theX(c) may be transfers or taxes. If we are alternatively concerned with the impactof sectoral growth on income inequality, then we may think of theX(c) as differentsources of income, or of the income of different socio-economic groups. By howmuch, then, is inequality affected by variations inλc?

We will consider two ways of measuring inequality, the Lorenz curve and theS-Gini inequality indices – of which the traditional Gini is again a special case.The derivative of the Lorenz curve ofX with respect toλc is given by:

∂LX(p)

∂λc

∣∣∣∣λc=1

=µX(c)

µX

(CX(c)

(p)− LX(p))

. (243)

Equation (243) therefore gives the change in the Lorenz curve per unit ofλc, thatis, per 100% proportional change in the value ofX(c). Say that we predict thatincome componentX(c) will increase by approximately by 10% over the nextyear38. Then we can predict that the Lorenz curveLX(p) will move by approx- E:19.8.49

38DAD: Curves|Lorenz & Concentration curves


imately10% of (243) over that same period. How big an impact this will be oninequality will depend of course on the size of the proportional change, on theimportance of the component (µX(c)

), and on the concentration of the componentrelative to that of total incomes (the differenceCX(c)

(p)− LX(p)).A similar result is obtained for the Gini indices39: E:19.8.50

∂IX(ρ)

∂λc

∣∣∣∣λc=1

=µX(c)

µX

(ICX(c)

(ρ)− IX(ρ))

. (244)

Thus, if for instance the removal of a subsidy or the advent of an external shock isforeseen to increase by 10% the price of a goodX(c), the Gini index can be pre-

dicted to move by approximately−[10% · µX(c)

/µX

(CX(c)

(p)− LX(p))]

. (The

negative sign comes from the fact that an increase in the price of a consumptiongood leads to a fall in the real value of the expenditures made on that good.) Theimpact per dollar of change in per capita income is then given by

1

µX

(ICX(c)

(ρ)− IX(ρ))

. (245)

We may also wish to assess the impact on inequality of a change inλc, per100% of mean income change. This is given by

µX∂LX(p)/∂λc

∂µX/∂λc

∣∣∣∣λc=1

= CX(c)(p)− LX(p) (246)

for the Lorenz curve and by

µX∂IX(ρ)/∂λc

∂µX/∂λc

∣∣∣∣λc=1

= ICX(c)(ρ)− IX(ρ) (247)

for the Gini indices. These expressions are simple to compute and have a niceinterpretation. Multiplying the above two expressions by the proportional impactthat some change inX(c) is predicted to have on totalper capitaincome gives thepredicted absolute change in inequality. For instance, if we predict that growth inrural areas will lift mean income in a country by 5%, then the Lorenz curve of total

incomeLX(p) will shift by approximately0.05(CX(c)

(p)− LX(p))

, whereX(c)

is rural income. If rural income is more concentrated among the poor than total

39DAD: Inequality|S-Gini index & Redistribution|Coefficient of concentration


income, this will push the Lorenz curves up; otherwise, growth in rural incomewill increase inequality.

Finally, we may prefer to know the elasticity of inequality with respect toµX ,when growth comes entirely fromX(c). It is given by

CX(c)(p)

LX(p)− 1 (248)

for the Lorenz curve and by

ICX(c)(ρ)

IX(ρ)− 1 (249)

for the Gini indices. Thus, a proportional increase in taxes that reduces totalmean net income by 1 percent will change the Gini index by1− ICX(c)

(ρ)/IX(ρ)percent, whereICX(c)

is the concentration index of taxesX(c). This will decreaseinequality if taxes are more concentrated than net income:ICX(c)

(ρ)/IX(ρ) > 1.The elasticity of the Lorenz curve and of the Gini indices with respect toµX(c)

when growth comes entirely from a proportional change inX(c) is finally givenby

µX(c)

µX

(CX(c)

(p)

LX(p)− 1

)(250)

and

µX(c)

µX

(ICX(c)

(ρ)

IX(ρ)− 1

). (251)

13.8.2 Tax and subsidy reform

As in the case of poverty (recall Section13.4), it is useful to assess the impactof a price reform (through consumption and production taxation) on inequality.Assume that we are interested in the effects of a revenue-neutral tax reform thatincreases the tax on a goodl for a benefit of a fall in the tax on a goodj. Recallthatγ is the MECF forj over that forl – the larger the value ofγ, the larger the fallin tj that we can generate for a given revenue-neutral increase intl. The impactof a marginal revenue-neutral increase in the price of goodl is then

∂LyR(p)

∂tl


=µxl(q)

µyR

[γ

(Cxj

(p)− LyR(p))− (

Cxl(p)− LyR(p)

)]

(252)


on the Lorenz curve and an impact

∂IyR(ρ)

∂tl


=µxl(q)

µyR

[γ

(ICxj

(ρ)− IyR(ρ))− (

ICxl(ρ)− IyR(ρ)

)]

(253)on the Gini indices40. Whenγ = 1, viz, when the marginal economic efficiency ofE:19.8.48taxingl andj is the same, expressions (252) and (253) reduce to a multiple of thedifference between the concentration curves and the concentration indices for thetwo goods. For instance, the change in the S-Gini inequality index is then givenby:

∂IyR(ρ)

∂tl


=µxl(q)

µyR

[ICxj

(ρ)− ICxl(ρ)

]. (254)

It is then better for inequality reduction to tax more the good that is less concen-trated among the poor, for the benefit of a reduction in the tax rate on the othergood, which is less concentrated among the rich.

We may also wish to express the above changes in inequality per 100% changein the value ofper capitareal income. This is then given by

µyR

∂µyR/∂tl

∂LyR(p)

∂tl


= γ(Cxj

(p)− LyR(p))−(

Cxl(p)− LyR(p)

)

(255)for the Lorenz curve and

µyR

∂µyR/∂tl

∂IyR(ρ)

∂tl


= γ(ICxj

(ρ)− IyR(ρ))−(

ICxl(ρ)− IyR(ρ)

)

(256)for the Gini indices.

13.9 References

The literature on the empirical effectiveness of targeting schemes has grownsubstantially in the last years. See for instanceBisogno and Chong (2001) (onthe effectiveness of proxy means tests),Hungerford (1996) (on the effectivenessof targeting of social expenditures in the US),Gueron (1990) (on the effective-ness of targeted employment programs in the US),Park, Wang, and Wu (2002)(on the effectiveness of targeting in China),Ravallion, van de Walle, and Gautam

40DAD: Curves|Lorenz & Concentration curves


(1995) (on the effect of the targeting of social programs on persistent and tran-sient poverty in Hungary),Ravallion (2002) (on the variability of targeting acrosseconomic cycles in Argentina),Schady (2002) (on the potential for geographictargeting in Peru), andMoffitt (1989) andSlesnick (1996) (on whether in-kindtransfers are efficient for poverty reduction).

The recent literature has also queried whether ”finer” geographical targetingschemes led to more equitable and more effective poverty reduction. Evidenceon this issue – which is linked to the broader context of the benefits and costsof decentralization – is discussedinter alia in Alderman (2002) (for Albania),Bigman and Srinivasan (2002) (for India), andRavallion (1999).

The targeting literature has often resorted to an analysis of programs’ ”target-ing errors” and how they vary with program reforms. These errors are variablycalled “leakage” and “undercoverage” errors, “E” and “F” mistakes, and “Type I”and “Type II” errors. Discussion and use of them can be found invan de Walleand Nead (1995) (a useful compendium of theoretical and empirical work on tar-geting; see in particular (Cornia and Stewart (1995) andGrosh (1995)), van deWalle (1998b) (on the virtues and costs of ”narrow and broad” targeting), and inWodon (1997b) (for use of ”ROC curves” – to study the performance of targetingindicators).

Work on the impact of marginal price changes on well-being and welfareincludes: Ahmad and Stern (1984), Ahmad and Stern (1991), Creedy (1999b),Creedy (2001), Newbery (1995) and Stern (1984), for the impact of indirectmarginal indirect tax reforms on some parametric social welfare functions, mak-ing use among other things of the ”distributional characteristics” of goods;May-eres and Proost (2001), for the impact of marginal indirect tax reforms in thepresence of an externality (peak car transport);Besley and Kanbur (1988), for theimpact of marginal changes in food subsidies on FGT poverty indices;Creedyand van de Ven (1997), Creedy (1998a) andCreedy (1998b), for the impact ofprice changes and inflation on well-being and social welfare;Liberati (2001),Mayshar and Yitzhaki (1995), Mayshar and Yitzhaki (1996), Yitzhaki and Thirsk(1990), Yitzhaki and Slemrod (1991) andYitzhaki and Lewis (1996), for the im-pact of marginal indirect tax reforms on classes of social welfare indices using”marginal” dominance analysis;Lundin (2001); for marginal dominance analysisfor a marginal tax reform affecting the importance of an externality (the presenceof carbon dioxide);Makdissi and Wodon (2002), for the use of CD curves in theanalysis of marginal poverty dominance; andYitzhaki (1997), for the impact oninequality of marginal price changes.

14 TARGETING IN THE PRESENCE OF REDISTRIBUTIVE COSTS193

14 Targeting in the presence of redistributive costs

The targeting schemes analyzed in Chapter13 assumed that there exist char-acteristics on which governments can condition benefit transfers. For instance,we modelled the impact on poverty on given 1$ to everyone that belonged tosome socio-demographic groupk. These transfers were therefore not dependenton levels of income, since we implicitly assumed that income levels were notobservable.

We will suppose now that the distribution of population characteristics (includ-ing the levels of original income) can be observed without costs (for expositionalsimplicity), but that there do exist costs to granting state support. The optimaltargeting rules that follow are different the above. They also differ from those ofthe traditional study of optimal income taxation, where labor supply and incomegeneration are endogenous but where redistributive imperfections are generallyruled out. Instead, assume that the behavior of agents is fixed (e.g., constrained bylabor market conditions) under alternative income support arrangements, exceptfor the feature that such agents may freely choose whether to participate in theincome support programmes. Given the plausible presence of redistributive costswhose size may vary with individuals, the state then wishes to minimize the valueof a poverty index, taking into account either the opportunity cost of governmentexpenditures or the constraint of an aggregate redistributive budget for povertyalleviation.

The existence of redistributive costs leads to policy criteria that weigh effi-ciency as well as redistributive objectives. It also has important implications forthe consideration of the principles of vertical and horizontal equity.

14.1 Poverty alleviation, redistributive costs and targeting

Redistributive costs can be administrative and arise generally from the diffi-culty of monitoring the true level of benefit entitlement. In that sense, they can beinterpreted as the certainty-equivalent costs of the presence of imperfect target-ing. The more difficult is it to ascertain accurately someone’s true entitlement, thegreater the expense of removing the associated imperfect targeting. Redistributivecosts can also be incurred by benefit recipients and they may then have to be de-ducted from the gross impact of income support to yield net poverty relief. Forexpositional simplicity, we assume here that all costs take the form of a welfareparticipation burden and that they are borne directly by the participating poor.

Assume that the poverty alleviation objectives of the state are to minimize the


poverty index,P (z) :

P (z) = n−1

n∑i=1

π(yi + NB i; z) (257)

whereyi is the initial income of individuali, z is the poverty line, andNB i is thenet benefit to individuali of the availability of a non-negative gross benefitB∗

i .As we shall define it below more precisely,NB i is no greater thanB∗

i since it isreduced by participating/claiming costs and administrative expenses in spendingB∗

i .In general,P can of course take a variety of shapes and can also be interpreted

as an element of a larger social welfare function, an element to whose maximiza-tion the government allocates a totalper capitabudgetB, such that

n−1

n∑i=1

Bi = B (258)

with Bi the level of gross benefitactuallyexpended on individuali.Let B∗

i then represent the benefitofferedto individual i, and denote byci thenon-negative cost toi of acceptingB∗

i . If B∗i is less thanci, then the benefit

awardedBi and the net benefitNB i will be zero. WhenB∗i ≥ ci, thenBi = B∗

i

andNB i = B∗i − ci. Define an indicator functionI[x] that the takes the value 1

whenx is true and 0 whenx is false. Then

Bi = I[NB i ≥ 0]B∗i . (259)

Costsci are only incurred whenB∗i is taken up. Think for instance ofci as an

administrative cost necessary to grant support toi. B∗i is then the level of gross

expenditures which the state would consider spending oni, Bi is the level of grossexpenditures actually spent oni, andNB i is the level of benefit net of administra-tive costs which eventually reaches the individual.

The government thus wishes to choose the variousB∗i , i = 1, . . . , n, to min-

imize (257), subject to (258). Note thatNB i andBi are not differentiable withrespect toB∗

i at the point at whichi just accepts state support,viz, whenB∗i = ci.

This causes no analytical difficulty since the optimum solutions for theB∗i never

have to lie at these corner points.Defineλ as the Lagrange multiplier associated to the budget constraint, and

π(1) as the non-negative derivative ofP with respect toy. For now, assume thatπ is continuous, differentiable and convex – we will discuss later the problem


for the important headcount case for whichπ does not fulfill these conditions.The government then wishes to ensure that the following condition is met at theoptimum values ofB∗

i andλ∗ for each of thei in receipt of state support:

π(1)(yi + NB i; z) = λ. (260)

The optimum value ofλ reflects the social opportunity cost of spending publicresources. Note that a benefit offerB∗

i below ci will not matter, for thenBi =NB i = 0, that is, it has neither a cost nor a benefit.

Whetheri should derive any net benefit at the optimum solution depends onits original incomeyi and upon the redistributive costci that it faces. Figure46illustrates this dependency. The straight lineλB∗

i displays the opportunity costin social welfare of grantingi a benefitB∗

i . The claim of such a benefit willbring a net benefitNB i that will decreaseπi (the contribution ofi to the povertyindex P ) once the claiming burden has been paid off. The shape of−πi above0 depends on the convexity of the functionπ (yi + NB i; z) and on the originalincomeyi. Individuals for which it is possible to find a level of expenditureB∗

i

for whichπ(yi; z)− π(yi + B∗i − ci; z) ≥ B∗

i will be granted eligibility. Whetherπ(yi; z)−π(yi +B∗

i − ci; z) eventually reachesλB∗i – and whether, therefore, the

poverty alleviation benefit of granting state support toi is worth its opportunitycost – will thus also hinge on the size ofci, the size of the redistributive costs.

Four cases are shown on Figure46 . Individual 1, who faces expensesc1,will happily be offered and claim benefitB∗

1 , with a net benefit reward ofNB1 =B∗

1 − c1. Individual 2, who faces the same redistributive cost but has a higheroriginal income, will barely be deemed eligible, just as is the case with individual3 with a lowery but a much higherc. Once qualified and recipients, however,individuals 2 and 3 will receive what may be a sizeable net and gross benefit,thus showing an important discontinuity in the function of optimal state support.From the above optimality condition (260), we may indeed note that, whenB∗

i isclaimed, the corresponding net benefit equalizes the post-benefit income (net ofredistributive costs) of all claimants. In other words, we have at the optimum that:

y1 + NB1 = y2 + NB2 = y3 + NB3 (261)

Individual 4, who enjoys a relatively largey and also faces high costs, doesnot benefit from the optimal program. Hence, all those individuals with:

• original income no less thany2 and costs greater thanc2,

• or original income greater thany2 and costs no less thanc2,


• or original income no less thany3 and costs greater thanc3,

• or original income greater thany3 and costs no less thanc3

ought not to receive income support. The greater the redistributive costci, the lessthe chance of receiving a positiveB∗

i , but the greater the optimalB∗i is if support

should be granted. Furthermore, the greater his original incomeyi, the less likelyan individual is to take up a positiveB∗

i and the smaller isB∗i if it is received.

14.2 Costly targeting

14.2.1 Minimizing the headcount

Consider now the case in whichP (z) = F (z) is the headcount.π (yi + NB i; z)is then discontinuous at the point at whichyi +NB i reachesz. This leads the stateto distributeBi in such a way as to raise toz as many of the individuals as possi-ble. In order to do this, it will grant income supportz − yi + ci first to that poorindividual for which that amount is lowest, then to that poor individual with thesecond lowestz − yi + ci, and so on, until the budget has run out. This relativelystraightforward recommendation is similar to Proposition I ofBesley and Coate(1992) in the absence of redistributive costs.

14.2.2 Minimizing the average poverty gap

Consider now the average poverty gap asP (z). It is continuous but not contin-uously differentiable everywhere inyi + NB i. Choosing to minimize the averagepoverty gap leads the state to choose benefit recipients in order to maximize thereturns in poverty gap reduction per unit of government expenditure. In otherwords, the state wishes to minimize the aggregate level of redistributive costs in-curred for a given total level of expenditures spent on the poor. Or, said againdifferently, the state attempts to fill as much as possible of the total poverty gaps,avoiding as much as possible in the process spending wasteful redistributive costs.

Because there are fixed costs to granting income support, once a desirablebenefit recipient has been identified, the state wishes to spend on him as much asis necessary to raise his net income toz. Thus, the government’s agency ought tocompute an ”efficiency” ratio(z − yi) / (z − yi + ci) of full poverty gap reductionover benefit expenditure for each individuali, and grant benefitB∗

i = z − yi + ci

first to that individuali with the greatest ”efficiency” ratio, thenB∗j = z − yj + cj

to that individualj with the second highest ratio,etc., until the budget is depleted.


Because some income support to some relatively poor individuals may yet arouserelatively high redistributive costs, the state may find it preferable to grant incomesupport to some richer individuals among the poor.

An individuali should then optimally benefit from state support if the absolutechange in his poverty gap does not fall below the opportunity cost of that change.For all such eligible individuals, the state also wishes to raise their net income tothe poverty line,z, with gross benefits or expenditures equal toBi = z − yi + ci.Hence, at the optimum, an individuali will be deemed eligible to state support if

z − yi ≥ λ∗ (z − yi + ci) , (262)

whereλ∗ is the opportunity cost of government resources. A person or a soci-ety may feel, for instance, that the benefit of a $1 reduction in the poverty gap isat the margin exactly worth a $2 decrease in tax raised, with a consequent valueof λ∗ = 0.5. Thus, for individuals to be optimally eligible, the social benefit ofpoverty gap reduction, net of the administrative and claiming costs, must exceedthe opportunity cost of gross state expenditures. Otherwise, state expenditureswould be better spent elsewhere than on income support, or taxes could be bene-ficially cut down.

The identification of an optimal set of claimants can thus be made on the basisof an opportunity cost,λ∗, and on the interaction ofz, yi andci. From (262), wesee that alli with

ci ≤ 1− λ∗

λ∗(z − yi) (263)

will be optimal recipients of income supportBi = z − yi + ci. A value ofλ∗

equal to 1 would eliminate alli with ci greater than zero. The lower the valueof λ∗, the lower the opportunity cost of government expenditures, and the easierit is for poor individuals to qualify for income support. Condition (263) abovethus explicitly defines a (convex) set of income support recipients with a borderfixed by a linear trade-off betweenci andyi. To locate precisely that border, theopportunity cost of government expenditures (λ∗) must be set. Doing this can bedone in at least three ways:

• through settingλ∗ directly, taking into consideration the social welfare valueof reducing the aggregate tax burden;

• through establishing a budget level that reflects the government’s politicalor economic ”capacity” to pay, and then by deriving the implied value ofλ∗;


• through identifying a point(yi, ci) that lies precisely on the border of theeligibility set, and then by calculating the impliedλ∗.

Let us illustrate the third way – which is both easy to follow and easy to inter-pret. At the borderline of eligibility, we note that:

λ∗ =z − yi

z − yi + ci

. (264)

Suppose, for instance, that we judge an individual withci/z = 0.25 andyi/z =0.5 to be deemed just barely eligible to income support. It follows from (264) thatλ∗ = 2/3. This says that a $2 decrease in the average poverty gap is deemed, atthe margin, socially worth a $3 increase inper capitataxes. With this information,the entire optimal set of eligible can be identified. The derived value ofλ∗ = 2/3says, for instance, that all those with no original income at all would yet receiveno state support if their redistributive costs exceeded 50% ofz. All those deemedeligible would, however, receiveBi = z− yi + ci, and would see their net incomeraised toz.

14.2.3 Minimizing a distribution-sensitive poverty index

Consider finally the case in which the optimal income support policy mustbe geared towards minimizing the average of the squared poverty gaps, namely,P (z; α = 2). As for the above, individuals found to be optimal beneficiaries ofstate support will be those whose fall in poverty exceeds the opportunity cost ofthe gross expenditures needed to decrease poverty,viz, those for whom we canfind aBi such that

(z − yi)2 − (z − yi −Bi + ci)

2 ≥ λ∗Bi, with z ≥ yi. (265)

For those eligible (recall (261)), we will have thatyi + B∗i − ci = e, wheree is

that constant to which the net income of all eligible individuals should be raised.Developing (265), we find that eligible individuals will fulfil the prerequisite that:

λci ≤ −e2 + 2ze− λe + yi2 − 2zyi + λyi. (266)

Because the return to decreasing the squared poverty gap decreases as net incomeapproachesz, redistributive policy will benefiti only if λ ≤ 2 (z − yi), the initialmarginal social welfare return to raisingi’s income. If this condition were notsatisfied,i would not receive income support even ifci were nil.


Equation (266) implicitly defines the set of the eligible based on their valuesof yi andci. Those with lowyi or low ci will be granted eligibility. The valuee to which the level of all eligible individuals’ income will be raised dependsimplicitly on the opportunity costλ∗. The optimality condition requires that themarginal welfare gain of increasingBi (whenBi = B∗

i ) is precisely equal to theopportunity costλ∗ of such additional expenditure. If the welfare gain were higherthan its opportunity cost, it would be preferable to increase support to the relevanti (instead of granting assistance to a new, additional recipient) since redistributivecostsci would then already have been ”sunk”. Hence, it must be that

λ = 2(z − e). (267)

Using (266) and (267), the border of the eligibility set can now be defined by

ci ≤ y2i − 2eyi + e2

i

2 (z − e). (268)

To define the set of optimally eligible individuals, we therefore need either toset directly the opportunity cost of state expenditures,λ∗, to agree on a povertyalleviation budgetB, to identify one of the border points of the eligibility set, orto rule on the valuee at which the net income of all eligible individuals should beraised. In everyone of these cases, a value judgement is expressed on the socialinterest of using costly redistributive tools. This value judgement determines theset of the eligible poor as well as the level of their post-transfer income.

Take the same ”borderline” individual as above, withci/z = 0.25 andyi/z =0.5. For such a border point, we finde/z = 0.809 andλ/z = 0.382. Using(268), it follows that whenyi = 0, for instance, allocation costs can go up toci/z = 1.71 as a proportion of the poverty line before eligibility to income sup-port is withdrawn. For all eligible individuals, net income will be raised to aproportione/z = 0.809 of the poverty line, with state expenditure oni equal toBi = 0.809z− yi + ci. Incomes will not be raised to the poverty line since, aboveyi + NB i = 0.809z, the marginal welfare gain of additional state expenditure islower than its opportunity cost.

Figure?? summarizes graphically these policy implications for theP (z; α =2) index by showing the set of the eligible as a function of their original incomeyand of the redistributive costsc that helping them incurs. The vertical axis showsthe level of expenditures which it is optimal to grant to individuals according againto their value ofy andc. For ease of reading, all variables are normalized by thepoverty linez.


14.2.4 Optimal redistribution

There are several important lessons to be gained from the above discussion,and in particular from Figure??. First, state support for the eligible poor com-pensates them fully for their lower original income and/or higher redistributiveexpenses. In other words, once government funds have been expended on them,they should be large enough to raisei’s net income to the level of all other eligibleindividuals.

Second, the case ofci = 0 is clearly a special case of the above. On Figure??, this special case shows above the line ofc = 0. In the more general frame-work in which redistributive costs are allowed, however, some well-known andlargely intuitive results do not hold any more. It is not true, for instance, thatthe state is indifferent as to the identity of the poor recipients with the sameyi:as seen above, there are clear directions on who among the poor should be tar-geted for poverty relief. Figure?? also shows that all individuals with zero costsare optimally eligible to state support regardless of their own resources. Withpositive and growing costs, however, eligibility quickly becomes restricted to thevery poor. As redistributive costs rise, the social gain of supporting those withrelatively high original incomes rapidly falls below the opportunity cost of stateresources. Hence, as long as there prevails at least some redistributive cost, not allindividuals should be raised to the same final net income, but an optimal selectionneeds to be made on the basis of original income and levels of redistributive costs.

This last result does not require variability in the redistributive costs acrossindividuals. The more positive the correlation between levels of original incomeand redistributive costs, the greater the chance that poor individuals would bedeemed entitled to state support. So long as there exists at least some burdento reaching the better off poor, the richer of them may, however,not appear inthe optimal set of benefit recipients. This can be seen on Figure?? for thoseindividuals withy/z at or slightly below 0.8, who become suddenly ineligible withsmall increases in theirc/z. This discontinuity of the optimal level of state supportas a function of original income also naturally occurs when using poverty indicesthat are discontinuous in income (such as the poverty headcount). Redistributivecosts introduce these discontinuities for continuous poverty indices too as well.

Third, the model above suggests some features of optimal redistribution policythat are somewhat disturbing, at least when considered in the context of the usualdiscussions of efficiency and equity. On account of the variability of redistribu-tive costs across individuals, some relatively richer individuals might be deemedoptimally eligible to income support whereas some poorer individuals might be


denied such support. Supporting the poorer and not the richer may generate agreater level of vertical equity and of redistribution, but this is clearly not neces-sarily optimal if individuals differ in ways (other than their original income) thatare relevant to the redistributive effectiveness of the state.

Finally, note in Figure?? that all eligible individuals will receive enough in-come support to raise their net income to0.809z. There are, however, many in-dividuals with original income less than0.809z who will not qualify for statesupport and whose final income will consequently have to remain below0.809z.Once optimal state support has been allocated, therefore, some of the originallypoorer individuals will enjoy a level of net income above that of formerly richerindividuals.

This reranking of individuals in the dimension of net incomes and welfare isespecially likely when richer individuals present high levels of redistributive costs.It will also occur among those richer and poorer individuals that face identicalredistributive expenses. Even more significantly, there are some originally richerindividuals with a relatively lowci that will be denied eligibility and end up worseoff than some initially poorer individuals with higherci. If deemed to be sociallyimportant, the explicit consideration of horizontal inequity as a social evil wouldthus necessarily need to put a constraint on these policies.

14.3 References

The literature on optimal income taxation is large and varied. A review canbe found inSlemrod (1990), Stern (1984) andTuomala (1990) – see alsoKan-bur, Keen, and Tuomala (1994a). The literature typically allows for labor supplyand income generation to be endogenous, but generally supposes the absence ofredistributive imperfections – seeStern (1982) for an exception to this.

Budgetary rules under the more specific objective of poverty reduction are dis-cussed inBourguignon and Fields (1990), Bourguignon and Fields (1997), Kanbur(1985), Chakravarty and Mukherjee (1998). Additional works on optimal incometaxation and optimal benefit provision includeBesley (1990) (for a comparisonof means testing and universal provision of public assistance),?) and Besleyand Coate (1995) (on the desirability of workfare constraints),Creedy (1996)(for a comparison of means testing and linear taxation for poverty reduction),Fortin, Truchon, and Beausejour (1990) (on comparing workfare and negative in-come tax systems),Glewwe (1992) (for designing benefit allocation rules whenincome is not observed),Haddad and Kanbur (1992) (for the potential role ofintra-household allocation issues),Immonen and et al. (1998) (for a comparison


of means testing and categorical benefit provision),Kanbur, Keen, and Tuomala(1994b) (for differences in the optimal rules implied by welfarist and non-welfaristsocial objectives),Keen (1992) (for the link between needs and optimal alloca-tions of benefits),Thorbecke and Berrian (1992) (for general-equilibrium optimalbudgetary rules),Viard (2001) (for a theory of optimal categorical transfer pay-ments), andWane (2001) (for optimal taxation when poverty generates negativeexternalities on society).

203

Part V

Estimation and inference fordistributive analysis

15 AN INTRODUCTION TODAD: A SOFTWARE FOR DISTRIBUTIVE ANALYSIS204

15 An introduction to DAD: A software for distribu-tive analysis

15.1 Introduction

DAD – which stands for ”Distributive analysis/Analyse distributive” – is de-signed to facilitate the analysis and the comparisons of social welfare, inequality,poverty and equity using micro (or disaggregated) data. It is freely distributedand its use does not require purchasing any commercial software.DAD’s featuresinclude the estimation of a large number of indices and curves that are useful fordistributive comparisons. It also provides various statistical tools to enable statis-tical inference. Many ofDAD’s features are useful for estimating the impact ofprograms (and reforms to these programs) on poverty and equity.

The first version ofDAD was launched in September 1998. It initially cameto life following a request by the Canadian International Development ResearchCentre (IDRC) to Universite Laval to support research then carried out in Africa inthe context of the IDRC’s programme on the Micro Impacts of Macro-economicand Adjustment Policies (MIMAP). Improved versions ofDAD subsequently ap-peared as errors and bugs were corrected and as attempts were made to make itmore reliable, more flexible and broader in scope. The current version is 4.3 –version 4.4 should be launched in June 2004.

Several factors motivated us in the process of buildingDAD. First, there seemedto be an ever increasing need for developing-country analysts to carry out povertyand inequality ”profiles”. Much of development policy is indeed now assessedthrough poverty criteria, and this is carried out among other things through theelaboration of poverty assessments, poverty reduction strategy papers (the nowwell-knownPRSP’s), poverty and social impact analyses,etc.. Much of these dis-tributive assessments had earlier typically been done by foreign consultants andby international organizations’ technical staff. Little was left in the form of na-tional capacity building and local empowerment following these largely externalexercises. Local researchers and national policy analysts typically felt alienatedby these poverty assessments which they often did not understand and which theycould not usually influence. To break that segregation between foreign experts andlocal policy makers and analysts, it seemed useful to introduce tools that wouldbenefit developing country analysts pedagogically and operationally.

Second, micro-data accessibility was increasingly becoming less of a problemto developing-country researchers. This followed what had occurred in more de-


veloped countries some 15 to 20 years earlier when data tapes and records startedto circulate widely in research centers and universities. This was made possi-ble in large part by the amazing increase in storage and processing speed thatthe computer revolution was creating. Developing-country analysts were gainingfrom the same advances, though with some lag due to tighter resource constraints.Furthermore, in addition to the computing and technical demands that handlinglarge data sets involved, developing country analysts often had had to deal withdata accessibility difficulties. This meantinter alia having to face skepticism andrent-seeking behavior from statistical agencies and international organization staffwhen requesting access to data that were supposed in principle to be public. Thatproblem had also become less severe by the end of the 1990’s, in part due to out-side pressure. To process and analyze these data then typically became the nextbarrier to break.

Third, much of distributive analysis was (and is still) handled as if it was notsubject to statistical imprecision. Indeed, a considerable amount of energy and re-sources seems to be wasted in discussions of poverty and inequality ”results” thatcannot be trusted on formal statistical grounds. Even changes in poverty head-counts of around 4% or 5% are often statistically insignificant within the usualstatistical precision criteria. Needless to say, the efforts deployed by analysts andpolicy makers to account for variations of less than 1% or 2% (as often occurs) inpoverty rates are typically a pure loss of resources. This unfortunate state of affairsneeded to be remedied by a much greater use of appropriate statistical techniques.Though conceptually relatively simple, the use of these techniques neverthelessrequired reading through some technical literature as well as writing tedious com-puter programmes.DAD was in large part written to help bypass these hurdles.Achieving this meant clearing the ground of statistically insignificant results andleaving more time and resources for the interpretation of those distributive find-ings that were statistically significant.

DADwas thus conceived to help policy analysts and researchers analyze povertyand equity using disaggregated data. An overriding operational objective was totry to makeDAD’s environment as accessible and as user friendly as possible.Carl Fortin, our co-author, convincingly argued from the start that we should pro-gramDAD in the Java programming language. An object-oriented language, Javacreated a new paradigm of platform independence: once written, Java applicationscould run on any operating system as well as on the internet. Conceived by Sunin 1995, Java could still be considered in 1998 to be an infant programming lan-guage. By now, however, it has become an important pillar of the programmingand internet industry. To makeDAD completely free of charge, we also chose not


to tie its use to statistical commercial softwares such as Excel, SPSS or STATA.We therefore opted to designDAD from scratch using some of Java’s packages asbuilding blocks.

To makeDAD as user friendly as possible, we use pop-up application windowsand spreadsheets as the main working tools. This enables users to visualize a lotof information at a glance, and to manage that information easily. Most of therelevant variables and options needed for running applications can be selectedfrom single application windows.DAD’s use of spreadsheets has the advantageof displaying the entire data sets to be used. Small data sets can easily be enteredmanually. Changes to cell values can be made directly on the spreadsheet. Theresults of operations on data vectors can be checked easily.DAD also allowsloading two data bases simultaneously, and to display each of these two data basesalternatively on the spreadsheet. This makes it easy to carry out applications witheither one or two data bases. That structure also enablesDAD to account forwhether the data bases are independent when it comes to computing standarderrors on distributive estimators that use information from two samples.

15.2 Loading, editing and saving databases inDAD

DAD’s databases are displayed on spreadsheets similar to those of SPSS, STATA,or Microsoft’s Excel – see Figure1. Every line in a sheet represents one obser-vation or one data ”record”. Typically, an observation consists of one of the sam-pling or statistical units that were drawn into a survey. In distributive analyses, asampling unit is often a household since it is households that are typically the lastsampling units in surveys. When observations represent households, there willthus be as many lines or observations in the data as there are households drawninto the household survey. The statistical units (or units of interest) are usually(for ethical reasons) the individuals. Even though the sampling units originallydrawn into the survey may have been the households, data sets are sometimes re-organized in such a way that each individual in a household is assigned its ownline of data. There will then be as many observations in a data set as there areindividuals found in the households.

A database used inDAD is then a matrix (a set of columns) whose length is thenumber of observations discussed above and whose width is the number of vari-ables contained in the database. Each column displays the values of a variable. Avariable has as many values as there are observations in the database. All columnsin DAD are therefore of the same length. Variable values can have a float format– indicating, for example, the level of household income – or an integer format –


Figure 1:The spreadsheet for handling and visualizing data inDAD.

showing for instance the socio-economic category to which a household belongs.There are several options for entering data intoDAD. The first one is to create

a new database inDAD and then enter the variable values manually. This can beuseful for exploratory or pedagogical purposes. Clearly, however, this option notconvenient for entering large databases intoDAD. A second option for readingexisting data bases intoDAD is done by using well-known copy/paste facilities.Before doing this, however, a new data base must be created inDAD and thenassigned a number of observations (or size) that corresponds to the length of thevariables that will be copied/pasted.

The third possibility for entering data intoDAD is typically more reliable(and also faster) than the first two and involves two steps. The first step savesthe database in an ASCII (or a text) format. The way in which this is donein practice depends on the software in which the data were previously handled.DAD’s Users Manual gives examples of such output procedures for several com-


mon commercial softwares. One fast alternative to this is offered by the useof STAT/TRANSFER (note however that this requires buying a license), whichtransforms databases rapidly from the most popular formats into an ASCII for-mat. Once the database is in ASCII format, it can easily be imported usingDAD’sData Import Wizard. The wizard ensuresinter alia that the imported databasedoes not contain missing or unreadable values. Once the data are read inDAD,they can be submitted to a number of arithmetical and logical operations, variablenames can be added or changed, and new variables can be created. Databasescan subsequently be saved inDAD’s preferred ASCII format (identified by theextension.daf).

As already mentioned, many ofDAD’s applications can use simultaneouslytwo databases. To use a second database, the user should first activate a secondfile by clicking on the buttonFile2, and then follow the same procedures as forloading a first file.

15.3 Inputting the sampling design information

The process of generating random surveys usually displays four importantcharacteristics:

• the base of sampling units (the base from which sample observations aredrawn) is stratified;

• sampling is multi-staged, generating clusters of observations;

• observations come with sampling weights, also called inverse probabilityweights;

• observations may have been drawn with or without replacement;

• observations often provide aggregate information on a number of units ofinterest (such as the different individuals that live in a household).

Recent versions ofDAD enable taking that structure into account in the estimationof the various distributive statistics as well as in the computation of the samplingdistributions of these statistics.

When a data file is first read or typed intoDAD, the survey design assignedto it by default is Simple Random Sampling. This supposes that the observationswere independently selected from a large base of sampling units. This, however,is rarely how surveys are designed and implemented. Once the data are loaded,


the exact sampling design structure can nevertheless be easily specified. This isdone using theSet Sample Design dialogue box. Specifying the sample designstructure can involve lettingDAD know about (up to) 5 vectors (see Figure2).

Figure 2:TheSet Sample Design window inDAD.

• STRATA: this specifies the name of the variable (in an integer format) thatcontains the Stratum identifiers.

• PSU: this specifies the name of the variable (in an integer format) that con-tains the identifiers for the Primary Sampling Units.

• LSU: this specifies the name of the variable (in an integer format) that con-tains the identifiers for the Last Sampling Units.

• SAMPLING WEIGHT: this specifies the name of the Sampling Weights vari-able.

• CORRECTION FACTOR: this providesDAD with a Finite Population Cor-rection variable.


15.4 Applications inDAD: basic procedures

Once data have been read intoDAD and that the sampling design has beenspecified, the field is wide open for the estimation of distributive statistics and forperforming distributive tests. For every application programmed inDAD, thereis a specific application window that facilitates the specification of variables, pa-rameters and options to generate the desired distributive statistics. For example,Figure33 shows the specific application window for computing the FGT povertyindex with one distribution. There is a separate specific window for the case oftwo distributions.

Most application windows, including that of Figure33, are divided into threepanels. The first panel is used to specify the relevant database variables neededfor the estimation. The second panel (generally at the bottom of the applicationwindow) specifies the parameter values and options to be used by the estimator –examples include the level of inequality aversion, the value of the poverty line andthe percentile to be considered as well as whether indices should be normalizedand whether statistical inference should be performed. The third panel activatesbuttons in order that various types of results may be generated. Some applicationwindows can also generate popping-up dialogue boxes. One example of this canbe found when clicking on theCompute linebutton in thePoverty|FGT applica-tion window. This serves to specify the manner in which the poverty line shouldbe (or was) estimated.

The following basic variables are typically required for carrying outDAD’scomputations.

• VARIABLE OF INTEREST. This is the variable that usually captures livingstandards. It can represent, for instance, expenditures per adult equivalent,calorie intake, normalized height-for-age scores for children,etc..

• SIZE VARIABLE . This refers to the ”ethical” of physical size of the ob-servation. For the computation of many distributive statistics, we will in-deed wish to take into account how many relevant individuals (or statisticalunits) are found in a given observation. We might, for instance, wish toestimate inequality across individuals or the proportion of children who arepoor. Individuals and children will then be respectively the statistical unitsof interest. Households do differ, however, in their size or in the numberof children they contain.DAD takes this into account through the use ofthe SIZE VARIABLE . When an observation represents a household, com-puting inequality across individuals requires specifying household size as


the SIZE VARIABLE , whereas computing poverty among children requiresputting the number of children in the household as theSIZE VARIABLE .Ifthe statistics of interest were the proportion of households in poverty, thenno SIZE VARIABLE would be needed.

• GROUP VARIABLE. (This should be used in combination withGROUP

NUMBER.) It is often useful to limit some distributive analysis to some pop-ulation subgroup. We might for example wish to estimate poverty withina country’s rural area or within the group of public workers. One way todo this is to setSIZE VARIABLE to zero for all of the observations that falloutside these groups of interest. Another way is by defining aGROUP VARI-ABLE whose values will allowDAD to identify which are the observationsof interest.

• GROUP NUMBER. GROUP NUMBERtellsDADon which value of theGROUP

VARIABLE to condition the computation of some distributive statistics. Thevalue forGROUP NUMBERshould be an integer. For example, rural house-holds might be assigned a value of 1 for some variable denoted asregion.SettingGROUP VARIABLE to region andGROUP NUMBERto 1 makesDADknow that we wish the distributive statistics to be computed only within thegroup the rural households.

• SAMPLING WEIGHT. Sampling weights are the inverse of the sampling rate.They are best specified once and for all using theSet Sample Design win-dow (as discussed above). Distributive statistics (but not necessarily theirsampling distribution and standard errors) will be left unchanged, however,if no variable is given for Sample Weight (inSet Sample Design window)and if the product of the sampling weight and size variables is subsequentlyspecified as theSIZE VARIABLE in the relevant application windows.

DAD’s applications with two distributions can be launched after having loadedtwo databases. Each time one launches an application that can support two distri-butions, the dialog box, shown in Figure3, opens to allow the user to specify thedesired number of distributions to be used as well as the name of the databases forthese distributions. The application window for two distributions is very similarto that for one. The main difference is the addition of a second panel to spec-ify the relevant variables to be used for the second distribution. The applicationfor two distributions generally serves to compute distributive differences acrossthe two distributions. For curve applications with two distributions, for instance,differences between the curves of the two distributions can usually be drawn.


Figure 3:Choosing between configurations of one or two distributions.

15.5 Curves

DAD has built-in tools that facilitate the use of curves to display distributiveinformation. Say, for instance, that we wish to graph a Lorenz curve. We cancompare it to the45 line to observe by much income shares differ from popula-tion shares. This is done by following these steps:

• From the main menu, select the submenu :Curve|Lorenz. Indicate that thenumber of distributions equals one.

• After choosing the application variables, click on the buttonGraph to drawthe first Lorenz curve.

• If you would like to draw another Lorenz curve for another variable of in-terest, return to the Lorenz application window , re-initialize the variable ofinterest and click again on the buttonGraph.

• When the graph window appears, click on the buttonDraw all to plot all ofthe curves.


• If you wish to draw the45 line, select (from the main menu of the graphwindow)Tools |Properties, and activate the optionDRAW THE 45 LINE

Figure13shows an example of Lorenz curves drawn byDAD.

Figure 4:Lorenz curves for two distributions

We can also compare two Lorenz curves to test for inequality dominanceof one distribution over the other. For this, we choose again the applicationCurves|Lorenz, but this time with two distributions.

DAD can also usually draw curves that show how the levels of some distribu-tive statistics vary with ethical parameters – such as inequality or poverty aversionparameters. Take for instance the Atkinson index of inequality. It may be infor-mative to check how fast it varies as a function ofε, its parameter of inequalityaversion. To do this, follow these steps:

• From the main menu, select the submenu :Inequality|Atkinson. Indicatethat the number of distribution equals one.


Figure 5:Differences in Lorenz curves drawn byDAD.

• After setting the application variables, click on the buttonRangeand spec-ify the desired range for the parameterε.

• Click on the buttonGraph to draw the curve that shows the Atkinson indexagainst the risk aversion parameterε.

• When the graph window appears, click on the buttonDraw to plot the curve.

15.6 Graphs

Recent versions ofDAD are quite flexible in terms of editing, saving and print-ing graphs. On most application windows, a buttonGraph is available to drawgraphs instantly. The type of graphs drawn depends on the application and onthe type ofGraph buttons selected. There are for instance twoGraph buttonsin the Poverty|FGT Index application window. Clicking on theGRAPH buttonplots estimates of the FGT index for a range of alternative poverty lines. Clickingon theGRAPH2 button draws instead estimates of the equally-distributed poverty


gap that is equivalent to the estimated FGT poverty index, and this for a range ofpoverty aversion parametersα.

Most of the options for editingDAD’s graphs can be accessed from theGraphProperties dialogue box – see Figure6. DAD’s graphs can also be saved in avariety of formats. Table1 lists some of them.

Curves are useful tools to check various types of distributive dominance. Table2 sums up some of the links between some of the applications and curves foundin DAD and the tests for various orders of social welfare, poverty and inequalitydominance.

Table 1: Available format to saveDAD’s graph.Extension Description

*.png Portable Network Graphic*.pmb Bitmat Image file*.tif Tag Image File Format*.jpg JPEG File Interchange Format*.pdf Portable Document Format*.ps Postscript

Figure 6:The dialogue box for graphical options


Table 2: Curves and stochastic dominance.Primal approach Dual approach

Order Social welfare1 Distribution|Distribution function Curves|Quantile2 Dominance|Poverty Dominances = 2 Curves|Generalized Lorenzs Dominance|Poverty Dominance

Poverty1 Dominance|Poverty Dominances = 1 Curves|Poverty Gap2 Dominance|Poverty Dominances = 2 Curves|CPGs Dominance|Poverty Dominance

Inequality1 Dominance|Inequality Dominances = 1 Curves|Normalised Quantile2 Dominance|Inequality Dominances = 2 Curves|Lorenzs Dominance|Inequality Dominance

15.7 Statistical inference: standard deviation, confidence in-tervals and hypothesis testing

DAD facilitates statistical inference in a number of original ways:

• DAD readily provides asymptotic standard errors on a large number of es-timators of distributive statistics, including estimators of inequality and so-cial welfare indices, normalized/un-normalized poverty indices, poverty in-dices with deterministic/estimated poverty lines, poverty indices with abso-lute/relative poverty lines, equally-distributed-equivalent incomes and povertygaps, quantiles, density functions, non-parametric regressions, points on alarge number of curves, crossing points of curves, critical poverty lines,differences in indices and curves, ratios of various statistics, various in-come/price/population impacts and elasticities, distributive decompositionsinto demographic/factor components, progressivity, redistribution and eq-uity indices, dominance statistics,etc.. It can be (and has typically formallybeen) shown that all of these estimators are asymptotically normally dis-tributed.

• DAD can calculate the sampling distribution of most of these estimatorstaking into account the sometimes complex design of the survey. This isdone as indicated in Section15.3. Existing (commercial) softwares can


sometimes take this design into account, but only for a sample number ofrelatively simple distributive statistics (such as simple sums and ratios).

Figure 7:STD option.

• DAD can provide at the click of a button estimates of confidence intervalsas well as test statistics andp-values for various symmetric and asymmetrichypothesis tests of interest.

• DAD can be used to simulate numerically the finite-sample sampling dis-tribution of most of the above-mentioned estimators using bootstrap pro-cedures. The bootstrap can be performed on the ordinary estimators or on(asymptotically) pivotal transforms of them. It is well known that boot-strapping on pivotal statistics leads to faster rates of convergence to thetrue sampling distribution than bootstrapping on untransformed non-pivotalstatistics. Pivotal bootstrapping is, however, usually more costly in time andresources since it requires estimates of the asymptotic distribution of the es-timators. This is not a problem forDAD, however, since the (sometimescomplex) asymptotic standard errors of these estimators are already pro-grammed into it. Moreover, as mentioned above, the asymptotic standard


errors and the pivotal statistics derived from them can be sample-designcorrected, providing one more degree of superior accuracy for the bootstrapprocedures available inDAD.

The Standard deviation, confidence interval and hypothesis testing dialogue boxis the main tool for tellingDAD what to do in terms of statistical inference. Thisbox is shown on Figure7.

15.8 References

For further information on Java’s development and structure, see Deitel andDeitel’s (2003)(Deitel and Deitel 2003) introductory book, or Chapter 1 of Lewisand Loftus (2000) (Lewis and Loftus 2000).

16 NON PARAMETRIC ESTIMATION INDAD 219

16 Non parametric estimation inDAD

16.1 Density estimation

16.1.1 Univariate density estimation

It is often useful to visualize the shapes of income distributions. There areessentially two main approaches to doing so, as well as a mixture of the two. Thefirst approach usesparametricmodels of income distributions. These models as-sume that the income distribution follows a known particular functional form, butwith unknown parameters. Popular examples of such functional forms include thelog-normal, the Pareto, and variants of the beta or gamma distributions. The mainstatistical challenge is then to estimate the unknown parameters of that functionalform, and to test whether a given functional form appears to estimate better theobserved distribution of income than another functional form.

The second approach does not posit a particular functional form and does notrequire the estimation of functional parameters. Instead, it lets the data entirely”speak for themselves”. It is therefore said to be non-parametric. The method ismost easily understood by starting with a review of the density estimation used bytraditional histograms. Histograms provide an estimate of the density of a variabley by counting how many observations fall into ”bins”, and by dividing that numberby the width of the bin times the number of observations in the sample. To see thismore precisely, denote the origin of the bins byy0 and the bins of the histogramby [y0 + mh, y0 + (m + 1)h] for positive or negative integersm. For instance, ifwe takem = 0, then the bin is described by the interval ranging from the originto the origin plush. The value of the histogram over each of these bins is thendefined by

f(y) =(# of yi in same bin asy)

n · (width of bin containingy). (269)

Such a histogram is shown on Figure40by the rectangles of varying heights overidentical widths, starting with originy0. For bins defined by[y0 + mh, y0 + (m +1)h], the bin width is a constant set toh, but we can also allow the widths tovary across the bins of the histogram. The choice ofh controls the amount ofsmoothingperformed by the histogram. A small bin width will lead to significantfluctuations in the value of the histogram across theh, and a very large widthwill set the histogram to the constanth−1. Choosing an appropriate value forsuch a smoothing parameter is in fact a pervasive preoccupation in non-parametric


estimation procedures, as we will discuss later. The choice of the origin can alsobe important, especially whenn is not very large. There can be, however, littleguidance on that latter choice, except perhaps when the nature of the data suggesta natural value fory0. One way to avoid choosing such ay0 is by constructingwhat will appear soon to be a ”naive” kernel density estimator, that is, one inwhich the pointy in f(y) is always at the center of the bin:

f(y) = (2hn)−1 (# of yi falling in [y − h, y + h]). (270)

This naive estimator can also be obtained from the use of a weight functionw(u), defined as:

w(u) =

0.5 if |u| < 10 otherwise

(271)

and by defining

f(y) = (nh)−1

n∑i=1

w

(y − yi

h

). (272)

This frees the density estimation from the choice ofy0. For the estimation ofthe naive density estimator, each observationyi provides a ”box” of width2hand height 0.5 that is centered atyi. f(y) over a range ofy is just the sum ofthesen boxes over that range. Because such boxes are not continuous, their sumf(y) will also be discontinuous, just like the traditional histogram. There will be”jumps” in f(y) at the pointsyi ± h, making this density estimator expositionallyunattractive. The naive estimator can also be improved statistically by choosingweighting functions that are smoother thanw(u) in 271. For this, we can thinkof replacing the weight functionw(u) by a general ”kernel function”K(u), suchthat

f(y) = (nh)−1

n∑i=1

K

(y − yi

h

). (273)

A smooth kernel estimate of the density function that generated the histogram isshown on Figure40.

In general, we would wish∫∞−∞ K(u)du = 1, since we would then have∫∞

−∞ f(y)dy = 1. For f(y) to qualify fully as a probability density function, we

would also requireK(u) ≥ 0 since we would then be guaranteed thatf(y) ≥ 0,although there are sometimes reasons to allow for negativity of the kernel func-tion. h is usually referred to as the window width, the bandwidth or the smoothing


parameter of kernel estimation procedures. There are also arguments to adjust thewindow width that applies to observationyi for the number of observations thatsurroundyi, makingh larger for areas where there are fewer observations. Thisis done for instance by the nearest neighbor and the adaptive kernel methods. Asin the use of the naive density estimator, each observation will provide a box or a”bump” to the density estimation off(y), and that bump will have a shape and awidth determined by the shape ofK(u) and the size ofh respectively41. E:19.5.1

The definition off(y) in (273) makes it inherit the continuity and differen-tiability properties ofK(u). It is often sound and convenient to choose a kernelfunction that is symmetric around 0, with

∫K(u)du = 1,

∫uK(u)du = 0 and

u2K(u)du = σ2K > 0. One such kernel function that has nice continuity and

differentiability properties is the Gaussian kernel, defined by

K(u) = (2π)−0.5 exp−0.5u2

. (274)

The ”bumps” provided by the Gaussian kernel have the familiar bell shapes, aresmoothly differentiable up to any desired level, and are such thatσ2

K = 1.

16.1.2 Statistical properties of kernel density estimation

The efficiency of non-parametric estimation procedures can be measured nat-urally by themean square error(MSE) that there is in estimating a function at apointy. The MSE in estimatingf(y) by f(y) is defined by

MSEy

(f)

= E

[(f(y)− f(y)

)2]

. (275)

The most common way of defining a measure ofglobalaccuracy simply sums themean square error across values ofy. This yields themean integrated square error(or MISE), a measure of the accuracy of estimatingf(y) over the whole range ofy:

MISE(f)

=

∫MSEy

(f)

dy. (276)

The relative efficiency of a particular choice of a kernel functionK(u) canthen be assessed relative to that choice of the kernel function which would mini-mize the MISE. The Gaussian kernel function has very good efficiency properties,although they are not quite as good as some other (less smooth) kernel functions,

41DAD:Density|Density function


such as the (efficiency-optimal) Epanechnikov, the biweight or the triangular ker-nels, which are described and discussed for instance inSilverman (1986) (see inparticular Table 3.1).

16.1.3 Choosing a window width

Even, however, if we were to agree on a particular shape for an observation-centered kernel function, there would still remain the question of which windowwidth to choose. Again, conditional on the choice of a particular form forK(u),we can choose the window width that minimizes the MISE. To see what this im-plies, note first that we can decompose the MSE aty as a sum of the square of thebias and of the variance that there is in estimatingf(y):

MSEy

(f)

=[E

(f(y)− f(y)

)]2

+ varf(y). (277)

For symmetric kernel functions, the bias can be shown to be approximately equalto

0.5h2σ2Kf (2)(y), (278)

where, as before,f (i)(y) stands for theith-order derivative off(y). The varianceequals:

n−1h−1f(y)cK , (279)

wherecK =∫

K(u)2du. Substituting (278) and (279) in (277) then gives:

MSEy

(f)

=(0.5h2σ2

Kf (2)(y))2

+ n−1h−1f(y)cK . (280)

Hence, considering (278), we find that the bias off(y) will be low if the kernelfunction has a low variance, since it is then the observations that are ”closer” toy that will count more, and it is those observations that provide the least biasedestimate of the density aty. But the bias also depends on the curvature off(y):in the absence of such a curvature, the density function is linear and the biasprovided by using observations on the left ofy is just (locally) outweighed by thebias provided by using observations on the right ofy.

Looking at (279), we findceteris paribusthat a flatter kernel (i.e., with a lowercK) decreases the variance off(y). A flatter kernel weights more equally theobservations found aroundy, and that reduces the variance of an estimator suchas (273). We also obtain the familiar result that the variance of the estimatordecreases proportionately with the size of the sample.


An increase inh plays an offsetting role on the precision off(y), as is shownby (280). Whenf (2)(y) 6= 0, a largeh increases the bias by making the estimatorstoo smooth: too much use is made of those observations that are not so close toy.Conversely, a largeh reduces the variance off(y) by making it less variable andless dependent on the particular value of those observations that are close toy.

Hence, in choosingh in an attempt to minimize MISE(f)

, a compromise needs

to be struck between the competing virtues of bias and variance reductions. Theprecise nature of this compromise will depend on the shape of the kernel functionas well as on the true population density function. For instance, if the Gaussiankernel is used and if the true density function is normal with varianceσ2, thenthe choice ofh that minimizes the MISE is given by (see for instanceSilverman(1986), p.45):

h∗ = 1.06σn−0.2. (281)

This value ofh∗ is conditional on bothK(u) andf(y) being normal density func-tions.Silverman (1986) also argues for a more robust choice ofh∗, given by

h∗ = 0.9An−0.2, (282)

whereA = min(standard deviation, interquartile range/1.34). This is because(282)

(. . . ) will yield a mean integrated square error within 10% of theoptimum for all thet-distributions considered, for the log-normal withskewness up to about 1.8, and for the normal mixture with separationup to 3 standard deviations. (. . . ) For many purposes it will certainlybe an adequate choice of window width, and for others it will be agood starting point for subsequent fine tuning. (Silverman (1986),p.48)

Further (asymptotic) results show that, under some mild assumptions — inparticular, that the density functionf(y) is continuous aty, and thath → 0 andnh → ∞ asn → ∞ — the kernel estimatorf(y) converges tof(y) asn → ∞.Whenh is chosen optimally, it is of the order ofn−1/5, and by (280) the MISE isthen of the order ofn−0.4. This is slightly lower than the analogous usual rate ofconvergence of parametric estimators, which isn−0.5.


16.1.4 Multivariate density estimation

Kernel estimation can also be used for multivariate density estimation. Letu, y andyi bed-dimensional vectors. We can estimate ad-dimensional densityfunction as:

f(y) = (nhd)−1

n∑i=1

K

(y − yi

h

). (283)

whereh is a window width common to all of the dimensions. The multivariateGaussian kernel is given byK(u) = (2π)−d/2 exp

(−0.5uTu). The issues of

kernel function and window width selections are similar to those discussed abovefor univariate density estimation. The approximately optimal window convergesat the raten−1/(d+4), and the optimal window width for the Gaussian kernel and amultivariate normal densityf(y) with unit variance is given by 4

n(2d+1)

1/(d+4).

16.1.5 Simulating from a nonparametric density estimate

Simulations from an estimated density are sometimes needed to compute es-timates of functionals of the unknown true density function. This is the case, forinstance, for the estimation inDAD of indices of classical horizontal inequity. Theestimation of such indices requires information on the net income distribution ofthose who have the same gross incomes, and such information cannot be gathereddirectly from sample observations of net and gross incomes since very few (ifany) exact equals can be observed in random samples of finite sizes. Another useof simulated distributions is for computing bootstrap estimates of the samplingdistribution of some estimators. The usual bootstrap procedure proceeds by con-ducting successive random sampling (with replacement) from the original sampleyin

i=1. This constrains the new samples to contain only those observationsyi

that were contained in the original sample. Those new samples could instead begenerated from a non-parametric estimate of the density of the original sampleof incomes, which would yield a bootstrap estimate that would be smoother andless dependent on the precise values that the observationsyi took in the originalsample.

Consider first the case of generatingJ independent realizations,y∗jJj=1, in

a univariate case, and suppose that a non-negative kernel with window widthhis used to estimatef(y). Also assume that observationi has sampling weightwi,and suppose for simplicity that these observations were drawn independently from


each other. The following simple algorithm is adapted slightly fromSilverman(1986), p.143. Forj = 1, . . . , J , we then:

Step 1 Choosei with replacement fromknk=1 with probabilitywkn

k=1/∑n

l=1 wl;

Step 2 Chooseε to have probability density functionK;

Step 3 Sety∗j = yi + hε.

Note that this algorithm does not even require computing directlyf(y).For the multivariate case, the above algorithm becomes just slightly more com-

plicated. For instance, for the estimation of classicalHI at gross incomex, weneed to generate a random sample of net incomes,y∗jJ

j=1, that follows the es-

timated kernel conditional densityf(y|x). For this, we use the original samplexi, yi, win

i=1 with sampling weightswi. Forj = 1, . . . , J , we then:

Step 1 Choosei with replacement fromknk=1 with probability

wkK

(xk−x

h

)∑n

l=1 wlK(

xl−xh

)n

k=1

;

Step 2 Chooseε to have probability density functionK;

Step 3 Sety∗j = yi + hε.

This gives a simulated sample of net incomesy∗jJj=1, conditional upon gross

income being exactly equal tox. A local index of classicalHI at x can then becomputed using this simulated sample, and global indices of classicalHI can beestimated simply by repeating this procedure for each of the observed values ofgross incomes,xln

l=1.Because they follow an estimated density function that is on average smoother

than the true one, the simulated samples generated by the above algorithms willhave a variance that is generally larger than both the variance observed in the sam-ple and the true population variance. Let for instance the sample variance of theyi be denoted asσ2

y. In the univariate case, the variance of the simulatedy∗j willequalσ2

y + h2σ2K . This can be a problem if, as is the case for the measurement of

indices of classicalHI , the quantity of interest is intimately linked to the disper-sion of income. There may also be a wish to constrain the simulated samples ofnet incomes to have precisely the same sample mean,µy, as the original sample.


Constraining the simulated samples to have the same mean and variance as for theoriginal sample can be done by translating and re-scaling the simulated samples.This involves replacingStep 3by

Step 3’ Sety∗j = µy + yi−µy+hε

(1+h2σ2K/σ2

y)0.5

in the univariate case. For the bivariate case, we also useStep 3’, but replaceµy

by E[y|x] andσ2y by σ2

y|x, which can be respectively computed as:

E[y|x] =

(n∑

i=1

wiK

(xi − x

h

))−1 n∑i=1

wiK

(xi − x

h

)yi (284)

and

σ2y|x =

(n∑

i=1

wiK

(xi − x

h

))−1 n∑i=1

wiK

(xi − x

h

) (yi − E[y|x]

)2

. (285)

Equation (284) is in fact an example of a kernel regression ofy onx, a procedureto which we now turn.

16.2 Non-parametric regression

The estimation of an expected relationship between variables is the secondmost important sphere of recent applications of kernel estimation techniques.Non-parametric regressions offer several useful applications in distributive anal-ysis. An example of such an application is the estimation of the relationship be-tween expenditures and calorie intake. Regressing calorie intake non parametri-cally on expenditure does not impose a fixed functional relationship between thosetwo variables along the entire range of calorie intake. On the contrary, it allows afair amount of flexibility by estimating the link between the two variables througha local weighting procedure. The local weighting procedure essentially considersat the expenditures of those individuals with a calorie intake in the ”region” ofthe specified minimum calorie intake. It weights those values with weights thatdecrease rapidly with the distance from the minimum calorie intake. Hence, thosewith calorie intakes far from the minimum specified level will contribute littleto the estimation of the expenditure needed to attain that minimum level. Theresults using this method are thus less affected by the presence of ”outliers” in


the distribution of incomes, and less prone to biases stemming from an incorrectspecification of the link between spending and calorie intake.

Basically, then, one is interested in estimating the predicted response,m(x),of a variabley at a given value of a (possibly multivariate) variablex, that is,

m(x) = E[y|x]. (286)

Alternatively, if the joint densityf(x, y) exists and iff(x) > 0, m(x) can also bedefined as:

m(x) =

∫yf(x, y)dy

f(x). (287)

The difficulty in estimating the functionm(x) is that we typically do not ob-serve in a sample a response ofy at that particular value ofx. Furthermore, evenif we did, there would rarely be other observations with exactly the same value ofx that would allow us to compute reliably the expected response in which we areinterested.

Let thenxi, yini=1 be a sample ofn observed realizations jointly ofx andy.

The response information that is provided by the sample can be expressed as:

yi = m(xi) + εi , whereE[εi] = 0. (288)

To estimatem(x), kernel regression techniques use a local averaging procedurethat involves weightsK(u) that are analogous to those used in section16.1 fordensity estimation. Recalling (273) and (287), this leads to the following Nadaraya-Watson non-parametric estimator ofm(x)42.: E:19.8.1

m(x) =(nhf(x)

)−1n∑

i=1

K

(x− xi

h

)yi

=

∑ni=1 K

(x−xi

h

)yi∑n

i=1 K(

x−xi

h

) . (289)

To reduce the bias of using neighboringyi, the kernel weightsK(

x−xi

h

)are typ-

ically inversely proportional to the distance betweenx andxi. They also dependon the window widthh.

As in the case of the kernel density estimators, the kernel smootherm(x) canbe shown to be consistent under relatively weak conditions, including thatm(x)

42DAD: Density|Non parametric regression


andf(x) are both continuous functions ofx, and thath → 0 andnh → ∞ asn → ∞ (see for instanceHardle (1990), Proposition 3.1.1). Again, the varianceof m(x) alone does not fully capture the convergence ofm(x) to m(x) since wemust also take into account the bias ofm(x), which comes from the smoothingof theyi in (289). Under suitable regularity conditions, including thath ∼ n−0.2,the asymptotic distribution of the kernel estimator(nh)0.5 (m(x)−m(x)) can beshown to be normal, with its center shifted by its asymptotic bias – seeHardle(1990), Theorem 4.2.1, for a demonstration. This asymptotic bias is a function ofthe form of the kernelK(u) and of the derivatives ofm(x) andf(x). It is givenby:

σ2K

(m(2)(x) + 2m(1)(x)f (1)(x)/f(x)

). (290)

This asymptotic bias can be estimated consistently using estimates ofm(2)(x),m(1)(x), f (1)(x) andf(x). This, however, complicates significantly the computa-tion of the sampling distribution ofm(x), and it can be avoided if we can expect(or can make) the bias to be small compared to the variance. This will be the caseif m(x) is relatively constant, or if we makeh fall just a bit faster than its optimalspeed ofn−0.2 – again, see the discussion of this inHardle (1990), pp.100–102.

The variance of(nh)0.5 (m(x)−m(x)) is given by:

σ2y|xcK/f(x). (291)

The conditional varianceσ2y|x can be estimated consistently as in (285). in the

case of kernel density estimation, note again that the smoothing process makesthe rate of convergence of the kernel estimatorm(x) to ben−0.4 instead of theusual slightly faster parametric convergence rate ofn−0.5.

16.3 References

This chapter draws significantly fromSilverman (1986) andHardle (1990), towhich readers are referred for more details and in-depth analysis. More recentcoverage of non-parametric estimation procedures can also be found in ???.

17 ESTIMATION AND STATISTICAL INFERENCE 229

17 Estimation and statistical inference

17.1 Sampling design

There exist in the population of interest a number of statistical units. For sim-plicity, we can think of these units as households or individuals. From an ethicalperspective, it is usually preferable to consider individuals as statistical units ofinterest, but for some purposes (such as the distribution of aggregate householdwellbeing) households may also be appropriate statistical units.

These statistical units are those for which we would like to observe socio-demographic information such as their household composition, labor activity, in-come or consumption. Since it is usually too costly to gather information on allstatistical units in a large population, one would typically be constrained to ob-tain information on only a sample of such units. Distributive analysis is thereforeusually done using survey data.

Since surveys are not censuses, we must take care to distinguish ”true” pop-ulation values from sample values. Sample differences across surveys are indeeddue both to true population differences and to sampling variability. Populationvalues are generally not observed (otherwise, we would not need surveys). Sam-ple values as such are rarely of interest: they would be of interest in themselvesonly if the statistical units which appeared by chance in a sample were also pre-cisely those which were of ethical interest, which is usually not the case. Hence,sample values matter in as much as they can helpinfer true population values. Thestatistical process by which such inference is performed is called statistical infer-ence. The sampling process should thus ideally be such that it can be used to makesome statistically-sensible distributive analysis at the level of the population, andnot solely for the samples drawn.

Sampling errors thus arise because distributive estimates are typically madeon the basis of only some of the statistical units of interest in a population. Thefact that we have no information on some of the population statistical units makesus infer with sampling error the population value of the distributive indicatorsin which we are interested. There is an important element of randomness in thevalue of this sampling error. The error made when relying solely on the informa-tion content of one sample depends on the statistical units present in that sample.The drawing of other samples would generate different sampling errors. Becausesamples are drawn randomly, the sampling errors that arise from the use of thesesamples are also random.

Since the true population values are unknown, the sampling error associated


with the use of a given sample is also unknown. Statistical theory does, how-ever, allow one to estimate the distribution of sampling errors from which actual(but unobserved) sampling errors arise. This nevertheless requires samples to beprobabilistic,viz, that there be a known probability distribution associated to thedistribution of statistical units in a sample. This also strictly means that there isabsence of unquantifiable and subjective criteria in the choice of units. If this werenot so, it would not be possible to assess reliably the sampling distribution of theestimators.

To draw a sample, a sampling base is used. A sampling base is made of all thesampling units (SU) from which a sample can be drawn. The base of samplingunits –e.g., the census of all households within in a country – is usually differentfrom the entire population of statistical units –e.g., the population of individuals,say. There are several reasons for this, an important one being that it is generallycost effective to seek information only within a limited number of clusters of sta-tistical units, grouped geographically or socio-economically. This also facilitatesthe collection of cluster-level (e.g.,village-level) information.

A process ofsimple random samplingdraws sample observations randomlyand independently from a base of sampling units, each with equal probability ofselection. Simple random sampling is rarely used in practice to generate house-hold surveys. Instead, a population of interest (a country, say) is often first dividedinto geographical or administrative zones and areas, called strata. The first stageof random selection then takes place from within a list of Primary Sampling Units(denoted as PSU’s) built for each stratum. Within each stratum, a number ofPSU’s are then randomly selected. PSU’s are often departments, villages,etc..This random selection of PSU’s provides ”clusters” of information.

Since the cost of surveying all statistical units un each of these clusters may beprohibitive, it may be necessary to proceed to further stages of random selectionwithin each selected PSU. For instance, within each department, a number ofvillages may be randomly selected, and within every selected village, a number ofhouseholds may also be randomly selected. The final stage of random selectionis done at the level of the last sampling units (LSU’s). These LSU’s are oftenhouseholds. Each selected LSU may then provide information on all individualsfound within that LSU. These individuals arenot selected – information on all ofthem appears in the sample. They therefore do not represent LSU’s in statisticalterminology.


17.2 Sampling weights

Sampling weights (also called inverse probability, expansion or inflation fac-tors) are the inverse of the sampling probabilities,viz, of the probabilities of asampling unit appearing in the sample. These sampling weights are SU-specific.The sum of these weights is an estimator of the size of the population of SU’s.

Samples are sometimes ”self-weighted”. Each sampling unit then has the samechance of being included in the survey. This arises, for instance, when the numberof clusters selected in each stratum is proportional to the size of each stratum,when the clusters are randomly selected with probability proportional to their size,and when an identical number of households (or LSU’s) across clusters is thenselected with equal probability within each cluster.

It is, however, common for the inclusion probability to differ across house-holds. One reason comes simply from the complexity of sample designs, whichmakes differential sampling weights occur frequently. Another reason is that thecosts of surveying SU’s vary, which makes it more cost effective to survey somehouseholds (e.g., urban ones) than others. Sampling precision can also be en-hanced with differential probabilities of household inclusion. The idea here is tosurvey with greater probability those households who contribute more to the phe-nomenon of interest. It leads to a sampling process usually called sampling with”probability proportional to size”.

Assume for instance that we are interested in estimating the value of a distribution-sensitive poverty index. The most important contributors to that index are obvi-ously the poor households, and more precisely the poorest among them. Anapriori suspicion might be that such poorest households are proportionately morelikely to be found in some areas than in others. Making inclusion probabilitieslarger for households in these more deprived areas will then enhance the samplingprecision of the estimator of the distribution-sensitive poverty index since it willgather more statistically informative data.

A reverse sample-design argument would apply for a survey intended to es-timate total income in a population. The most important contributors to total in-come are the richest households, and it would thus be sensible to sample themwith a greater probability. Yet one more consequence of the principle of ”proba-bility proportional to size” is the desirability of sampling with greater probabilitythose households of larger sizes. Distributive analysis is normally concerned withthe distribution of individual well-being.Ceteris paribus, larger-size householdscontribute more information towards such assessment, and should therefore besampled with a greater probability (roughly speaking, with a probability propor-


tional to their size).Omitting sampling weights in distributive analysis will systematically bias

both the estimators of the values of indices and points on curves as well as theestimation of the sampling variance of these estimators43. Including such weightsE:19.2.1will usually make the analysis free of asymptotic biases. To see this, we followDeaton (1998), p.45, and letY be the population total of thex’s, with a populationof sizeN . An estimator of that population total is then given by

Y =N∑

i=1

tiwixi, (292)

whereti is the number of times uniti appears in a random sample of sizen. Letπi be the probability that uniti is selected each time an observation is drawn.Households with a low value ofπi will have a low probability of being selectedin the survey, relative to others with a higherπi. Then, E[ti] = nπi = wi

−1 isthe expected number of times uniti will appear in the sample, or, for largen, it isroughly speaking the probability of being in the sample. Hence,

E[Y

]=

N∑i=1

E[ti]wixi =N∑

i=1

xi = Y (293)

andY is therefore an unbiased estimator ofY . An analogous argument applies toshow thatN =

∑Ni=1 tiwi is an unbiased estimator of population sizeN .

17.3 Stratification

The sampling base is usually stratified in a number of strata. The basic advan-tage of stratification is to use prior information on the distribution of the popula-tion, and to ”partition” it in parts that are thought to differ significantly from eachother. Sampling then draws information systematically from each of those partsof the population. With stratification, no part of the sampling base therefore goesunrepresented in the final sample.

To be more specific, a variable of interest, such as householdper capitain-come, often tends to be less variable within some stratum than across an entirepopulation. This is because households within the same stratum typically shareto a greater extent than within the entire population some socio-economic charac-teristics – such as geographical locations, climatic conditions, and demographic

43DAD:Poverty|FGT index


characteristics – that are determinants of the incomes of these households. Strati-fication helps generate systematic sample information from a diversity of ”socio-economic areas”.

Because information from a ”broader” spectrum of the population leads onaverage to more precise estimates, stratification generally decreases the samplingvariance of estimators. For instance, suppose at the extreme that household in-come is the same for all households in a given stratum, and this, for all strata.In this case, supposing also that the population size of each stratum is known inadvance, it is sufficient to draw only one household from each stratum to knowexactly the distribution of income in the population.

17.4 Clustering (or multi-stage sampling)

Multi-stage sampling implies that SU’s end up in a sample only subsequentlyto a process of multi-stage selection. ”Groups” (or clusters) of SU’s are first ran-domly selected within a population (which may be stratified). This is followedby further sampling within the selected groups, which may be followed by yetanother process of random selection within the subgroups selected in the previousstage.

The first stage of random selection is done at the level of primary samplingunits (PSU). An important assumption would seem to be that first-stage samplingbe random and with replacement for the selection of a PSU to be done indepen-dently from that of another. There are many cases, however, in which this is nottrue.

1. First-stage sampling is typically made without replacement.

This will not matter in practice for the estimation of the sampling varianceif there is multi-stage sampling, that is, if there is an additional stage ofsampling within each selected PSU. The intuitive reason is that selecting aPSU only reveals random and incomplete information on the population ofstatistical units within that PSU, since not all of these statistical units appearin the sample when their PSU is selected. Selecting that same PSU oncemore (in a process of first-stage sampling with replacement) does thereforereveal additional information, information different from that provided bythe first-time selection of that PSU. This extra information is roughly ofequal value to that which would have been revealed if a process of samplingwithout replacement had forced the selection of a different PSU.


Hence, in the case of multi-stage sampling, first-stage sampling withoutreplacement does not extract significantly more information than first-stagesampling with replacement. It does not therefore practically lead to lessvariable estimators than a process of first-stage sampling with replacement.

If, however, there is no further sampling after the initial selection of PSU’s,then a finite population correction (FPC) factor should be used in the com-putation of the sampling variance. This would generate a better estimate ofthe true sampling variance. If FPC factors are not used, then the samplingvariance of estimators will tend to be overestimated. This means that it willbe more difficult to establish statistically significant differences across dis-tributive estimates, making the distributive analysis more conservative andless informative than it could have been.

2. Sampling is oftensystematic.

Systematic sampling can be done in various ways. For instance, a completelist of N sampling units is gathered. Lettingn be the number of samplingunits that are to be drawn, a ”step”s is defined ass = N/n. A first samplingunit is randomly chosen within the firsts units of the sampling list. Let therank of that first unit bek ∈ 1, 2, · · · , s. Then− 1 subsequent units withranksk + s, k + 2s, k + 3s,...,k + ns, then complete the sample.

If the order in which the sampling units appear in the sampling list is ran-dom, then such systematic sampling is equivalent to pure random sampling.If, however, this is not the case, then the effect of such systematic samplingon the sampling variance of the subsequent distributive estimators dependson how the sampling units were ordered in the sampling list in the first place.

(a) For instance, a ”cyclical” ordering makes sampling units appear in cy-cles. ”Similar” sampling units then show up in the sampling list atroughly fixed intervals. Suppose for illustrative purposes that the sizeof these intervals is the same ass. Then, systematic sampling will leadto a gathering of information on similar units (e.g., with similar in-comes), thus reducing the statistical information that is extracted fromthe sample. This will reduce the sampling precision of estimators, andincrease their sampling variance.

(b) A cyclical ordering of sampling units suggests that there is more sampling-unit heterogeneity around a given sampling unit than across the whole


sampling base (since information around sampling units is simply cycli-cally repeated across the sampling base). A more frequent phenomenonarises when adjacent sampling units show less heterogeneity than thatshown by the entire sampling base. A typical occurrence of this iswhen sampling units are ordered geographically in a sampling list.Households living close to each other appear close to each other inthe list. Villages far away from each other are also far away in thesampling list. Since geographic proximity is often associated withsocio-economic resemblance, the farther from each other in the list aresampling units, the more likely will they also differ in socio-economiccharacteristics.Systematic sampling will then force units from across the entire sam-pling list to appear in the sample. Representation fromimplicit stratawill thus be compelled into the sample. This will lead to a samplingfeature usually calledimplicit stratification. Pure random samplingfrom the sampling list will not force such a systematic extraction ofinformation, and will therefore lead to more variable estimators.By how far implicit stratification reduces sampling variability dependson the degree of between-stratum heterogeneity which stratification al-lows to extract, just as for explicit stratification. The larger the hetero-geneity of units far from each other, the larger the fall in the samplingvariability induced by the systematic sampling’s implicit stratification.One way to account for and to detect the impact of implicit stratifica-tion in the estimation of sampling variances is to group pairs of adja-cent sampling units into implicit strata. Assume again thatn samplingunits are selected systematically from a sampling list. Then, createn/2 implicit strata and compute sampling variances as if these wereexplicit strata. If these pairs did not really constitute implicit strata(because, say, the ordering in the sampling list had in fact been estab-lished randomly), then this procedure will not affect affect much theresulting estimate of the sampling variance. But if systematic sam-pling did lead to implicit stratification, then the pairing of adjacentsampling units will reduce the estimate of the sampling variance –since the variability within each implicit stratum will be found to besystematically lower than the variability across all selected samplingunits.

Generally, variables of interest (such as incomes) vary less within a cluster


than between clusters. Hence,ceteris paribus, multi-stage selection reduces the”diversity” of information generated compared to SRS and leads to a less infor-mative coverage of the population. The impact of clustering sample observationsis therefore to tend to decrease the precision of estimators, and thus to increasetheir sampling variance.Ceteris paribus, the lower the within-cluster variabilityof a variable of interest, the larger the loss of information that there is in samplingfurther within the same clusters.

To see this, suppose the extreme case in which household income happensto be the same for all households in a cluster, and this, for all clusters. In suchcases, it is clearly wasteful to adopt multi-stage sampling: it would be sufficient todraw one household from each cluster in order to know the distribution of incomewithin that cluster. More information would be gained from sampling from otherclusters.

17.5 Impact of stratification, clustering, weighting and sam-pling without replacement on sampling variability

There are two modelling approaches to thinking about how data were initiallygenerated. The first one, which is also the more traditional in the sampling designliterature, is the finite population approach. The second approach is the super-population one: the actual population is a sample drawn from all possible popu-lations, the infinite super-population. This second approach sometimes presentsanalytical advantages, and it is therefore also regularly used in econometrics.

To illustrate the impact of stratification and clustering on sampling variabil-ity, consider therefore the following ”super-population model”, based onDeaton(1998), p.56. Then,

xhij = µ + αh︸︷︷︸stratum effect

+ βhi︸︷︷︸cluster effect

+ εhij︸︷︷︸household effect

. (294)

For simplicity, assume that thexhij are drawn from the same numbern of clustersin each of theL strata, and that the same number of LSU (or ”households”)m isselected in each of the clusters. The indiceshij then stand for:

• h = 1, ..., L: stratumh

• i = 1, ..., n: clusteri (in stratumh)


• j = 1, ..., m: householdj (in clusteri of stratumh).

For simplicity, also assume thatαh is distributed with mean 0 and varianceσ2α,

that βhi is distributed with mean 0 and varianceσ2β, and thatεhij is distributed

with mean 0 and varianceσ2ε . Assume moreover that these three random terms

are distributed independently from each other.

17.5.1 Stratification

Say that we wish to estimate mean incomeµ. The estimator,µ, is given by

µ = (Lmn)−1L∑

h=1

n∑i=1

m∑j=1

xhij. (295)

Let

µh = (mn)−1n∑

i=1

m∑j=1

xhij (296)

be the estimator of the mean of stratumh. Clearly, E[µh] = µ + αh and E[µ] = µsince by (295) and (296)

E[µ] = (Lmn)−1L∑

h=1

n∑i=1

m∑j=1

E[xhij] = (Lmn)−1(Lmn)µ = µ. (297)

and

E[µh] = (mn)−1n∑

i=1

m∑j=1

E[xhij|αh] = (mn)−1mn (µ + αh) = µ + αh. (298)

Because of the independence of sampling across strata, we also have that

var(µ) = var

(L−1

L∑

h=1

µh

)= L−2

L∑

h=1

var (µh) . (299)

The sampling variability ofµ is thus a simple average of the sampling variancesof theL strata’sµh.

Stratification can in fact be thought of as an extreme case of clustering, withthe number of selected clusters corresponding to the number of population clus-ters, and with sampling being done without replacement to ensure that all popula-tion clusters will appear in the sample. Suppose instead that one were to selectL


strata randomly and with replacement, to make it possible that not all of the stratawill be selected. This is in a sense what happens when stratification is droppedand clustering is introduced. Using (295) and (296), we then have that

µ = L−1

L∑

h=1

thµh (300)

whereth is a random variable showing the number of times stratumh was selected.Then, denotingµh = µ + αh, we have approximately that

µ ∼= µ + L−1

L∑

h=1

((th − E [th]) µh + (µh − µh) E [th]) . (301)

and thus that

var(µ) ∼= L−2 var

(L∑

h=1

αhth +L∑

h=1

(µh − αh)

)(302)

sinceµ∑L

h=1 th = µ and E[th] = 1. Assuming independence betweenµh andthand between theµh, we have that

var(µ) ∼= L−2

var

(L∑

h=1

αhth

)+

L∑

h=1

var (µh)

. (303)

Sinceth follows a multinomial distribution, withvar(th) = (L−1)/L andcov(th, ti) =−1/L, we find that

var

(L∑

h=1

αhth

)=

L∑

h=1

α2h var (th) +

L∑

h=1

∑

i6=h

αhαi cov(th, ti) = L

L∑

h=1

α2h = Lσ2

α.(304)

Hence, using (302) and (304), we obtain

var(µ) ∼= L−2

L∑

h=1

var (µh) + L−1σ2α. (305)

The last term in (305) is the effect upon sampling variability of removingstratification. The larger this term, the greater the fall in sampling variability thatoriginates from stratification.


17.5.2 Clustering

Let us now investigate the effect of clustering on the sample variance, that is,onvar (µh). We find:

var (µh) = var

((mn)−1

n∑i=1

m∑j=1

xhij

)

= (mn)−2 var

βh1 + . . . + βh1︸︷︷︸m times

+ . . . + βhn + . . . + βhn︸︷︷︸m times︸︷︷︸

n times

+n∑

i=1

m∑j=1

εhij

= (mn)−2 var

(m

n∑i=1

βhi +n∑

i=1

m∑j=1

εhij

)

=σ2

β

n+

σ2ε

mn. (306)

The first line of (306) follows by the definition ofµh, and the second line followsfrom (294) – note thatαh is fixed for all of thexhij in the same stratumh. The lastline of (306) is obtained from the sampling independence betweenβhi andεhij.

Hence, for aper-stratumgiven number of observationsmn, it is better to havea largen to reduce sampling variability, namely, it is better to draw observationsfrom a large number of clusters. The larger the cross-cluster variabilityσ2

β, themore important it is to have a large number of clusters in order to keepvar (µh)low. Ceteris paribus, for a given sample size and for a givenσ2

β + σε, the sam-pling variance of distributive estimators is smaller the smaller the between-clusterheterogeneity,σ2

β, but the larger the within-cluster heterogeneity,σ2ε .

17.5.3 Finite population corrections

Sampling without replacement imposes that all of the selected sampling unitsare different. It therefore extracts on average more information from the samplingbase than sampling with replacement, and ensures that the samples drawn are onaverage closer to the population of sampling units. Sampling without replacementtherefore increases the precision of sample estimators. To account for this increasein sampling precision, a FPC factor can be used, although it complicates slightlythe estimation of the variance of the relevant estimators.


Assume simple random sampling ofn sampling units from a population ofNsampling units. Thus, we have thatwi = N/n for all of then sample observations.To illustrate the derivation of an FPC factor in this simplified case, we followDeaton (1998), p.42-44, andCochrane (1977). An estimatorY of the populationtotalY of thex’s is given by

Y =N

n

N∑i=1

tixi (307)

where the random variableti indicates whether – and how many times – the pop-ulation uniti was included in the sample. Taking the variance of (307), we find:

var(Y

)=

(N

n

)2(

N∑i=1

x2i var(ti) +

N∑i=1

N∑

j 6=i

xixj cov(ti, tj)

). (308)

Using (307) and (308), the distinction between simple random samplingwithandwithout replacement is analogous to the distinction between a binomial anda multinomial distribution for theti. With sampling without replacement, theprobability that any one population unit appears in the final sample is equal ton/N , i.e., E[ti] = n/N . Sinceti then takes either a 0 or a 1 value, it thus followsa binomial distribution with parametern/N . The variance ofti is then given byE[t2i ]− (n/N)2 = n/N − (n/N)2 = n/N (1− n/N) . The covariancecov(ti, tj)can be found by noting that E[titj] = P(ti = tj = 1) = n/N · (n − 1)/(N − 1),and thus that

cov(ti, tj) = − n(N − n)

N2(N − 1). (309)

Substitutingvar(ti) andcov(ti, tj) into (308), and defining

S2 = (N − 1)−1

N∑i=1

(xi −N−1Y

)2, (310)

we find

var(Y

)= N2(1− f)

S2

n(311)

where1− f = (N − n)/N is an FPC factor.


Take now the case of simple random sampling with replacement. We can thenexpressti for any given population uniti as a sum ofn independent drawstij,with j = 1, . . . , n, each onetij indicating whether observationi was selected indrawj. Thus:

ti =n∑

j=1

tij. (312)

Since for any drawj, E[tij] = 1/N , the expected value ofti is againn/N , butti may not take values greater than 1. The drawstij being independent, and eachdraw having a binomial distribution with parameter1/N , we have that

var(ti) =n∑

j=1

var (tij) =n∑

j=1

1

N

(1− 1

N

)=

n

N

(1− 1

N

), (313)

which is the variance of a multinomial distribution with parametersn and1/N .It can be checked that the covariancecov(ti, tj) is given by−n/N2. Substitutingvar(ti) andcov(ti, tj) into (308) again, we now find

var(Y

)= N2 (N − 1)

N

S2

n. (314)

This is larger than (311): the difference between the two results equals

N2 (n− 1)

N

S2

n(315)

and depends on the magnitude ofn relative toN . The larger the value ofn relativeto N , the greater the sampling precision gains that there are in samplingwithoutreplacement.

17.5.4 Impact of weighting on sampling variance

We follow once more the approach ofDeaton (1998), pp.45-49, andCochrane(1977). Suppose that we are again interested in estimating the variance of theestimatorY of a totalY , but for simplicity assume that sampling is done withreplacement so that we can for now ignore FPC factors.Y is now defined as:

Y =N∑

i=1

tiwixi. (316)


Taking its variance, we find

var(Y

)=

N∑i=1

w2i x

2i var(ti) +

N∑i=1

N∑

j 6=i

wiwjxixj cov(ti, tj). (317)

ti follows once more a multinomial distribution, but now withvar(ti) = nπi(1−πi) andcov(ti, tj) = −nπiπj. Substituting this into (317), we find

var(Y

)= n−1

(N∑

i=1

x2i

πi

− Y 2

). (318)

To estimate (318), we can substitute population values by sample values andthus use the estimator

var(Y

)= n−1

N∑i=1

tiwix2

i

πi

−(

N∑i=1

tiwixi

)2 . (319)

Denote asyi = wixi, i = 1, . . . , n the n sample values ofwixi, and let y =n−1

∑ni=1 yi. Then, (319) leads to

var(Y

)=

n

n− 1

N∑i=1

(yi − y)2 , (320)

with the difference that a familiarn/(n − 1) small-sample correction factor hasbeen introduced in (320) to correct for the small-sample bias in estimating thevariance of theyi.

17.5.5 Summary

The above calls to mind the importance for statistical offices of making avail-able sample design information. This includes providing

• the sampling weights;

• stratum and PSU (cluster) identifying variables;

• information on the presence or not of systematic sampling (and thus of im-plicit stratification), including the relationship between the numbering ofsampling units and the original ordering of these units in the sampling base;


• the finite population correction factors, namely, the size of the samplingbases, when appropriate.

Equipped with this information, distributive analysts can provide reliable esti-mates of the sampling precision of their estimators44. E:19.9.1

17.6 Formulae for computing standard errors of distributiveestimators with complex sample design

We provide in this section a detailed account of the computation of samplingvariances inDAD, taking full account of the sampling design. Let:

• h = 1, . . . , L: the list of the strata (e.g. the geographical regions)

• i = 1, . . . , Nh: the list of primary sampling units (PSU;e.g., villages) instratumh

• Nh: the population number of PSU in a stratumh

• nh: the number of selected PSU in a stratumh

• Mhi: the population number of last sampling units (LSU) (e.g., households)in PSUhi

• mhi: the number of selected LSU in the PSUhi (for instance, the numberof households from villagehi that appear in the sample)

• qhij: the number of observations in selected LSUhij (e.g., the number ofhousehold members in a householdhij whose socio-economic informationis recorded in the survey, with each household member providing 1 line ofinformation in the data file).

• whij: the sampling weight of LSUhij

• M =∑L

h=1

∑Nh

i=1 Mhi: the population number of LSU (e.g., the number ofhouseholds in the population)

• m =∑L

h=1

∑nh

i=1 mhi: the number of selected LSU (e.g., the number ofselected households that appear in the sample)

44DAD: Edit|Set simple design


• Xhijk: the value of the variable of interest (e.g., adult-equivalent income)for statistical unithijk in the population

• Shijk: the size of statistical unithijk in the population (e.g., if the statisticalunit is a household, thenShijk may be the number of persons in householdhij, or alternatively the number of adult equivalents)

• Y =∑L

h=1

∑Nh

i=1

∑Mhi

j=1

∑qhij

k=1 ShijkXhijk: the population total of interest

• xhijk: the value ofX (the variable of interest) that appears in the sample forsample observationhijk

• shijk: the size of selected sample observationhijk

• Y =∑L

h=1

∑nh

i=1

∑mhi

j=1

∑qhij

k=1 whijshijkxhijk: the estimated population totalof interest

• M =∑L

h=1

∑nh

i=1

∑mhi

j=1 whij: the estimated population number of LSU

• yhij =∑qhij

k=1 shijkxhijk: the relevant sum in LSUhij

• yhi =∑mhi

j=1 whijyhij: the relevant sum in PSUhi

• yh = n−1h

∑nh

i=1 yhi: the relevant mean in stratumh

The sampling covariance of two totals,Y andZ (Z being defined similarly toY ) is then estimated by

covSD(Y , Z) =L∑

h=1

nh2(1− fh)

Syzh

nh

(321)

where

Syzh = (nh − 1)−1

nh∑i=1

(yhi − yh) (zhi − zh) (322)

– note the similarity with (310) and (311) – and wherefh is a function of a user-specified FPC factor,fpch, for stratumh, such that,

• if a fpch is not specified by the user, thenfh = 0;

• if fpch ≥ nh, thenfh = nh/fpch;


• if fpch ≤ 1, thenfh = fpch.

Recall that settingfh 6= 0 is useful only when the sampling design is of theform either of simple random sampling or of stratified random sampling with nosubsequent sub-sampling within the PSU’s selected. In both cases, sampling musthave been done without replacement.

The varianceVSD of Y is obtained from (321) simply by replacing(zhi − zh)by (yhi − yh).

An often-used indicator of the impact of sampling design on sampling vari-ability is called the design effect,deff . The design effect is the ratio of the design-based estimator of the sampling variance (VSD ) over the estimate of the samplingvariance assuming that we have obtained a simple random sample ofm LSU with-out replacement. Denote this latter estimate asVSRS . Then,

deff =VSD

VSRS

. (323)

For such a simple sampling design, we would have that

Y =M

m

L∑

h=1

nh∑i=1

mhi∑j=1

yhij (324)

and, recalling (311), the sampling variance ofY would then equal

VSRS =

(M

m

)2

var

(L∑

h=1

nh∑i=1

mhi∑j=1

yhij

)= M2(1− f)

var(y)

m, (325)

wherevar(y) is the variance of the populationyhij, and wheref = m/M if a FPCfactor is specified for the computation ofVSD , andf = 0 otherwise.VSRS canthen be estimated as follows:

VSRS = M2(1− f)

(1

m− 1

L∑

h=1

nh∑i=1

mhi∑j=1

whij

M

(yhij − Y /M

)2)

. (326)

Some of the above variables often take familiar forms and names:

• xhijk can be thought of as an ”individual-level” variable, such as height,health status, schooling, or own consumption. This variable is called the


”variable of interest” inDAD. If xhijk is indeed individual-specific, thenshijk will not exceed 1 in most reasonable instances. Individual outcomesare, however, not always observed. Even if they are, we may sometimesbelieve that there is equal sharing in the household to which individualsbelong. In those cases,xhijk will typically take the form of adult-equivalentincome or other household-specific measure of living standard.

• shijk gives the ”size” of the sample observationhijk. This size may bepurely demographic, such as the number of individuals in the unit whoseliving standard is captured byxhijk. It may also be 1 even ifhijk repre-sents a household, if we are interested in a household count for distributiveanalysis. Butshijk may also be an ethical size, which depends on norma-tive perceptions on how important the unit is in terms of some distributiveanalysis. Examples of such sizes include the number of adult-equivalentsin the unit (if, say, we wish to assign individuals an ethical weight that isproportional to their ”needs”), the number of families, the number of adults,the number of workers, the number of children, the number of citizens, thenumber of voters,etc..

• qhij is the number of sample observations or statistical units provided bythe last sampling unit. This LSU may contain a grouping of households, ofvillages,etc... More commonly for the empirical analysis of poverty andequity, a LSU represents a household.

17.7 References

General references on estimation and inference taking into account survey de-sign includeAsselin (1984) andCochrane (1977). Applications to economic anal-ysis are discussed and presented inDeaton (1998), Howes and Lanjouw (1998)(focussing on poverty analysis), andZheng (2002) (with a specific focus on Lorenzcurves). Alternative approaches to taking into account survey design can be foundinter alia in Cowell (1989) (modelling sampling weights as jointly distributedwith living standards), and inBiewen (2002b) andSchluter and Trede (2002a)(for dependence across members of the same PS – households in their case).Ken-nickell and Woodburn (1999) illustrates the impact of a consistent estimation ofsurvey weights in the US Surveys of Consumer Finances for the analysis of thedistribution of wealth.

18 STATISTICAL INFERENCE IN PRACTICE 247

18 Statistical inference in practice

Assessing statistically the extent of poverty and equity in a distribution, orchecking for distributive differences, usually involves three steps. First, one for-mulates distributive hypotheses of interest, such as that the poverty headcount isless than 20%, or that tax equity has not decreased over time, or that inequal-ity is greater in one country than in another. Second, one computes distributivestatistics, weighting observations by their sampling weights and (when appropri-ate) by a size variable. Third, one uses these statistics to test the hypotheses ofinterest. This can involve testing these hypotheses directly, or building confidenceintervals of ranges in which we can confidently locate the true population valueof the distributive indices of interest. This third step may allow for the effects ofsurvey design on the sampling distributions of distributive indices and test statis-tics, and may also involve performing numerical simulations of such samplingdistributions, if the circumstances make it desirable to do so.

18.1 Asymptotic distributions

Under the null hypothesis thatµ = µ0, and under some generally mild regu-larity conditions, all of the estimatorsµ and associated test statistics consideredin this book and programmed inDAD can be shown to be asymptotically nor-mally distributed with meanµ0 and asymptotic sampling varianceσ2

µ. This can beformally stated as:

µ ∼ N(µ0, σ

2µ

). (327)

The parameterσ2µ is unknown, but we can typically estimate it consistently byσ2

µ

– this is indeed usually readily provided byDAD. Asymptotically, we can thenalso write that:

µ ∼ N(µ0, σ

2µ

), (328)

which also implies that

µ− µ0

σµ

∼ N(0, 1), (329)

a statistics that does not depend on unknown (or ”nuisance”) parameters, and thatis therefore ”pivotal”. Many of the results that follow rely implicitly on this result.


In the simplest cases, the estimators of interest can be expressed as a straight-forward sum of variable values across observations. Take for instance the case ofan estimatorα1 estimated using a sampleyin

i=1 of n observations ofyi:

α1 = n−1

n∑i=1

yi (330)

This is of course just the sample mean of theyi’s. As is well known, theasymptotic sampling distribution ofα1 is given by

α1 ∼ N(α1, n−1σ2

y) (331)

whereα andσ2y are respectively the population mean and the variance ofy. That

variance can be estimated consistently by the sample variance of theyi’s.Unfortunately, most of the distributive estimators do not take the simple form

(330). Instead, they often take the following general form:

θ = g (α1, α2, . . . , αK) , (332)

where

• αk is expressible as a sum of observations ofyk,i: αk =∑n

i=1 yk,i,

• θ can be expressed as a continuous functiong of theα′s,

• andyk,i is usually somek-specific transform of the income of individual orhouseholdi.

The sampling distribution ofθ will depend on the functiong and on the jointsampling distribution of the estimatorsαk, k = 1, . . . , K. This joint samplingdistribution is usually easily estimated by considering the joint distribution of theαk (recall (330)).

DAD then generally usesRao (1973)’s linearization approach to derive thestandard error of these distributive indices. Defineα = (α1, α2, . . . , αK)′ and letG be the gradient ofg with respect to theα’s:

G =

(∂θ

∂α1

,∂θ

∂α2

, . . . ,∂θ

∂αK

). (333)


A linearization ofθ then yields

θ ∼= θ + G((α1, α2, . . . , αK)′ − (α1, α2, . . . , αK)′) (334)

The sampling variance ofθ can then be shown to be asymptotically equal to

GV G′ (335)

whereV is the asymptotic covariance matrix of theαk, given by

V =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

var(α1) cov(α1, α2) ... cov(α1, αK)

cov(α2, α1) var(α2) ... cov(α2, αK

......

.. ....

cov(αK , α1) cov(αK , α2) ... var(αK)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(336)

The gradient elements∂θ∂α1

, ∂θ∂α2

, . . . , can be estimated consistently using the es-

timates ∂θ∂α1

, ∂θ∂α2

, . . . of the true derivatives. The elements of the covariance matrixcan also be estimated consistently using the sample data, replacing for instance

var(α) by var(α). Note that it is at the level of the estimation of these covarianceelements that the full sampling design structure is taken into account (see Section17.6).

18.2 Hypothesis tests

The outcome of an hypothesis test is a statistical decision: the conclusion ofthe test will either be torejecta null hypothesis,H0, in favor of an alternative,H1,or to fail to reject it. Most hypothesis tests involving an unknown true populationparameterµ fall into three special cases:

1. H0 : µ = µ0 againstH1 : µ 6= µ0

2. H0 : µ ≤ µ0 againstH1 : µ > µ0

3. H0 : µ ≥ µ0 againstH1 : µ < µ0

The ultimate statistical decision may be correct or incorrect. Two types of errorcan occur:


1. The first one, a Type I error, occurs when we rejectH0 when it is in facttrue.

2. The second one, a Type II error, occurs when we fail to rejectH0 whenH0

is in fact false.

The power of the test of an hypothesisH0 versusH1 is the probability of rejectingH0 in favor ofH1 whenH1 is true.

Let α be the level of statistical significance in which we are interested.α isoften referred to as the size of an hypothesis test. It is the probability of makinga Type I error that we are willing to tolerate, namely, the probability that we maywrongly reject a null hypothesis. Typical values ofα are 0.01, 0.025 , 0.05 and0.1. Letz(p) be thep-quantile of the standardized normal distribution. That is,if F is a standard normal distribution function, thenF (z(p)) ≡ p. Let µ0 be anestimate ofµ, that is, the value ofµ computed from the sample at hand, and definez0 asz0 = (µ0 − µ0)/σµ. The rules of rejection and non-rejection of the usualtypes of hypothesis tests are then as follows:

1. (Two-sidedH1) RejectH0 : µ = µ0 in favor ofH1 : µ 6= µ0 if and only if :

µ0 < µ0 − σµz(1− α/2) or µ0 > µ0 − σµz(α/2) (337)

Note that (337) is equivalent to:

z0 < z(α/2) or z0 > z(1− α/2). (338)

Note also that the size of such a test isα since, under the null hypothesis,we have that

P (z0 < z(α/2) or z0 > z(1− α/2)) = α. (339)

2. (Lower-boundedH1) RejectH0 : µ ≤ µ0 in favor of H1 : µ > µ0 if andonly if:

µ0 < µ0 − σµz(1− α). (340)

Again, (340) is equivalent to

z0 > z(1− α). (341)

3. (Upper-boundedH1) RejectH0 : µ ≥ µ0 in favor of H1 : µ < µ0 if andonly if :

µ0 > µ0 − σµz(α), (342)

which is equivalent toz0 < z(α). (343)


18.3 p-values and confidence intervals

Table3 sums up the confidence intervals andp-values for each of the threetypes of hypothesis tests considered above.

Thep-value of an hypothesis test is the smallest significance level for whichH0 would be rejected in favor of someH1. Roughly speaking, ap-value thusindicates the maximum probability that an error is made when one rejects a nullhypotheses in favor of the alternative hypothesis. It therefore gives us the ”risk”that there is of rejecting a null hypothesis.

A p-value is typically compared to some subjective error probability thresh-olds such as 1%, 5% or 10%. If thep-value exceeds these thresholds, we do notreject the null hypothesis; if thep-value lies beneath the threshold, we reject thenull hypothesis in favor of the alternative hypothesis.

A confidence interval (or, more generally, a confidence set)υ(1 − α) is arange of values that is constructed using the sample data and that has a specifiedprobability(1−α) of containing the true parameter of interestµ. The probabilityvalue1−α associated with a confidence interval is known as the confidence level.More formally, letΥ be the ”parameter space” ofµ, that is, the range of all of thepossible values thatµ could possibly take. A confidence intervalυ(1− α) is thenan estimate ofµ in the sense that there should be a high probability1 − α thatµis in that intervalυ(1− α).

More precisely, a confidence level(1− α) is the smallest probability thatµ isin υ(1− α):

1− α ≤ min P (µ ∈ υ(1− α) |µ ∈ Υ) . (344)

Typical confidence levels are 0.9, 0.95 and 0.99. Note thatυ(1 − α) is a randomvariable since it depends on the particular sample drawn from the population.Roughly speaking, a1 − α confidence level is then the proportion of the timesthat a confidence intervalυ(1 − α) will include the unknown parameter whenindependent samples are taken repeatedly from the same population, and that a1 − α confidence interval is calculated for each sample. As for hypothesis tests,confidence intervals can be two sided, lower bounded or upper bounded.

The width of a confidence interval thus gives us some idea about how uncertainwe are about the true unknown parameter. In fact, building confidence intervalsprovides more information than carrying out simple hypothesis tests of the typesdescribed above. This is because confidence intervals provide a range of plausiblevalues for the unknown parameter. Looking at Table3, it can also be seen thatthere is a nice symmetry between the results of hypothesis tests and the confidenceintervals that correspond to those tests. Indeed, the confidence intervals of Table


Table 3: Confidence intervals andp values associated to the usual hypothesistests

Case Confidence interval p-value H1 is:

1 [µ0 − σµz(1− α/2), µ0 − σµz(α/2)] 2[1− F (|z0|)] two-sided2 [µ0 − σµz(1− α), +∞] 1− F (z0) lower-bounded3 [−∞, µ0 − σµz(α)] F (z0) upper-bounded

3 include all of the hypothesizedH0 values that cannot be rejected in favor of thecorresponding two-sided, lower-bounded or upper-boundedH1 hypotheses. Saiddifferently, choosing anyµ0 value inside of these confidence intervals willnotlead to the rejection ofH0, but choosing any value ofµ0 outside of these intervalswill lead to the rejection ofH0 in favor ofH1.

18.4 Statistical inference using a non-pivotal bootstrap

The technique of the bootstrap (BTS), inspired in large part by?), is beingapplied with increasing frequency in the applied economics literature. BTS is amethod for estimating the sampling distribution of an estimator which proceedsby re-sampling repetitively one’s initial data. For each simulated sample, onerecalculates the value of this estimator and then uses that BTS distribution to carryout statistical inference. In finite samples, neither the asymptotic nor the BTSsampling distribution is necessarily superior to the other. In infinite samples, theyare usually equivalent. The following steps summarize a typical BTS procedure:

- Draw n observations with replacement from the initial sample by takinginto account the precise way in which the original sample was drawn (repli-cating, for instance, as closely as possible the survey design).

- Repeat the previous stepB − 1 independent times.

- Assess the sampling distribution of the estimator (for instance, its samplingvariance) using the distribution of itsB simulated values.

Let the vectorV be made ofB estimates ofµ, each one computed from one ofB simulated (or bootstrap) samples. The vectorV is the main tool for capturingthe sampling distribution of the estimatorµ. Thus, we have:

V = µ1, µ2, . . . , µB, (345)


whereµi is the estimate ofµ computed from theith bootstrap sample. For two-sided tests and confidence intervals with significance levelα or confidence level1−α, the number of simulations should be chosen so thatα(B+1)/2 is an integer(to facilitate the computation of critical test values). Letµ∗(p) be thep-quantileof the vectorV : we then have thatp = B−1

∑Bi=1 I(µi ≤ µ∗(p)). The rules of

rejection and non-rejection are then:

1. RejectH0 : µ = µ0 in favor ofH1 : µ 6= µ0 if and only if :

µ0 > µ∗(1− α/2) or µ0 < µ∗(α/2). (346)

2. RejectH0 : µ ≤ µ0 in favor ofH1 : µ > µ0 if and only if:

µ0 < µ∗(α). (347)

3. RejectH0 : µ ≥ µ0 in favor ofH1 : µ < µ0 if and only if :

µ0 > µ∗(1− α). (348)

Table4 summarizes the confidence intervals andp-values for each of the threeusual types of hypothesis tests, using non-pivotal bootstrap statistics. The inter-pretation and the use of these statistics are analogous to what we saw in Section18.3.


Tabl

e4:

Con

fiden

cein

terv

als

andp

valu

esfo

rth

eus

ualh

ypot

hesi

ste

sts,

usin

gno

n-pi

vota

lboo

tstr

apst

atis

-tic

sC

ase

Con

fiden

cein

terv

alp-

valu

eH

1is

:

1[µ∗ (

α/2

),µ∗ (

1−

α/2

)]2B

−1m

in

(B ∑ i=

1

I(µ

i≤

µ0),

B ∑ i=1

I(µ

i≥

µ0))

two-

side

d

2[µ∗ (

α),

+∞

]B−1

B ∑ i=1

I(µ

i≥

µ0)

low

er-b

ound

ed

3[−∞

,µ∗ (

1−

α)]

B−1

B ∑ i=1

I(µ

i≤

µ0)

uppe

r-bo

unde

d


18.5 Hypothesis tests and confidence intervals using pivotal boot-strap statistics

Let µ andσµibe respectively the average of theµi in V and the estimate of the

asymptotic standard deviation ofµ computed from theith bootstrap sample. Letti be the following asymptoticallypivotalstatistics:

ti =µi − µ

σµi

. (349)

ti is asymptotically pivotal since it follows asymptotically a standardizedN(0, 1)normal distribution which is free of ”nuisance” parameters,i.e., parameters thatare unknown.

Let the vectorV then be defined as:

V = t1, t2, · · · , tB (350)

and lett∗(p) bep-quantile of the vectorV . The rules of rejection and non-rejectionof the usual null hypotheses are then as follows:

1. RejectH0 : µ = µ0 in favor ofH1 : µ 6= µ0 if and only if :

z0 < t∗(α/2) or z0 > t∗(1− α/2). (351)

2. RejectH0 : µ ≤ µ0 in favor ofH1 : µ > µ0 if and only if:

z0 > t∗(1− α). (352)

3. RejectH0 : µ ≥ µ0 in favor ofH1 : µ < µ0 if and only if :

z0 < t∗(α). (353)

Table5 summarizes the confidence intervals andp-values associated to each ofthe three usual types of hypothesis tests, using pivotal bootstrap statistics. Again,these statistics can be interpreted and used basically as above in Section18.3.


Tabl

e5:

Con

fiden

cein

terv

als

andp

valu

esfo

rth

eus

ualh

ypot

hesi

ste

sts,

usin

gpi

vota

lboo

tstr

apst

atis

tics

Cas

eC

onfid

ence

inte

rval

p-va

lue

H1

is:

1[µ

0−

σµt∗

(1−

α/2

),µ

0−

σµt∗

(α/2

)]2B

−1m

in

(B ∑ i=

1

I(t

i≤

z 0),

B ∑ i=1

I(t

i≥

z 0))

two-

side

d

2[µ

0−

σµt∗

(1−

α),

+∞

]B−1

B ∑ i=1

I(t

i≥

z 0)

low

er-b

ound

ed

3[−∞

,µ0−

σµt∗

(α)]

B−1

B ∑ i=1

I(t

i≤

z 0)

uppe

r-bo

unde

d


18.6 References

Much of the statistical inference literature for distributive analysis has focusedon deriving the sampling distribution of inequality and poverty indices. SeeCow-ell (1999) andDavies, Green, and Paarsch (1998) for overall reviews, as well asAaberge (2001b) for cross-country evidence of the role of sampling variability,Barrett and Pendakur (1995) for generalized Gini indices,Beach and Al. (1994)for decile means,Bishop, Chakraborti, and Thistle (1990) for Sen’s welfare index,Bishop, Chakraborti, and Thistle (1991a) for relative Gini-based relative depriva-tion indices,Bishop, Chow, and Zheng (1995b) for decomposable poverty indices,Bishop, Formby, and Zheng (1997) for Sen’s poverty index,Bishop, Formby, andZheng (1998) for Gini-based progressivity indices,Chotikapanich and Griffiths(2001) for approximating S-Gini indices using grouped data,Davidson and Duc-los (2000b) for various classes of poverty indices with deterministic and estimatedpoverty lines,Duclos (1997a) for linear progressivity and vertical equity indices,Kakwani (1993) for additive poverty indices,Ogwang (2000) for the Gini index,Preston (1995) for poverty indices with estimated poverty lines,Rongve (1997)for poverty indices with known poverty lines,Rongve and Beach (1997) for theuse of approximations to inequality indices,Thistle (1990) for two classes of in-equality indices,Van de gaer, Funnell, and McCarthy (1999) andZheng and Cush-ing (2001) for comparing inequality across statistically dependent incomes,Xu(1998) for theP (z, ρ = 2) poverty index, andZheng (2001b) for poverty indiceswith estimated poverty lines.

The second major area of statistical inference research in distributive analysishas dealt with the sampling distribution of tools for stochastic dominance. This in-cludesAnderson (1996) for integrals of distribution functions,Bahadur (1966) forquantiles,Beach and Davidson (1983) for the Lorenz curve,Bishop, Chakraborti,and Thistle (1989) for Generalized Lorenz curves,Bishop and Formby (1999)for a review,Dardanoni and Forcina (1999) for different inference approachesto ordering Lorenz curves,Davidson and Duclos (1997) for Lorenz and con-centration curves,Davidson and Duclos (2000b) for primal and dual dominancecurves,Klavus (2001) for an application to health care financing in Finland,Maa-soumi and Heshmati (2000) for an application to Swedish distributions,Xu (1997)for Generalized Lorenz curves,Xu and Osberg (1998) for ”deprivation curves”,Zheng, John, Formby, James, and Chow (2000) for mean-normalized dominancecurves, andBishop, Chow, and Formby (1994b), andZheng (1999b) for marginaldominance analysis using Lorenz and quantile curves.

Issues, methods and applications dealing with the multiple hypothesis tests


associated to inferring stochastic dominance orderings can be foundinter alia inBarrett and Donald (2003) for simulations of the distribution of statistics neededfor complete sets of hypothesis tests,Beach and Richmond (1985) for the jointsampling distribution of some of these statistics,Bishop, Formby, and Thistle(1992) andBishop, Chakraborti, and Thistle (1994a) for applications of the union-intersection approach,Kaur, Prakasa Rao, and Singh (1994) for testing second-order dominance,Kodde and Palm (1986) for Wald criteria for joint testing ofequality and inequality hypotheses, andWolak (1989) for testing multivariate in-equality constraints.

For general references to the bootstrap, seeEfron and Tibshirani (1993) and?).Specific applications of the bootstrap and other re-sampling simulation methodsto distributive analysis can be foundinter alia in Biewen (2000) (for inequalityindices),Biewen (2002a) (for a demonstration of the consistency of bootstrap-ping inequality, poverty and mobility indices),Mills and Zandvakili (1997) (forinequality indices),Palmitesta, Provasi, and Spera (2000) (for the Gini family ofinequality indices),Xu (2000) (for iterated bootstrapping of the S-Gini indices),Karagiannis and Kovacevic’ (2000) andYitzhaki (1991) for jackknife calculationsof the variance of the Gini.

For the use of the ”influence function” in protecting against the possible pres-ence of contaminated data, seeCowell and Victoria Feser (1996b) (for inequalityindices),Cowell and Victoria Feser (1996a) (for poverty indices), andCowell andVictoria Feser (2002) (for social welfare rankings).

Other statistically relevant works can be found (among others) inElbers, Lan-jouw, and Lanjouw (2003) andHentschel and et al. (2000) for ”poverty mapping”– the estimation of small-area statistics on poverty and inequality using variousdata sources;Breunig (2001) for a bias correction to the estimation of the coef-ficient of variation; andLerman and Yitzhaki (1989) for the impact of using ag-gregated data in the estimation of inequality indices and in making social welfarerankings.

To generate estimation and statistical inference results usingDAD, the analystdoes not need to specify the functional forms of the distribution of the populationof interest. Said differently, to estimate, for instance, poverty and equity indices,or to generate the asymptotic standard deviations of such indices, we do not needto tell DAD that the incomes we are studying are distributed according to a normal,a Pareto, or a beta distribution for instance. In that sense, all ofDAD’s results are”distribution free”.

In some circumstances, it may however be useful to do distributive analysisconditional on some distributional assumption. Examples of such analysis can be


found inChotikapanich and Griffiths (2002) (estimation of Lorenz curves),Cow-ell, Ferreira, and Litchfield (1998) (density estimation in Brazil),Cheong (2002)(estimation of US Lorenz curves),Horrace, Schmidt, and Witte (1995) (samplingvariability of order statistics using parametric and non-parametric approaches),Ogwang and Rao (2000) (parametric models of Lorenz curves),Ryu and Slottje(1999) (parametric approximations of Lorenz curves),Sarabia, Castillo, and Slot-tje (1999) andSarabia, Castillo, and Slottje (2001) (general methods for buildingparametric models of Lorenz curves), andSchluter and Trede (2002b) (parametricestimation of tails of Lorenz curves).

19 EXERCISES. 260

19 Exercises.19.1 Household size and living standards19.1.1Using the file AGGR-7 [276], compute the poverty headcount by using the vari-able EXPCAP and a poverty line of 373 FCFA.answer

19.1.2Then, find the poverty line which you must use with the variable TTEXP to obtainthe same estimate of poverty as that obtained in question19.1.1. answer

19.1.3Using for the variable EXPCAP the poverty line used in question19.1.1, and forthe variable TTEXP the poverty line found in question19.1.2, decompose povertyacross household size GSIZE using EXPCAP and TTEXP. Discuss.answer

19.1.4Using again the same file AGGR-7 [276], decompose poverty across the sex of thehousehold head SEX by using EXPCAP and TTEXP and their associated povertyline used in questions19.1.1and19.1.2. Discuss.answer

19.2 Choice of aggregating weights and poverty analysis.19.2.1Using the file AGGR-7 [276], compute total poverty in Cameroon without usingthe SIZE variable and by using it. Discuss.

19.2.2Using the file AGGR-7 [276], decompose total poverty in Cameroon according tothe categories captured by GSIZE without using the SIZE variable and by usingit. Discuss.

19.2.3Using the file DECB-8 [276], decompose total poverty in Cameroon according tothe categories captured REGION without using the SIZE variable and by using it.Discuss.

http://132.203.59.36/DAD/answers/ans11/ans11.htm




19 EXERCISES. 261

19.3 Absolute and relative poverty19.3.1Using the file DECB-8 [276], compute the average of EXPEQ for the whole ofCameroon and for each of the two regions of REGION.

19.3.2Then, compute half of these averages for the whole of Cameroon and for eachof its two regions in REGION. (These statistics are subsequently used as relativepoverty thresholds.)

19.3.3Finally, compute the poverty headcount for the whole of Cameroon and for eachof its two regions using as poverty lines :

a- a national absolute threshold of 373 FCFA;

b- the national relative threshold;

c- the relative thresholds for each of the two regions.

Check whether using an estimate of the national relative threshold (asopposed to a known or deterministic national relative threshold) hasan impact on the standard error of the national headcount.

19.4 Estimating poverty lines19.4.1Computing a food poverty line with a ”FEI-inspired” method.With LINE-6[276], draw a non-parametric regression of FDEQ on CALEQ for an interval ofCALEQ of 0 to 4000 calories. Find the level of food expenditures that is expectedto yield an intake of 2400 calories per day.

19.4.2Computing a CBN poverty line. With LINE-6 [276], draw a non-parametric re-gression of EXPEQ on FDEQ for an interval of FDEQ which includes the foodpoverty line estimated in19.4.1. Find the level of total expenditures expected at alevel of food expenditures equal to the food poverty line estimated in19.4.1.

19.4.3Using the results of19.4.1and19.4.2, compute the share of food expenditures inthe total expenditures of those whose level of food expenditures equals the foodpoverty line. By dividing the food poverty threshold by this share, estimate aglobal poverty line.

19 EXERCISES. 262

19.4.4A second method of estimation of the share of food expenditures in total expen-ditures. With LINE-6, draw a non-parametric regression of FDEQ on EXPEQfor an interval of EXPEQ of 0 to 500 FCFA. Find the level of food expendituresexpected at a level of total expenditures equal to the food poverty line estimatedin 19.4.1.

19.4.5Using the results of19.4.4, compute the share of food expenditures in the totalexpenditures of those whose level of total expenditures equals the food povertyline estimated in19.4.1. By dividing the food poverty line by this share, estimatea second global poverty line.

19.4.6A third method for the estimation of the non-food poverty line.With LINE-6,draw a non-parametric regression of EXPEQ on FDEQ for an interval of FDEQwhich includes the food poverty line estimated in19.4.1. Find the level of totalexpenditures expected at a level of food expenditures equal to the food povertyline estimated in19.4.1.

19.4.7Using the results of19.4.6, compute the expected non-food expenditures of thosewhose level of total expenditures equals the food poverty line estimated in19.4.1.By adding these expected non-food expenditures to the food poverty line, estimatea third global poverty line.

19.4.8Computation of a global poverty line according to the FEI method. With LINE-6[276], draw a non-parametric regression of EXPEQ on CALEQ for an intervalranging from 0 to 4000 calories for CALEQ. Estimate the global poverty line thatcorresponds to 2400 calories per day.

19.5 Descriptive data analysis

19.5.1Density functions. With DECB-8 [276], estimate the density of LEXPEQ for

the whole country and for each region (by using REGION).

19.6 Decomposing poverty

19 EXERCISES. 263

19.6.1With AGGR-7 [276], decompose poverty across SEX. Then check the calcula-tions of the absolute and relative decompositions provided byDAD by separatelycalculating the poverty indices for each group in SEX. Reconstruct manually thedecomposition to verify thatDAD gives the correct decomposition results.

19.6.2Using DECA-7 [276], decompose total poverty according to the socio-economiccategories AGE and EDUC.

19.6.3Using DECB-8 [276], decompose total poverty according to the socio-economiccategories SECT, TYPE and OCCUP.

19.7 Poverty dominance

19.7.1Using DECA-7 [276], plot the first-order dominance curves separately for thosewho have a primary level and a superior level of education (see variable EDUC)and for poverty lines varying between 0 and 1000 FCFA What do these curvesshow?

19.7.2Using DECA-7 [276], plot the first-order dominance curves separately for thefemale-headed and for the male-headed households, for poverty lines varying be-tween 0 and 300 FCFA. What do these curves indicate? Find the relevant ”criticalthresholds” and comment.

19.7.3Repeat19.7.1and19.7.2for second- and third-order dominance.

19.7.4Compute the FGT poverty index forα = 0 and for poverty lines equal to 150, 250and 300 FCFA, separately for the female- and male-headed households.

19.7.5Repeat19.7.4for second- and third-order dominance.

19.7.6Using DECB-8 [276], draw poverty gap curves separately for the two groups iden-tified by the variable REGION.

19 EXERCISES. 264

19.7.7Using DECA-7 [276], draw poverty gap curves separately for the female-headedand the male-headed households.

19.7.8Using DECB-8 [276], draw CPG curves separately for the two groups identifiedby the variable REGION.

19.7.9Using DECA-7 [276], draw CPG curves separately for the female-headed and themale-headed households.

19 EXERCISES. 265

19.8 Fiscal incidence, growth, equity and poverty

19.8.1Use the file ”CAN4”[280] to predict the level of taxes paid and benefits receivedby individuals at different gross incomes X. For this, use the window ”non-parametricregression”, and choose alternatively for thex axis the ”level” or the ”percentile”of gross incomes. What do these regressions indicate?

19.8.2Use the file ”CAN6”[280] to draw the Lorenz Curve for gross income (X) and netincome (N) in 1990 Canada.

a- What does the difference between the two Lorenz curves indi-cate?

b- Then, draw a concentration curve for each of the three transfersB1, B2 et B3 and the tax T. What can you say about the TRprogressivity and the ”equity” of the distribution of the tax andbenefits?

c- Would a proportional increase in the benefit B1 combined witha proportional decrease in B3 of the same absolute magnitudebe good for inequality, poverty and social welfare?

d- Would a proportional increase in the benefit B2 financed by abalanced-budget proportional increase in the tax T be good forinequality, poverty and social welfare?

19.8.3Use the same file ”CAN6”[280] to check the IR progressivity of each of the threebenefits and the tax T. For this, you can draw concentration curves for X combinedseparately with each of the three transfers B1, B2 and B3 and the tax T. What canyou say about the IR progressivity and the ”equity” of the distribution of the taxand benefits? How does it compare with the TR progressivity results?

19.8.4Using the file ”CAN4”[280], compute the concentration indices of each of B andT, and compare them to the Gini index of gross income X. Then, compute anestimate of TR progressivity of the tax and benefit system in Canada.

19.8.5Compare the Lorenz curve for N with the concentration curve for N (using X asthe ranking variable). What does this tell you?

19 EXERCISES. 266

19.8.6Express the total redistribution exerted by the Canadian tax and transfer system asvertical equity minus horizontal inequity (reranking), using Gini and concentra-tion indices.

19.8.7Draw the conditional standard deviation of benefits B and taxes T at various valuesof gross income X.

19.8.8Draw the conditional standard deviation of net income N at various values of grossincome X. What does this indicate?

19.8.9Draw the share of total taxes T paid by those at differentlevelsof gross income X,and at differentranksof X. Compute this as the ratio of expected taxes over meangross incomeµX . Do the same for total benefits B. Compare this to the share oftotal gross income, computed asX overµX . What does this say?

19.8.10Compute the average tax rate paid by individuals at differentlevelsof gross in-come X and at differentranksof X. Estimate this as the expected tax paid atXoverX. What does this say about tax progressivity in Canada?

Use the file PERHE-12 [281] for exercises19.8.11to 19.8.17.

19.8.11Compare the Lorenz curve ofper capita total expenditures (EXPCAP), usingSIZE, and of total expenditures (TTEXP). Which type of expenditures is moreequally distributed? Why?

a- To understand better why, add to the graph a concentration curveof total expenditures, usingper capitaexpenditures as the rank-ing variable, and WHHLD to count observations; this will indi-cate the concentration of total expenditures among the pooresthouseholds, ranked byper capitaexpenditures.

b- To complete your understanding, add a concentration curve forhousehold size, using WHHLD as the aggregating weight andEXPACP as the ranking variable; this will indicate the concen-tration of individuals among the poorest households, as rankedby EXPCAP. Does this help you understand the difference be-tween the above two Lorenz curves?

19 EXERCISES. 267

19.8.12Predict the proportion of individuals who visited a public health center and a pub-lic hospital in a given month. For this, use the variables CENTRO and HOSPIT,who indicate the proportion of individuals in a household who visited these in-stitutions. Make this prediction at different percentiles of the distribution ofpercapitatotal expenditures.

19.8.13Graph again the concentration curve of total expenditures (TTEXP) using WHHLDand EXPCAP to rank individuals. Compare it to the concentration curve amonghouseholds (thus use WHHLD) of their use of health centres and public hospitals,which is given respectively by NCENTRO and NHOSPIT, and use EXPCAP torank households. What does this suggest?

19.8.14Add to the previous graph the concentration curve of individuals in households.What does this information add to your equity judgement?

19.8.15Draw on a new graph the concentration curve of total expenditures (TTEXP) us-ing WHHLD and EXPCAP to rank households. Compare this to the concentrationcurves for access to piped water (PUBWAT) and to sewerage (PUBSEW), usingWHHLD as the aggregating weight to draw the curves and EXPCAP as the rank-ing variable. That is, find out the concentration of access to piped water andsewerage among various proportions of poorest households, and compare that totheir share in total expenditures. What do you find?

19.8.16Add to your previous graph the concentration curves for the number of individualswho have piped water (NPUBWAT) and who have sewerage (NPUBSEW), usinghousehold weighting and EXPCAP as the ranking variable. How do you interpretthe differences you obtain with the results of question19.8.15?

19.8.17Redo the previous analysis of the incidence of access to piped water (PUBWAT)and to sewerage (PUBSEW), but this time use individual weighting (which isusually considered to be the best descriptive choice from a normative or ethi-cal perspective). Thus, draw the Lorenz curve ofper capitatotal expenditures(EXPCAP) using individual weighting, WIND. Compare this to the concentra-tion among individuals of the access to piped water (PUBWAT) and to sewerage

19 EXERCISES. 268

(PUBSEW), using WIND as the aggregating weight to draw the curves and EXP-CAP as the ranking variable. That is, find out the concentration of access to pipedwater and sewerage among various proportions of poorest individuals, and com-pare that to their share of the population of the total expenditures.

Use the file PERED-16 [281] for exercises19.8.18to 19.8.22.

19.8.18Make a new graph again of the concentration curve of total expenditures (TTEXP)using WHHLD and EXPCAP to rank households. Compare it to the concentra-tion curve of the number of children at various levels of public education, NPUB-PRIM, NPUBSEC and NPUBUNIV using the same aggregating weights. Is ed-ucation enrolment equitably distributed according to this? What happens to ourunderstanding of the ”picture” if we add the concentration curve for the numberof children NCHILD?

19.8.19Now add the Lorenz curve ofper capitatotal expenditures EXPCAP using NCHILDas the size variable. Compare it to the concentration curve of the enrolment of chil-dren at various levels of public education, which is given by PUBPRIM, PUBSECand PUBUNIV, using NCHILD and EXPCAP as the ranking variable. Has yourequity judgement evolved?

19.8.20Redraw the Lorenz curve ofper capitatotal expenditures, now using WCH0612as the aggregating weight, and compare it to the concentration curve of PUBPRIMusing the same aggregating weight.

19.8.21Redraw the Lorenz curve ofper capitatotal expenditures now using WCH1318 asthe aggregating weight, and compare it to the concentration curve of PUBPSECusing the same aggregating weight.

19.8.22Test the hypothesis that a small increase in secondary school fees combined witha decrease in primary school fees of the same total magnitude would not changethe distribution of well-being in Peru.

Use the file SENESAM [283] to do exercises19.8.23to 19.8.26.

19 EXERCISES. 269

19.8.23Using EXPEQ as ranking variable, predict the proportion of children between 7and 12 at different levels of living standards who attend primary school. Comparethese results to those you obtain when you separate children into boys and girls.

19.8.24Draw the conditional standard deviation of primary school attendance separatelyfor boys and girls at various values of EXPEQ. What does this indicate?

19.8.25Draw the conditional standard deviation of primary school attendance separatelyfor each of the three STRATA, and this, at various values of EXPEQ. Explainwhat you find.

19.8.26Compare the Lorenz curve for EXPEQ with the concentration curve for EXPEQ(using TTEXP as the ranking variable). What does this suggest?

Use the file ESPMEN [288] to do exercises19.8.27to 19.8.53. When needed,use EXPEQ as the variable of interest, the headcount as the poverty index, and apoverty line of 60000 FCFA per adult equivalent.

19.8.27Draw the concentration curves of FDEQ, NFDEQ, HEALTHEQ and SCHEXPEQfor the population of individuals (i.e., setting the size variable to SIZE) and usingEXPEQ as the ranking variable. How do these curves compare to the Lorenz curvefor EXPEQ?

19.8.28Compare the Lorenz curves of EXPEQ and INCOMEQ. What do you find? Howdo you explain this?

19.8.29Compare the Lorenz curves of EXPEQ for each of the 3 values of DEPT.

19.8.30Draw the CD curve (normalized by the mean of the variables but not by thepoverty lines) of FDEQ and NFDEQ for different poverty lines and for c=1. Whatdoes it tell you?

19 EXERCISES. 270

19.8.31Compute the Gini inequality index for TTEXP, EXPEQ and INCOMEQ. Do thisfor values ofρ equal to 1, 2 and 3. Then, draw these indices for each of thesevariables on a graph forρ ranging from 1 to 5.

19.8.32Decompose inequality in EXPEQ as a sum of inequality in each of its four com-ponents, FDEQ, NFDEQ, HEALTHEQ and SCHEXPEQ.

19.8.33Draw the share of total SCHEXPEQ of those at differentlevelsof EXPEQ, andat differentranksof EXPEQ. Compute this as the ratio of expected SCHEXPEQconditional on some value of EXPEQ over that value of EXPEQ. over Do thesame for HEALTHEQ. What do you find?

19.8.34What would the impact on poverty be if we were to transfer 1000 FCFA (per adultequivalent) to each individual in the population?

19.8.35Where, among the different DEPT, would the impact of group-targeting an equalamount to all be the greatest for the same overall budget spent by the government?Does this result depend on the choice of the poverty line?

19.8.36Where, among the different REGION, would the impact of group-targeting anequal amount to all be the greatest for the same overall budget spent by the gov-ernment?

19.8.37Assume that some form of government targeting can raise everyone‘s EXPEQ bythe same proportion in a particular area. Per FCFA of overallper capitaincreasein EXPEQ, for which targeted DEPT would aggregate poverty reduction be thelargest? Check this for the headcount and for the average poverty gap indices.

19.8.38Assume that some form of government targeting can raise everyone‘s EXPEQ bythe same proportion in a particular area. Per FCFA of overallper capitaincreasein EXPEQ, for which targeted ZONE would aggregate poverty reduction be thelargest?

19 EXERCISES. 271

19.8.39Say that food prices are about to increase by about 5%, due to the removal of foodsubsidies. In which group within DEPT will poverty increase the most?

19.8.40Using CD curves, check whether the ZONE for which the impact of an increasein food prices will be the largest depends on the choice of the poverty line and onthe choice of poverty index (focus on first-order poverty indices).

19.8.41The government wishes to determine whether increasing the price of HEALTHEQ,for the benefit of a fall in the price of SCHEXPEQ, would be good for poverty.

a- Compare the distributive cost/benefit of changing the price ofeach of HEALTHEQ and SCHEXPEQ.

b- Check whether the reform is good for poverty for ratios of MCPFranging from 0.5 to 2.0.

19.8.42Find the impact on poverty of those within ZONE=1 of a predicted increase of 3%in expenditures EXPEQ.

19.8.43Find the impact on national poverty of a predicted increase of 3% in the expendi-tures EXPEQ of those within ZONE=1. Compare your results to those obtainedfor ZONE=2. Do this for FGT indices withα=0, 1 and 2.

19.8.44Find the impact on national poverty of a predicted increase of 3% in everyone’sexpenditures FDEQ. Compare your results to those for a 3% increase in every-one’s NFDEQ. Do this for the FGT indices withα=0, 1 and 2.

19.8.45Per FCFA of growth in overallper capita EXPEQ, in which of ZONE=1 orZONE=2 is growth in expenditures EXPEQ conducive to greater poverty reduc-tion?

19.8.46Per FCFA of growth in overallper capitaEXPEQ, which of growth in FDEQ orin NFDEQ leads to greater poverty reduction? Graph this for a range of povertylines and for all poverty indices of the second-order (α=1, ors=2).

19 EXERCISES. 272

19.8.47What is the elasticity of poverty with respect to EXPEQ? Compute this for thedifferent DEPT.

19.8.48The government wishes to determine whether increasing the price of HEALTHEQby 5%, for the benefit of a revenue-neutral fall in the price of SCHEXPEQ, wouldbe good for inequality reduction. Assume a ratio of MCPF=1. Find out the impacton the Lorenz curve and on the Gini coefficient.

19.8.49Say that food prices are about to increase by about 10%, due to the removal offood subsidies. What is the predicted impact on the Gini index and on L(p=0.5)?

19.8.50Find the impact on the Gini index of inequality of a predicted increase of 3% inthe expenditures EXPEQ.

19.8.51Find the impact on the Gini index and onL(p = 0.5) of a predicted increase of3% in everyone’s expenditures FDEQ. Compare your results to those for a similar3% increase in NFDEQ.

19.8.52The government wishes to determine whether increasing the price of NFDEQ, forthe benefit of a fall in the price of FDEQ, would be good for poverty.

i Assess this for a ratio of the MCPF of NFDEQ over that of FDEQ equalto 1, for a range of poverty lines and for all distribution-sensitive povertyindices (second-order,α=1 ors=2).

ii Up to which ratio of MCPF can we go and still declare the reform to begood for poverty?

iii Are these conclusions also valid for the goal of inequality reduction?Use the file ESPSANT [289] to do exercises19.8.53to 19.8.54.. Wen needed, usethe headcount as a poverty index and set the poverty line to 60000 FCFA per adultequivalent.

19.8.53Using EXPEQ as the ranking variable, predict the proportion of individuals atdifferent levels of living standards whose households make use of public healthservices. Do this separately for the different values of SEX.

19 EXERCISES. 273

19.8.54Compute the proportion of EXPEQ that is spent on HEALTHEQ by individualsat differentlevelsandrank of EXPEQ. What does this suggest?

Use the file SCOL [290] to do exercises19.8.55to 19.8.57. When needed, use theheadcount as the poverty index and set the poverty line to 60000 FCFA per adultequivalent.

19.8.55Predict the proportion of children below 14 at different values of EXPEQ that at-tend primary school. Compare the results you obtain across the different values ofZONE. How do these results compare with those for attending secondary school?

19.8.56Compare the concentration curve (among children below 14) of attendance atprimary school, secondary school, public primary school and public secondaryschool, using EXPEQ as the ranking variable. Draw this for various proportionsof the poorest children. Compare this concentration curve with the Lorenz curvefor EXPEQ. Discuss your results.

19.8.57Draw the concentration curves of UNI and SUP. Compare this to the Lorenz curvefor EXPEQ for the same population.

19 EXERCISES. 274

19.9 Impact of a complex sample design on the estimated stan-dard deviation of various distributive statistics.

19.9.1Load the file Burkina94 [291] and initialize its sampling design (SD). Do thisfirst only by specifying the variable WEIGHT as sampling weight.

a- Compute the mean of total expenditure per adult equivalent (EX-EPQ) with the size variable equal to SIZE. Why does STD1differ from STD2? What is a sufficient condition so that bothstandard deviations be equal?

b- Now use both variables WEIGHT and STRATA to reinitializethe SD of this file. Compute, again, the mean of total expen-diture per adult equivalent when the size variable is SIZE, andcompare between the STD’s of question a. What can be saidabout the impact of stratification on STD1?

c- Now use variables WEIGHT, STRATA and PSU to reinitializethe SD of this file. Compute, again, the mean of total expen-diture per adult equivalent when the size variable is SIZE, andcompare between the STD’s of questions a and b. What can yousay about the impact of PSU’s on STD1?

d- By using the GSE variable to specify the social professionalgroup, compute the mean of total expenditure per adult equiv-alent when the size variable is SIZE and for groups 1,2, and 6.How does the sampling variability differ across these estimates?

19.10 Impact of equivalence scales and choice of statistical units19.10.1Load the file Senegal95 [283] and compute the mean of total expenditure peradult equivalent (EXEPQ) after initializing the sampling design with variablesSTRATA, PSU and WEIGHT.

19.10.2Well-being, in a household, can be represented alternatively by:

a- Total expenditure of household “EXP”

b- Total expenditure per capita “EXPCAP”

c- Total expenditure per adult equivalent “EXPEQ”

19 EXERCISES. 275

19.10.3Using the variable SIZE to set the size variable, compute the headcount and theaverage poverty gap indices when the poverty line equals 140000 FCFA and whenthe variable of interest is alternatively EXPEQ, EXPCAP and EXP. Explain whythe results differ.

19.10.4When sample observations represent households, three size variables can be usedin combination with the variable of interest EXPEQ:

a- 1 for all households

b- The number of persons in the household (SIZE)

c- The number of adult equivalents in the household (EQUI)

Compute the FGT index forα = 0, 1 for every one of these three alternativedefinitions of the size variable and explain the difference.

19.10.5Compute the Gini and Atkinson (withε = 0.5) indices of inequality for:

a- Total expenditure when the size variable equals 1 for all house-holds.

b- Expenditure per capita when the size variable equals SIZE.

c- Expenditure per equivalent adult when the size variable equalsSIZE.

d- Expenditure per equivalent adult when the size variable equals1.

Comment on the differences between these results.

19 EXERCISES. 276

CAMEROON 96, LINE-6, AGGR-7, DECA-7, and DECB-8

The files LINE-6, AGGR-7, DECA-7, and DECB-8 are made of a sub-sample of1000 observations drawn from a survey (theEnquete Camerounaise aupres desmenagesor ECAM) on the expenditures and the incomes of households in 1996Cameroon. The file CAMEROON96 is made of approximately 1700 householdsfrom the same survey. The ECAM is a nationally representative survey, with sam-ple selection using two-stage stratified random sampling. The first stage consistsin the selection of 150 PSUs (“ılot”) within each of the six strata, and the secondstage consists in the selection of households within each PSU.

Yaounde Douala Cities Rural (3 strata) Total(<50000 inhabitants)

Households 336 384 360 630 1710PSU (ılot) 42 48 30 30 150

In Yaounde and Douala, PSU’s are systematically selected with equal prob-abilities. The number of PSU’s drawn by stratum is proportional to the numberof urban households found in 1987 in that stratum. In a second stage, 8 house-holds are drawn in every PSU (with equal probabilities), using a list of householdsestablished during an enumeration of that PSU.

For the other cities, at the first stage, one city is selected for every one of theten provinces. Enumeration zones are then drawn with probability proportional tothe number of households originally listed in 1987. Households are then drawn asabove.

In every one of the three rural strata, two PSU’s were selected within the semi-urban area and 8 in the rural area. PSU’s were again drawn with probability pro-portional to the number of households enumerated in 1987. Within each selectedPSU, 21 households were systematically selected from a household list.

19 EXERCISES. 277F

igur

e8:

19 EXERCISES. 278

Variables for CAMEROON 96

STRATA: Stratum of the household

1. Yaounde

2. Douala

3. Cities

4. Rural: Forest

5. Rural: Hauts-Plateaux

6. Rural: Savane

PSU: PSU of the household

WEIGHT: Sampling weight

SIZE: Household size

NADULT: Number of adults

NCHILD: Number of children

EXPEQ: Total expenditures per adult equivalent per day

F EXP: Food expenditures per adult equivalent per day

NF EXP: Non food expenditures per adult equivalent per day

INS LEV: Education level of the head of the household

1. Primary

2. Professional Training

3. Secondary 1st cycle

4. Secondary 2nd cycle

5. Superior

6. Not responding

Variables for LINE-6, AGGR-7, DECA-7, and DECB-8

TTEXP: total expenditures of the household (all monetary data are in Francs CFAper day)

EXPEQ: total expenditures of the household, per adult equivalent

EXPCAP: total expenditures of the household, per capita

FDEQ: food expenditures of the household, per adult equivalent

NFDEQ: non- food expenditures of the household, per adult equivalent

19 EXERCISES. 279

WHHLD: sampling weight of the household

SIZE: household size

CALEQ: calories of household, per adult equivalent.

SEXE: sex of household head (man, woman , not reported)

REGION: region (urban, rural)

AGE: age of household head (less than 35 years; 35-50 years; more than 50 years)

EDUC: Education level of household head (primary; vocational training; secondaryfirst cycle; secondary second cycle; superior; other)

TYPE: household type (one adult; single parent; nuclear; wider nuclear household)

OCCUP: usual occupation of household head (independent with employees; inde-pendent without employees; unqualified employee; manager or qualifiedemployee; traders; others)

SECT: sector of occupation (formal, informal, other)

GSIZE: indicator of household size (1 person; 2 persons; 3-4 persons; 5-7 persons;8 or more)

19 EXERCISES. 280

CAN4 and CAN6

CAN4 andCAN6 come from the Canadian Surveys of Consumer Finance. Theycontain the following variables:

X: Yearly gross income per adult equivalent.

T: Income taxes per adult equivalent.

B1: Transfer 1 per adult equivalent.



B: Sum of transfers B1, B2 and B3

N: Yearly net income per adult equivalent (X minus T plus B)

19 EXERCISES. 281

PERHE-12 and PERED-16

PERHE-12 and PERED-16 contain a sample of 3623 household observationsdrawn from the 1994 Peru LSMS survey.

Variables for PERHE-12

TTEXP: total expenditures of household (constant June 1994 soles per year).

EXPCAP: total expenditures,per capita(constant June 1994 soles per year).

WHHLD: household aggregation weight.

SIZE: household size.

CENTRO: proportion of individuals in the household who used a public health centerin last month.

HOSPIT: proportion of individuals in the household who used a public hospital in lastmonth

PUBWAT: household has piped water.

PUBSEW: household has sewerage

NCENTRO: number of individuals in the household who used a public health center inlast month

NHOSPIT: number of individuals in the household who used a public hospital in lastmonth

NPUBWAT: number of individuals in household who have access to piped water

NPUBSEW: number of individuals in household who have access to sewerage

19 EXERCISES. 282

Variables for PERHE-16

EXPCAP: total expenditures,per capita(constant June 1994 soles per year).

WHHLD: household aggregation weight.

WIND: individual aggregation weight.

WCHILD: child aggregation weight.

WCH0612: aggregation weight for children between 6 and 12.


NCHILD: number of children in household (18 and below)

NCHILD0612: number of children between 6 and 12.

NCHILD1318: number of children between 13 and 18.

NPUBPRIM: number of household members in public primary school

NPUBSEC: number of household members in public secondary school

NPUBUNIV: number of household members in public post-secondary school

PUBPRIM: number of household members in public primary school as a proportion ofNCHILD

PUBSEC: number of household members in public secondary school as a proportionof NCHILD


TTEXP: total expenditures of household (constant June 1994 soles per year).

19 EXERCISES. 283

SENEGAL 95 and SENESAM

SENEGAL 95 is drawn from a nationally representative survey carried out in1995 Senegal (theEnquete senegalaise aupres des menages), with sample selec-tion using a multi-stage stratified random sampling procedure. The country wasfirst split in five strata. The first sampling stage consisted in the selection of PSU’s(enumeration areas, orSecteurs d,enumeration (SE)) from a 1990 ”Master Sam-ple” list with probability proportional to the number of households in the PSU’s.396 SE were thus selected in the urban area and 204 in the rural area. Censusdistricts were then selected within each SE. In a final stage, 15 households weresystematically selected within each of the urban census districts, and similarly 24households were systematically selected within each of the rural census districts.

STRATA Householdsin census

SE inMasterSample

CensusDis-tricts inESAM

# of house-holds inESAM

+ URBAN 333343 396 132 1980+ DAKAR 187799 218 74 1110- High socio-economic level 69065 82 28 420- Medium socio-economic

level52768 63 22 330

- Low socio-economic level 59946 73 24 360+ Other urban areas 145544 178 58 870RURAL 450276 204 55 1320TOTAL 783319 600 187 3300

19 EXERCISES. 284

Figure 9:

19 EXERCISES. 285

Fig

ure

10:ti

tle.

19 EXERCISES. 286

Variables for SENEGAL 95


1. Dakar

(a) High social economic level

(b) Medium social economic level

(c) Low social economic level.

2. Other cities

3. Rural area

PSU: PSU of the household

WEIGHT: Household sampling weight


EQUI: Number of adult equivalents in the household

EXP: Total household expenditures

EXPEQ: Total expenditures per adult equivalent

EXPCAP: Total expenditures per capita

INS LEV: Education level of the head of the household

Variables for SENESAM

WEIGHT: aggregation weight.

STRATA: survey strata.



WMCH712: aggregation weight for boys between 7 and 12.

WFCH712: aggregation weight for girls between 7 and 12.

TTEXP: total household expenditures

EQUI: number of adult equivalents in household

EXPEQ: total expenditures per adult equivalent

SCHEXP: household school expenditures

19 EXERCISES. 287

SCHEXPEQ:school expenditures per adult equivalent

PSCH712:proportion of children between 7 and 12 in school.

PMSCH712:proportion of boys between 7 and 12 in school.

PFSCH712:proportion of girls between 7 and 12 in school.

19 EXERCISES. 288

ESPMEN: this is drawn from Senegal‘s ESP (Enquete Senegalaise Prioritaire)


STRATA: survey strata.


REGION: region.

DEPT: geographical department.

ZONE: geographical zone.

TTEXP: total household expenditures (include health and education)



FDEQ: food expenditures per adult equivalent

NFDEQQ: non-food expenditures per adult equivalent

INCOMEQ: income per adult equivalent

HEALTHEQ: health expenditures per adult equivalent.

SCHEXPEQ:education expenditures per adult equivalent.

AGE: age of household head

19 EXERCISES. 289

ESPSANT: this is drawn from Senegal‘s ESP (Enquete Senegalaise Prioritaire):





TTEXP: total household expenditures (include health and education)



HEALTHEQ: health expenditures per adult equivalent.

HEALTHUSE: household uses public health services.

SEX: sex of household head

19 EXERCISES. 290

ESPSCOL: this is drawn from Senegal‘s ESP (Enquete Senegalaise Prioritaire):







NCHILD: number of children

PRIM: proportion of children below 14 going to primary school

SEC: proportion of children below 14 going to secondary school

UNI: proportion of household members attending university.

SUP: proportion of household members attending superior education.

SEX: sex of household head.

PUBPRIM: proportion of children below 14 going to public primary school

PUBSEC: proportion of children below 14 going to public secondary school

19 EXERCISES. 291

Burkina 94

Burkina 94 is drawn from a nationally representative survey (Enquete Prior-itaire) carried out in 1994 Burkina Faso with sample selection using two-stagestratified random sampling. Seven strata were formed. Five of these strata wererural and two were urban. Enumeration areas (PSU’s, orzones de denombrement)were sampled in a first stage from a list computed from the 1985 census. This first-stage sampling within strata 7 (Ougadougou-Bobo-Dioulasso) was made withequal probability and without replacement. First-stage sampling within the other6 strata was made with probability proportional to the size (estimated from the1985 census) of each PSU and without replacement. 20 households were thensystematically sampled within each of the selected PSU’s in a second stage.

Figure 11:The sampling design of Burkina Faso’s Enquete Prioritaire (1994)

Variables for BURKINA 94

WEIGHT: Sampling weight



PSU: Enumeration area of the household

GSE: Social economic group of the household head

1. wage-earning (public sector)

2. wage-earning (private sector)

3. Artisan or trading

4. Others activities

5. Farmers (crop)

6. Farmers (food)

19 EXERCISES. 292

7. Inactive

SEXE: Sex of household head

1. Male

2. Female

EXP: Total household expenditures

EXPEQ: Total expenditures per adult equivalent

SMW: Size * Weight

MEAN PSU: Mean expenditure (EXPEQ) in a PSU

MEAN STR: Mean expenditure (EXPEQ) in a stratum

REFERENCES 293

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20 GRAPHS AND TABLES 341

20 Graphs and tables


Figure 12: Capabilities, achievements and consumption

CF

F C

T

A

U1

B

zT

zC

TC

TC

T

C

S1

Y1

T


Figure 13: Lorenz curve

L(p

)

p0

.51

1

F(

)µ

Q(0

.5)/

µ

p

L(p

)

0


Figure 14: Generalized Lorenz curve

GL

(p)

p1

µ

p

GL

(p)

0

µ


Figure 15: Capabilities and achievements under varying preferences

T

CF

F C

T

A

U1

B

zT

zC

D

U2

E

TC

TC

T

C

S1

Y1


Figure 16: Capability set and achievement failure

CF

F C

T

A

U1

zT

zC

TC

TC

T

C

Y1

TT

CT’

B’

S1

’


Figure 17: Minimum consumption needed to escape capability poverty

CF

F C

T

A

U1

B

zT

zC

TC

TC

T

C

S1

Y1

T

S2

Y2


Figure 18: The cumulative poverty gap (CPG) curveG

(p;z

)

µg (z

)

F(z

)p

01

AB


Figure 19: Engel curves and cost-of-basic-needs baskets

1234

0.2

50.5

0.7

51.0

p

x (

p)

1 x (

p)

2

x (

p)

2

x (

p)

1

z (p

)F

z (p

)F


Figure 20: Food preferences and the cost of a minimum calorie intake

U2

U1

U0

Y0

Y1

Y2 x

1

x2

Min

imum

calo

rie c

onstr

ain

t


Figure 21: Food, non-food and total poverty lines

Food

exp.

Total

exp.

z F

45o

z F

A

B

D

E

O

G


Figure 22: Expenditure and calorie intake

Expenditure

Calorie intakez k

z


Figure 23: Subjective poverty linesMinimum subjective income

Inco

me

. ..

.

.

..

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

..

.

..

z*

z*

a

b

45

o


Figure 24: Estimating a subjective poverty line with discrete subjective in-formation

Un

der

th

e

po

ver

ty l

ine?

..

..

..

..

..

..

..

...

..

..

..

..

..

..

....

1 0in

com

e

....

. ..

. .

. ..

....

....

...

. .

...

...

. .

..

. ..

.

.

. .

.

. .

.

z*

"fal

se p

oo

r" "fal

se r

ich

"


Figure 25: s-order poverty dominance

D (

)s

y

y

D (

)s

y

D (

)s

yA

B

zz

s


Figure 26: Quantile curve for discrete distribution

Q(p

)

p0

.33

0.6

61

.0

10

20

30


Figure 27: Quantile curve for a continuous distribution

y=Q(p)

p=

F(y)

ymax

1

µ

Q(p)

0.5

Q(0.5)

µ

F( )µ


Figure 28: Incomes and poverty at different percentilesQ(p)

p1

ymax

q

g(q;z)

µ g(z)

z

Q (p;z)*

Q(p)

F(z)

Q(q)


Figure 29: Price adjustments and well-being with two commodities

mea

t

fish

U1

U3

U2

AB

qm

qc

D


Figure 30: Equivalence scales and reference well-being

x (

y,q

)

y

r

yc

yd

ye

yf

cd

ef

x0

x1

one

man

a co

uple


Figure 31: Atkinson social evaluation functions and the cost of inequality

0y

ξµ

y1

2

C

U(y

; )ε

W(

)ε

y

U(y

; )ε

U(y

) 2

U(y

) 1


Figure 32: Inequality aversion and the cost of inequality

0y

ξµ

y1

2

U(y

; )ε

W(

)ε

y

U(y

; )ε

0

U(y

; )ε1

0ξ1


Figure 33: Application window for estimating the FGT poverty index – onedistribution.


Figure 34: Homothetic social evaluation functions

y2

y1

45

0

WA

WB

y2

y2B A

y1

y1

AB

ξ Aµ AG

F

E

D

O


Figure 35: Social utility and incomesU(y; )ε

y/µ

ε =0.5

ε =0

ε =1

1

-1


Figure 36: Marginal social utility and incomes

U

(y;

)(1

)ε 1

y/ µ

1

ε =0

.5

ε =0

ε =1

ε =2

ε =1

ε =2

ε =0

.5


Figure 37: Mean income and inequality for constant social welfareξµ

I1

µ 0

µ 1

ξ =µ1 ξ =µ

0


Figure 38: Inequality and social welfare dominance

0

0

1

1

L(p

)G

L(p

)

1

µ

µ

A

B

L (p)

L (p)A

B

A

B

GL (p)

GL (p)

0

0

1

1

L(p

)G

L(p

)

1

µ

µ

A

B

L (p)

L (p)A

B

A

B

GL (p)

GL (p)

0

1

1

GL

(p)

µ

µ

A

B

A

B

GL (p)

GL (p)

0

0

1

1

L(p

)G

L(p

)

1

µ

µA

B

L (p)

L (p)A

B

A

BGL (p)

GL (p)

Case 2

Case 3 Case 4

0 1

1

L (p)

L (p)A

B

Case 1


Figure 39: Primal stochastic dominance curves

F(y

)

yz

D (

z)2 y

’

z-y

’

d(F

(y’)

)


Figure 40: Histograms and density functionsh

isto

gra

md

ensi

ty

f(y

)

yy 0


Figure 41: Contribution of poverty gaps to FGT indices

pF

(z)

1

Q(p

)/z

g(p

;z)/

z

(g(p

;z)/

z)2

(g(p

;z)/

z)3


Figure 42: The relative contribution of the poor to FGT indices

p F(z)

1

1/F(z)

g(p) /P(z; =2)2

α

g(p) /P(z; =1) α

F( (z)) µg


Figure 43: Socially-representative poverty gaps for the FGT indices

g(p

;z)

p

ξ(z;

=1

)α

F(z

)

ξ(z;

=2

)α

ξ(z;

=3

)α

ξF

( (z

; =

3))

αξ

F(

(z;

=1

))α


Figure 44: Elasticity of the poverty headcount with respect to thepoverty line

f(y

)F

(z)

zy

f(z)

z’

f(z’

)


Figure 45: Growth elasticity of the poverty headcount

f(y

)F

(z)

zy

f(z)

z’

f(z’

)


Figure 46: Targeting and redistributive costs

B* iλB

* i

c 1,2

c3,4

-1

B* 1

Povert

y A

llevia

tion a

nd R

edis

trib

utive C

osts

- (

y;z

)π

π

-2π

-3π

-4π


Figure 47: The weighting function ω(p; ρ)


Figure 48: The weighting function κ(p; ρ)


Figure 49: The sets of poverty indices that belong to the classesΠi(z+), i =1, 2, 3

Π (z )1 +

Π (z )2 +

Π (z )3 +


Figure 50: The sets of distributions that are ordered by the dominance con-ditions ∆Di(y) ≥ 0, y ≤ z+, and i = 1, 2, 3

∆D (y)>0, y<z +3_ _

∆D (y)>0, y<z +2_ _

∆D (y)>0, y<z +1_ _


Figure 51: Classes of poverty indices and upper bounds for poverty lines

Π (z )1 +

Π (z )1 ++Π (z )

2 ++


Table 6: Equivalent income and price changesq1 = q2 = 1

i yi x1 x2 U y/z1 150.00 50.00 100.00 79.37 0.502 210.00 70.00 140.00 111.12 0.703 300.00 100.00 200.00 158.74 1.004 380.00 126.67 253.33 201.07 1.275 500.00 166.67 333.33 264.57 1.676 510.00 170.00 340.00 269.86 1.707 550.00 183.33 366.67 291.02 1.838 600.00 200.00 400.00 317.48 2.009 800.00 266.67 533.33 423.31 2.6710 1000.00 333.33 666.67 529.13 3.33

q1 = 1 and q2 = 3i yi x1 x2 U yR

i y/z yRi /zr y/700

1 160.00 53.33 35.56 40.70 76.92 0.26 0.26 0.232 200.00 66.67 44.44 50.88 96.15 0.32 0.32 0.293 500.00 166.67 111.11 127.19 240.37 0.80 0.80 0.714 630.00 210.00 140.00 160.26 302.87 1.01 1.01 0.905 1100.00 366.67 244.44 279.82 528.82 1.76 1.76 1.576 1240.00 413.33 275.56 315.43 596.13 1.99 1.99 1.777 1300.00 433.33 288.89 330.70 624.97 2.08 2.08 1.868 1500.00 500.00 333.33 381.57 721.12 2.40 2.40 2.149 1600.00 533.33 355.56 407.01 769.20 2.56 2.56 2.2910 2770.00 923.33 615.56 704.64 1331.68 4.44 4.44 3.96

20 GRAPHS AND TABLES 383Ta

ble

7:E

stim

ated

pove

rty

lines

inC

amer

oon

acco

rdin

gto

diffe

rent

met

hods

(Fra

ncs

CFA

/day

/adu

lteq

uiva

lent

)F

EIf

ood

pove

rty

line

Low

erno

n-fo

odpo

vert

ylin

eLo

wer

tota

lpov

erty

line

Upp

erno

n-fo

odpo

vert

ylin

eU

pper

tota

lpov

erty

line

Cam

eroo

n25

611

737

327

853

4Y

aoun

de33

714

348

041

274

9D

oual

a40

818

158

958

899

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SYMBOLS 388

Symbols

1− f : FPC factor, 229A: hypothetical distribution, 52,

80, 130AD(z): absolute deprivation, 78B: benefit, 111B: hypothetical distribution, 52,

80, 130Bi: level of gross benefit expended

on individuali, 183B∗

i : benefit offered toi, 182C: number of goods, 170C: number of income components,

172CB(p): concentration curve of ben-

efit B at rankp, orderedin terms ofX, 158

CF (ε): cost of inequality subse-quent to a flat tax, 125

CN(ε): cost of inequality in thedistribution of net income,125

CN(p): concentration curve forNat rankp, ordered in termsof X, 105

CN(p): concentration curve forNat rankp, ordered in termsof X), 156

CT (p): concentration curve for taxesT at rankp, ordered in termsof X, 104

CU : cost of inequality in the dis-tribution of locallyp-expected

SYMBOLS 389

utilities of net incomes, 125Ds

A(z): stochastic dominance curveof orders atz and for dis-tributionA, 139, 146

F (y): distribution function, 52, 168F (z): proportion of individuals un-

derneath the poverty linez, 139, 167

FN |x(·): distribution function ofNconditional onX being equalto x, 103

FX,N(·, ·): joint distribution func-tion of gross and net in-comes, 103

G(p; z): Cumulative Poverty Gap,77, 145

GCxc(p): generalized concentra-tion curve forxc, 158, 169

GL(p): Generalized Lorenz curve,65, 151

I: index of inequality correspond-ing to the social welfarefunctionW , 60

I(2): standard Gini index, 54I(ρ): S-Gini inequality index, 54–

56, 107, 124I(ρ, ε): Atkinson-Gini inequality

index, 60I(θ): Generalized entropy inequal-

ity index, 66I(k; θ): inequality within subgroup

k, 66ICX(c)

(ρ): concentration index forX(c), 106, 179

IT (ρ): S-Gini indices ofTR-progressivity,119

IV (ρ): S-Gini indices ofIR-progressivityand vertical equity, 119

I[x]: indicator function, 183I∗(z; ρ.ε): index of inequality in

censored income, 77IN(ε): Atkinson inequality index

for N , 64, 124K: number of mutually exclusive

population subgroups, 75K(u): the multivariate Gaussian

Kernel, 213LA(p): Lorenz curve forA, 50–

52, 105, 107, 156, 177M : the population number of last

sampling units (LSU), 232Mhi: the population number of last

sampling units (LSU), 232N : net income, 103N(p): p-quantile of net income,

124, 157, 159Nh: the population number of pri-

mary sampling units (PSU)in a stratumh, 232

Nj: observationj of N , 119P (k; z; α): FGT poverty index of

subgroupk, 75, 163P (z): poverty index, 137, 182P (z; α): Foster-Greer-Thorbecke(FGT)

poverty index, 164P (z; α = 1): average poverty gap,

140P (z; ε): Clark, Hemming and Ulph

poverty index, 70P (z; ρ): S-Gini poverty index, 71P (z; ρ, ε): Atkinson-Gini poverty

index, 70PC(z; ε): Chakravarty poverty in-

dex, 71PW (z): Watts poverty index, 71Q(k; p; z): p-quantile in a group

k, 164Q(p): p-quantile, 51, 103, 110, 137,

141Q∗(p; z): p-quantile censored atz,

SYMBOLS 390

70QN(q|p): q-quantile function for

net incomes conditional onap-quantile ofX, 104

R(q): per capita net commoditytax revenues, 170

RD(z, ρ): relative deprivation incensored income, 78

Shijk: the size of statistical unithijk in the population, 233

T : taxes net of transfers, 103T (X): deterministic portion of tax

T atX, 103T(j): componentj of total taxT ,

121U : utility function, 92U(y; ε): utility function of y with

parameterε, 63, 149V (y, q, ϑ): indirect utility function,

31V (y, q; ϑ): indirect utility function,

167W : social welfare function, 59, 149W (ε): Atkinson social welfare func-

tion, 62, 63, 124W (ρ): S-Gini social welfare func-

tion, 64W (ρ, ε): Atkinson-Gini social wel-

fare function, 62WN(ε): Atkinson social welfare func-

tion with locallyp-expectednet incomes, 123

WU(ε): Atkinson social welfare func-tion with locallyp-expectedutilities of net incomes, 123

X: gross income, 103, 157, 172X(p): p-quantile of gross income,

122, 157, 159Xj: observationj of X, 119X(c): income componentc of total

incomeX, 172X(c): type c of total expenditure

X, 106Xhijk: the value of the variable of

interest for statistical unithijk, 233

Y : food budget, 91Y : population total of thex’s, 221Y : the population total of interest,

233∆P (z): difference in poverty in-

dices, 139∆f(y): difference in the densities

of income, 139Γ(z): average poverty gap, 143Ωs: class ofs-order social welfare

indices, 150, 172Πs(z): class ofs-order poverty in-

dices, 137, 172Θ: set of possible taste parame-

ters, 168Υs(l): class ofs-order inequality

indices, 153Υ: a parameter space, 240Ξg(z): cost of inequality in poverty

gaps, 76Ξg(z; α): cost of inequality in poverty

for FGT indices, 76α: parameter of inequality aver-

sion in measuring poverty,76

α = s− 1: ethical parameter, 152Πs(z+): class of duals-order poverty

indices, 144ε: parameter of relative inequality

aversion, 56η(k): equal amount to each mem-

ber of a groupk, 163η: some positive value, 98γ: efficiency cost of taxingj rel-

SYMBOLS 391

ative to that of taxingl,170

M : estimator of the population num-ber of LSU, 233

N : estimator of population size N,221

Y : estimator of Y, 221Y : estimator of the population to-

tal of interest, 233µ: estimator ofµ, 226VSRS : estimator of the sampling

variance under simple ran-dom sampling, 234

κ(p): weight used in linear indices,53

κ(p; ρ): weight used in S-Gini in-dices, 53, 55

λ(k): proportional factor for groupk, 164

λ: Lagrange multiplier, 183λ: proportional factor, 55, 176λ∗: social opportunity cost of spend-

ing public funds, 183λc: proportional factor for compo-

nentc, 172CD c(z; α): consumption dominance

curve of a componentc,169

IC (ρ): S-Gini indices of concen-tration, 106

IR(ρ): S-Gini indices of redistri-bution, 119

LP(X): Liability Progression atX, 108

NB i: net benefit of state supporti, 182

RP(X): Residual Progression atX, 108

RR(ρ): S-Gini indices of rerank-ing, 119

TR: tax or tranfer, 156deff : design effect, 234pc: proportion, 83E[ti]: the expected number of times

unit i will appear in thesample, 221

µ(k): mean income in groupk,166

µ(k): mean income in subgroupk, 66

µg(z): average poverty gap, 72,131

µF : average income under a welfare-neutral flat tax, 125

µT : mean tax, 157µX : mean of variableX, 50, 157,

159, 173, 178, 226ν: stochastic tax determinant, 103ω(p): weight on income used in

linear indices, 59, 144, 150ω(p; ρ): weight on income used in

S-Gini indices, 54, 55B: government supportper capita

budget, 183B(p): expected benefit at rankp,

158D

s(lµ): normalized stochastic dom-

inance curve, 154N(p): expected net income of those

at rankp in the distribu-tion of gross incomes, 119

P (z; α): normalized FGT indices,155, 169

Q(p): income shares or normal-ized quantiles, 159

Q(p) = Q(p)/µ: income sharesor normalized quantiles, 153

T (X)/X: expected net tax of thoseat rank X in the distribu-tion of gross incomes, 157

SYMBOLS 392

T (p): expected net tax rate of thoseat X in terms of gross in-comes, 110

U(p): expected utility at rankp,124

X(c)(p): expected value of incomecomponentc at rankp inthe distribution of total in-comeX, 172

φ(k): average population shares,82

g(p; z): normalized poverty gap,155

yh: mean in stratumh, 233δ(p): expected relative deprivation

at rankp, 58I(θ): contribution of between sub-

group inequality to total in-equality, 67

CD c(z; α): normalizedCD curves,171

φ(k): share of the population foundin subgroupk, 66, 75, 83

π: poverty function, 183π(Q(p); z): contribution ofQ(p)

to poverty index, 137π(i)(Q(p); z): i-th order derivative

of π(Q(p); z) with respectto Q(p), 137

πi: probability that uniti is se-lected, 221

ρ: ethical parameter, 53, 55ρ(p): consumption ratiox2(p)/x1(p),

90σ2

T (p): conditional variance ofTat gross incomeX(p), 122

σ2y|x: variance ofy conditional on

some valuex, 217var(µ): variance ofµ, 226var(y): variance of the population

yhij, 234ε: random term, 214εG(z; α): Gini elasticity of FGT

poverty indices, 176εy(k; z; α): elasticity of total poverty

with respect to total incomewith growth from compo-nentk, 174

εy(z; α): elasticity of total povertywith respect to total income,174

ς: some positive value, 98ϑ: taste parameters, 31, 167f(y): estimator off(y), 212VSD : the design-based estimator

of the sampling variance,234

ξ(ρ, ε): equally distributed equiv-alent(EDE) income, 60

ξ∗(z; ρ, ε): equally distributed equiv-alent (EDE) income of cen-sored income, 70

ξg(z): equally distributed (EDE)poverty gap, 76

ξg(z; α): equally distributed equiv-alent (EDE) poverty gapfor the un-normalized FGTindices, 74, 76

ξN(ε): equally distributed equiv-alent (EDE) incomes forWN(ε), 124

ξN(ε): equally distributed equiv-alent (EDE) incomes forWN(ε), 124

ξU(ε): equally distributed equiv-alent (EDE) incomes forWU(ε), 124

ζ+(s): upper bound of range ofpoverty lines over which

SYMBOLS 393

dominance holds at orders, 146

c: equivalence scale parameter, 140c: good, 168c: redistributive costs, 188ci: cost toi of acceptingB∗

i , 183e: expenditure function, 32e: net income for optimal recipi-

ents of state support, 187f(Q(p)): density of income atp-

quantile, 51f(k; z): density of income atz for

groupk, 164f(y): density function, 167, 212f (i)(y): i-order derivative of func-

tion f , 211fh: function of a user-specified FPC

factor, 233g(p; z): poverty gap, 143g(p; z): poverty gap at percentile

p, 73h∗: optimal bandwidth, 212l: proportion of mean as relative

poverty line, 155lµ: proportion of mean as relative

poverty line, 155l+: upper bound of range of pro-

portions over dominancemust be checked, 153, 157

m: the number of LSU, 232mhi:the number of selected LSU

in the PSUhi, 232nh: the number of selected PSU in

a stratumh, 232p: percentile, 50–52pi = i/n: percentile correspond-

ing to ordered observationi, 50

q: percentile, 52q: price vector, 31, 32, 167

qR: reference prices, 31, 32, 167qc: price of goodc, 168qhij: the number of observations

in selected LSUhij, 232r: number of randomly selected

individuals from a popu-lation, 55

r: percentile, 52s: class of indices, 132s: order of stochastic dominance,

132, 140, 223t: average tax as a proportion of

average gross income, 105t: population subgroup, 83t: vector of tax rates , 170t(X): expected tax atX as a pro-

portion ofX, 108th: random variable, 227ti: number of times uniti appears

in a random sample of sizen, 221

tl: tax onl, 171tl: tax rate on goodl, 170, 171t(j): average taxT(j) over average

gross incomeX, 121w(u): weight function, 209wi: the sampling weight of obser-

vationi, 213whij: the sampling weight of LSU

hij, 232x0 : reference level, 37xc: consumption of commodityc,

91, 169xc(q): consumption of commod-

ity c with pricesq, 169xc(y, q): expected consumption of

goodc at incomey whenfacing prices q, 168

xc (y, q; ϑ): consumption of goodc at incomey and prefer-

SYMBOLS 394

encesϑ , when facing pricesq, 168

xhijk: the value ofX that appearsin the sample for sampleobservationhijk, 233

y: income, 167y: total nominal expenditure, 31yR: Equivalent consumption ex-

penditure, 32yR: real income, 167yi: income of individuali, 50, 135,

182yhijk: sum ofy in LSU hij, 233yhi: relevant ofy in PSUhi, 233z: poverty line, 137, 139, 167z∗: poverty line, 100z+: upper bound of range of poverty

lines over which dominancemust be checked, 137

zk: minimum calorie intake rec-ommended for a healthylife, 95

zF : minimal food expenditure nec-essary for living in goodhealth, 89

zT : total poverty line, 89zNF : required non-food expendi-

tures, 89

AUTHORS 395

Authors

Aaberge, Rolf, 40, 42, 68, 69, 115,246

Achdut, L., 115Ahmad, E., 181Alderman, Harold, 180Altshuler, Rosanne, 114Amiel, Yoram, 136, 137Anand, S., 84Anand, S. , 40Anderson, Gordon, 246Ankrom, Jeff, 127Araar, Abdelkrim, 41, 69Arkes, Jeremey, 40Aronson, J. Richard, 127, 128Asselin, Louis-Marie, 235Atkinson, A. B., 68, 127, 147, 160Auerbach, A. J., 127Auersperg, M., 69

Bahadur, R. R., 246Banks, James, 42Barrett, C. R., 69Barrett, Garry F., 41, 246, 247Barrington, Linda, 102Baum, S. R., 113Baum, Sandy, 113Beach, C., 246, 247Beausejour, L., 190Beblo, Miriam, 69Beck, John H., 137Ben Porath, Elchanan, 69Bergmann, Barbara R., 101Berliant, M. C., 127Berrebi, Z. M., 69Berrian, David, 190Bershadker, Andrew, 41Besley, T., 181, 185, 190

Bhorat, Haroon, 41Bidani, Benu, 84, 101Biewen, Martin, 235, 247Bigman, David, 180Bigsten, A., 85Bishop, J. A., 68, 114, 127, 148,

160, 246, 247Bishop, John A, 160Bisogno, Marcelo, 180Bjorklund, Anders, 40Blackburn, McKinley L., 69, 101Blacklow, Paul, 41Blackorby, C., 42, 68, 69, 84, 113,

136Blanchflower, David G., 102Blank, Rebecca M., 114Blum, W. J., 112Blundell, Richard, 41, 42Bodier, Marceline, 41Borg, Mary O’Malley, 114, 127Bosch, A., 42Bossert, Walter, 69, 136Bourguignon, Francois, 68, 69, 190Bradbury, Bruce, 42Breunig, Robert, 247Buhmann, B., 38, 42Burkhauser, Richard V., 40, 42Burton, Peter S, 41

Callan, T., 102Calonge, S., 115, 128Cancian, Maria, 115Cantillon, Sara, 41Carlson, Marcia, 41Caspersen, Erik, 114Cassady, K., 113Castillo, Enrique, 248

AUTHORS 396

Chakraborti, Subhabrata, 246, 247Chakraborty, A. B., 69Chakravarty, S. R., 84Chakravarty, Satya R., 68, 69, 84,

127, 136, 190Champernowne, D. G., 68Chatterjee, Srikanta, 115Chen, Shaohua, 84, 85, 160Cheong, Kwang Soo, 248Chern, Wen S., 41Chew, S. H., 69Chong, Alberto, 180Chotikapanich, Duangkamon, 246,

248Chow, K. Victor, 114, 148, 160,

246Christiansen, T., 115, 128Christiansen, V., 137Citoni, G., 115, 128Clark, A. E, 69Clark, S., 84Clarke, Lynda, 41Coate, S., 185, 190Cochrane, William G., 229, 230,

235Coder, J., 40Coder, John, 84Cogneau, Denis, 41Conniffe, Denis, 42Constance, F. C., 84Corneo, Giacomo, 136Cornia, G. A., 181Coronado, Julia Lynn, 40Coulombe, Harold, 84Coulter, Fiona A. E., 42Cowell, Frank A., 42, 63, 68, 69,

136, 137, 235, 246–248Creedy, John, 40, 114, 127, 136,

181, 190Crossley, Thomas F, 41

Cushing, Brian J., 148, 246Cutler, D. M., 38, 41

Dagum, Camilo, 69Dalton, H., 68Danziger, S. P., 41, 68Danzinger, Sheldon, 84Dardanoni, Valentino, 113, 127, 246Dasgupta, Partha, 40, 68, 160Datt, G. , 80Datt, Gaurav, 85, 160Davidovitz, Liema, 137Davidson, N. R., 246Davidson, Russell, 84, 114, 147,

160, 246Davies, Hugh, 41Davies, James B., 113, 136, 160,

246Davis, J. A., 69, 117De Gregorio, Jose, 41De Janvry, Alain, 85De Vos, K., 84De Vos, Klaas, 41, 42, 101, 102Deaton, Angus, 84Deaton, Angus S., 41, 221, 225,

229, 230, 235Decoster, Andre, 114, 127Deininger, Klaus, 85Deitel, H. M., 207Deitel, P. J., 207Del Rio, Coral, 69, 147Desai, M., 41Deutsch, Joseph, 69Dilnot, A. W., 114Dolan, Paul, 136Dollar, David, 85Donald, Stephen G., 247Donaldson, David, 41, 42, 68, 69,

84, 113, 136, 147Doorslaer, E. Van, 115, 128

AUTHORS 397

Dorosh, P. A., 115Duclos, Jean-Yves, 40, 42, 69, 84,

113, 114, 127, 128, 147,160, 246

Duro, Juan Antonio, 69

Ebert, Udo, 41, 42, 69Efron, Bradley , 247Elbers, Chris, 247Epstein, L. G., 69Erbas, S. Nuri, 41Essama Nssah, B, 85Essama Nssah, Boniface, 68Esteban, Joan, 69Estivill, P., 102

Feldstein, M., 118, 127Fellman, Johan, 112, 114Ferreira, Francisco H. G., 248Ferreira, M. L., 42Festinger, Leon, 69, 117Fields, Gary S., 41, 69, 190Finke, Michael S., 41Fishburn, P. C., 136, 160Fisher, Gordon M., 101, 102Fisher, S., 147Fleurbaey, Marc, 42Fong, Christina, 136Forcina, Antonio, 246Formby, John P., 40, 68, 113, 114,

127, 148, 160, 246, 247Formby, John P. , 113Formby, John, P., 246Fortin, B., 190Foster, James E., 68, 69, 84, 101,

147, 160Fournier, Martin, 69Fox, Jonathan J., 41Frick, Joachim R., 40Fritzell, J., 40

Fullerton, Don, 40Funnell, Nicola, 246

Garner, Thesia I., 42, 102Gautam, Madhur, 180Gerdtham, U., 115, 128Gerfin, M., 115, 128Gibson, John, 40Gilboa, Itzhak, 69Giles, C., 114Gillespie, W. Irwin, 114Glass, Thomas, 40Glennerster, Howard, 102Glewwe, P., 84Glewwe, Paul, 42, 190Goedhart, T., 99, 101Goerlich Gisbert, Francisco J., 69Gottschalk, Peter, 68Gouveia, Miguel, 160Gravelle, J. G., 114Green, David A., 246Greer , J., 84Greer, J., 101Gregoire, Philippe, 84Griffiths, William, 246, 248Groleau, Yves, 160Grootart, C., 84Grosh, Margaret E., 181Gross, L., 115, 128Gruner, Hans Peter, 136Gueron, J. M., 180Gustafsson, Bjorn, 40, 84, 114, 115

Hardle, Wolfgang, 217Haddad, Lawrence, 41, 190Hagenaars, A., 84Hagenaars, A. J. M., 40, 41, 101Haggblade, S., 115Hagnere, Cyrille, 42Hainsworth, G. B., 68

AUTHORS 398

Hakinnen, U., 115, 128Hamming, R. , 84Hanratty, Maria J., 114Harding, Ann, 40Hattem, V., 115, 128Hauser, R., 40Hausman, P., 102Haveman, Robert H., 41Hayes, K., 113Heady, Christopher, 114Hentschel, Jesco, 31, 40Hentschel, Jesko, 247Heshmati, Almas, 246Hey, J. D. , 58, 69Hill, Carolyn J, 84Hills, John, 114Horrace, W. C., 248Howard, R., 114Howes, Stephen, 235Hoy, Michael, 113, 136, 160Huang, Jikun, 40Hungerford, Thomas L., 180Hurn, Stan, 136Hyslop, Dean R., 69

Iceland, John, 84Idson, Todd, 41Immonen, Ritva, 190

Jantti, Markus, 114Jakobsson, U., 112James, S., 246Jansen, E. S., 137Jantti, Markus, 68, 84Jeandidier, B., 102Jenkins, Stephen P., 40–42, 69, 126,

127, 147Jensen, Robert T., 41Johnson, David, 68Johnson, Paul, 42, 114, 127, 128

Jorgenson, Dale W., 41Joshi, Heather, 41Juster, F. Thomas, 41

Kahen Jr., H., 112Kakwani, Nanak C., 68, 69, 84,

113, 127, 128, 136, 246Kanbur, Ravi, 41, 84, 181, 190Kanbur, S. M. R., 41, 190Kaplow, Louis, 127Karagiannis, Elias, 247Katz, L., 38, 41Kaur, A., 247Kay, J. A., 114Keen, Michael, 113, 190Keeney, Mary, 115Kelkar, Ujwala R., 69Kennickell, Arthur B., 235Khetan, C. P., 114Kiefer, D. W., 113Kim, Hoseong, 40King, M. A., 118, 127Klasen, Stephan, 41Klavus, Jan, 246Knaus, Thomas, 69Kodde, D. A. , 247Kolm, S. C., 68, 136, 160Kovacevic’, Milorad, 247Kraay, Aart, 85Kroll, Yoram, 137Kuester, Kathleen A., 41Kundu, A., 84

Lambert, Peter J., 58, 68, 69, 113,114, 126–128, 147

Lancaster, Geoffrey, 42Lanjouw, Jean Olson, 41, 235, 247Lanjouw, Peter, 31, 40–42, 115,

247Latham, Roger, 113

AUTHORS 399

Layte, Richard, 41Lazear, E. P., 41Le Breton, Michel, 113Lee, J., 41Leibbrandt, Murray, 115Lerman, Robert I., 69, 115, 127,

247Lewbel, A., 42Lewis, J., 181, 207Liberati, Paolo, 181Lipton, Michael, 26, 84Litchfield, Julie A., 248Liu, P. W., 113Loftus W., 207Lokshin, Michael, 102, 136Loomis, J. B., 114Lundberg, Shelly J., 41Lundin, Douglas, 181Lyon, A. B., 114Lyssiotou, Panayiota, 42

Maasoumi, Esfandiar, 246Madden, David, 102Makdissi, Paul, 160, 181Makonnen, Negatu, 40Marshall, Judith, 41Mason, Paul M., 114Mayeres, Inge, 181Mayshar, Joram, 181McCarthy, Tom, 246Mcclements, L. D., 38McKay, Andrew, 84Meenakshi, J. V., 42Mehran, F., 113Melby, Ingrid, 42Mercader Prats, Magda, 42, 114Merton, R. K., 69Merz, Joachim, 42Metcalf, Gilbert E., 114Michael, R. T., 41, 84

Michael, Robert T, 84Micklewright, J., 68Milanovic, B., 84Milanovic, Branko, 68, 69, 113,

114Miller, Cynthia, 41Mills, Jeffrey A., 113, 247Mitrakos, Theodore, 41, 114Moffitt, R., 180Mookherjee, D. , 69Morduch, Jonathan, 84, 115Morris, C. N., 114Morrisson, Christian, 68Mosler, Karl, 136Moyes, Patrick, 41, 42, 69, 113,

160Muffels, R., 102Mukherjee, Diganta, 190Mukhopadhaya, Pundarik, 115Muliere, Pietro, 136, 160Muller, Christophe, 41Musgrave, R. A., 112, 116, 126Myles, John, 84

Narayan, Deepa, 41Nead, Kimberly, 181Nelissen, Jan H. M., 114Newbery, David M., 181Nicol, Christopher J., 42Nolan, B., 127Nolan, Brian, 41, 69Norregaard, J., 114Norris, C. N., 114Novak, E. Shawn, 114Nozick, R., 136

O’higgins, M., 114O’Leary, Nigel C, 40O’Neill, Donal, 41Ogwang, Tomson, 246, 248

AUTHORS 400

Ok, Efe A., 68, 136Okun, A. M., 112, 136Orshansky, M., 102Osberg, L., 246Osberg, Lars, 84Oswald, Andrew J., 69, 102

Paarsch, Harry J., 246Palm, F. C., 247Palmitesta, Paola, 247Papapanagos, Harry, 113Park, Albert, 180Parker, Simon C., 40, 69Pattanaik, Prasanta K, 84Paul, Satya, 69Pechman, J. A., 112Pen, Jan, 160Pena, Christine, 41Pendakur, Krishna, 41, 42, 84, 246Persson, M. , 114Pfahler, Wilhelm, 113Pfahler, Wilhelm, 113Phipps, Shelley A., 41, 42Picot, Garnett, 84Plotnick, Robert D., 126, 127Poddar, S. N., 114Podder, Nripesh, 69, 115Pollak, Robert A., 41, 42Polovin, Avraham, 137Poupore, John G., 40Pradhan, Menno, 102Prakasa Rao, B. L. S., 247Pratt, J. W., 160Preston, I., 114Preston, Ian, 41, 246Price, Donald I., 114Proost, Stef, 181Propper, C., 40Provasi, Corrado, 247

Quisumbing, Agnes R, 41

Radner, Daniel B., 42Rady, Tamer, 84Rainwater, L., 38Ramos, Xavier, 126, 127Rao, R. C., 237Rao, U. L. Gouranga, 248Rao, Vijayendra, 41Ravallion, Martin, 13, 26, 40, 42,

76, 80, 84, 85, 101, 102,115, 136, 160, 180

Rawls, J., 136Ray, Ranjan, 41, 42Reed, Deborah, 115Renwick, Trudi J., 101Revier, C. F., 114Reynolds, M., 113Richmond, J., 247Richter, Kaspar, 41Robinson, Angela, 136Rodgers, J L, 84Rodgers, J R, 84Rongve, Ian, 246Rosen, H. S., 127Rossi, A. S., 69Rothschild, M. , 68Rowntree, S., 84, 89Rozelle, Scott D., 40Ruggeri, G. C., 114Ruggles, P., 114Ruiz Castillo, Javier, 41, 42, 69,

147Runciman, W. G., 58, 69, 117Ryu, Hang K., 248

Sa Aadu, J., 127Sadoulet, Elisabeth, 85Sahn, David E., 41, 115, 160Salas, Rafael, 136Salles, Maurice, 69Sarabia, Jose Maria, 248

AUTHORS 401

Sastry, D. V. S., 69Saunders, Peter, 40, 68, 87Sayers, Chera L., 41Scarsini, M., 160Schady, Norbert R., 180Schluter, Christian, 235, 248Schmaus, G., 38, 114Schmidt, P., 248Schokkaert, Erik, 127, 137Schultz, T. Paul, 69Schwab, R. M., 114Schwartz, Amy Ellen, 114Schwarz, H. B., 114Schwarze, Johannes, 40, 69Seaks, T. G., 113Sefton, Tom, 41Sen, A. K., 40Sen, Amartya K., 14, 15, 17, 18,

40, 58, 62, 68, 84, 136,160

Sengupta, Manimay, 84Shah, A., 41Shapiro, Stephen L., 114Shi, Guanming, 160Shi, Li, 40, 84, 115Shimeles, A., 85Shipp, Stephanie, 68Shneyerov, Artyom A., 69Shorrocks, Anthony F., 69, 84, 113,

115, 136, 147, 160Sicular, Terry, 115Siddiq, Fazley K., 40Silber, Jacques, 69, 115Silver, H., 69, 87Silverman, B. W., 211, 212, 214,

217Simler, Kenneth R., 115Singh, H., 247Skoufias, Emmanuel, 41Slemrod, J., 181, 190

Slesnick, Daniel T., 41, 180Slitor, R. E., 112Slottje, Daniel J., 113, 248Smeeding, Timothy M., 38, 40, 42,

68, 84Smith, T. E., 84Smith, W. James, 68, 113, 160Smolensky, E., 113Sotomayor, Orlando J., 115Spencer, B. D., 147Spera, Cosimo, 247Squire, Lyn, 85Srinivasan, P. V., 180Stanovnik, Tine, 102Starret, D., 68, 160Stephenson, G., 114Stern, N., 181, 190Stern, N. H., 181Stewart, F., 181Stifel, David C., 41, 160Stiglitz, J. E., 68Stodder, James, 137Stranahan, Harriet A., 127Strauss, R. P., 127Streeten, Paul, 16, 40Subramanian, S., 69Suits, D. B., 113Sutherland, Holly, 41Sweetman, Olive, 41Sykes, D., 113Szulc, A., 84

Tabi, Martin, 113, 114Takayama, N., 84Tam, Mo Yin S., 136Tavares, Jose, 160Thin, T., 112Thirsk, W., 181Thistle, Paul D., 113, 160, 246,

247

AUTHORS 402

Thon, D., 84Thorbecke, E., 101Thorbecke, Erik, 84, 190Tibshirani, Robert J., 247Townsend, Peter, 18, 40, 87Trannoy, Alain, 42, 113Trede, Mark, 235, 248Trippeer, Donald R., 127Truchon, M. , 190Tsakloglou, Panos, 41, 114Tsui, Kai yuen, 69, 85Tuomala, M., 190Tuomala, Matti, 190

Ulph, D., 84

V. Justin, 114, 127, 181Van Camp, Guy, 114, 127Van de gaer, Dirk, 246Van de Ven, Justin, 127Van De Walle, Dominique, 24, 115,

180, 181Van den Bosch, Karel, 40, 102Van Doorslaer, Eddy, 114, 115, 128Van Praag, B. M. S., 101Van Wart, D., 114Vermaeten, Arndt, 114Vermaeten, Frank, 114Viard, Alan D., 190Vickrey, W., 112Vickson, R. G., 160Victoria Feser, Maria Pia, 247

Wagstaff, Adam, 115, 128Wagstaff, Adam , 115Wales, Terence J., 41Walton, Michael, 41Wane, Waly, 190Wang, Qingbin, 160Wang, Sangui, 180Wang, You Qiang, 69

Watts, H. W., 84Weinberg, Daniel H., 40Weymark, J. A., 69, 84, 147Whelan, Christopher T., 69Whitmore, G. A., 160Wildasin, D., 170Willig, R. D., 136Wissen, P., 114Witte, A. D., 248Wodon, Quentin T., 101, 115, 181Wolak, F. A., 247Wolff, Edward N., 41Wolfson, M., 40Woodburn, R. Louise, 235Woolard, Christopher, 115Woolard, Ingrid, 115Woolley, Frances R, 41Worswick, Christopher, 41Wu, Guobao, 180

Xu, Kuan, 84, 246, 247

Yaari, Menahem E., 54, 69, 160Yao, Shujie, 115Yates, Judith, 40Yfantopoulos, J., 102Yitzhaki, Shlomo, 58, 69, 115, 127,

160, 181, 247Yoo, Gyeongjoon, 69Younger, Stephen D., 115

Zaidi, M Asghar, 42, 101Zaidi, M. Asghar, 41Zandvakili, Sourushe, 69, 113, 247Zeager, Lester A., 148Zhang, Renze, 136Zheng, Buhong, 40, 84, 101, 136,

147, 148, 160, 235, 246Zheng, Yi, 160Zoli, Claudio, 136

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