- 1 - 9/26/04PP253/ARE253 Alain de Janvry & Elisabeth SadouletFall 2004InequalityPositive analysis: Indicators and determinants1. Describing and measuring inequality1.1. Describing inequality: Graphic representation of inequality with the Lorenz curve (Figure 1)Objective: Represent inequality in income, consumption, wealth, and/or landholdings.Cumulative % of total population ranked by income level1100100Cumulative % of total income1100µAµBLorenz curveCumulative share * mean incomeGeneralized Lorenz curveFigure 1. Lorenz curvesNote 1: If two Lorenz curves cross, inequality comparisons require additional criteria.Note 2: Generalized Lorenz curve shows the role of both inequality and average income level on the fractionof mean income held by any percentile of the population.1.2. Measuring inequality: alternative indicatorsDesirable properties of inequality indicators (Dalton):- Anonymity principle: Permutations of people should not affect the inequality measure.- Dalton transfer principle: A transfer from a richer to a poorer person should reduce inequality.- Population principle: The inequality index should be unaffected by population size.- Relative income principle: Index should be unaffected by changes in absolute income levels, onlyby relative incomes.An inequality index is said to be Lorenz-consistent if it satisfies these four properties.Define: n = number of persons in the populationri = income rank of household i, 1 ≤ ri≤ nyi = income of household iµ = average income.σ = standard deviation of income.Y = total income of the population.For group data:k = 1, ..., m groupsnk = number of households in group k.µk = average income in group k.• Coefficient of variation: CV=σµ.CV index is Lorenz-consistent.• Gini coefficient: G =AA + B=2nµcov(y , r ).Gini index is Lorenz-consistent.• Theil entropy index: T =yiYi =1n∑lnyi/ Y1 / n⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Limits: equality = 0 ≤ T ≤ ln n = maximum inequality.Does not satisfy the population principle.• Income shares and Kuznets ratiosIncome shares: Share of income of the poorest 20% (say) in total income.Kuznets ratios: Ratio of income of richest 20% (say) to poorest 40% (say)Does not satisfy the transfer principle.Two useful properties of indicators are:- Decomposable in between and within sub-populations inequality (regions, socio-economic groups). Giniis not decomposable, Theil and CV are decomposable- Possibility to compute the index, even with some negative income yi. Possible with Gini, and CV, not withTheil.2. Decomposition of inequality indices2.1. Decomposition of the coefficient of variation by sources of incomeµiµi∑CViCVρ= wii∑ci= 1µi = mean income from source iCVi = CV of income source iwi = weight of income source i or share of source i in average income = µiµ.ci = relative concentration coefficientρ = corr(yi, y)2.2. Decomposition of the Gini coefficient by sources of income G =2nµcov y,r( )=µiµi∑2nµicov yi, ri( )⎡ ⎣ ⎢ ⎤ ⎦ ⎥ cov yi, r( )cov yi,ri( )= wiGiRiwii∑GiGRi= wii∑gi= 1wi = share of source i in average income,Gi = Gini coefficient of income source i,Ri = correlation of income source to overall inequality relative to correlation of incomesource to within source inequality,gi = relative concentration coefficientIf gi > 1, i-th source increases inequality;If gi < 1, i-th source decreases inequality.wigi = share of total inequality contributed by income source i.- 2 - 9/26/04Example: Decomposition of inequality measures, Egypt, 1986-87Sources of incomeAgricultureNon AgricultureRemittancesTotalWeight of income sourcewi = µi/µ0.578 0.326 0.096 1.000Decomposition of coefficient of variationOverall CV CV 0.392Corr(yi, y) * CV of income sourceρi CVi0.267 0.335 1.340Relative concentration coefficients CVci = ρi CVi/CV0.681 0.855 3.418Decomposition of CV wi ci0.393 0.279 0.328 1.000Decomposition of Gini coefficientGini of income source Gi0.509 0.675 0.932Ratio of correlationsRi0.626 0.161 0.924Overall Gini G 0.302Relative concentration coefficients Gini gi = Ri Gi/G 1.054 0.359 2.848Decomposition of Gini wi gi0.609 0.117 0.273 1.000Source: R. Adams, IFPRI Research Report No. 86, 1991.Comments:- Role of wi: Agricultural is the most important source of income.- Role of gi: Remittances (gi > 1) contribute to increase total inequality; non-agriculture (gi < 1) contributesto reduce total inequality; agriculture is about neutral (gi near 1).- Role of Gi: Remittances have the highest source Gini (as few households get them, and they are verylarge).- Role of Ri: Remittance income is highly correlated to total inequality, increasing inequality.Conclusion:- Agriculture makes the highest contribution to inequality (60.9%, measured by wi gi) due to its high incomeshare (wi). Remittances contributes 27.3% of total inequality in spite of its low income share because it has alarge Gini (Gi) and a high correlation with overall income inequality (Ri).3. Relationship between level of income (GNPpc) and inequality: Empirical evidence on the Kuznetscurve (inverted-U).From cross section data (relationship reflects the “Latin America effect”)Kuznets, AER, 1955.Paukert, International Labor Review, 1973. 56 countries. Confirms Kuznets inverted U.Ahluwalia, JDE, 1976. 60 countries. Confirms Kuznets.Adelman and Fuwa, Economie Appliquée, 1994. Confirm Kuznets, but weakening between the1970s (growth) and the 1980s (debt crisis and recession).Jha, World Development, 1996. Confirms Kuznets.Anand and Kanbur, Journal of Development Economics, 1993. Reject Kuznets based on choice offunctional form.From time series data for each countryBowman, World Development, 1997. Rejects KuznetsFrom panel data: time series data across countriesDeininger and Squire, World Bank Economic Review, 1996; JDE, 1998. Work with panel data andcountry fixed effects. Reject Kuznets.Conclusion: Does not seem to holdPolicy implication:If holds: Growth takes care of inequalityIf does not hold: Need special policy interventions to reduce inequality.4. Role of GDPpc growth on inequality- Growth is neutral on inequality: Dollar and Kraay (2000); Deininger and Squire. Elasticity of incomeof poor with respect to aggregate income = 1. Is this pro-poor growth?- de Janvry and Sadoulet (1999) for Latin America: growth does not reduce inequality, but recessionincreases inequality creating a ratchet effect. Hence, importance of “socially
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