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## What is inequality and how we measure it

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**What is inequality and how we measure it**Milanovic, “Global inequality and its implications” Lectures 1 & 2**Absolute vs. relative**• Is conception of inequality based on absolute or relative income distances? • Does inequality increases if all incomes go up by the same percentage? (stay the same, go up, even go down; Dalton) • How about when they all go up by the same constant? • Is inequality anonymous? If poor and rich swap places (note: this is pro-poor growth) will inequality be less or the same?**Relative and absolute inequality**• Relative inequality is about ratios; absolute inequality is about differences. • State A: two incomes $1,000 and $10,000 per year • State B: these rise to $2,000 and $20,000 • Ratio is unchanged but the absolute gain to the rich is twice as large in state B • 40% of participants in experiments view inequality in absolute terms (Amiel and Cowell).**Growth and inequality**Relative inequality Absolute inequality The tide rises all boats by the same proportion of their initial income; no Δ in relative inequality But absolute income differences increase Source: Ravallion (2003)**Important definitions to keep in mind**• Welfare aggregates: expenditures, consumption, income (net or gross) • Who is the recipient: household or individual? • What is the ranking criterion: income per capita, household income, or income per equivalent unit?**Issues to keep in mind**• Survey issues: non-compliance (refusal to participate), underreporting, top-coding. Researchers can do nothing about these. • Income: valuation of home consumption, imputed rent, self-employment income, property income; net or gross income. Researchers can do very little about that. • Coverage and classification of expenditures • Distinguish consumption and expenditures (use of imputation; treatment of bulky purchases like cars)**Survey non-compliance**• Distinguish from income underreporting • Both stronger among the rich than the poor; underestimate of inequality • If survey non-compliance increases in income (as empirical studies show) => poverty HC overestimated, inequality probably underestimated (although we cannot establish Lorenz dominance) • We believe that non-compliance increases in income because (mean) richer areas generally show higher % of refusal to participate • US inequality may be underestimated by as much as 4 Gini points or 10% (Korinek, Mistiaen, Ravallion, 2006)**Income vs. expenditures (or consumption)?**• Income: gives actual economic power • Expenditures or consumption: give actual standard of living • Savings (as % of income) generally larger for higher income households => inequality of income greater than inequality of expenditures • Income can be negative; C cannot be => inequality of income greater than inequality of expenditures • So at both ends, income gives higher inequality (would also give greater poverty)**Welfare metric: Income vs. expenditure or**consumptionIncome and expenditure per capita by percentile (people ranked by YPC) People at the bottom (up to 30th percentile) dissave; people at the top (richest 30 percent) save**This despite high correlation in general between income and**expenditures(so high ρ can sometimes be misleading) South African 1998 expenditure and income per capita (in logs)**Expenditure-to-income ratio across ventiles**Blue line: the story as before. Red line: high C households dissave. Gini for YPC = 28.4; Gini for XPC = 26.7 Data Poland Heide; see XYratios.xls file**The consumption-income ratios:overall net dissaving (asset**sales); or more likely, better reporting of consumption than income in HS Overall C/Y=1.12 Overall C/Y ratio = 1.09 Blue line: the same story as before; Red line: C-rich people underreport their income Source: Serbia LSMS 2002; file poorAZ.xls**Where in terms of YPC distribution, are high C people who**report C/Y ratio>2? Graph shows where in YPC distribution are people from the 20th (highest) C ventile whose reported C/Y is greater than 2. They are across all income distribution even among those who are income poor. Source: Serbia LSMS 2002;**Actual distributions and functional forms: Actual income**distribution (Malaysia 1997 YPC) and log-normal curve imposed on it**D(p|yp) and D(h|yh):Mexico 2002**Expressed in terms of either mean per capita or mean per HH income.**Equivalence scales**• The basic idea: to reach the same degree of utility, people may not need the same amount of income • But we know nothing about how individuals “convert” income into utility (no inter-personal comparisons) • What we know (or suppose): (i) cost of food is less for children than for adults; (ii) people who live together share public goods (“it’s cheaper in per capita terms for two people to live together than individually”; think of heating costs)**Equivalence scale is then needed to adjust household income**for components (i) and (ii) • Instead of dividing total household income (Y) by number of people (n), we have y*=Y/nΘwhere y* = “true” welfare of each individual in household and Θ = a parameter that (broadly speaking) expresses economies of size**The Barten model**• WITH PUBLIC AND PRIVATE GOODS ONLY • where y*=“true” income or consumption (welfare) per household member at the optimum, Y=total household income or consumption, n= number of household members, ρ = share of spending on food (economies of size=0). • = the (reverse) of the economy of size in the consumption of housing. (Note that if housing were a pure public good, would be equal to 0, and the entire “utility” from the public good would be consumed by each household member). • = the (reverse of) the overall level of “publicness” in consumption. reflects both the composition of consumption (between the public and private goods), and the economies of size in the consumption of public good. • is a technological parameter, is an overall calculated elasticity.**2. Including children too**• 3. Finally, simplify (so that new theta includes both public-private and child-adult components)**Malaysia 1995: Sensitivity of inequality measures on the**assumptions regarding economies of scale and size (theta)**Sensitivity of inequality measures to equivalence scales and**income vs. expenditure welfare indicator Income Expenditure Source: South Africa 1994-95**Sensitivity of inequality measures to equivalence scales and**income vs. expenditure welfare indicator (cont.) Income Expenditure Source: Hungary 1993**Generally, Gini (and other inequality measures) go down as**equivalence scales increase (means that larger households “gain” some utility because of economies of size, and also probably because they have more children) • But this is not always the case as illustrated on the examples of South Africa and Hungary • If YPC does not fall much with HH size, then Gini might not change much as equivalence scales increase**Welfarist approach (Dalton) to inequality vs. measurement**only (Gini) The methods of Italian writers…are not…comparable to his [Dalton’s] own, inasmuch as their purpose is to estimate, not the inequality of economic welfare, but the inequality of incomes and wealth, independently of all hypotheses as to the functional relations between these quantities and economic welfare or as to the additive character of the economic welfare of individuals. Corrado Gini, Measurement of Inequality of Incomes, Economic Journal, March 1921.**Inequality measurement axioms**• 1. If all incomes are multiplied by a constant (Y1=Y*C), inequality does not change. • 2. Increase of all incomes by a constant (Y1=Y+C), reduces inequality (follows from 1). New distribution is Lorenz-superior. • 3. If number of recipients is multiplied (at each income level) by a constant, inequality does not change • 4. Progressive transfer (which does not change the rankings of individuals) reduces inequality (Dalton’s axiom). (Dalton improvement = income of the poor ↑ by at least as much as income of the rich goes down. • 5. Symmetry or anonymity: if two people swap positions, inequality does not change. • 6. Inequality measure lies in [0,1] domain.**Measures of inequalityDesirable properties and how different**measures satisfy them.**Gini decomposability**• By income source • By recipient Where π=share (of recipients) in total income, p=share in total population, s=share (of income source) in total income, μ=mean, L=overlap term and Ri=cov(xi,r(y))/cov(xi,r(xi) source correlation coefficient with total income**Gini calculation from grouped data**Where fi=frequency of i-th group, qi=cumulative share of income received by the bottom i groups Often, the “true” Gini is approximated by the following heuristic formula: True Gini = 1/3 Gini (min) + 2/3 Gini (max)**EXAMPLE. Romania 1998 (Integrated Household Survey results**as reported in Statistical Yearbook 1999). The very lowest and the very top interval (both in italics) are assumed. The results are as follows (using Kakwani’s formulas). Gini minimum is 26.09, Gini maximum is 27.51 (a difference of 5.4 percent). The heuristic Gini would then be 27.04. A very simple formula (approximation; when N, it is exact; practically, good as soon as N>10 or 12):**Lorenz curves that intersect (with almost the same Gini)**Source: thepast.xls (lorenz2)**Generalized Lorenz curve: real ($PPP) income at the same**percentile levels**Another example of a generalized Lorenz curve: France vs.**United States**Second order dominance: real ($PPP) income at the same**cumulative percentile levels**Several often-used ‘functional’ distributions**• Lognormal (the most popular) • Pareto (the oldest; good fit for highest incomes) • Where yl=minimum possible value of y, α=Pareto constant (=1.5) • Singh-Maddala**Gini distribution**• Where Yt=total income, Yy=aggregate income up to income level y, γ (gamma)=parameter, C=constant.**For each distribution, one can calculate corresponding**Ginis, Lorenz curves and any measure of inequality • Often used as approximations to empirical (true) distributions, or a way to estimate distribution if we have only a few data points (e.g., if only published group data are available)