Session 5 Review Today Inequality measures Four basic axioms Lorenz Lorenz inconsistent measures Today Additional Properties Lorenz consistent measures

Inequality Measures Notation Definition x is the income distribution xi is the income of the ith person n=n(x) is the population size. D is the set of all distributions of any population size Definition An inequality measure is a function I from D to R which, for each distribution x in D indicates the level I(x) of inequality in the distribution.

Four Basic Properties Definition Ex Symmetry (Anonymity) Idea We say that x is obtained from y by a permutation of incomes if x = Py, where P is a permutation matrix. Ex Symmetry (Anonymity) If x is obtained from y by a permutation of incomes, then I(x)=I(y). Idea All differences across people have been accounted for in x

Replication Invariance (Population Principle) Def We say that x is obtained from y by a replication if the incomes in x are simply the incomes in y repeated a finite number of times Ex Replication Invariance (Population Principle) If x is obtained from y by a replication, then I(x)=I(y). Idea Can compare across different sized populations

Scale Invariance (Zero-Degree Homogeneity) Def We say that x is obtained from y by a proportional change (or scalar multiple) if x=αy, for some α > 0. Ex Scale Invariance (Zero-Degree Homogeneity) If x is obtained from y by a proportional change, then I(x)=I(y). Idea Relative inequality

Def Ex Transfer Principle Idea We say that x is obtained from y by a (Pigou-Dalton) regressive transfer if for some i, j: i) yi < yj ii) yi – xi = xj – yj > 0 iii) xk = yk for all k different to i,j Ex Transfer Principle If x is obtained from y by a regressive transfer, then I(x) > I(y). Idea Mean preserving spread increases measured inequality

Def Any measure satisfying the four basic properties (symmetry, replication invariance, scale invariance, and the transfer principle) is called a relative inequality measure.

Lorenz Curve (1905) Lorenz Curve Cumul Pop p Inc Lx(p) 1/3 1/15 2/3 Ex Lorenz Curve 1. Order incomes lowest to highest to obtain ordered version Ex Find the cumulative share of population up to each i = 1,…,n: p = i/n 3. Find the cumulative share of the income up to each i 4. Plot pairs and connect the dots Cumul Pop p Inc Lx(p) 1/3 1/15 2/3 7/15 1 15/15

Note At any population share p, the Lorenz curve Lx gives the share of income received by the lowest p of income recipients (often stated in % terms)

Lorenz Curve Def Let F(s) be a cumulative distribution function (cdf) and let QF(p) = F-1(p) be its quantile function. Then the Lorenz Curve associated with F is given by:

Lorenz Curve Source: Foster (1985), p. 17

Characteristics Starts at (0,0); ends at (1,1). Always convex (because population is ordered from poorest to richest) and increasing (if incomes positive). Q/ Lorenz curve of a perfectly equal distribution? Q/ Lorenz curve of a perfectly unequal distribution?

Lorenz Criterion Def Note Ex Given two distributions x and y, we say that x Lorenz-dominates y if and only if: Lx(p) > Ly(p) for all p, with > for some p Note x is unambiguously less unequal than y Ex

Note Lorenz criterion generates a relation that is incomplete: when Lorenz curves cross, the Lorenz criterion cannot decide between the two distributions. But when it does hold, the ranking is considered unambiguous Shorrocks and Foster (1987) provide additional conditions by which two distributions can be ranked when Lorenz curves cross. Still it does not eliminate all the incompleteness.

Q/ How does the Lorenz criterion fare with respect to the four basic axioms? A/ Satisfies them all!

The Lorenz Curve and the Four Axioms Symmetry and Replication invariance satisfied since permutations and replications leave the curve unchanged. Proportional changes in incomes do not affect the LC, since it is normalized by the mean income. Only shares matter. So it is scale invariant. A regressive transfer will move the Lorenz curve further away from the diagonal. So it satisfies transfer principle.

Lorenz Consistency (Foster, 1985) Def An inequality measure I: D→R is Lorenz consistent whenever the following hold for any x and y in D: (i) if x Lorenz dominates y, then I(x) < I(y), and (ii) if x has the same Lorenz curve as y, then I(x) = I(y). Theorem An inequality measure I(x) is Lorenz consistent if and only if it satisfies symmetry, replication invariance, scale invariance and the transfer principle, ie, if it is a relative inequality measure.

Note If Lorenz curves don’t cross, then all relative measures follow the Lorenz curve. If Lorenz curves cross, then some relative measure of inequality might be used to make the comparison. But the judgment may depend on the chosen measure.

Classifying Inequality Measures Note By motivation Various ways to classify By motivation Positive vs. normative Statistical By properties Lorenz consistent vs. inconsistent Decomposable Some well known measures are not Lorenz consistent

Lorenz inconsistent measures Violate one or more of the four basic axioms Usually Transfer Scale invariance

Range Ex x = (8,6,1) I(x) = (8-1)/5 = 7/5 Idea Gap between the highest and the lowest income measure in number of means. Note Ignores the distribution between the extremes: Violates transfer principle

90/10 ratio Ex x = (8,6,1) I(x) = 8 Idea Ratio of the 90th percentile income and the 10th percentile income. Note Ignores the distribution apart from the two points: Violates transfer principle

Kuznets Ratio where I(x;p,r) = Sr(x)/sp(x) sp(x) = Lx(p) share of income earned by the poorest p share of the population Sr(x) = 1 – Lx(1-r) share of income earned by the richest r share of the population Ex If p = r = 1/3, x = (8,6,1). Then I(x) = 8 Note The ratios derived from points along the Lorenz Curve. Like the range, it ignores the distribution between the cutoffs and therefore violates the transfer principle

Relative Mean Deviation Idea The average departure of incomes from the mean, measured in means Note Not sensitive to certain transfers. Violates transfer principle.

Variance Note By squaring the gaps of each income to the mean, the bigger gaps receive a higher weight; it satisfies the transfer principle However, the variance goes up four times when incomes are doubled; it violates scale invariance.

Variance of Logarithms where Note Applies variance to the distribution of log-incomes. It is scale invariant (why?) But violates transfer principle when relatively high incomes are involved. Can be incredibly Lorenz inconsistent (See Foster and Ok, 1998).

Additional Properties Q/ How to discern among Lorenz consistent (or relative) measures? A/ Additional axioms for inequality measures

Def Let = (μ,…, μ) denote the smoothed distribution which gives everyone in x the mean μ of x. Normalization If x = , then I(x) = 0. Idea If all incomes are same, then the measure of inequality is zero.

I has no “jumps” Continuity For any sequence xk, if xk converges to x, then I(xk) converges to I(x) Idea I has no “jumps” A small change in any income should not result in an abrupt change in inequality

Motivation Transfer sensitivity How should these progressive transfers (from x to x’, and from x to x”) be regarded by an inequality measure? x=(2,4,6,8) x' =(3,3,6,8) x" =(2,4,7,7) Transfer sensitivity A transfer-sensitive inequality measure places greater emphasis on transfers at the lower end of the distribution; so I(x’) < I(x”) Formalized by Shorrocks and Foster (1987).

Subgroup Consistency Note Suppose that x’ and x share means and population sizes, while y’ and y also share means and population sizes. If I(x’) > I(x) and I(y’) = I(y), then I(x’,y’) > I(x,y). Note Important for policy design & evaluation, regional vs. national Yet, there are arguments against; see, for example Foster & Sen (1997), p.160.

Additive Decomposability Suppose that x and y are any two distributions, then where the final term is the inequality of the smoothed distribution in which persons in x receive the mean of x and persons in y receive the mean of y. Note Very useful for policy design & evaluation, regional vs. national vs. global. Implies subgroup consistency (why?)

Session 5 Review Today Inequality measures Four basic axioms Lorenz Lorenz inconsistent measures Today Additional Properties Lorenz consistent measures

Lorenz Consistent Inequality Measures We now consider measures that satisfy the four basic axioms First positive, then normative.

Squared Coefficient of Variation Note The variance of the (mean-)normalized distribution, thus it is scale invariant. Not transfer sensitive. A transfer has the same impact regardless where it takes place. Is additively decomposable, with weights on within-group term:

Gini Coefficient Several equivalent definitions Idea G = E|s – t|/(Es + Et) or the expected difference between two incomes drawn at random, over twice the mean Ex differences x = (6,1,8) Then μ = 5 and G = (28/9)/10 = .311

Gini Coefficient Idea G = (μ – S)/μ where S = ΣΣmin(xi,xj)/n2 is the expected value of the min of two incomes drawn randomly. Ex min’s x = (6,1,8). Then μ = 5 and S = 31/9 G = (5-S)/5 = 14/45 = .311

Gini Coefficient Note Ex incomes weighted by ranks; also: x = (6,1,8). Then μ = 5 and S = (1*8 + 3*6 + 5*1)/9 G = (5-S)/5 = 14/45 = .311

Relationship with the Lorenz Curve Note Relationship with the Lorenz Curve O C D A

Note Not subgroup consistent (examples in OEI) Not decomposable in ‘traditional way’. Instead it has an alternative ‘breakdown’ into three terms: where R is a non-negative residual term that balances the equation. It indicates the extent to which the subgroups’ distributions overlap.

Theil’s first measure Q/ What does this mean? where |x| = Σxi and si = xi/|x| is i’s income share Shannon’s entropy measure Note Se(1,0,…,0) = 0 Se(1/n,…,1/n) = ln(n) So T(x1,…,xn) = max Se – Se(s1,…,sn)

Properties Lorenz consistent Additively decomposable with weights being or income share of subgroup

Theil’s second measure Q/ What does this mean? where g = (Πxi)1/n is the geometric mean Note If all incomes are equal, then D = 0 since μ = g Otherwise D > 0 since μ > g

Properties Lorenz consistent Additively decomposable with weights being or population share of subgroup

Generalized Entropy Measures When α = 1, it is the Theil first measure. When α = 0, it is the Theil second measure, also known as Mean Logarithmic Deviation When α = 2, it is a multiple of the Squared Coefficient of Variation.

Note Lorenz Consistent. Parameter α is indicator of ‘inequality aversion’ (more averse as α falls). It also indicates the measure’s sensitivity to transfers at different parts of the distribution With α =2, it is ‘transfer neutral’. With α <2, it favours transfers at the lower end of the distribution (includes both Theil’s measures). With α >2 it shows a kind of ‘reverse sensitivity’ stressing transfers at higher incomes. (These are less used)

They are additively decomposable with weights being: Only Theil’s Measures (α=1 and α=0) have weights that sum up to 1.

Normative Measures Dalton (1920), Kolm (1968), Atkinson (1970), Sen (1973) Core Idea Inequality represents the loss in welfare from unequal incomes Q/ Given a social welfare function W(x), how to measure inequality?

Dalton (1920) Used utilitarian social welfare with identical strictly concave, increasing utilities. Measured inequality as maximum welfare minus actual welfare, all over maximum welfare. Atkinson (1970) Noted the dependence on the way utility is measured noted it need not be scale invariant defined a new approach that did not have these problems

Atkinson (1970) Core Concept Equally Distributed Equivalent Income (EDE) The income level which, if assigned to all individuals produces the same social welfare than the observed distribution.

Atkinson’s Approach J = Total income JK is set of all possible distributions I1, I2, I3 ane three social welfare levels A = Actual distribution D = Equally Distributed Equivalent Income E = Mean Income I = (E – D)/E

Atkinson’s normative measure Interpretation

Atkinson’s Class All the members in the family are Lorenz consistent. Parameter α is a measure of ‘inequality aversion’ or relative sensitivity of transfers at different income levels. The lower is α, the higher is the aversion to inequality and more weight is attached to transfers at the lower end of the distribution. Each member is subgroup consistent but it is not additively decomposable.

Atkinson’s Class Note where and