Inequality and Poverty

Inequality and Poverty Public Economics: University of Barcelona Frank Cowell http://darp.lse.ac.uk/ub June 2005

Issues to be addressed • Builds on lecture 2 • “Distributional Equity, Social Welfare” • Extension of ranking criteria • Parade diagrams • Generalised Lorenz curve • Extend SWF analysis to inequality • Examine structure of inequality • Link with the analysis of poverty

Major Themes • Contrast three main approaches to the subject • intuitive • via SWF • via analysis of structure • Structure of the population • Composition of inequality and poverty • Implications for measures • The use of axiomatisation • Capture what is “reasonable”? • Find a common set of axioms for related problems

Inequality and Poverty Overview... Inequality rankings Inequality measurement Relationship with welfare rankings Inequality and decomposition Poverty measures Poverty rankings

Inequality rankings • Begin by using welfare analysis of previous lecture • Seek inequality ranking • We take as a basis the second-order distributional ranking • …but introduce a small modification • The 2nd-order dominance concept was originally expressed in a more restrictive form.

Yet another important relationship • The share of the proportion q of distribution F is given by L(F;q) := C(F;q) / m(F) • Yields Lorenz dominance, or the “shares” ranking G Lorenz-dominates Fmeans: • for every q, L(G;q) ³L(F;q), • for some q, L(G;q) > L(F;q) • The Atkinson (1970) result: For given m, G Lorenz-dominates F Û W(G) > W(F) for all WÎW2

The Lorenz diagram 1 0.8 L(.; q) 0.6 L(G;.) proportion of income Lorenz curve for F 0.4 L(F;.) 0.2 practical example, UK 0 0 0.2 0.4 0.6 0.8 1 q proportion of population

Official concepts of income: UK original income + cash benefits gross income - direct taxes disposable income - indirect taxes post-tax income + non-cash benefits final income What distributional ranking would we expect to apply to these 5 concepts?

Impact of Taxes and Benefits. UK 2000/1. Lorenz Curve

Assessment of example • We might have guessed the outcome… • In most countries: • Income tax progressive • So are public expenditures • But indirect tax is regressive • So Lorenz-dominance is not surprising. • But what happens if we look at the situation over time?

“Final income” – Lorenz

1.0 0.9 0.8 0.7 0.6 0.5 0.0 0.1 0.2 0.3 0.4 0.5 “Original income” – Lorenz • Lorenz curves intersect • Is 1993 more equal? • Or 2000-1?

Inequality ranking: Summary • Second-order (GL)-dominance is equivalent to ranking by cumulations. • From the welfare lecture • Lorenz dominance equivalent to ranking by shares. • Special case of GL-dominance normalised by means. • Where Lorenz-curves intersect unambiguous inequality orderings are not possible. • This makes inequality measures especially interesting.

Inequality and Poverty Overview... Inequality rankings Inequality measurement • Intuition • Social welfare • Distance Three ways of approaching an index Inequality and decomposition Poverty measures Poverty rankings

An intuitive approach • Lorenz comparisons (second-order dominance) may be indecisive • But we may want to “force a solution” • The problem is essentially one of aggregation of information • Why worry about aggregation? • It may make sense to use a very simple approach • Go for something that you can “see” • Go back to the Lorenz diagram

The best-known inequality measure? 1 0.8 proportion of income 0.6 Gini Coefficient 0.5 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 proportion of population

The Gini coefficient • Equivalent ways of writing the Gini: • Normalised area above Lorenz curve • Normalised difference between income pairs.

Intuitive approach: difficulties • Essentially arbitrary • Does not mean that Gini is a bad index • But what is the basis for it? • What is the relationship with social welfare? • The Gini index also has some “structural” problems • We will see this in the next section • What is the relationship with social welfare? • Examine the welfare-inequality relationship directly

SWF and inequality • Issues to be addressed: • the derivation of an index • the nature of inequality aversion • the structure of the SWF • Begin with the SWF W • Examine contours in Irene-Janet space

xj xi Equally-Distributed Equivalent Income • The Irene &Janet diagram • A given distribution • Distributions with same mean • Contours of the SWF • Construct an equal distribution E such that W(E) = W(F) • EDE income • Social waste from inequality • Curvature of contour indicates society’s willingness to tolerate “efficiency loss” in pursuit of greater equality • E • F O m(F) x(F)

Welfare-based inequality • From the concept of social waste Atkinson (1970) suggested an inequality measure: Ede income x(F) I(F) = 1 – —— m(F) Mean income • Atkinson assumed an additive social welfare function that satisfied the other basic axioms. W(F) = òu(x) dF(x) • Introduced an extra assumption: Iso-elastic welfare. x1 - e– 1 u(x) = ————, e ³ 0 1 – e

The Atkinson Index • Given scale-invariance, additive separability of welfare • Inequality takes the form: • Given the Harsanyi argument… • index of inequality aversion ebased on risk aversion. • More generally see it as a stament of social values • Examine the effect of different values of e • relationship between u(x) and x • relationship between u′(x) and x

Social utility and relative income U  = 0 4 3  = 1/2 2  = 1 1  = 2  = 5 0 1 2 3 4 5 x / m -1 -2 -3

Relationship between welfare weight and income  =1 U'  =2  =5 4 3 2  =0 1  =1/2 x / m  =1 0 0 1 2 3 4 5

A further look at inequality • The Atkinson SWF route provides a coherent approach to inequality. • But do we need to approach via social welfare • An indirect approach • Maybe introduces unnecessary assumptions, • Alternative route: “distance” and inequality • Consider a generalisation of the Irene-Janet diagram

x j x k x i The 3-Person income distribution Income Distributions With Given Total ray of Janet's income equality Karen's income 0 Irene's income

x j x k x i Inequality contours • Set of distributions for given total • Set of distributions for a higher (given) total • Perfect equality • Inequality contours for original level • Inequality contours for higher level 0

A distance interpretation • Can see inequality as a deviation from the norm • The norm in this case is perfect equality • Two key questions… • …what distance concept to use? • How are inequality contours on one level “hooked up” to those on another?

Another class of indices • Consider the Generalised Entropy class of inequality measures: • The parameter a is an indicator sensitivity of each member of the class. • a large and positive gives a “top -sensitive” measure • a negative gives a “bottom-sensitive” measure • Related to the Atkinson class

Inequality and a distance concept • The Generalised Entropy class can also be written: • Which can be written in terms of income shares s • Using the distance criterion s1−a/ [1−a] … • Can be interpreted as weighted distance of each income shares from an equal share

The Generalised Entropy Class • GE class is rich • Includes two indices from Henri Theil: • a = 1:  [ x / m(F)] log (x / m(F)) dF(x) • a = 0: –  log (x / m(F)) dF(x) • For a < 1 it is ordinally equivalent to Atkinson class • a = 1 – e . • For a = 2 it is ordinally equivalent to (normalised) variance.

Inequality contours • Each family of contours related to a different concept of distance • Some are very obvious… • …others a bit more subtle • Start with an obvious one • the Euclidian case

GE contours: a = 2

GE contours: a < 2 a = 0.25 a = 0 a = −0.25 a = −1

GE contours: a limiting case a = −∞ • Total priority to the poorest

GE contours: another limiting case a = +∞ • Total priority to the richest

By contrast: Gini contours • Not additively separable

Distance: a generalisation • The responsibility approach gives a reference income distribution • Exact version depends on balance of compensation rules • And on income function. • Redefine inequality measurement • not based on perfect equality as a norm • use the norm income distribution from the responsibility approach • Devooght (2004) bases this on Cowell (1985) • Cowell approach based on Theil’s conditional entropy • Instead of looking at distance going from perfect equality to actual distribution... • Start from the reference distribution

Inequality and Poverty Overview... Inequality rankings Inequality measurement Structural issues Inequality and decomposition Poverty measures Poverty rankings

Why decomposition? • Resolve questions in decomposition and population heterogeneity: • Incomplete information • International comparisons • Inequality accounting • Gives us a handle on axiomatising inequality measures • Decomposability imposes structure. • Like separability in demand analysis first, some terminology

(4) (3) (6) (5) (2) (1) (i) (ii) (iii) (iv) • The population • Attribute 1 A partition • Attribute 2 • One subgroup population share pj income share sj Ij subgroup inequality

What type of decomposition? • Distinguish three types of decomposition by subgroup • In increasing order of generality these are: • Inequality accounting • Additive decomposability • General consistency • Which type is a matter of judgment • More on this below • Each type induces a class of inequality measures • The “stronger” the decomposition requirement… • …the “narrower” the class of inequality measures

1:Inequality accounting This is the most restrictive form of decomposition: accounting equation weight function adding-up property

2:Additive Decomposability As type 1, but no adding-up constraint:

3:General Consistency The weakest version: population shares increasing in each subgroup’s inequality income shares

A class of decomposable indices • Given scale-invariance and additive decomposability, • Inequality takes the Generalised Entropy form: • Just as we had earlier in the lecture. • Now we have a formal argument for this family. • The weight wj on inequality in group j is wj = pjasj1−a

What type of decomposition? • Assume scale independence… • Inequality accounting: • Theil indices only (a = 0,1) • Here wj = pj or wj = sj • Additive decomposability: • Generalised Entropy Indices • General consistency: • moments, • Atkinson, ... • But is there something missing here? • We pursue this later

What type of partition? • General • The approach considered so far • Any characteristic used as basis of partition • Age, gender, region, income • Induces specific class of inequality measures • ... but excludes one very important measure • Non-overlapping in incomes • A weaker version • Partition just on the basis of income • Allows one to include the "missing" inequality measure • Distinction between them is crucial for one special inequality measure

Inequality and Poverty