Frank Cowell: TU Lisbon – Inequality & Poverty Inequality Measurement July 2006 Inequality measurement Measurement Technical University of Lisbon Frank Cowell

Frank Cowell: TU Lisbon – Inequality & Poverty Issues to be addressed Builds on lecture 3 Builds on lecture 3  “Distributional Equity, Social Welfare” Extension of ranking criteria Extension of ranking criteria  Parade diagrams  Generalised Lorenz curve Extend SWF analysis to inequality Extend SWF analysis to inequality Examine structure of inequality Examine structure of inequality Link with the analysis of poverty Link with the analysis of poverty

Frank Cowell: TU Lisbon – Inequality & Poverty Major Themes Contrast three main approaches to the subject Contrast three main approaches to the subject  intuitive  via SWF  via analysis of structure Structure of the population Structure of the population  Composition of Inequality measurement  Implications for measures The use of axiomatisation The use of axiomatisation  Capture what is “reasonable”?  Find a common set of axioms for related problems

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Inequality rankings Inequality measures Inequality axioms Inequality decomposition Inequality in practice Inequality measurement Relationship with welfare rankings

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality rankings Begin by using welfare analysis of previous lecture Begin by using welfare analysis of previous lecture Seek inequality ranking Seek inequality ranking We take as a basis the second-order distributional 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. The 2nd-order dominance concept was originally expressed in a more restrictive form.

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality rankings Using the welfare analysis above… Using the welfare analysis above… Seek an inequality ranking Seek an inequality ranking Take as a basis the 2nd-order distributional ranking Take as a basis the 2nd-order distributional ranking  …but introduce a small modification  Normalise by dividing by the mean  Away of forcing an “iso-inequality” path as mean income changes The 2nd-order dominance concept was originally expressed in this more restrictive form… The 2nd-order dominance concept was originally expressed in this more restrictive form…

Frank Cowell: TU Lisbon – Inequality & Poverty Yet another important relationship The share of the proportion q of distribution F is given by The share of the proportion q of distribution F is given by L(F;q) := C(F;q) /  (F) Yields Lorenz dominance, or the “shares” ranking Yields Lorenz dominance, or the “shares” ranking For given , G Lorenz-dominates F  W(G) > W(F) for all W  W 2 The Atkinson (1970) result: The Atkinson (1970) result: G Lorenz-dominates F  means:   for every q, L(G;q)  L(F;q),   for some q, L(G;q) > L(F;q)

Frank Cowell: TU Lisbon – Inequality & Poverty The Lorenz diagram proportion of income proportion of population L(G;.) L(F;.) L(.; q) q Lorenz curve for F practical example, UK

Frank Cowell: TU Lisbon – Inequality & Poverty Application of ranking The tax and -benefit system maps one distribution into another... The tax and -benefit system maps one distribution into another... Use ranking tools to assess the impact of this in welfare terms. Use ranking tools to assess the impact of this in welfare terms. Typically this uses one or other concept of Lorenz dominance. Typically this uses one or other concept of Lorenz dominance.

Frank Cowell: TU Lisbon – Inequality & Poverty original income + cash benefits gross income - direct taxes disposable income - indirect taxes post-tax income + non-cash benefits final income Official concepts of income: UK What distributional ranking would we expect to apply to these 5 concepts?

Frank Cowell: TU Lisbon – Inequality & Poverty Impact of Taxes and Benefits. UK 2000/1. Lorenz Curve

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

Frank Cowell: TU Lisbon – Inequality & Poverty “Final income” – Lorenz

Frank Cowell: TU Lisbon – Inequality & Poverty “Original income” – Lorenz   Lorenz curves intersect   Is 1993 more equal?   Or ?

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality ranking: Summary Second-order (GL)-dominance is equivalent to ranking by cumulations. Second-order (GL)-dominance is equivalent to ranking by cumulations.  From the welfare lecture Lorenz dominance equivalent to ranking by shares. 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. Where Lorenz-curves intersect unambiguous inequality orderings are not possible. This makes inequality measures especially interesting. This makes inequality measures especially interesting.

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

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Inequality rankings Inequality measures Inequality axioms Inequality decomposition Inequality in practice Inequality measurement Three ways of approaching an index Intuition Social welfare Distance

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

Frank Cowell: TU Lisbon – Inequality & Poverty proportion of income proportion of population Gini Coefficient The best-known inequality measure?

Frank Cowell: TU Lisbon – Inequality & Poverty Equivalent ways of writing the Gini: Equivalent ways of writing the Gini: 1. Normalised area above Lorenz curve The Gini coefficient 2. Normalised difference between income pairs.

Frank Cowell: TU Lisbon – Inequality & Poverty Intuitive approach: difficulties Essentially arbitrary 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? What is the relationship with social welfare? The Gini index also has some “structural” problems The Gini index also has some “structural” problems  We will see this in the next section What is the relationship with social welfare? What is the relationship with social welfare?  Examine the welfare-inequality relationship directly

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Inequality rankings Inequality measures Inequality axioms Inequality decomposition Inequality in practice Inequality measurement Three ways of approaching an index Intuition Social welfare Distance

Frank Cowell: TU Lisbon – Inequality & Poverty SWF and inequality Issues to be addressed: Issues to be addressed:  the derivation of an index  the nature of inequality aversion  the structure of the SWF Begin with the SWF W Begin with the SWF W Examine contours in Irene-Janet space Examine contours in Irene-Janet space

Frank Cowell: TU Lisbon – Inequality & Poverty Equally-Distributed Equivalent Income O xixi xjxj   The Irene &Janet diagram   A given distribution   Distributions with same mean   Contours of the SWF E (F)(F) (F)(F) F   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

Frank Cowell: TU Lisbon – Inequality & Poverty Atkinson assumed an additive social welfare function that satisfied the other basic axioms. Atkinson assumed an additive social welfare function that satisfied the other basic axioms.  (F) I(F) = 1 – ——  (F) Mean income Ede income Welfare-based inequality x  1 -  – 1 u(x) = ————,   1 –  Introduced an extra assumption: Iso-elastic welfare. Introduced an extra assumption: Iso-elastic welfare. From the concept of social waste Atkinson (1970) suggested an inequality measure: From the concept of social waste Atkinson (1970) suggested an inequality measure: W(F) =  u(x) dF(x)

Frank Cowell: TU Lisbon – Inequality & Poverty The Atkinson Index Given scale-invariance, additive separability of welfare Given scale-invariance, additive separability of welfare Inequality takes the form: Inequality takes the form: Given the Harsanyi argument… Given the Harsanyi argument…  index of inequality aversion  based on risk aversion. More generally see it as a stament of social values More generally see it as a stament of social values Examine the effect of different values of  Examine the effect of different values of   relationship between u(x) and x  relationship between u′(x) and x

Frank Cowell: TU Lisbon – Inequality & Poverty Social utility and relative income      U x / 

Frank Cowell: TU Lisbon – Inequality & Poverty Relationship between welfare weight and income  =1/2  =0  =1       U' x / 

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Inequality rankings Inequality measures Inequality axioms Inequality decomposition Inequality in practice Inequality measurement Three ways of approaching an index Intuition Social welfare Distance

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

Frank Cowell: TU Lisbon – Inequality & Poverty Another class of indices Consider the Generalised Entropy class of inequality measures: Consider the Generalised Entropy class of inequality measures: The parameter  is an indicator sensitivity of each member of the class. The parameter  is an indicator sensitivity of each member of the class.   large and positive gives a “top -sensitive” measure   negative gives a “bottom-sensitive” measure Related to the Atkinson class Related to the Atkinson class

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality and a distance concept The Generalised Entropy class can also be written: The Generalised Entropy class can also be written: Which can be written in terms of income shares s Which can be written in terms of income shares s Using the distance criterion s 1−  / [1−  ] … Using the distance criterion s 1−  / [1−  ] … Can be interpreted as weighted distance of each income shares from an equal share Can be interpreted as weighted distance of each income shares from an equal share

Frank Cowell: TU Lisbon – Inequality & Poverty The Generalised Entropy Class GE class is rich GE class is rich Includes two indices from Henri Theil: Includes two indices from Henri Theil:   = 1:  [ x /  (F)] log (x /  (F)) dF(x)   = 0: –  log (x /  (F)) dF(x) For  < 1 it is ordinally equivalent to Atkinson class For  < 1 it is ordinally equivalent to Atkinson class   = 1 – . For  = 2 it is ordinally equivalent to (normalised) variance. For  = 2 it is ordinally equivalent to (normalised) variance.

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours Each family of contours related to a different concept of distance Each family of contours related to a different concept of distance Some are very obvious… Some are very obvious… …others a bit more subtle …others a bit more subtle Start with an obvious one Start with an obvious one  the Euclidian case

Frank Cowell: TU Lisbon – Inequality & Poverty GE contours:  2

Frank Cowell: TU Lisbon – Inequality & Poverty GE contours:  2  25   −  − 

Frank Cowell: TU Lisbon – Inequality & Poverty GE contours: a limiting case  −∞ Total priority to the poorest Total priority to the poorest

Frank Cowell: TU Lisbon – Inequality & Poverty GE contours: another limiting case Total priority to the richest Total priority to the richest  +∞

Frank Cowell: TU Lisbon – Inequality & Poverty By contrast: Gini contours Not additively separable Not additively separable

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Inequality rankings Inequality measures Inequality axioms Inequality decomposition Inequality in practice Inequality measurement A fundamentalist approach

Frank Cowell: TU Lisbon – Inequality & Poverty A further look at inequality The Atkinson SWF route provides a coherent approach to inequality. The Atkinson SWF route provides a coherent approach to inequality. But do we need to approach via social welfare But do we need to approach via social welfare  An indirect approach  Maybe introduces unnecessary assumptions, Alternative route: “distance” and inequality Alternative route: “distance” and inequality

Frank Cowell: TU Lisbon – Inequality & Poverty The 3-Person income distribution 0 Irene's income Janet's income Karen's income i x k x x j ray of equality Income Distributions With Given Total

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours 0 i x k x x j   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

Frank Cowell: TU Lisbon – Inequality & Poverty Themes Cross-disciplinary concepts Cross-disciplinary concepts Income differences Income differences Reference incomes Reference incomes Formal methodology Formal methodology

Frank Cowell: TU Lisbon – Inequality & Poverty Methodology Exploit common structure Exploit common structure  poverty  deprivation  complaints and inequality  see Cowell (2005) Cowell (2005)Cowell (2005) Axiomatic method Axiomatic method  minimalist approach  characterise structure  introduce ethics

Frank Cowell: TU Lisbon – Inequality & Poverty “Structural” axioms Take some social evaluation function  Take some social evaluation function  Continuity Continuity Linear homogeneity Linear homogeneity Translation invariance Translation invariance

Frank Cowell: TU Lisbon – Inequality & Poverty Structural axioms: illustration x1x1 x3x3 x2x2 D for n=3 D for n=3 An income distribution An income distribution Perfect equality Perfect equality Contours of “Absolute” Gini Contours of “Absolute” Gini Continuity Continuity  Continuous approach to I = 0 Linear homogeneity Linear homogeneity  Proportionate increase in I Translation invariance Translation invariance  I constant D for n=3 D for n=3 An income distribution An income distribution Perfect equality Perfect equality Contours of “Absolute” Gini Contours of “Absolute” Gini Continuity Continuity  Continuous approach to I = 0 Linear homogeneity Linear homogeneity  Proportionate increase in I Translation invariance Translation invariance  I constant 0 1 x *

Frank Cowell: TU Lisbon – Inequality & Poverty Amiel-Cowell (1999) approach B C Irene's income Janet's income xixi xjxj 0 ray of equality   The Irene &Janet diagram   A distribution   Possible distributions of a small increment   Does this direction keep inequality unchanged?   Or this direction?   Consider the iso- inequality path.   Also gives what would be an inequality- preserving income reduction   A “fair” tax? ll ll A

Frank Cowell: TU Lisbon – Inequality & Poverty xixi xjxj Scale independence   Example 1.   Equal proportionate additions or subtractions keep inequality constant   Corresponds to regular Lorenz criterion

Frank Cowell: TU Lisbon – Inequality & Poverty xixi xjxj x 2 Translation independence   Example 2.   Equal absolute additions or subtractions keep inequality constant

Frank Cowell: TU Lisbon – Inequality & Poverty xixi xjxj Intermediate case   Example 3.   Income additions or subtractions in the same “intermediate” direction keep inequality constant

Frank Cowell: TU Lisbon – Inequality & Poverty xixi xjxj x 2 Dalton’s conjecture   Amiel-Cowell (1999) showed that individuals perceived inequality comparisons this way.   Pattern is based on a conjecture by Dalton (1920)   Note dependence of direction on income level

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Inequality rankings Inequality measures Inequality axioms Inequality decomposition Inequality in practice Inequality measurement Structural issues

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

Frank Cowell: TU Lisbon – Inequality & Poverty A partition population share subgroup inequality income share j j s j I j (ii) (i) (iii) (iv) The population The population Attribute 1 Attribute 1 One subgroup One subgroup Attribute 2 Attribute 2 (1) (2) (3) (4) (5) (6)

Frank Cowell: TU Lisbon – Inequality & Poverty What type of decomposition? Distinguish three types of decomposition by subgroup Distinguish three types of decomposition by subgroup In increasing order of generality these are: In increasing order of generality these are:  Inequality accounting  Additive decomposability  General consistency Which type is a matter of judgment 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

Frank Cowell: TU Lisbon – Inequality & Poverty adding-up property weight function 1:Inequality accounting This is the most restrictive form of decomposition: accounting equation

Frank Cowell: TU Lisbon – Inequality & Poverty 2:Additive Decomposability As type 1, but no adding-up constraint:

Frank Cowell: TU Lisbon – Inequality & Poverty population shares 3:General Consistency The weakest version: income shares increasing in each subgroup’s inequality

Frank Cowell: TU Lisbon – Inequality & Poverty A class of decomposable indices Given scale-invariance and additive decomposability, Given scale-invariance and additive decomposability, Inequality takes the Generalised Entropy form: Inequality takes the Generalised Entropy form: Just as we had earlier in the lecture. Just as we had earlier in the lecture. Now we have a formal argument for this family. Now we have a formal argument for this family. The weight  j on inequality in group j is  j =  j  s j 1−  The weight  j on inequality in group j is  j =  j  s j 1− 

Frank Cowell: TU Lisbon – Inequality & Poverty What type of decomposition? Assume scale independence… Assume scale independence… Inequality accounting: Inequality accounting:  Theil indices only (   Here  j =  j or  j = s j Additive decomposability: Additive decomposability:  Generalised Entropy Indices General consistency: General consistency:  moments,  Atkinson,... But is there something missing here? But is there something missing here?  We pursue this later

Frank Cowell: TU Lisbon – Inequality & Poverty What type of partition? General 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 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 Distinction between them is crucial for one special inequality measure

Frank Cowell: TU Lisbon – Inequality & Poverty Different (equivalent) ways of writing the Gini: Different (equivalent) ways of writing the Gini: Normalised area under the Lorenz curve Normalised area under the Lorenz curve The Gini coefficient Normalised pairwise differences Normalised pairwise differences A ranking-weighted average A ranking-weighted average But ranking depends on reference distribution But ranking depends on reference distribution proportion of income proportion of population Gini Coefficient

Frank Cowell: TU Lisbon – Inequality & Poverty Partitioning by income... x*x* N1N1 N2N2 0 x ** N1N1 x' x   Case 2: effect on Gini differs in subgroup and population x'x   Case 1: effect on Gini is same in subgroup and population   Non-overlapping income groups   Overlapping income groups   Consider a transfer:Case 1   Consider a transfer:Case 2 x

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Inequality rankings Inequality measures Inequality axioms Inequality decomposition Inequality in practice Inequality measurement Performance of inequality measures

Frank Cowell: TU Lisbon – Inequality & Poverty Non-overlapping decomposition Can be particularly valuable in empirical applications Can be particularly valuable in empirical applications Useful for rich/middle/poor breakdowns Useful for rich/middle/poor breakdowns Especially where data problems in tails Especially where data problems in tails  Misrecorded data  Incomplete data  Volatile data components

Frank Cowell: TU Lisbon – Inequality & Poverty Choosing an inequality measure Do you want an index that accords with intuition? Do you want an index that accords with intuition?  If so, what’s the basis for the intuition? Is decomposability essential? Is decomposability essential?  If so, what type of decomposability? Do you need a welfare interpretation? Do you need a welfare interpretation?  If so, what welfare principles to apply?

Frank Cowell: TU Lisbon – Inequality & Poverty Atkinson and Brandolini. (2004) Absolute vs Relative measures

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality measures and US experience

Frank Cowell: TU Lisbon – Inequality & Poverty References Amiel, Y. and Cowell, F. A. (1999) Thinking about Inequality, Cambridge University Press, Cambridge, Chapter 7. Amiel, Y. and Cowell, F. A. (1999) Thinking about Inequality, Cambridge University Press, Cambridge, Chapter 7. Amiel, Y. and Cowell, F. A. (1999) Amiel, Y. and Cowell, F. A. (1999) Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, Atkinson, A. B. (1970) Atkinson, A. B. and Brandolini. A. (2004) “Global World Inequality: Absolute, Relative or Intermediate?” Paper presented at the 28th General Conference of the International Association for Research on Income and Wealth. August 22. Cork, Ireland. Atkinson, A. B. and Brandolini. A. (2004) Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Chapter 2, Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Chapter 2, Cowell, F. A. (2000) Cowell, F. A. (2000) Cowell, F. A. (2006) “Theil, Inequality Indices and Decomposition,” Research on Economic Inequality, 13, Cowell, F. A. (2006) “Theil, Inequality Indices and Decomposition,” Research on Economic Inequality, 13, Cowell, F. A. (2006) Cowell, F. A. (2006) Piketty, T. and E. Saez (2003) “Income inequality in the United States, ,” Quarterly Journal of Economics, 118, Piketty, T. and E. Saez (2003) Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Shorrocks, A. F. (1983) Shorrocks, A. F. (1983) Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4, Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4,