Frank Cowell: TU Lisbon – Inequality & Poverty Inequality: Advanced Topics July 2006 Inequality and Poverty Measurement Technical University of Lisbon Frank Cowell

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Introduction Deprivation Complaints Inequality: Advanced Topics Themes and methodology

Frank Cowell: TU Lisbon – Inequality & Poverty Purpose of lecture We will look at recent theoretical developments in distributional analysis We will look at recent theoretical developments in distributional analysis Consider some linked themes Consider some linked themes  alternative approaches to inequality  related welfare concepts Use ideas from sociology and philosophy Use ideas from sociology and philosophy Focus on the way modern methodology is applied Focus on the way modern methodology is applied

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Introduction Deprivation Complaints Deprivation, complaints, inequality An alternative approach

Frank Cowell: TU Lisbon – Inequality & Poverty Responsibility Standard approach to case for redistribution Standard approach to case for redistribution  Use reference point of equality  How effective is tax/benefit system in moving actual distribution toward reference point? Does not take account of individual responsibility Does not take account of individual responsibility  Role of individual actions  The responsibility “cut”  Dworkin (1981a, 1981b) Distinguish between Distinguish between  things that are your fault  things for which you deserve compensation

Frank Cowell: TU Lisbon – Inequality & Poverty Responsibility and redistribution Should affect the evaluation of distributions Should affect the evaluation of distributions  Both case for redistribution... ... and effectiveness of taxation. Need to differentiate between Need to differentiate between  characteristics for which people can be held responsible  characteristics for which people should not Assume that these characteristics are known and agreed... Assume that these characteristics are known and agreed...

Frank Cowell: TU Lisbon – Inequality & Poverty Each person i has a vector of attributes a i : Each person i has a vector of attributes a i :  Attributes partitioned into two classes  R-attributes: for which the individual is responsible  C-attributes: for which the individual may be compensated Situation before intervention: Situation before intervention:  Determined by income function f  f maps attributes into incomes f(a i )  Only person i’s attributes involved Situation after intervention: Situation after intervention:  Determined by distribution rule F  Has similar flavour to claims problem Basic structure

Frank Cowell: TU Lisbon – Inequality & Poverty Distribution rule Also assume that the rule F is anonymous Also assume that the rule F is anonymous The rule F: The rule F:  depends on whole profile of attributes  maps the attributes into income of i. Assume feasibility: Assume feasibility: Profile of attributes But what other principles should the rule F satisfy? But what other principles should the rule F satisfy?

Frank Cowell: TU Lisbon – Inequality & Poverty Responsibility: Principle EIER Bossert and Fleurbaey (1996) Bossert and Fleurbaey (1996) Equal Income for Equal Responsibility Equal Income for Equal Responsibility  Focus on distribution itself  Full compensation

Frank Cowell: TU Lisbon – Inequality & Poverty Responsibility: Principle ETEC Equal Transfers for Equal C-attributes Equal Transfers for Equal C-attributes  Focus on changes in distribution  Strict Compensation

Frank Cowell: TU Lisbon – Inequality & Poverty A difficulty Fleurbaey (1995a,b) In this special case......a natural redistribution mechanism For large populations... For large populations... EIER and ETEC are incompatible except for... EIER and ETEC are incompatible except for... Additive separability: Additive separability: Consider two compromise approaches

Frank Cowell: TU Lisbon – Inequality & Poverty Compromise (1) Insist on Full compensation (EIER) Weaken ETEC Egalitarian-equivalent mechanisms Every agent has a post-tax income equal to   the pre-tax income earned given reference compensation characteristics plus...   a uniform transfer Reference profile

Frank Cowell: TU Lisbon – Inequality & Poverty Compromise (2) Insist on strict compensation (ETEC) Weaken EIER Conditionally egalitarian mechanisms Conditionally egalitarian mechanisms Every agent k is guaranteed the average income of a hypothetical economy Every agent k is guaranteed the average income of a hypothetical economy  In this economy all agents have characteristics equal to reference profile Reference profile

Frank Cowell: TU Lisbon – Inequality & Poverty Application The responsibility approach gives a reference income distribution The responsibility approach gives a reference income distribution  Exact version depends on balance of compensation rules  And on income function f. Redefine inequality measurement Redefine inequality measurement  not based on perfect equality as a norm  use the norm income distribution from the responsibility approach bases this on Devooght (2005) bases this on Cowell (1985) Devooght (2005)Cowell (1985)  Cowell approach based on Theil’s conditional entropy  Instead of looking at information content in going from perfect equality to actual distribution...  Start from the reference distribution

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Introduction Deprivation Complaints Deprivation, complaints, inequality An economic interpretation of a sociological concept

Frank Cowell: TU Lisbon – Inequality & Poverty Individual deprivation The Yitzhaki (1979) definition The Yitzhaki (1979) definitionYitzhaki (1979)Yitzhaki (1979) Equivalent form Equivalent form In present notation In present notation Use the conditional mean Use the conditional mean

Frank Cowell: TU Lisbon – Inequality & Poverty Deprivation: Axiomatic approach 1 The Better-than set for i The Better-than set for i Focus Focus  works like the poverty concept

Frank Cowell: TU Lisbon – Inequality & Poverty Deprivation: Axiomatic approach 2 Normalisation Normalisation Additivity Additivity  works like the independence axiom

Frank Cowell: TU Lisbon – Inequality & Poverty Bossert-D’Ambrosio (2004) This is just the Yitzhaki individual deprivation index This is just the Yitzhaki individual deprivation index There is an alternative axiomatisation There is an alternative axiomatisation  Ebert-Moyes (Economics Letters 2000)  Different structure of reference group

Frank Cowell: TU Lisbon – Inequality & Poverty Aggregate deprivation Simple approach: just sum individual deprivation Simple approach: just sum individual deprivation Could consider an ethically weighted variant Could consider an ethically weighted variant   Chakravarty and Chakraborty (1984) Chakravarty and Chakraborty (1984)  Chakravarty and Mukherjee (1999b) Chakravarty and Mukherjee (1999b) Chakravarty and Mukherjee (1999b) As with poverty consider relative as well as absolute indices… As with poverty consider relative as well as absolute indices…

Frank Cowell: TU Lisbon – Inequality & Poverty Aggregate deprivation (2) An ethically weighted relative index An ethically weighted relative index  Chakravarty and Mukherjee (1999a) Chakravarty and Mukherjee (1999a) Chakravarty and Mukherjee (1999a) One based on the generalised-Gini One based on the generalised-Gini  Duclos and Grégoire (2002) Duclos and Grégoire (2002) Duclos and Grégoire (2002)

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Introduction Deprivation Complaints Deprivation, complaints, inequality Reference groups and distributional judgments Model Inequality results Rankings and welfare

Frank Cowell: TU Lisbon – Inequality & Poverty The Temkin approach Larry Temkin (1986, 1993) approach to inequality Larry Temkin (1986, 1993) approach to inequality  Unconventional  Not based on utilitarian welfare economics  But not a complete “outlier” Common ground with other distributional analysis Common ground with other distributional analysis  Poverty  deprivation Contains the following elements: Contains the following elements:  Concept of a complaint  The idea of a reference group  A method of aggregation

Frank Cowell: TU Lisbon – Inequality & Poverty What is a “complaint?” Individual’s relationship with the income distribution Individual’s relationship with the income distribution The complaint exists independently The complaint exists independently  does not depend on how people feel  does not invoke “utility” or (dis)satisfaction Requires a reference group Requires a reference group  effectively a reference income  a variety of specifications  see also Devooght (2003) Devooght (2003)Devooght (2003)

Frank Cowell: TU Lisbon – Inequality & Poverty Types of reference point BOP BOP  The Best-Off Person  Possible ambiguity if there is more than one  By extension could consider the best-off group AVE AVE  The AVErage income  Obvious tie-in with conventional inequality measures  A conceptual difficulty for those above the mean? ATBO ATBO  All Those Better Off  A “conditional” reference point

Frank Cowell: TU Lisbon – Inequality & Poverty Aggregation The complaint is an individual phenomenon. The complaint is an individual phenomenon. How to make the transition from this to society as a whole? How to make the transition from this to society as a whole? Temkin makes two suggestions: Temkin makes two suggestions: Simple sum Simple sum  Just add up the complaints Weighted sum Weighted sum  Introduce distributional weights  Then sum the weighted complaints

Frank Cowell: TU Lisbon – Inequality & Poverty The BOP Complaint Let r(x) be the first richest person you find in N. Let r(x) be the first richest person you find in N. Person r (and higher) has income x n. Person r (and higher) has income x n. For “lower” persons, natural definition of complaint: For “lower” persons, natural definition of complaint: Similar to fundamental difference for poverty: Similar to fundamental difference for poverty: Now we replace “p” with “r” Now we replace “p” with “r”

Frank Cowell: TU Lisbon – Inequality & Poverty BOP-Complaint: Axiomatisation Use same structural axioms as before. Plus… Use same structural axioms as before. Plus… Monotonicity: income increments reduce complaint Monotonicity: income increments reduce complaint Independence Independence Normalisation Normalisation

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Introduction Deprivation Complaints Deprivation, complaints, inequality A new approach to inequality Model Inequality results Rankings and welfare

Frank Cowell: TU Lisbon – Inequality & Poverty Implications for inequality Broadly two types of axioms with different roles. Broadly two types of axioms with different roles. Axioms on structure: Axioms on structure:  use these to determine the “shape” of the measures. Transfer principles and properties of measures: Transfer principles and properties of measures:  use these to characterise ethical nature of measures

Frank Cowell: TU Lisbon – Inequality & Poverty A BOP-complaint class The Cowell-Ebert (SCW 2004) result The Cowell-Ebert (SCW 2004) result Similarity of form to FGT Similarity of form to FGT Characterises a family of distributions … Characterises a family of distributions …

Frank Cowell: TU Lisbon – Inequality & Poverty The transfer principle Do BOP-complaint measures satisfy the transfer principle? Do BOP-complaint measures satisfy the transfer principle?  If transfer is from richest, yes  But if transfers are amongst hoi polloi, maybe not Cowell-Ebert (SCW 2004): Cowell-Ebert (SCW 2004): Look at some examples that satisfy this Look at some examples that satisfy this

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours To examine the properties of the derived indices… To examine the properties of the derived indices… …take the case n = 3 …take the case n = 3 Draw contours of T  –inequality Draw contours of T  –inequality Note that both the sensitivity parameter  and the weights w are of interest… Note that both the sensitivity parameter  and the weights w are of interest…

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours (  =2) w 1 =0.5 w 2 =0.5 Now change the weights…

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours (  =2) w 1 =0.75 w 2 =0.25

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours (  = 1) w 1 =0.75 w 2 =0.25

Frank Cowell: TU Lisbon – Inequality & Poverty By contrast: Gini contours

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours (  = 0) w 1 =0.5 w 2 =0.5 Again change the weights… Again change the weights…

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours (  = –1) w 1 =0.75 w 2 =0.25

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours (  = –1) w 1 =0.5 w 2 =0.5

Frank Cowell: TU Lisbon – Inequality & Poverty Special cases If    then inequality just becomes the range, x n – x 1. If    then inequality just becomes the range, x n – x 1. If   –  then inequality just becomes the “upper- middle class” complaint: x n –x n-1. If   –  then inequality just becomes the “upper- middle class” complaint: x n –x n-1. If  = 1 then inequality becomes a generalised absolute Gini. If  = 1 then inequality becomes a generalised absolute Gini. “triangles” “Y-shapes” Hexagons

Frank Cowell: TU Lisbon – Inequality & Poverty Which is more unequal? A B

Frank Cowell: TU Lisbon – Inequality & Poverty Focus on one type of BOP complaint A B

Frank Cowell: TU Lisbon – Inequality & Poverty Orthodox approach A B

Frank Cowell: TU Lisbon – Inequality & Poverty T  – inequality

Frank Cowell: TU Lisbon – Inequality & Poverty The “sequence” Temkin’s seminal contributions offer an intuitive approach to considering changes in inequality. Temkin’s seminal contributions offer an intuitive approach to considering changes in inequality. Take a simple model of a ladder with just two rungs. Take a simple model of a ladder with just two rungs. The rungs are fixed, but the numbers on them are not. The rungs are fixed, but the numbers on them are not. Initially everyone is on the upper rung. Initially everyone is on the upper rung. Then, one by one, people are transferred to the lower rung. Then, one by one, people are transferred to the lower rung.  Start with m = 0 on lower rung  Carry on until m = n on lower rung What happens to inequality? What happens to inequality?  Obviously zero at the two endpoints of the sequence  But in between?

Frank Cowell: TU Lisbon – Inequality & Poverty The “sequence” (2) For the case of T  –inequality we have For the case of T  –inequality we have This is increasing in m if  > 0 This is increasing in m if  > 0 For other cases there is a degenerate sequence in the same direction For other cases there is a degenerate sequence in the same direction

Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Introduction Deprivation Complaints Deprivation, complaints, inequality A replacement for the Lorenz order? Model Inequality results Rankings and welfare

Frank Cowell: TU Lisbon – Inequality & Poverty Rankings Move beyond simple inequality measures Move beyond simple inequality measures The notion of complaint can also be used to generate a ranking principle that can be applied quite generally. The notion of complaint can also be used to generate a ranking principle that can be applied quite generally. This is rather like the use of Lorenz curves to specify a Lorenz ordering that characterises inequality comparisons. This is rather like the use of Lorenz curves to specify a Lorenz ordering that characterises inequality comparisons. Also similar to poverty rankings with arbitrary poverty lines. Also similar to poverty rankings with arbitrary poverty lines.

Frank Cowell: TU Lisbon – Inequality & Poverty Cumulative complaints Define cumulative complaints Define cumulative complaints Gives the CCC Gives the CCC  cumulative-complaint contour  Just like TIP / Poverty profile Use this to get a ranking principle Use this to get a ranking principle i/n r(x) / n K(x)K(x)

Frank Cowell: TU Lisbon – Inequality & Poverty Complaint-ranking The class of BOP-complaint indices The class of BOP-complaint indices Define complaint ranking Define complaint ranking Like the generalised-Lorenz result Like the generalised-Lorenz result

Frank Cowell: TU Lisbon – Inequality & Poverty Social welfare again Temkin’s complaints approach to income distribution was to be viewed in terms of “better” or “worse” Temkin’s complaints approach to income distribution was to be viewed in terms of “better” or “worse” Not just “less” or “more” inequality. Not just “less” or “more” inequality. Can incorporate the complaint-inequality index in a welfare-economic framework: Can incorporate the complaint-inequality index in a welfare-economic framework: Linear approximation: Linear approximation: Total income Inequality

Frank Cowell: TU Lisbon – Inequality & Poverty Welfare contours (φ=1) A’s income B’s income

Frank Cowell: TU Lisbon – Inequality & Poverty Welfare contours (φ<1) A’s income B’s income

Frank Cowell: TU Lisbon – Inequality & Poverty Welfare contours (φ>1) A’s income B’s income Meade’s “superegalitarianism”

Frank Cowell: TU Lisbon – Inequality & Poverty The ATBO Complaint Again, a natural definition of complaint: Again, a natural definition of complaint: Similar to fundamental difference for deprivation: Similar to fundamental difference for deprivation: Use this complaint in the Temkin class Use this complaint in the Temkin class Get a form similar to Chakravarty deprivation Get a form similar to Chakravarty deprivation

Frank Cowell: TU Lisbon – Inequality & Poverty Summary: complaints “Complaints” provide a useful basis for inequality analysis. “Complaints” provide a useful basis for inequality analysis. Intuitive links with poverty and deprivation as well as conventional inequality. Intuitive links with poverty and deprivation as well as conventional inequality. BOP extension provides an implementable inequality measure. BOP extension provides an implementable inequality measure. CCCs provide an implementable ranking principle CCCs provide an implementable ranking principle

Frank Cowell: TU Lisbon – Inequality & Poverty References (1) Bossert, W. and M. Fleurbaey (1996) “Redistribution and compensation,” Social Choice and Welfare, 13, Bossert, W. and C. D’Ambrosio (2004) “Reference groups and individual deprivation”. Working Paper , Département de sciences économiques, Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal (Québec) H3C 3J7, Canada. Bossert, W. and C. D’Ambrosio (2004) “Reference groups and individual deprivation”. Working Paper , Département de sciences économiques, Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal (Québec) H3C 3J7, Canada. Bossert, W. and C. D’Ambrosio (2004) Bossert, W. and C. D’Ambrosio (2004) Chakravarty, S. R. and A. B. Chakraborty (1984) “On indices of relative deprivation,” Economics Letters, 14, Chakravarty, S. R. and A. B. Chakraborty (1984) Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and their meaning in terms of social satisfaction.” Theory and Decision 47, Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and their meaning in terms of social satisfaction.” Theory and Decision 47, Chakravarty, S. R. and D. Mukherjee (1999a) Chakravarty, S. R. and D. Mukherjee (1999a) Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by deprivation orderings,” Social Indicators Research 46, Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by deprivation orderings,” Social Indicators Research 46, Chakravarty, S. R. and D. Mukherjee (1999b) Chakravarty, S. R. and D. Mukherjee (1999b) Cowell, F. A. (1985) “The measurement of distributional change: an axiomatic approach.” Review of Economic Studies, 52, Cowell, F. A. (1985) Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” Distributional Analysis Discussion Paper, 84, STICERD, LSE, Houghton St., London, WC2A 2AE. Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” Distributional Analysis Discussion Paper, 84, STICERD, LSE, Houghton St., London, WC2A 2AE. Cowell, F. A. (2005) Cowell, F. A. (2005) Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice and Welfare 23, Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice and Welfare 23, Cowell, F. A. and U. Ebert (2004) Cowell, F. A. and U. Ebert (2004) Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and empirics,” Economics and Philosophy, 19, , Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and empirics,” Economics and Philosophy, 19, , Devooght, K. (2003) Devooght, K. (2003) Devooght, K. (2005) “To each the same and to each his own. A proposal to measure responsibility-sensitive income inequality,” Working paper, University of Kortrijk. Devooght, K. (2005)

Frank Cowell: TU Lisbon – Inequality & Poverty References (2) Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and the measurement of poverty,” Review of Income and Wealth 48, Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and the measurement of poverty,” Review of Income and Wealth 48, Duclos, J.-Y. and P. Grégoire (2002) Duclos, J.-Y. and P. Grégoire (2002) Dworkin, R. (1981a) “What is equality? Part I: Equality of welfare.” Philosophy and Public Affairs, 10, Dworkin, R. (1981b) “What is equality? Part I: Equality of resources.” Philosophy and Public Affairs, 10, Dutta, B. and D. Ray (1989) “A concept of egalitarianism under participation constraints” Econometrica, 57, Dutta, B. and D. Ray (1989) Fleurbaey, M. (1995a) “Equal opportunity or equal social outcome?” Economics and Philosophy 11, Fleurbaey, M. (1995b) “Equality and responsibility,” European Economic Review, 39, Fleurbaey, M. (1995b) Fleurbaey, M. (1995c) “Three solutions to the compensation problem,” Journal of Economic Theory, 65, Fleurbaey, M. (1995c)

Frank Cowell: TU Lisbon – Inequality & Poverty References (3) Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s index of individual deprivation. Economics Letters 68, Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s index of individual deprivation. Economics Letters 68, Ebert, U. and P. Moyes (2000) Ebert, U. and P. Moyes (2000) Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-Greer- Thorbecke poverty orderings,” Journal of Public Economic Theory 4, Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-Greer- Thorbecke poverty orderings,” Journal of Public Economic Theory 4, Ebert, U. and P. Moyes (2002) Ebert, U. and P. Moyes (2002) Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, Foster, J. E., Greer, J. and Thorbecke, E. (1984) Foster, J. E., Greer, J. and Thorbecke, E. (1984) Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis of UK poverty trends,” Oxford Economic Papers, 49, Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis of UK poverty trends,” Oxford Economic Papers, 49, Jenkins, S. P. and Lambert, P. J. (1997) Jenkins, S. P. and Lambert, P. J. (1997) 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) Temkin, L. S. (1986) “Inequality.” Philosophy and Public Affairs 15, Temkin, L. S. (1986) “Inequality.” Philosophy and Public Affairs 15, Temkin, L. S. (1986) Temkin, L. S. (1986) Temkin, L. S. (1993) Inequality. Oxford: Oxford University Press. Temkin, L. S. (1993) Inequality. Oxford: Oxford University Press. Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly Journal of Economics 93, Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly Journal of Economics 93, Yitzhaki, S. (1979) Yitzhaki, S. (1979)