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7 March 2006 Frank Cowell: EC513 Public Economics EC513 PhD Public Economics 2005/6 http://darp.lse.ac.uk/EC513.htm Deprivation, Complaints and Inequality Frank Cowell: EC513 Public Economics Overview... Deprivation, complaints, inequality Introduction Themes and methodology Poverty Deprivation Complaints Frank Cowell: EC513 Public Economics Purpose of lecture We will look at recent theoretical developments in distributional analysis Consider some linked themes alternative approaches to inequality related welfare concepts Use ideas from sociology and philosophy Focus on the way modern methodology is applied Frank Cowell: EC513 Public Economics Themes Cross-disciplinary concepts Income differences Reference incomes Formal methodology Frank Cowell: EC513 Public Economics Methodology Exploit common structure poverty deprivation complaints and inequality see Cowell (2005) Axiomatic method minimalist approach characterise structure introduce ethics Frank Cowell: EC513 Public Economics “Structural” axioms Take some social evaluation function F... Continuity Linear homogeneity Translation invariance Frank Cowell: EC513 Public Economics Structural axioms: illustration x2 x* D for n=3 An income distribution Perfect equality Contours of “Absolute” Gini Continuity Continuous approach to I = 0 Linear homogeneity Proportionate increase in I Translation invariance I constant • 1 0 • x1 x3 Frank Cowell: EC513 Public Economics Overview... Deprivation, complaints, inequality Introduction An alternative approach Poverty Deprivation Complaints Frank Cowell: EC513 Public Economics Poverty concepts Given poverty line z Headcount p(x,z)/n Poverty gap a reference point fundamental income difference Foster et al (1984) poverty index Cumulative poverty gap Frank Cowell: EC513 Public Economics TIP / Poverty profile G(x,z) Cumulative gaps versus population proportions Proportion of poor TIP curve TIP curves have same interpretation as GLC (Shorrocks 1983) i/n 0 p(x,z)/n TIP dominance implies unambiguously greater poverty Frank Cowell: EC513 Public Economics Poverty: Axiomatic approach Characterise an ordinal poverty index P(x ,z) See Ebert and Moyes (2002) Use some of the standard axioms we introduced for analysing social welfare Apply them to n+1 incomes – those of the n individuals and the poverty line Show that given just these axioms… …you are bound to get a certain type of poverty measure. Frank Cowell: EC513 Public Economics Poverty: The key axioms Adapt standard axioms from social welfare Strengthen two other axioms anonymity independence monotonicity income increments reduce poverty scale invariance translation invariance Also need continuity Plus a focus axiom Frank Cowell: EC513 Public Economics A closer look at the axioms Let D denote the set of ordered income vectors The focus axiom is Scale invariance now becomes Define the number of the poor as Independence means: Frank Cowell: EC513 Public Economics Ebert-Moyes (2002) Gives two types of FGT measures “relative” version “absolute” version Additivity follows from the independence axiom Frank Cowell: EC513 Public Economics Brief conclusion Poverty indexes can be constructed from scratch Use standard axioms Exploit the poverty line as a reference point Impose structure Frank Cowell: EC513 Public Economics Overview... Deprivation, complaints, inequality Introduction An economic interpretation of a sociological concept Poverty Deprivation Complaints Frank Cowell: EC513 Public Economics Individual deprivation The Yitzhaki (1979) definition Equivalent form In present notation Use the conditional mean Frank Cowell: EC513 Public Economics Deprivation: Axiomatic approach 1 The Better-than set for i Focus works like the poverty concept Frank Cowell: EC513 Public Economics Deprivation: Axiomatic approach 2 Normalisation Additivity works like the independence axiom Frank Cowell: EC513 Public Economics Bossert-D’Ambrosio (2004) This is just the Yitzhaki individual deprivation index There is an alternative axiomatisation Ebert-Moyes (Economics Letters 2000) Different structure of reference group Frank Cowell: EC513 Public Economics Aggregate deprivation Simple approach: just sum individual deprivation Could consider an ethically weighted variant Chakravarty and Chakraborty (1984) Chakravarty and Mukherjee (1999b) As with poverty consider relative as well as absolute indices… Frank Cowell: EC513 Public Economics Aggregate deprivation (2) An ethically weighted relative index Chakravarty and Mukherjee (1999a) One based on the generalised-Gini Duclos and Grégoire (2002) Frank Cowell: EC513 Public Economics Overview... Deprivation, complaints, inequality Introduction Reference groups and distributional judgments Poverty Deprivation Complaints •Model •Inequality results •Rankings and welfare Frank Cowell: EC513 Public Economics The Temkin approach 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 Poverty deprivation Contains the following elements: Concept of a complaint The idea of a reference group A method of aggregation Frank Cowell: EC513 Public Economics What is a “complaint?” Individual’s relationship with the income distribution The complaint exists independently does not depend on how people feel does not invoke “utility” or (dis)satisfaction Requires a reference group effectively a reference income a variety of specifications see also Devooght (2003) Frank Cowell: EC513 Public Economics Types of reference point BOP AVE The Best-Off Person Possible ambiguity if there is more than one By extension could consider the best-off group The AVErage income Obvious tie-in with conventional inequality measures A conceptual difficulty for those above the mean? ATBO All Those Better Off A “conditional” reference point Frank Cowell: EC513 Public Economics Aggregation The complaint is an individual phenomenon. How to make the transition from this to society as a whole? Temkin makes two suggestions: Simple sum Just add up the complaints Weighted sum Introduce distributional weights Then sum the weighted complaints Frank Cowell: EC513 Public Economics The BOP Complaint Let r(x) be the first richest person you find in N. Person r (and higher) has income xn. For “lower” persons, natural definition of complaint: Similar to fundamental difference for poverty: Now we replace “p” with “r” Frank Cowell: EC513 Public Economics BOP-Complaint: Axiomatisation Use same structural axioms as before. Plus… Monotonicity: income increments reduce complaint Independence Normalisation Frank Cowell: EC513 Public Economics Overview... Deprivation, complaints, inequality Introduction A new approach to inequality Poverty Deprivation Complaints •Model •Inequality results •Rankings and welfare Frank Cowell: EC513 Public Economics Implications for inequality Broadly two types of axioms with different roles. Axioms on structure: use these to determine the “shape” of the measures. Transfer principles and properties of measures: use these to characterise ethical nature of measures Frank Cowell: EC513 Public Economics A BOP-complaint class The Cowell-Ebert (SCW 2004) result Similarity of form to FGT Characterises a family of distributions … Frank Cowell: EC513 Public Economics 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): Look at some examples that satisfy this Frank Cowell: EC513 Public Economics Inequality contours To examine the properties of the derived indices… …take the case n = 3 Draw contours of T–inequality Note that both the sensitivity parameter and the weights w are of interest… Frank Cowell: EC513 Public Economics Inequality contours (=2) •Now change the weights… w1=0.5 w2=0.5 Frank Cowell: EC513 Public Economics Inequality contours (=2) w1=0.75 w2=0.25 Frank Cowell: EC513 Public Economics Inequality contours ( = 1) w1=0.75 w2=0.25 Frank Cowell: EC513 Public Economics By contrast: Gini contours Frank Cowell: EC513 Public Economics Inequality contours ( = 0) Again change the weights… w1=0.5 w2=0.5 Frank Cowell: EC513 Public Economics Inequality contours ( = –1) w1=0.75 w2=0.25 Frank Cowell: EC513 Public Economics Inequality contours ( = –1) w1=0.5 w2=0.5 Frank Cowell: EC513 Public Economics Special cases “triangles” If then inequality just becomes the range, xn– x1 . “Y-shapes” If – then inequality just becomes the “uppermiddle class” complaint: xn–xn-1 . If = 1 then inequality becomes a generalised absolute Gini. Hexagons 0 Frank Cowell: EC513 Public Economics 0 Which is more unequal? A 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 B 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 Frank Cowell: EC513 Public Economics 0 Focus on one type of BOP complaint A 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 B 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 Frank Cowell: EC513 Public Economics 0 Orthodox approach A 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 B 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 inequality Frank Cowell: EC513 Public Economics T – inequality 22 21 20 19 A: (2,5,9,20,30) B: (2,6,9,19,30) 18 17 16 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Frank Cowell: EC513 Public Economics The “sequence” Temkin’s seminal contributions offer an intuitive approach to considering changes in inequality. Take a simple model of a ladder with just two rungs. The rungs are fixed, but the numbers on them are not. Initially everyone is on the upper 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? Obviously zero at the two endpoints of the sequence But in between? Frank Cowell: EC513 Public Economics The “sequence” (2) For the case of T–inequality we have This is increasing in m if > 0 For other cases there is a degenerate sequence in the same direction Frank Cowell: EC513 Public Economics Overview... Deprivation, complaints, inequality Introduction A replacement for the Lorenz order? Poverty Deprivation Complaints •Model •Inequality results •Rankings and welfare Frank Cowell: EC513 Public Economics Rankings Move beyond simple inequality measures 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. Also similar to poverty rankings with arbitrary poverty lines. Frank Cowell: EC513 Public Economics Cumulative complaints Define cumulative complaints Gives the CCC cumulative-complaint contour Just like TIP / Poverty profile K(x) Use this to get a ranking principle i/n r(x) / n Frank Cowell: EC513 Public Economics Complaint-ranking The class of BOP-complaint indices Define complaint ranking Like the generalised-Lorenz result Frank Cowell: EC513 Public Economics Social welfare again Temkin’s complaints approach to income distribution was to be viewed in terms of “better” or “worse” Not just “less” or “more” inequality. Can incorporate the complaint-inequality index in a welfare-economic framework: Total income Inequality Linear approximation: B’s income Frank Cowell: EC513 Public Economics Welfare contours (φ=1) A’s income B’s income Frank Cowell: EC513 Public Economics Welfare contours (φ<1) A’s income B’s income Frank Cowell: EC513 Public Economics Welfare contours (φ>1) Meade’s “superegalitarianism” A’s income Frank Cowell: EC513 Public Economics The ATBO Complaint Again, a natural definition of complaint: Similar to fundamental difference for deprivation: Use this complaint in the Temkin class Get a form similar to Chakravarty deprivation Frank Cowell: EC513 Public Economics Summary: complaints “Complaints” provide a useful basis for inequality analysis. Intuitive links with poverty and deprivation as well as conventional inequality. BOP extension provides an implementable inequality measure. CCCs provide an implementable ranking principle Frank Cowell: EC513 Public Economics References (1) Bossert, W. and C. D’Ambrosio (2004) “Reference groups and individual deprivation”. Working Paper 2004-10, Département de sciences économiques, Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal (Québec) H3C 3J7, Canada. Chakravarty, S. R. and A. B. Chakraborty (1984) “On indices of relative deprivation,” Economics Letters, 14, 283-287 Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and their meaning in terms of social satisfaction.” Theory and Decision 47, 89-100 Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by deprivation orderings,” Social Indicators Research 46, 125-135.. Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” Distributional Analysis Discussion Paper, 84, STICERD, LSE, Houghton St., London, WC2A 2AE. Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice and Welfare 23, 71-89. Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and empirics,” Economics and Philosophy, 19, 241 - 263, Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and the measurement of poverty,” Review of Income and Wealth 48, 471-492. Frank Cowell: EC513 Public Economics References (2) Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s index of individual deprivation. Economics Letters 68, 263-270. Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-GreerThorbecke poverty orderings,” Journal of Public Economic Theory 4, 455473. Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, 761-776 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, 317-327. Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Temkin, L. S. (1986) “Inequality.” Philosophy and Public Affairs 15, 99-121. Temkin, L. S. (1993) Inequality. Oxford: Oxford University Press. Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly Journal of Economics 93, 321.324.