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Frank Cowell: HMRC-HMT Economics of Taxation 14 December 2011 HMRC-HMT Economics of Taxation 2011 http://darp.lse.ac.uk/HMRC-HMT 9.1 Distributional Analysis and Methods Frank Cowell: HMRC-HMT Economics of Taxation Distributional Analysis and Methods Overview... The basics How to represent problems in distributional analysis SWF and rankings Inequality measures Evidence 2 Frank Cowell: HMRC-HMT Economics of Taxation Distributional analysis Covers a broad class of economic problems Similar techniques inequality social welfare poverty rankings measures Four basic components need to be clarified “income” concept “income receiving unit” concept usually equivalised, disposable income but could be other income types, consumption, wealth… usually the individual but could also be family, household a distribution method of assessment or comparison See Cowell (2000, 2008, 2011), Sen and Foster (1997) 3 xj A representation with 3 incomes Income distributions with given total X Feasible income distributions given X Equal income distributions Janet's income Frank Cowell: HMRC-HMT Economics of Taxation 1: “Irene and Janet” approach xi + xj + xk = X particularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria 0 xi 4 Frank Cowell: HMRC-HMT Economics of Taxation 2: The parade Plot income against proportion of population xF(x) Parade in ascending order of "income" / height 1 Related to familiar statistical concept distribution function F(∙) F(x0) Pen (1971) especially useful in cases where it is appropriate to adopt a parametric model of income distribution x0.8 x0.2 0 x 0 proportion of the population 0.2 x0 q 0.8 1 5 Frank Cowell: HMRC-HMT Economics of Taxation Distributional Analysis and Methods Overview... The basics How to incorporate fundamental principles SWF and rankings Inequality measures Evidence 6 Frank Cowell: HMRC-HMT Economics of Taxation Social-welfare functions A standard approach to a method of assessment Basic tool is a social welfare function (SWF) Maps set of distributions into the real line I.e. for each distribution we get one specific number In Irene-Janet notation W = W(x) Properties will depend on economic principles Simple example of a SWF: Total income in the economy W = Si xi Perhaps not very interesting Consider principles on which SWF could be based 7 Frank Cowell: HMRC-HMT Economics of Taxation Another fundamental question What makes a “good” set of principles? There is no such thing as a “right” or “wrong” axiom. However axioms could be appropriate or inappropriate Use a simple framework to list some of the basic axioms Need some standard of “reasonableness” For example, how do people view income distribution comparisons? Assume a fixed population of size n. Assume that individual utility can be measured by x Income normalised by equivalence scales Rules out utility interdependence Welfare is just a function of the vector x := (x1, x2,…,xn ) Follow the approach of Amiel-Cowell (1999) Appendix A 8 Frank Cowell: HMRC-HMT Economics of Taxation SWF axioms Anonymity. Suppose x′ is a permutation of x. Then: W(x′) = W(x) Population principle. W(x) W(y) W(x,x,…,x) W(y,y,…,y) Decomposability. Suppose x' is formed by joining x with z and y' is formed by joining y with z. Then : W(x) W(y) W(x') W(y') Monotonicity. W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn) Transfer principle. (Dalton 1920) Suppose xi< xj then, for small : W(x1,x2..., xi+ ,..., xj ,..., xn) > W(x1,x2,..., xi,..., xn) Scale invariance. W(x) W(y) W(lx) W(ly) 9 Frank Cowell: HMRC-HMT Economics of Taxation Classes of SWFs Anonymity and population principle: Introduce decomposability can write SWF in either Irene-Janet form or F form may need to standardise for needs etc get class of Additive SWFs W : W(x) = Si u(xi) or equivalently W(F) = u(x) dF(x) If we impose monotonicity we get W1 W : u(•) increasing If we further impose the transfer principle we get W2 W1: u(•) increasing and concave 10 Frank Cowell: HMRC-HMT Economics of Taxation An important family Take the W2 subclass and impose scale invariance. Get the family of SWFs where u is iso-elastic: x 1–e – 1 u(x) = ————— , e 1–e has same form as CRRA utility function Parameter e captures society’s inequality aversion. Similar interpretation to individual risk aversion See Atkinson (1970) 11 Frank Cowell: HMRC-HMT Economics of Taxation Ranking and dominance Introduce two simple concepts first illustrate using the Irene-Janet representation take income vectors x and y for a given n First-order dominance: y[1] > x[1], y[2] > x[2], y[3] > x[3] Each ordered income in y larger than that in x Second-order dominance: y[1] > x[1], y[1]+y[2] > x[1]+x[2], y[1]+y[2] +…+ y[n] > x[1]+x[2] …+ x[n] Each cumulated income sum in y larger than that in x Need to generalise this a little represent distributions in F-form (anonymity, population principle) q: population proportion (0 ≤ q ≤ 1) F(x): proportion of population with incomes ≤ x m(F): mean of distribution F 12 Frank Cowell: HMRC-HMT Economics of Taxation 1st-Order approach Basic tool is the quantile, expressed as Q(F; q) := inf {x | F(x) q} = xq “smallest income such that cumulative frequency is at least as great as q” Use this to derive a number of intuitive concepts interquartile range, decile-ratios, semi-decile ratios graph of Q is Pen’s Parade Also to characterise the idea of 1st-order (quantile) dominance: “G quantile-dominates F” means: Illustrate using for every q, Q(G;q) Q(F;q), Parade: for some q, Q(G;q) > Q(F;q) A fundamental result: G quantile-dominates F iff W(G) > W(F) for all WW1 13 Frank Cowell: HMRC-HMT Economics of Taxation Parade and 1st-order dominance Plot quantiles against proportion of population Q(.; q) Parade for distribution F again Parade for distribution G G In this case G clearly quantile-dominates F But (as often happens) what if it doesn’t? Try second-order method F 0 q 1 14 Frank Cowell: HMRC-HMT Economics of Taxation 2nd-Order approach Basic tool is the income cumulant, expressed as C(F; q) := ∫ Q(F; q) x dF(x) “The sum of incomes in the Parade, up to and including position q” Use this to derive a number of intuitive concepts the “shares” ranking, Gini coefficient graph of C is the generalised Lorenz curve Also to characterise the idea of 2nd-order (cumulant) dominance: “G cumulant-dominates F” means: Illustrate using for every q, C(G;q) C(F;q), GLC: for some q, C(G;q) > C(F;q) A fundamental result (Shorrocks 1983): G cumulant-dominates F iff W(G) > W(F) for all WW2 15 Plot cumulations against proportion of population C(.; q) GLC for distribution F GLC for distribution G m(G) m(F) C(G; . ) C(F; . ) 0 q Intercept on vertical axis is at mean income cumulative income Frank Cowell: HMRC-HMT Economics of Taxation GLC and 2nd-order dominance 0 1 16 Frank Cowell: HMRC-HMT Economics of Taxation 2nd-Order approach (continued) A useful tool: the share of the proportion q of distribution F is L(F;q) := C(F;q) / m(F) “income cumulation at q divided by total income” Yields Lorenz dominance, or the “shares” ranking: “G Lorenz-dominates F” means: for every q, L(G;q) L(F;q), for some q, L(G;q) > L(F;q) Illustrate using Lorenz curve: Another fundamental result (Atkinson 1970): For given m, G Lorenz-dominates F iff W(G) > W(F) for all W W2 17 Frank Cowell: HMRC-HMT Economics of Taxation Lorenz curve and ranking Plot shares against proportion of population 1 Perfect equality Lorenz curve for distribution F 0.8 Lorenz curve for distribution G L(.; q) 0.6 L(G;.) 0.4 L(F;.) 0.2 In this case G clearly Lorenz-dominates F So F displays more inequality than G But (as often happens) what if it doesn’t? No clear statement about inequality (or welfare) is possible without further information 0 0 0.2 0.4 proportion of population 0.6 q 0.8 1 18 Frank Cowell: HMRC-HMT Economics of Taxation Distributional Analysis and Methods Overview... The basics Three ways of approaching an index SWF and rankings Inequality measures Evidence 19 Frank Cowell: HMRC-HMT Economics of Taxation 1: Intuitive inequality measures Perhaps borrow from other disciplines… A standard measure of spread… But maybe better to use a normalised version coefficient of variation Comparison between these two is instructive variance Same iso-inequality contours for a given m. Different behaviour as m alters Alternative intuition based on Lorenz approach Lorenz comparisons (2nd-order) may be indecisive problem is essentially one of aggregation of information so use the diagram to “force a solution” 20 Frank Cowell: HMRC-HMT Economics of Taxation 1 Gini coefficient Redraw Lorenz diagram A “natural” inequality measure…? L(.; q) normalised area above Lorenz curve can express this also in I-J terms q 0 1 Also (equivalently) represented as normalised difference between income pairs: In F-form: In Irene-Janet terms: 21 Frank Cowell: HMRC-HMT Economics of Taxation 2: SWF and inequality The Irene &Janet diagram A given distribution Distributions with same mean xj Contours of the SWF Construct an equal distribution E such that W(E) = W(F) Equally-Distributed Equivalent income Social waste from inequality contour: x values such that W(x) = const •E O x(F) m(F) •F xi Curvature of contour indicates society’s willingness to tolerate “efficiency loss” in pursuit of greater equality 22 Frank Cowell: HMRC-HMT Economics of Taxation Welfare-based inequality From the concept of social waste Atkinson (1970) suggested an inequality measure: x(F) I(F) = 1 – —— m(F) Atkinson further assumed additive SWF, W(F) = u(x) dF(x) isoelastic u So inequality takes the form 23 Frank Cowell: HMRC-HMT Economics of Taxation 3: “Distance” and inequality SWF route provides a coherent approach to inequality But do we need to use an approach via social welfare? it’s indirect maybe introduces unnecessary assumptions Alternative route: “distance” and inequality Can see inequality as a deviation from the norm norm in this case is perfect equality …but what distance concept to use? 24 Frank Cowell: HMRC-HMT Economics of Taxation Generalised Entropy measures Defines a class of inequality measures, given parameter a : GE class is rich. Some important special cases for a < 1 it is ordinally equivalent to Atkinson (a= 1 – e) a= 0: – log (x / m(F)) dF(x) (mean logarithmic deviation) a= 1: [ x / m(F)] log (x / m(F)) dF(x) (the Theil index) or a = 2 it is ordinally equivalent to (normalised) variance. Parameter a can be assigned any positive or negative value indicates sensitivity of each member of the class a large and positive gives a “top-sensitive” measure a negative gives a “bottom-sensitive” measure each agives a specific distance concept 25 Frank Cowell: HMRC-HMT Economics of Taxation Inequality contours Each a defines a set contours in the I-J diagram each related to a different concept of distance For example the Euclidian case other types a=.25 a= a=−.25 a=−1 a=2 26 Frank Cowell: HMRC-HMT Economics of Taxation Distributional Analysis and Methods Overview... The basics Attitudes and perceptions SWF and rankings Inequality measures Evidence 27 Frank Cowell: HMRC-HMT Economics of Taxation Views on distributions Do people make distributional comparisons in the same way as economists? Summarised from Amiel-Cowell (1999) examine proportion of responses in conformity with standard axioms both directly in terms of inequality and in terms of social welfare Anonymity Population Decomposability Monotonicity Transfers Scale indep. Inequality Num Verbal SWF Num Verbal 83% 58% 57% 35% 51% 66% 66% 58% 54% 47% - 54% 53% 37% 55% 33% - 72% 66% 40% 31% 47% 28 Frank Cowell: HMRC-HMT Economics of Taxation Inequality aversion Are people averse to inequality? evidence of both inequality and risk aversion (Carlsson et al 2005) risk-aversion may be used as proxy for inequality aversion? (Cowell and Gardiner 2000) What value for e? affected by way the question is put? (Pirttilä and Uusitalo 2010) evidence on risk aversion is mixed high values from survey evidence (Barsky et al 1997) much lower from savings analysis (Blundell et al 1994) from happiness studies 1.0 to 1.5 (Layard et al 2008) related to the extent of inequality in the country? (Lambert et al 2003) perhaps a value of around 0.7 – 2 is reasonable see also HM Treasury (2003) page 94 29 Frank Cowell: HMRC-HMT Economics of Taxation Conclusion Axiomatisation of welfare can be accomplished using just a few basic principles Ranking criteria can provide broad judgments But may be indecisive, so specific SWFs could be used What shape should they have? How do we specify them empirically? Several axioms survive scrutiny in experiment but Transfer Principle often rejected 30 Frank Cowell: HMRC-HMT Economics of Taxation References (1) Amiel, Y. and Cowell, F.A. (1999) Thinking about Inequality, Cambridge University Press Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, 244-263 Barsky, R. B., Juster, F. T., Kimball, M. S. and Shapiro, M. D. (1997) “Preference parameters and behavioral heterogeneity: An Experimental Approach in the Health and Retirement Survey,” Quarterly Journal of Economics, 112, 537-579 Blundell, R., Browning, M. and Meghir, C. (1994) “Consumer Demand and the LifeCycle Allocation of Household Expenditures,” Review of Economic Studies, 61, 57-80 Carlsson, F., Daruvala, D. and Johansson-Stenman, O. (2005) “Are people inequality averse or just risk averse?” Economica, 72, Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Chapter 2, 87166 * Cowell, F.A. (2008) “Inequality: measurement,” The New Palgrave, second edition * Cowell, F.A. (2011) Measuring Inequality, Oxford University Press Cowell, F.A. and Gardiner, K.A. (2000) “Welfare Weights”, OFT Economic Research Paper 202, Office of Fair Training, Salisbury Square, London 31 Frank Cowell: HMRC-HMT Economics of Taxation References (2) Dalton, H. (1920) “Measurement of the inequality of incomes,” The Economic Journal, 30, 348-361 HM Treasury (2003) The Green Book: Appraisal and Evaluation in Central Government (and Technical Annex), TSO, London Lambert, P. J., Millimet, D. L. and Slottje, D. J. (2003) “Inequality aversion and the natural rate of subjective inequality,” Journal of Public Economics, 87, 10611090. Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) “The marginal utility of income,” Journal of Public Economics, 92, 1846-1857. Pen, J. (1971) Income Distribution, Allen Lane, The Penguin Press, London Pirttilä, J. and Uusitalo, R. (2010) “A ‘Leaky Bucket’ in the Real World: Estimating Inequality Aversion using Survey Data,” Economica, 77, 60–76 Sen, A. K. and Foster, J. E. (1997) On Economic Inequality (Second ed.). Oxford: Clarendon Press. Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 32