Ordinal multidimensional inequality: theory and application to the 2x2 case

Sonne-Schmidt, Christoffer and Tarp, Finn and Peter, Lars (2011): Ordinal multidimensional inequality: theory and application to the 2x2 case. Published in:

Abstract

This paper develops a method for ordinal comparison of multidimensional inequality. In our model, one distribution is more unequal than another when the distributions have common arithmetic median and the first can be obtained from second by one or more elementary shifts in population density that increase inequality. For the benchmark 2x2 case (i.e. the case of two binary outcome variables), we derive an empirical approach to making ordinal inequality comparisons. As an illustration, we apply the model to childhood poverty in Mozambique.

• Item Type: MPRA Paper
• Language: English
• Keywords: Qualitative data, multidimensional first order dominance, multidimensional inequality, ordinal comparison
• Subjects: D - Microeconomics > D6 - Welfare Economics > D63 - Equity, Justice, Inequality, and Other Normative Criteria and Measurement; I - Health, Education, and Welfare > I3 - Welfare, Well-Being, and Poverty > I32 - Measurement and Analysis of Poverty; O - Economic Development, Innovation, Technological Change, and Growth > O1 - Economic Development > O15 - Human Resources ; Human Development ; Income Distribution ; Migration

References

Alkire S, 2002. Dimensions of Human Development. World Development 30, 2, 181—205.

Alkire S, Foster JE, 2008. Counting andMultidimensional Poverty Measurement. Oxford Poverty and Human Development Initiative (OPHI) Working Paper 7.

Alkire S, Santos ES, 2010. Acute Multidimensional Poverty: A New Index for Developing Countries, Oxford Poverty and Human Development Initiative (OPHI) Working Paper 38.

Allison RA, Foster JE, 2004. Measuring health inequality using qualitative data. Journal of Health Economics 23, 505-524.

Anderson G, 2008. The empirical assessment of multidimensional welfare, inequality and poverty: sample weighted multivariate generalizations of the Kolmogorov Smirnov two sample test for stochastic dominance. Journal of Economic Inequality 6, 73-87.

Arndt C, Distante R, Hussain MA, Østerdal LP, et al., 2011. Ordinal Welfare Comparisons with Multiple Discrete Indicators: A First Order Dominance Approach and Application to Child Poverty. Discussion Paper No. 11-13, Department of Economics, University of Copenhagen.

Apouey B, 2007.Measuring health polarization with self assessed health data. Health Economics 16, 875—894.

Atkinson AB, Bourguignon F, 1982. The comparison of multidimensioned distributions of economic status. Review of Economic Studies 12, 183-201.

Atkinson AB, 2003. Multidimensional deprivation: contrasting social welfare and counting approaches. Journal of Economic Inequality 1, 53-65.

Barnett V, 1976. The ordering of multivariate data. Journal of the Royal Statistical Society. Series A (General), 139, 318-355.

Bhattacharya B, Dykstra RL, 1994. Statistical Inference for Stochastic Ordering. In: M Shaked M, JG Shanthikumar (Eds.), Stochastic orders and their applications, Academic Press.

Boland PJ, Proschan F, 1988. Multivariate arrangement increasing functions with applications in probability and statistics. Journal of Multivariate Analysis, 25, 286-298.

Bourguignon F, 1989. Family size and social utility: Income distribution dominance criteria. Journal of Econometrics 42, 67-80.

Bourguignon F, Chakravarty S, 2003. The measurement of multidimensional poverty. Journal of Economic Inequality 1, 25-49.

Buvinic M, Gupta GR, 1997. Female-Headed Households and Female-Maintained Families: Are They Worth Targeting to Reduce Poverty in Developing Countries. Economic Development and Cultural Change, 45, 2, 259-280.

Crawford I, 2005. A nonparametric test of stochastic dominance in multivariate distributions. DP 12/05, Department of Economics, University of Surrey.

Davidson R, Duclos Jy, 2006. Testing for Restricted Stochastic Dominance. IZA Discussion Paper No. 2047.

Dardanoni V and Forcina A, 1999. Inference for Lorenz curve orderings. Econometrics Journal 2, 49-75.

DHS. Demographic and Health Survey. Macro International Inc. United States. Email: [e-mail]. Web: http://www.measuredhs.com.

Decancq K, 2011. Elementary multivariate rearrangements and stochastic dominance on a Fréchet class. Draft.

Duclos J-Y, Sahn DE, Younger SD, 2006. Robust multidimensional poverty comparisons. Economic Journal 116, 943-968.

Duclos J-Y, Sahn DE, Younger SD, 2007. Robust multidimensional poverty comparisons with discrete indicators of well-being. In: Inequality and poverty re examined. Editors: SP Jenkins, J Micklewright. Oxford University Press.

Duclos J-Y, Sahn DE, Younger SD, 2011. Partial multidimensional inequality orderings. Journal of Public Economics 95, 225-238.

Efron B, Tibshirani RJ, 1993. An Introduction to the Bootstrap. Chapman & Hall.

Epstein L, Tanny SM, 1980. Increasing generalized correlation: a definition and some economic consequences. Canadian Journal of Economics 13, 16-34.

Grusky D, Kanbur R. 2006. The Conceptual Foundations of Poverty and Inequality Measurement. In: D Grusky and R Kanbur (Eds.) Poverty and Inequality, Stanford University Press.

Gravel N, Moyes P, 2006. Ethically robust comparisons of distributions of two individual attributes. Working paper 0605, IDEP.

Gravel N, Moyes P, 2011. Bidimensional Inequalities with an Ordinal Variable. In: M Fleurbaey, M Salles and JA Weymark (Eds.). Social Ethics and Normative Economics, Social Ethics and Normative Economics: Essays in Honour of Serge-Christophe Kolm, Studies in Choice and Welfare, 101-127.

Gravel N, Moyes P, Tarroux B, 2009. Robust International Comparisons of Distributions of Disposable Income and Regional Public Goods. Economica 76, 432-461.

Gravel N, Mukhopadhyay A, 2010. Is India better off today than 15 years ago? A robust multidimensional answer. Journal of Economic Inequality 8, 173-195.

Erreygers G, 2006. Beyond the Health Concentration Index: An Atkinson Alternative for the Measurement of the Socioeconomic Inequality of Health. Working paper.

Hamada K, 1974. Comment on Hadar and Russel (1974). In M Balch, D McFadden and S Wu (Eds.), Essays on Economic Behavior under Uncertainty, p. 150-153, North-Holland, Amsterdam.

Haldane JBS, 1948. Note on the median of a multivariate distribution, Biometrica 35, 414-415.

Hayford Jf, 1902. What is the center of an area or the center of a population? Journal of American Statistical Organization 8, 47-58.

Handa S, 1996. Expenditure behavior and children’s welfare: An analysis of female headed households in Jamaica. Journal of Development Economics 50, 165-187.

Howes S, 1993. Inferring population rankings from sample data. A new test for inferring dominance from sample data. Discussion paper, STICERD, London School of Economics.

IAF, 2002. Inquérito aos Agregados Familiares (Household Survey). Web: http://www.ine.gov.mz/inqueritos_dir/iaf/.

INE. Instituto Nacional de Estatística (National Institut of Statistics). Maputo, Mozambique. Web: www.ine.gov.mz.

Kakwani NC, Wagstaff A, van Doorslaer E, 1997. Socioeconomic inequalities in health: measurement, computation and statistical inference. Journal of Econometrics 77, 87-104.

Kamae T, Krengel U, O’Brien GL, 1977. Stochastic inequalities on partially ordered spaces. The Annals of Probability 5, 899-912.

Kaur A, Prakasa Rao BLS.and Singh H, 1994. Testing for Second-Order Stochastic Dominance of Two Distributions. Econometric Theory 10, 849-866

Lambert PJ, 2001. The distribution and redistribution of income. Third edition. Manchester University Press, Manchester.

Lindelow M, 2006. Sometimes more equal than others: how health inequalities depend on the choice of welfare indicator. Health Economics 15, 263-279.

Lehmann EL, 1955. Ordered families of distributions. The Annals of Mathematical Statistics 26, 399-419.

Maasoumi E, 1999. Multidimensioned approaches to welfare analysis. In: J Silber (Ed.), Handbook of Income Inequality Measurement, Kluwer.

Madden D, 2010. Ordinal and cardinal measures of health inequality: an empirical comparison. Health Economics 19, 243—250.

McCaig B, Yatchew A, 2007. International welfare comparisons and nonparametric testing of multivariate stochastic dominance. Journal of Applied Econometrics 22, 951-969.

MISAU. Ministério da Saúde (Ministry of Health). Maputo, Mozambique. Web: http://www.misau.gov.mz/.

Müller A, Stoyan D, 2002. Comparison methods for stochastic models and risks. John Wiley and Sons, New York, NY.

Muller C, Trannoy A, 2011. A dominance approach to the appraisal of the distribution of well-being across countries. Journal of Public Economics 95, 239-246.

Naga RHA, Yalcin T, 2008. Inequality measurement for ordered response health data, Journal of Health Economics 27, 1614-1625.

Orazem PF, King OM, 2007. Schooling in Developing Countries: The Roles of Supply, Demand and Government Policy. In: Handbook of Development Economics, vol. 4. Elsevier Science B.V., Amsterdam.

Parker S, Rubalcava L, Teruel G, 2007. Evaluating Conditional Schooling and Health Programs. In: Handbook of Development Economics, vol. 4. Elsevier Science B.V., Amsterdam.

Quirk JD, Saposnik R, 1962. Admissibility and measurable utility functions. Review of Economic Studies 29, 140-146.

Robertson T, Wright F, Dykstra RL, 1988. Order Restricted Statistical Inference. John Wiley, New York, NY.

Savaglio E, 2006. Three Approaches to the Analysis of Multidimensional Inequality. In: Inequality and Economic Integration. Editors: F Farina, E Savaglio, Routledge.

Sen A, 1985. Commodities and Capabilities, Elsevier, Amsterdam.

Sen A, 1993. Capability and Well-being, in M. Nassbaum and A. Sen (eds), The Quality of Life, Clarendon Press, Oxford.

Sen A, 1997. On Economic Inequality. Expanded Edition with a substantial annexe by James E. Foster and Amartya Sen. Clarendon Press, Oxford.

Sen A, 2006. Conceptualizing and measuring poverty. In: D Grusky and R Kanbur (Eds.) Poverty and Inequality, Stanford University Press.

Shaked M, Shanthikumar JG, 2007. Stochastic Orders. Springer.

Small CG, 1990. A survey of multidimensional medians. International Statistical Review 38, 263-277.

Sonne-Schmidt C, Tarp F, Østerdal LP, 2008. Ordinal comparison of multidimensional deprivation: theory and application. Discussion Paper no. 08-33, Department of Economics, University of Copenhagen.

Strassen V, 1965. The existence of probability measures with given marginals. The Annals of Mathematical Statistics 36, 423—439.

Strauss J, Thomas D, 1995. Human resources: Empirical modeling of household and family decisions. In: Srinivasan, T.N., Behrman, J.R. (Eds.), Handbook of Development Economics, vol. 3A. North-Holland, Amsterdam, pp. 1883—2023.

Tchen A, 1980. Inequalities for distributions with given marginals. The Annals of Probability 8, 814-827.

Trannoy A, 2006. Multidimensional Egalitarianism and the dominance approach: a lost paradise? In: Inequality and Economic Integration, F Farina and E Savaglio (eds.). Routledge, London.

Tsui K, 1999. Multidimensional inequality and multidimensional generalized entropy measures: an axiomatic derivation. Social Choice and Welfare 16, 145-157.

UNICEF, 2006. Childhood Poverty in Mozambique: A Situation and Trends Analysis. United Nations, Mozambique. Available at: http://www.eldis.org/fulltext/childpoverty_Moz.pdf.

Wagstaff A, van Doorslaer E, Paci P, 1989. Equity in the finance and delivery of health care: some tentative cross-country comparisons. Oxford Review of Economic Policy 5, 89-112.

Wagstaff A, Paci P, van Doorslaer E, 1991. On the measurement of inequalities in health. Social Science and Medicine 33, 545-557.

Weymark JA, 2006, The normative approach to measurement of multidimensional inequality. In: Inequality and Economic Integration. Editors: F Farina, E Savaglio, Routledge.

Østerdal LP, 2010. The mass transfer approach to multivariate discrete first order stochastic dominance: direct proof and implications. Journal of Mathematical Economics 46, 1222-1228.