Introduction to Unidimensional Poverty and Inequality Measures

Instructor: María Emma Santos, Research Officer OPHI

Class Objectives:

  • The Lorenz Curve and Lorenz Dominance. Atkinson’s theorem.
  • The Generalized Lorenz Curve and Generalized Lorenz Dominance. Generalization ofAtkinson’s theorem.
  • The four basic properties of unidimensional inequality measures. Lorenz consistent inequality measures.
  • Other properties (Transfer Sensitivity, Consistency and Decomposability).
  • Unidimensional inequality measures and their properties.
  • Conceptual differences between the measurement of inequality and the measurement of poverty.
  • The two steps in poverty measurement: identification and aggregation.
  • Properties of unidimensional poverty measures.
  • Unidimensional poverty measures and their properties


Presentation on Inequality
Presentation on Poverty

Problem Set:
Problem Set

Answer Keys:
Answer Key 1 (pdf)
Answer Key 2 (.do)

Reading List:
Basic Readings:
FIELDS, G. (2001): Distribution and Development. New York: Russell Sage
Foundation and Cambridge, MA: MIT. Chapters 2 and 4.
FOSTER, J. (2008): “Inequality Measurement” in David Clark (ed.) The Elgar
Companion to Development Studies. Cheltenham: Edward Elgar.
FOSTER, J. E. (2006): “Poverty Indices” in A. de Janvry and R. Kanbur (eds),
Poverty, Inequality and Development: Essays in Honor to Erik Thorbecke.
New York: Springer Science.
RAY, D. (1998): Development Economics. New Jersey: Princeton. Chapters 6 and 8.

More Advanced Readings:

FOSTER, J. E. and A. Sen (1997): “On Economic Inequality: After a Quarter
Century”. Annex to the Expanded Edition of A. Sen. On Economic Inequality.
Oxford: Clarendon Press. Sections A.2 to A.6.

Further readings:
On Inequality:
ATKINSON, A. B. (1970): “On the Measurement of Inequality”, Journal of Economic Theory. 2: 244-263.
BOURGUIGNON, F. (1979): “Decomposable Income Inequality Measures”, Econometrica. 47: 901-20.
COWELL, F. A. (1988): “Inequality Decomposition – Three Bad Measures”, Bulletin of Economic Research. 40: 309-312
COWELL, F. A. (2000): “Measurement of Inequality”, in Anthony B. Atkinson and Francois Bourguignon. Handbook of Income Distribution, Volume 1. Amsterdam: North-Holland. (This is a survey)
DALTON, H. (1920): “The Measurement of Inequality of Incomes”, Economic Journal. 30(119): 348-61.
FOSTER, J. and A. SHNEYEROV. (1999): “A General Class of Additively Decomposable Inequality Measures”, Economic Theory. 14: 89-111.
FOSTER, J. and A. SHNEYEROV. (2000): “Path Independent Inequality Measures”, Journal of Economic Theory. 91: 199-222.
FOSTER, J. E. (1985): “Inequality Measurement” in H. Peyton Young (ed.) Fair Allocation. Providence, RI: American Mathematical Society. (It provided the first proof that “Lorenz consistency” is equivalent to the four basic axioms of inequality measurement.)
FOSTER, J. E. and E. A. OK. (1999): “Lorenz Dominance and the Variance of Logarithms”, Econometrica. 67: 901-908.
FOSTER, J. E. (1983): “An Axiomatic Characterization of the Theil Measure of Income Inequality”, Journal of Economic Theory. 31: 105-121.
SHORROCKS, A. F. (1980): “The Class of Additively Decomposable Inequality Measures”, Econometrica. 48(3): 613-25.
SHORROCKS, A. F. (1982): “Inequality Decomposition by Factor Components”, Econometrica. 50(1): 193-211.
SHORROCKS, A. F. (1983): “Ranking Income Distributions”, Economica. 50(197): 3-17.
SHORROCKS, A. F. (1988): “Aggregation Issues in Inequality Measurement” in W. Eichorn (ed.) Measurement in Economics. New York: Physica-Verlag.
THEIL, H. (1967) Economics and Information Theory. Amsterdam: North-Holland. 9
WEYMARK, J. A. (1981): “Generalized Gini Inequality Indices”, Mathematical Social Sciences. 1: 409-30.

On Poverty:
ATKINSON, A. B. (1987): “On the Measurement of Poverty”, Econometrica. 55: 749-764.
CLARK, S., HEMMING, R., and D. ULPH. (1981): “On Indices for the Measurement of Poverty”, Economic Journal. 91: 515-26.
FOSTER, J. E. (1998) “Absolute versus Relative Poverty” American Economic Review, Papers and Proceedings. 88: 335-341.
FOSTER, J. E. and A. SHORROCKS. (1988): “Poverty Orderings and Welfare Dominance”, Social Choice and Welfare. 5: 179-198.
FOSTER, J. E. and A. SHORROCKS. (1991): “Subgroup Consistent Poverty Indices”, Econometrica. 59: 687-709.
FOSTER, J.E., GREER, J., and E. THORBECKE. (1984): “A Class of Decomposable Poverty Indices”, Econometrica. 52: 761-6.
RAVALLION, M. (1996): “Issues in Measuring and Modelling Poverty”, Economic Journal. 106: 1328-1343.
RAVALLION, M. (1992): “Poverty Comparisons, A Guide to Concepts and Methods”, Living Standards Measurement Study, Working Paper 88, World Bank, Washington D.C.
SEN, A. (1976): “Poverty: An Ordinal Approach to Measurement”, Econometrica 44: 219-231.

Suggestions on applied research on the topic:
There is a huge literature of empirical application of unidimensional inequality and poverty measures to different countries and regions.
The World Bank has produced a lot of applied research on these topics. The most common measures such as the FGT Indices as well as the Gini Coefficient are periodically estimated and released through the World Development Indicators. There is also a big amount of World Bank Policy Research Working Papers done on specific countries. Two recent World Bank Development Reports are worth noting: the World Development Report 2000/2001, Attacking Poverty: Opportunities, Empowerment and Security and the World Development Report 2006: Equity and Development. For all these information please visit the World Bank website. These documents can be found either on ‘Data and Research’ or on ‘Publications’.

Examples of applications of inequality and poverty measurement to specific cases include:

ARIMAH, B. (2004): “Poverty Reduction and Human Development in Africa”, Journal of Human Development. 5: 399-415.
BOROOAH, V. K., GUSTAFSSON, B. and SHI, L. (2006): “China and India: Income Inequality and Poverty North and South of the Hymalayas”, Journal of Asian Economics. 17: 797-817.
DEATON, A. and DREZE, J. (2002): “Poverty and Inequality in India: A Re- Examination”, Economic and Political Weekly. September 7, 2002. 
DERCON, S. and KRISHNAN, P. (1998): “Changes in Poverty in Rural Ethiopia 1989-1995: Measurement, Robustness Tests and Decomposition”, Working Paper 98-7, Centre for the Study of African Economies, Institute of Economics and Statistics, University of Oxford. 
OZLER, B. (2007): “Not separate, Not Equal: Poverty and Inequality in Post- Apartheid South Africa”, Economic Development and Cultural Change. 55: 487-529.
SZEKELY, M. and HILGERT, M. (1999): “What’s Behind the Inequality We Measure: An Investigation Using Latin American Data”, Research Working Paper 409, Inter American Development Bank (IADB).
SZÉKELY, M., LUSTIG, N., MEJÍA, C., José, A. and M. CUMPA. (2000): “Do We Know How Much Poverty There Is?”, Research Working Paper No. 437, Inter American Development Bank (IADB). 

Interesting Softwares:
You can estimate poverty and inequality measures and other distributional analysis tools using Stata by writing your own do-files (short programs that allow you to perform the estimations you choose). We will be practicing this during the summer school. There are also ado files that you can download and run as commands in Stata. These are programs that other people have written to compute different things and made available on line. However, there are other useful software tools that are worth mentioning: You can replicate the poverty estimates with an interactive computational tool available at the World Bank website called Pvocal.  This tool only works with macro data.
For estimations using micro-data, there are two -free downloadable- packages: DAD and DASP. Both are specifically designed for distributional analysis. They allow you to compute the most popular indices and curves for the analysis of poverty, inequality and social welfare. DASP has some advantages over DAD in terms of the maximal number of variables it can deal with and in supporting missing values.

Additional useful explanatory references are:

ABDELKRIM, A. (2006) “DASP, Distributive Analysis Stata Package” (PPT)
ZHANG, Q. (2003) “DAD, an Innovative Tool for Income Distribution Analysis”, Journal of Economic Inequality. 1: 281–284.