Measurement of the Bipolarization Events

Authors: Stefan V. Stefanescu

Abstract:

We intend to point out the differences which exist between the classical Gini concentration coefficient and a proposed bipolarization index defined for an arbitrary random variable which have a finite support. In fact Gini's index measures only the "poverty degree" for the individuals from a given population taking into consideration their wages. The Gini coefficient is not so sensitive to the significant income variations in the "rich people class" . In practice there are multiple interdependent relations between the pauperization and the socio-economical polarization phenomena. The presence of a strong pauperization aspect inside the population induces often a polarization effect in this society. But the pauperization and the polarization phenomena are not identical. For this reason it isn't always adequate to use a Gini type coefficient, based on the Lorenz order, to estimate the bipolarization level of the individuals from the studied population. The present paper emphasizes these ideas by considering two families of random variables which have a linear or a triangular type distributions. In addition, the continuous variation, depending on the parameter "time" of the chosen distributions, could simulate a real dynamical evolution of the population.

Keywords: Bipolarization phenomenon, Gini coefficient, income distribution, poverty measure.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1076076

References:

[1] A. Agresti, An introduction to categorical data analysis. New York: Wiley Series in Probability and Statistics, 1996.
[2] C. D'Ambrosio, "Household characteristics and the distribution of income in Italy - An application of social distance measures," The Review of Income and Wealth, vol. 47, no. 1, pp. 43-64, 2001.
[3] A. B. Atkinson, "On the measurement of inequality." Journal of Economic Theory, no. 2, pp. 244-263, 1970.
[4] D. J. Bartolomew, The statistical approach to social measurement. London: Academic Press, 1996.
[5] N. Bhattacharya, and B. Mahalanobis, "Regional disparities in household consumption in India," Journal of the American Statistical Association, vol. 62, pp. 143-161, 1967.
[6] F. Bourguignon, "Decomposable income inequality measures," Econometrica, vol. 47, pp. 901-920, 1979.
[7] A. Chateauneuf, T. Gajdos, and P. H. Wilthien, "The principle of strong diminishing transfer," Journal of Economic Theory, vol. 103, pp. 311- 333, 2002.
[8] F. A. Cowell, "On the structure of additive inequality measures," Review of Economic Studies, vol. 47, pp. 521-531, 1980.
[9] F. A. Cowell, and S. P. Jenkins. "How much inequality can we explain ? A methodology and an application to the United States," The Economic Journal, vol. 105, pp. 421-430, 1995.
[10] J. B. Davies, and A. Shorrocks, "Optimal grouping of income and wealth data," Journal of Econometrics, vol. 42, pp. 97-108, 1989.
[11] J. Esteban, and D. Ray, "On the measurement of polarization," Econometrica, vol. 62 , no. 4, pp. 819-852, 1994.
[12] J. Esteban, and D. Ray, "Conflict and distribution," Journal of Economic Theory, vol. 87, pp. 379-415, 1999.
[13] J. Esteban, C. Gradin, and D. Ray, "Extension of a measure of polarization with application to the income distribution of five OECD countries," Maxwell School of Citizenship and Public Affairs, Syracuse University, New York, Working paper no. 218, November 1999.
[14] B. S. Everitt, S. Landau, and M. Leese, Cluster analysis. London: Arnold, 2001.
[15] J. Foster, and A. K. Sen, On economic inequality. Oxford: Clarendon Press, 1997.
[16] J. E. Foster, and A. A. Shneyerov, "Path independent inequality measures," Journal of Economic Theory, Oxford, vol. 91, no. 2, pp. 199- 222, 2000.
[17] M. Fournier, "Inequality decomposition by factor component - A new approach illustrated on the Taiwanese case," CERDI, Université d-Auvergne, November 1999.
[18] J. Gastwirth, "The estimation of a family of measures of economic inequality," Journal of Econometrics, no. 3, pp. 61-70, 1975.
[19] C. Gradin, "Polarization by sub-populations in Spain, 1973-1991," Review of Income and Wealth, vol. 46, no. 4, pp. 457-474, December 2000.
[20] P. E. Hart, "The comparative statistics and dynamics of income distributions," Journal of the Royal Statistical Society, vol. 139, pp. 108- 125, 1976.
[21] N. C. Kakwani, "Statistical inference in the measurement of poverty," Review of Economics and Statistics, vol. 75, no. 3, pp. 632-639, 1993.
[22] C. Kleiber, and S. Kotz, Statistical size distributions in economics and actuarial sciences. New Jersey: John Wiley & Sons - Interscience, 2003.
[23] J. O. Lanjouw, and P. Lanjouw, "How to compare apples and oranges - Poverty measurement based on different definitions of consumption," Review of Income and Wealth, vol. 47, no. 1, pp. 25-42, 2001.
[24] A. W. Pedersen, "Inequality as relative deprivation - Theoretical issues and implications for empirical research," preprint ISA RC19 Conference 2001, "Old a new social inequalities", University of Oviedo, September 2001.
[25] G. Pyatt, "On the interpretation and disaggregation of Gini coefficient," The Economic Journal, vol. 86, pp. 243-255, 1976.
[26] F. Schmid, "A general class of poverty measures," Statistical Papers, vol. 34, pp. 189-211, 1993.
[27] A. K. Sen, On economic inequality. Oxford: Oxford University Press, 1973.
[28] A. F. Shorrocks, "The class additively decomposable inequality measures," Econometrica, vol. 48, pp. 613-625, 1980.
[29] A. F. Shorrocks, "Inequality decomposition by population subgroups," Econometrica, vol. 52, pp. 1369-1385, 1984.
[30] P. Stefanescu, and S. Stefanescu, "Extending the Gini index to measure inequality and poverty," Economic Computation and Economic Cybernetics Studies and Research, vol. 35, no. 1-4, pp. 145- 154, 2001.
[31] P. Stefanescu, and S. Stefanescu, "Comparing Gini index with three other concentration measures," Economic Computation and Economic Cybernetics Studies and Research, vol. 36, no. 1-4, pp. 173-184, 2002.
[32] P. Stefanescu, and S. Stefanescu, "The polarization index for bounded exponential distributions," Economic Computation and Economic Cybernetics Studies and Research, vol. 40, no. 3-4, pp. 211-218, 2006.
[33] P. Stefanescu, and S. Stefanescu, "The properties of a polarization index for bounded exponential distributions," U. P. B. - Scientific Bulletin, Series A : Applied Mathematics and Physics, vol. 68, no. 4, pp. 9-20, 2006.
[34] P. Stefanescu, and S. Stefanescu, "The Monte Carlo estimation of a polarization index for an arbitrary distribution," Economic Computation and Economic Cybernetics Studies and Research, vol. 41, no. 3-4, pp. 181-192, 2007.
[35] S. Stefanescu, "Measuring the socio-economic bipolarization phenomenon," Romanian Journal of Economic Forecasting, vol. 9, no. 1, pp. 149-161, 2008.
[36] K. Y. Tsui, "Multidimensional inequality and multidimensional entropy measures - An axiomatic derivation," Social Choice and Welfare, vol. 16, no 1, pp. 145-157, 1999.
[37] B. Wilfling, "Lorenz ordering of power function order statistics," Statistics & Probability Letters, vol. 30, pp. 313-319, 1996.
[38] M. Wolfson, "When inequalities diverge," American Economic Review, vol. 84, no. 2, pp. 353-358, 1994.
[39] M. Wolfon, "Divergent inequalities - Theory and empirical results," The Review of Income and Wealth, vol. 43, no. 4, pp. 401-422, 1997.
[40] S. Yitzhaki, "Economic distance and overlapping of distributions," Journal of Econometrics, vol. 61, pp. 147-159, 1994.
[41] X. Zhang, and R. Kanbur, "What difference do polarization measures make ? - An application to China," Journal of Development Studies, vol. 37, pp. 85-98, 2001.
[42] C. Zoli, "Intersecting generalized Lorenz curves and the Gini index," Social Choice and Welfare, vol. 16, pp. 183-196, 1999.