# An Alternative Functional Form for the Lorenz Curve with Empirical Applications

## Abstract

Given that the Lorenz curve is widely used for analyzing income distribution and inequality, this study introduces an alternative functional form for the Lorenz curve that is constructed based on the weighted average of the exponential function and the functional form implied by Pareto distribution. Using the data on the Gini index and the decile income shares of Thailand and other 4 countries with different income inequality, socioeconomic, and regional backgrounds, this study shows that the alternative functional form meets required criteria for a good functional form suggested by Dagum (1977). Moreover, this study compares the performance of the alternative functional form to that of Kakwani (1980). The results show that the performance of the alternative functional form is comparable to that of Kakwani (1980). However, the alternative functional form has an advantage in that the Gini index can be conveniently computed since it has an explicit mathematical solution whereas, for the Kakwani (1980)’s functional form, the Gini index is computed by using the numerical integration since its closed-form expression does not exist. Furthermore, this study finds that when the values of cumulative normalized rank of income are low, the Kakwani (1980)’s functional form does not always satisfy the monotonic increasing condition for the Lorenz curve. Thus, when applying any functional form for the Lorenz curve to analyze and formulate policy at the lower tail of income distribution, the shape of the estimated Lorenz curve should be considered together with the values of goodness-of-fit statistics and the estimated Gini index.

## References

Abraham, R. G., van den Bergh, S., & Nair, P. (2003). A new approach to galaxy morphology. I. Analysis of the Sloan Digital Sky Survey early data release. The Astrophysical Journal, 588(1), 218-229.

Aggarwal, V. (1984). On optimum aggregation of income distribution data. Sankhyā: The Indian Journal of Statistics, Series B, 46(3), 343-355.

Basmann, R. L., Hayes, K., Slottje, D., & Johnson, J. (1990). A general functional form for approximating the Lorenz curve. Journal of Econometrics, 92, 727-744.

Bertoli-Barsotti, L., & Lando, T. (2019). How mean rank and mean size may determine the generalised Lorenz curve: With application to citation analysis. Journal of Informetrics, 13, 387-396.

Chakrabarti, B. K., Chakraborti, A., Chakravarty, S. R., & Chatterjee, A. (2013). Econophysics of income and wealth distributions. New York: Cambridge University Press.

Cheong, K. S. (2002). An empirical comparison of alternative functional forms for the Lorenz curve. Applied Economics Letters, 9, 171-176.

Chotikapanich, D. (1993). A comparison of alternative functional forms for the Lorenz curve. Economics Letters, 41, 129-138.

Coulter, P. B. (1989). Measuring inequality: A methodological handbook. Boulder: Westview Press.

Cromley, G. A. (2019). Measuring differential access to facilities between population groups using spatial Lorenz curves and related indices. Transactions in GIS, 23(6), 1332-1351.

Dagum, C. (1977). A new model of personal income distribution: Specification and estimation. In D. Chotikapanich (Ed.), Modeling income distributions and Lorenz curves. Economic studies in equality, social exclusion and well-being, vol 5. (pp. 3-25). New York: Springer.

Das, A. K. (2014). Quantifying photovoltaic power variability using Lorenz curve. Journal of Renewable and Sustainable Energy, 6, 033124.

Delbosc, A., & Currie, G. (2011). Using Lorenz curves to access public transport equity. Journal of Transport Geography, 19, 1252-1259.

Eliazar, I. I. (2018). A tour of inequality. Annals of Physics, 389, 306-332.

Eliazar, I. I., & Sokolov, I. M. (2012). Measuring statistical evenness: A panoramic overview. Physica A: Statistical Mechanics and its Applications, 391, 1323-1353.

Fellman, J. (2018). Income inequality measures. Theoretical Economics Letters, 8, 557-574.

Graczyk, P. P. (2007). Gini coefficient: A new way to express selectivity of kinase inhibitors against a family of kinases. Journal of Medicinal Chemistry, 50(23), 5773-5779.

Gupta, M. R. (1984). Functional form for estimating the Lorenz curve. Econometrica, 52, 1313-1314.

Helene, O. (2010). Fitting Lorenz curves. Economics Letters, 108, 153-155.

Jordá V., Sarabia J. M., & Jäntti, M. (2021). Inequality measurement with grouped data: Parametric and non-parametric methods. Journal of the Royal Statistical Society: Statistics in Society, Series A, 184, 964-984.

Kakwani, N. C. (1980). On a class of poverty measures. Econometrica, 48, 437-446.

Kakwani N. C., & Podder N. (1973). On the estimation of Lorenz curves from grouped observations. International Economic Review, 14, 278-292.

Kakwani, N. C., & Podder, N. (1976). Efficient estimation of the Lorenz curve and associated inequality measures from grouped observations. Econometrica, 44, 137-148.

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association, 9(70), 209-219.

Ogwang, T., & Rao, U.L.G. (1996). A new functional form for approximating the Lorenz curve. Economics Letters, 52, 21-29.

Ogwang, T., & Rao, U.L.G. (2000). Hybrid models of the Lorenz curve. Economics Letters, 69, 39-44.

Ortega, P., Martín, G., Fernández, A., Ladoux, M., & García, A. (1991). A new functional form for estimating Lorenz curves. Review of Income and Wealth, 37, 47-452.

Paul, S., & Shankar, S. (2020). An alternative single parameter functional form for Lorenz curve. Empirical Economics, 59, 1393-1402.

Pavkova, K., Currie, G., Delbosc, A., & Sarvi, M. (2016). Selecting tram links for priority treatments – The Lorenz curve approach. Journal of Transport Geography, 55, 101-109.

Rao, U. L. G., & Tam, A. Y-P. (1987). An empirical study of selection and estimation of alternative models of the Lorenz curve. Journal of Applied Statistics, 14, 275-280.

Rasche, R. H., Gaffney, J. M., Koo, A. Y. C., & Obst, N. (1980). Functional forms for estimating the Lorenz curve. Econometrica, 48, 1061-1062.

Rohde, N. (2009) An alternative functional form for estimating the Lorenz curve. Economics Letters, 105, 61-63.

Ryu, H., & Slottje, D. (1996). Two flexible functional forms for approximating the Lorenz curve. Journal of Econometrics, 72, 251-274.

Sarabia, J. M. (1997). A hierarchy of Lorenz curves based on the generalized Tukey’s lambda distribution. Econometric Reviews, 16, 305-320.

Sarabia, J. M., Castillo, E., & Slottje, D. (1999). An ordered family of Lorenz curves. Journal of Econometrics, 91, 43-60.

Sarabia, J. M., Castillo, E., & Slottje, D. (2001). An exponential family of Lorenz curves. Southern Economic Journal, 67, 748-756.

Sarabia, J. M., & Pascual, M. (2002). A class of Lorenz curves based on linear exponential loss functions. Communications in Statistics – Theory and Methods, 31, 925-942.

Sarabia, J. M., Prieto, F., & Sarabia, M. (2010). Revisiting a functional form for the Lorenz curve. Economics Letters, 107, 249-252.

Sarabia, J. M., Prieto, F., & Jordá, V. (2015). About the hyperbolic Lorenz curve. Economics Letters, 136, 42-45.

Sarabia, J. M., Jordá, V., & Trueba, C. (2017). The lame class of Lorenz curves. Communications in Statistics – Theory and Methods, 46, 5311-5326.

Sazuka N., & Inoue, J. (2007). Fluctuations in time intervals of financial data from the view point of the Gini index. Physica A: Statistical Mechanics and its Applications, 383, 49-53.

Sitthiyot, T., Budsaratragoon, P., & Holasut, K. (2020). A scaling perspective on the distribution of executive compensation. Physica A: Statistical Mechanics and its Applications, 543, 123556.

Sitthiyot, T., & Holasut, K. (2020). A simple method for measuring inequality. Palgrave Communications, 6, 112.

Tanak, A. K., Mohtashami Borzadaran, G. R., & Ahmadi, J. (2018). New functional forms of Lorenz curves by maximizing Tsallis entropy of income share function under the constraint on generalized Gini index. Physica A: Statistical Mechanics and its Applications, 511, 280-288.

The Office of the National Economic and Social Development Council. (2021). Poverty and Income Distribution Statistics [dataset]. Retrieved from https://www.nesdc.go.th/main.php?filename=social

The United Nations University World Institute for Development Economics Research (UNU-WIDER). (2020). World Income Inequality Database (WIID), Version date: 6 May 2020 [dataset]. Retrieved from https://www4.wider.unu.edu/

Wang, Z., & Smyth, R. (2015). A hybrid method for creating Lorenz curves. Economics Letters, 133, 59-63.

Zhou, X., Yan, D., & Jiang, Y. (2015). Application of Lorenz curve and Gini index in the analysis of load feature in HVAC systems. Procedia Engineering, 121, 11-18.

## Downloads

## Published

## How to Cite

*Thailand and The World Economy*,

*41*(1), 106–125. Retrieved from https://so05.tci-thaijo.org/index.php/TER/article/view/263126

## Issue

## Section

## License

Copyright (c) 2023 Thailand and The World Economy

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.