# Parametric Lorenz Curves: Models and Applications

• José María Sarabia
Chapter
Part of the Economic Studies in Equality, Social Exclusion and Well-Being book series (EIAP, volume 5)

## Abstract

The Lorenz curve (LC) is an important instrument for analyzing the size of distribution of income or wealth and inequality. Finding an appropriate functional form is an important practical and theoretical problem. In this chapter we study parametric models for the LC and some important applications.

The basic properties that a function should satisfy in order to be a genuine LC are discussed. Next, we study the different ways for generating parametric families of LCs, as well as some of their basic properties, including their relationship with the underlying income distribution function. The basic parametric models proposed in the literature are studied, including the Pareto, lognormal and other important families of LCs.

Some general strategies to obtain extensions and generalizations of the basic parametric models are presented. One of the main applications of LCs is the study of inequality. We begin studying different measures of inequality together with their expressions in terms of the LC. These measures include the Gini index and some of their generalizations proposed by Kakwani (1980) and Yitzhaki (1983). Their corresponding expressions for the proposed parametric families of LCs will be obtained. The Lorenz ordering is also studied. The Lorenz ordering is a partial order that allows the comparison of two distributions when its corresponding LCs do not intersect. Some basic properties of this order are studied, including the effect of transformations, its relations with other partial orderings and their application to important parametric income distributions. The recent proposal of multivariate versions of the LC are studied. Finally, some applications of the Lorenz curve are presented.

## Keywords

Income Distribution Gini Index Pareto Distribution Stochastic Dominance Lorenz Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Aggarwal, V. (1984) On Optimal Aggregation of Income Distribution Data, Sankhya B, 46, 343-35.Google Scholar
2. Aggarwal, V. and R. Singh (1984) On Optimum Stratification with Proportional Allocation for a Class of Pareto Distributions, Communications in Statistics: Theory and Methods, 13, 3107-3116.
3. Aitchison, J. and J. A. C. Brown (1957) The Lognormal Distribution, Cambridge University Press, Cambridge.Google Scholar
4. Arnold, B. C. (1983) Pareto Distributions, International Cooperative Publishing House, Fairland, Maryland USA.Google Scholar
5. Arnold, B. C. (1986) A Class of Hyperbolic Lorenz Curves, Sankhya B, 48, 427-436.Google Scholar
6. Arnold, B. C. (1987) Majorization and the Lorenz Curve: A Brief Introduction, Lecture Notes in Statistics, 43, Springer-Verlag, Berlin.Google Scholar
7. Arnold, B. C. (2007) The Lorenz Curve: Evergreen after 100 Years, Advances in Income Inequality and Concentration Measures, Routledge, New York.Google Scholar
8. Arnold, B. C., C. A. Robertson, P. L. Brockett and B. Y. Shu (1987) Generating Ordered Families of Lorenz Curves by Strongly Unimodal Distributions, Journal of Business and Economic Statistics, 5, 305-308.
9. Atkinson, A. B. (1970) On the Measurement of Inequality, Journal of Economic Theory, 2, 244-263.
10. Basmann, R. L., K. L. Hayes, D. J. Slottje and J. D. Johnson (1990) A General Functional Form for Approximating the Lorenz Curve, Journal of Econometrics, 43, 77-90.
11. Beach, C. M. and R. Davidson (1983) Distribution-Free Statistical Inference with Lorenz Curves and Income Shares, Review of Economics and Statistics, 50, 723-735.
12. Bishop, J. A., S. Chakravorty and P. D. Thistle (1989) Asymptotically DistributionFree Statistical Inference for Generalized Lorenz Curves, Review of Economics and Statistics, 71, 725-727.
13. Burrell, Q. L. (2005) Symmetry and Other Transformation of Lorenz/Leimkuhler Representations of Informetric Data, Information Processing and Management, 41, 1317-1329.
14. Castillo, E., A. S. Hadi and J. M. Sarabia (1998) A Method for Estimating Lorenz Curves, Communications in Statistics: Theory and Methods, 27, 2037-2063.
15. Chotikapanich, D. (1993) A Comparison of Alternative Functional Forms for the Lorenz Curve, Economic Letters, 41, 129-138.
16. Cronin, D. C. (1979) A Function for the Size Distribution of Income: A Further Comment, Econometrica, 47, 773-774.
17. Dagum, C. (1977) A New Model for Personal Income Distribution: Specification and Estimation, Economie Appliqu ée, 30, 413-437.Google Scholar
18. Dasgupta, P., A. K. Sen and D. Starret (1973) Notes on the Measurement of Inequality, Journal of Economic Theory, 6, 180-187.
19. Davies, J. and M. Hoy (1994) The Normative Significance of Using Third-Degree Stochastic Dominance in Comparing Income Distributions, Journal of Economic Theory, 64, 520-530.
20. Davies, J. and M. Hoy (1995) Making Inequality Comparisons when Lorenz Curves Intersect, American Economic Review, 85, 980-986.Google Scholar
21. Davies, J. B., D. A. Green and H. J. Paarsch (1998) Economic Statistics and Social Welfare Comparisons. A Review., Handbook of Applied Economic Statistics, 1-38, Marcel Dekker, New York.Google Scholar
22. Dorfman, R. (1979) A Formula for the Gini Coefficient, Review of Economics and Statistics, 61, 146-149.
23. Fellman, J. (1976) The Effect of Transformations on the Lorenz Curve, Econometrica, 44, 823-824.
24. Gastwirth, J. L. (1971) A General Definition of the Lorenz Curve, Econometrica, 39, 1037-1039.
25. Gupta, M. R. (1984) Functional Form for Estimating the Lorenz Curve, Econometrica, 52, 1313-1314.
26. Hadar, J. and W. R. Russell (1969) Rules for Ordering Uncertain Prospects, American Economic Review, 59, 25-34.Google Scholar
27. Hanoch, G. and H. Levy (1969) The Efficiency Analysis of Choices Involving Risk, Review of Economic Studies, 36, 335-346.
28. Holm, J. (1993) Maximum Entropy Lorenz Curves, Journal of Econometrics, 44, 377-389.
29. Jacobson, A. A. D. M. and D. M. Kammen (2005) Letting the (Energy) Gini out of the Bottle: Lorenz Curves of Cumulative Electricity Consumption and Gini Coefficients as Metrics of Energy Distribution and Equity, Energy Policy, 33, 1825-1832.
30. Kakwani, N. (1980) On a class of poverty measures, Econometrica, 48, 437-446.
31. Kakwani, N. and N. Podder (1973) On estimation of lorenz curves from grouped observations, International Economic Review, 14, 278-292.
32. Kakwani, N. C. (1984) Welfare Ranking in Income Distribution, Innequality Measurement and Policy, vol. 3 of Advances in Econometrics, 191-215, JAI Press, Gleenwitch, Conn.Google Scholar
33. Kakwani, T. and N. Podder (1976) Efficient Estimation of the Lorenz Curve and Associated Inequality Measures from Grouped Observations, Econometrica, 44-1, 137-149.
34. Kleiber, C. (1996) Dagum vs. Singh-Maddala Income Distributions, Economics Letters, 53, 265-268.
35. Kleiber, C. (1999) On the Lorenz Order within Parametric Families of Income Distributions, Sankhya¯ B, 61, 514-517.Google Scholar
36. Kleiber, C. and S. Kotz (2003) Statistical Size Distributions in Economics and Actuarial Sciences, John Wiley, Hoboken, NJ.Google Scholar
37. Koshevoy, G. (1995) Multivariate Lorenz Majorization, Social Choice and Welfare, 12, 93-102.
38. Koshevoy, G. and K. Mosler (1996) The Lorenz Zonoid of a Multivariate Distribution, Journal of American Statistical Association, 91, 873-882.
39. Ma, Z., J. Shi, G. Wang and Z. He (2006) Temporal Changes in the Inequality of Early Growth of Gunninghamia Lanceolata (lamb.) Hook: A Novel Application of the Gini Coefficient and Lorenz Asymmetry., Genetica, 126, 343-663.
40. McDonald, J. B. (1984) Some Generalized Functions for the Size Distribution of Income, Econometrica, 52, 647-663.
41. Mehran, F. (1976) Linear Measures of Income Inequality, Econometrica, 44, 805-809.
42. Mosler, K. (2002) Multivariate Dispersion, Ccentral Regions and Depth. the Lift Zonoid Approach, Lecture Notes in Statistics, 165, Springer-Verlag, Berlin.Google Scholar
43. Moyes, P. (1987) A New Concept of Lorenz Domination, Economic Letters, 23, 203-207.
44. Muliere, P. and M. Scarsini (1989) A Note on Stochastic Dominance and Inequality Measures, Journal of Economic Theory, 49, 314-323.
45. Ogwang, T. and U. L. G. Rao (1996) A New Functional Form for Approximating the Lorenz Curve, Economic Letters, 52, 21-29.
46. Ogwang, T. and U. L. G. Rao (2000) Hybrid Models of the Lorenz Curve, Economic Letters, 69, 39-44.
47. Ortega, P., A. Martín, A. Fern ández, M. Ladoux and A. Garc á (1991) A New Func-tional Form for Estimating Lorenz Curves, Review of Income and Wealth, 37, 447-452.
48. Pakes, A. G. (1981) On Income Distributions and Their Lorenz Curves, Tech. rep., Department of Mathematics, University of Western Australia.Google Scholar
49. Rao, U. L. G. and A. Y. P. Tam (1987) An Empirical Study of Selection and Esti-mation of Alternative Models for the Lorenz Curve, Journal of Applied Statistics, 14, 275-280.
50. Rasche, R. H., J. Gaffney, A. Y. C. Koo and N. Obst (1980) Functional Forms for Estimating the Lorenz Curve, Econometrica, 48, 1061-1062.
51. Rothschild, M. and J. E. Stiglitz (1970) Increasing Risk: I. A Definition, Journal of Economic Theory, 2, 225-253.Google Scholar
52. Ryu, H. K. and D. J. Slottje (1996) Two Flexible Functional Form Approaches for Approximating the Lorenz Curve., Journal of Econometrics, 72, 251-274.
53. Ryu, H. K. and D. J. Slottje (1999) Handbook on Income Inequality Measurement, chap. Parametric Approximations of the Lorenz Curve, pp. 291-314, Kluwer, Boston.Google Scholar
54. Sadras, V. and R. Bongiovanni (2004) Use of Lorenz Curves and Gini Coefficients to Asses Yield Inequality within Paddocks, Field Crops Research, 90, 303-310.
55. Sarabia, J. M. (1997) A Hierarchy of Lorenz Curves Based on the Generalized Tukey’s Lambda Distribution, Econometric Reviews, 16, 305-320.
56. Sarabia, J. M. and E. Castillo (2005) About a Class of Max-Stable Families with Applications to Income Distributions, Metron, 63, 505-527.Google Scholar
57. Sarabia, J. M., E. Castillo, M. Pascual and M. Sarabia (2005) Mixture Lorenz Curves, Economics Letters, 89, 89-94.
58. Sarabia, J. M., E. Castillo and D. Slottje (2001) An Exponential Family of Lorenz Curves, Southern Economic Journal, 67, 748-756.
59. Sarabia, J. M., E. Castillo and D. J. Slottje (1999) An Ordered Family of Lorenz Curves, Journal of Econometrics, 91, 43-60.
60. Sarabia, J. M., E. Castillo and D. J. Slottje (2002) Lorenz Ordering between Mcdonalds Generalized Functions of the Income Size Distribution, Economics Letters, 75, 265-270.
61. Sarabia, J. M. and M. Pascual (2002) A Class of Lorenz Curves Based on Linear Exponential Loss Functions, Communications in Statistics: Theory and Methods, 31, 925-942.
62. Sen, A. K. (1976) Poverty: An Ordinal Approach to Measurement, Econometrica, 44, 219-231.
63. Shalit, H. and S. Yitzhaki (1984) Mean-Gini, Portfolio Theory and the Pricing of Risky Assets, Journal of Finance, 39, 1449-1468.
64. Shorrocks, A. F. (1983) Ranking Income Distributions, Economica, 50, 3-17.
65. Shorrocks, A. F. and J. E. Foster (1987) Transfer Sensitive inequality Measures, Review of Economic Studies, 54, 485-497.
66. Singh, S. K. and G. S. Maddala (1976) A Function for the Size Distribution of Incomes, Econometrica, 44, 963-970.
67. Slottje, D. J. (1990) Using Grouped Data for Constructing Inequality Indices: Para-metric vs. Non-parametric Methods, Economic Letters, 32, 193-197.
68. Taguchi, T. (1972a) On the Two-Dimensional Concentration Surface and Ex-tensions of Concentration Coefficient and Pareto Distribution to the Two-Dimensional Case-I, Annals of the Institute of Statistical Mathematics, 24, 355-382.
69. Taguchi, T. (1972b) On the Two-Dimensional Concentration Surface and Ex-tensions of Concentration Coefficient and Pareto Distribution to the Two-Dimensional Case-II, Annals of the Institute of Statistical Mathematics, 24, 599-619.
70. Taillie, C. (1981) Lorenz ordering within the Generalized Gamma Family of Income Distributions, in C. Taillie, G. P. Patil and B. Balderssari (eds.) Statistical Distributions in Scientific Work, vol. 6, pp. 181-192, Reidel, Boston.Google Scholar
71. Thistle, P. D. (1989) Ranking Distributions with Generalized Lorenz Curves, Southern Economic Journal, 56, 1-12.
72. Villase ñor, J. A. and B. C. Arnold (1989) Elliptical Lorenz Curves, Journal of Econometrics, 40, 327-338.
73. Wilfling, B. (1996) Lorenz Ordering of Generalized Beta-II Income Distributions, Journal of Econometrics, 71, 381-388.
74. Wilfling, B. and W. Kr ämer (1993) Lorenz Ordering of Singh-Maddala Income Distributions, Economic Letters, 43, 53-57.
75. Yitzhaki, S. (1982) Stochastic Dominance, Mean Variance and Gini’s Mean Difference, American Economic Review, 72, 178-185.Google Scholar
76. Yitzhaki, S. (1983) On an Extension of the Gini Inequality Index, International Economic Review, 24, 617-628.
77. Zenga, M. (1984) Proposta per un Indice di Concentrazione Basato sui Rapporti fra Quantili di Popolazione e Quantili di Reddito, Giornale degli Economisti e Annali di Economia, 48, 301-326.Google Scholar