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Lorenz Densities and Lorenz Curves

In: Income Modeling and Balancing

Author

Listed:
  • Thomas Kämpke

    (Research Institute for Applied Knowledge Processing (FAW/n))

  • Franz Josef Radermacher

    (University of Ulm)

Abstract

Lorenz curves and Lorenz densities are introduced for real-valued random variables with finite and strictly positive expectation. Gastritic’s definition of a Lorenz curve is used. Basic properties of Lorenz curves are given as well as approximation results. Interestingly, when a sequence of distribution functions converges to a limit distribution function, the corresponding sequence of Lorenz curves need not converge to the Lorenz curve of the limit distribution function. Yet, convergence can be ensured under sufficient conditions. The characterizations of the function sets that are equal to either all Lorenz curves or all Lorenz densities are stated both. Examples of Lorenz curves are given including the Lorenz curve of the Cantor distribution. Some principles to derive Lorenz curves from other Lorenz curves are shown and finally, inequality measures based on Lorenz curves are given, with the Gini index being the most prominent example.

Suggested Citation

  • Thomas Kämpke & Franz Josef Radermacher, 2015. "Lorenz Densities and Lorenz Curves," Lecture Notes in Economics and Mathematical Systems, in: Income Modeling and Balancing, edition 127, chapter 0, pages 29-54, Springer.
    • Handle: RePEc:spr:lnechp:978-3-319-13224-2_3
    DOI: 10.1007/978-3-319-13224-2_3
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