Abstract
We axiomatically characterize the Theil ordering of income inequality. In addition to the uncontroversial axioms of anonymity, homogeneity, replication invariance, strong directedness, and a standard continuity property, we appeal to both an independence and a decomposability axioms. These two axioms are ordinal implications of Theil decomposability, the central axiom in previous characterizations of the Theil index. To the best of our knowledge, the present is the first fully ordinal characterization of this index.
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Notes
Foster (1983) main result further shows that continuity can be dispensed with by strengthening the Directness axiom and assuming the Pigou–Dalton principle of transfers instead.
This restriction consists of the index applied only to two-person societies, with one person earning a proportion \(t\) of the total income and the other one earning the remaining proportion \(1-t\).
These simple societies are ones where a proportion \(1-t\) of the population has no income at all and the remaining proportion shares all of society’s income evenly.
We denote by \(\succ \) and \(\sim \) the asymmetric and symmetric parts of \(\succcurlyeq \).
We adopt the convention that \(0\ln (0)=0\).
A related property is the unit-consistency axiom introduced by Zheng (2007a). It guarantees that, as long as income is measured in the same unit in all societies, inequality rankings are independent of this unit. We are not sure as to the extent to which substituting this weaker axiom for homogeneity would affect our results. Examples of unit-consistent measures may be found in Zheng (2007a, b) and del Rio and Alonso-Villar (2010).
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We thank the Spanish Ministerio de Educación y Ciencia (project SEJ2009-11213) for research support. We also thank Mikel Bilbao for his invaluable help.
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Lasso de la Vega, C., Urrutia, A. & Volij, O. An axiomatic characterization of the Theil inequality ordering. Econ Theory 54, 757–776 (2013). https://doi.org/10.1007/s00199-012-0739-6
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DOI: https://doi.org/10.1007/s00199-012-0739-6