Abstract
This chapter describes the standard approach to normative inequality measurement. The key idea is to interpret inequality as a welfare loss, when social welfare is measured by a conventional social welfare function. We focus on the of Atkinson family of inequality indices, which extends the initial ideas of Dalton by applying some of the notions that are common in expected utility theory. Atkinson uses the notion of equally distributed equivalent income to evaluate the actual distribution and to assess the size of the welfare loss. Those inequality indices are derived from a utilitarian social welfare function applied to individuals with identical cardinal utility functions. Atkinson generates a family of indicators that depend on a single parameter, to be understood as a measure of inequality aversion.
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Notes
- 1.
Those types of comparability reflect the class of invariant transformations in utilities that are admissible. Under ordinal level comparability, any monotone increasing function that is common to all agents is admissible. Under cardinal unit comparability, any linear transformation with common slope for all agents is admissible.
- 2.
We are implicitly assuming here that all utilities are different. When there are ties in the minimum values, one chooses the option that is preferred for the first agent in the ranking from less to more utility who is not indifferent.
- 3.
- 4.
Dalton’s original formulation is given by \( \frac{{\displaystyle \sum_{i=1}^nu\left({y}_i\right)}}{n.u\left(\mu \right)} \), which takes on value 1 for zero inequality. We use this formulation to satisfy the principle of normalisation.
- 5.
Indeed, any linear transformation will also do because utilities are cardinal.
- 6.
- 7.
See Pratt (1964) and Lambert (1993, Ch. 4, Th. 4.2).
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Villar, A. (2017). Normative Inequality Indices. In: Lectures on Inequality, Poverty and Welfare. Lecture Notes in Economics and Mathematical Systems, vol 685. Springer, Cham. https://doi.org/10.1007/978-3-319-45562-4_4
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