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Theory and Decision

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07.06.2017

Multidimensional Pigou–Dalton transfers and social evaluation functions

verfasst von: Marcello Basili, Paulo Casaca, Alain Chateauneuf, Maurizio Franzini

Erschienen in: Theory and Decision | Ausgabe 4/2017

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Abstract

We axiomatize, in the multidimensional case, a social evaluation function that can accommodate a natural Pigou–Dalton principle and correlation increasing majorization. This is performed by building upon a simple class of inframodular functions proposed by Müller and Scarsini under risk.

Vorheriger Artikel On the consistency of choice

Nächster Artikel Asymmetric Choquet random walks and ambiguity aversion or seeking

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Note that throughout the paper we assume \(n\ge 3\). Indeed, to obtain an additive representation in a parsimonious way through the classical independence axiom as in Debreu’s theorem (1960), it is known that \(n>2\) is required, which in any case appears to make sense for applications.

Since this paper has been performed, we are aware of a similar Pigou–Dalton principle introduced by (Bosmans et al., 2009). Nevertheless, main differences persist between the two papers: our definition is model-free and our main motivation is to link this principle to inframodularity.

Let \((X, Y) \in \mathbb {R}^{m} \times \mathbb {R}^{m}, X \wedge Y=(\ldots , \min (x_{i}, y_{i}), \ldots ), X \vee Y=(\ldots , \max (x_{i}, y_{i}), \ldots )\). Correlation Increasing Majorization stipulates the meaningful requirement that replacing two individuals endowed initially with X and Y by individuals endowed with \(X \wedge Y\) and \(X \vee Y\) increases inequality. Since an inframodular function u is submodular, i.e., \(u(X)+u(Y) \ge u(X \wedge Y)+u(X \vee Y)\), one gets this property.

In 2010, the HDI functional has changed its additive form to a multiplicative form as introduced above. Section 6 includes a discussion about this ‘new’ HDI. Details are in Zambrano (2014).

See the Proof of Theorem 2, in Sect. 3

Indeed, \(X^{(p)}\downarrow X\) and \(X^{(p)}\uparrow X\) means that the sequences \(X^{(p)}\) are, respectively, decreasing (increasing) with respect to the pointwise order in \(\mathbb {R}^{m}\) while converging towards X.

Indeed, below \(\mathbb {Z}, \mathbb {N}\) and \(\mathbb {Q}\) (respectively, \(\mathbb {Z^{*}}, \mathbb {N^{*}}\) and \(\mathbb {Q^{*}}\)) denote as usually the set of integers, non-negative integers, rational numbers (respectively, non-null elements in \(\mathbb {Z}, \mathbb {N}\) and \(\mathbb {Q}\)).

See, e.g., Kolm 1976a, b; Atkinson 1970 or else Aczél 1966 to get this result.

Each column represents an individual.

For more details, see UNDP (1990).

Zambrano (2014) has a discussion about this new index, its computation and axiomatization. See also Herrero et al. (2010).

This exam is called ENEM (Exame Nacional do Ensino Médio—National high school exam). This exam is non-mandatory and has been used both as an admission test for enrollment in federal universities and educational institutes, as well as for certification for a high school degree.

Kovacevic (2010) offers a good review and discussion about the importance of the inequality to evaluate the human development.

Note that \(u_{j}\) increasing comes from our monotonicity axiom A.3.

Aczél, J. (1966). Lectures on functional equations and their applications. New York: Academic Press.

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Bosmans, K., Lauwers, L., & Ooghe, E. (2009). A consistent multidimensional Pigou–Dalton transfer principle. Journal of Economic Theory, 144(3), 1358–1371.CrossRef

Bourguignon, F., & Chakravarty, S. R. (2003). The measurement of multidimensional poverty. The Journal of Economic Inequality, 1(1), 25–49.CrossRef

Debreu, G. (1960). Topological methods in cardinal utility theory. In K. J. Arrow, S. K, & P. Suppes (Eds.), Mathematical methods in the social sciences (pp. 16–26). Stanford: Stanford University Press.

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Titel: Multidimensional Pigou–Dalton transfers and social evaluation functions
verfasst von: Marcello Basili
Paulo Casaca
Alain Chateauneuf
Maurizio Franzini
Publikationsdatum: 07.06.2017
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 4/2017
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-017-9605-0

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