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Social Choice and Welfare

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28.11.2016 | Original Paper

Inequality and isoelastic equivalence scales: restrictions and implications

verfasst von: Udo Ebert, Patrick Moyes

Erschienen in: Social Choice and Welfare | Ausgabe 2/2017

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Abstract

Comparisons of well-being across heterogenous households necessitate that households’ incomes are adjusted for differences in size and composition: equivalence scales are commonly used to achieve this objective. Equivalence scales with constant elasticity with respect to family size have been argued to provide a good approximation to a large variety of scales (see, e.g., Buhmann et al., Equivalence scales, well-being, inequality and poverty: sensitivity estimates across ten countries using the Luxembourg Income Study (LIS) database. Rev Income Wealth 34:115–142, 1988) and they therefore play a prominent role in empirical work. Focusing on inequality of well-being, we first show that, if one requires that the index of inequality is—in addition to standard properties—invariant to modifications of the relative (marginal) distributions of needs and income across households, then the equivalence scales must be isoelastic. In addition, if all households’ members have the same preferences and if households maximise the sum of their members’ utilities, then the only preferences consistent with isoelastic scales are of the Cobb–Douglas type.

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Adjustments for differences in needs by means of equivalence scales may be considered too specific an approach and an alternative procedure has been proposed by Atkinson and Bourguignon (1987) (see also Bourguignon 1989; Jenkins and Lambert 1993; Bazen and Moyes 2003; Ebert 2010; Moyes 2012; Gravel and Moyes 2012). While this approach has mainly focused on the derivation of quasi-orderings like the sequential Lorenz dominance criterion for making comparisons of living standards across heterogenous populations, it is equally possible to use multidimensional (cardinal) indices (see, e.g., Maasoumi 1999; Ebert 1995; Gravel et al. 2009 among others).

The so-called square root scale replaces the former Oecd-modified scale proposed by Hagenaars et al. (1994) that assigned a value of 1 to the household head, of 0.5 to each additional adult and of 0.3 to each child. While the levels of poverty, inequality or welfare are sensitive the use of one scale rather than another, trends over time and rankings of countries are much less affected (see, e.g., Burniaux et al. 1998).

This makes only sense if the equivalence scales are independent of household income which implicitly amounts to imposing strong restrictions on the preferences of the household’s members (see, Blackorby and Donaldson 1993; Blundell and Lewbel 1991). Admittedly, this assumption is debatable and there indeed is ample empirical evidence that it is violated in practice (see, e.g., Donaldson and Pendakur 2004; Koulovatianos et al. 2005a, b).

Under certain conditions, it is indeed possible to establish the existence of a single critical value of the size elasticity for which the poverty ranking of household-size groups switches (see Lanjouw and Ravallion 1995 for details).

Admittedly, for this requirement to make sense, one has to assimilate neediness with household size: for more on this, see Sect. 2.

We refer the reader to Apps and Rees (2009, Chapter 3) that provides an expositional survey of the different models of household behaviour available in the literature.

The difficulty facing researchers in practice is precisely that most microdata bases provide limited information—usually households’ incomes and compositions—that does not allow one to uncover the actual distribution of well-being among the households’ members.

In general, the fact that all individuals in the society have the same preferences does not imply that they have the same utility function. Since, in our model, individuals are all alike, it is natural to assume that they have also the same (cardinal) utility function.

This is consistent with the maximisation of a symmetric, monotone non-decreasing and quasi-concave social welfare function \(F({U}(x_{1},G),\ldots ,{U}(x_{n},G))\), where n is the number of persons in the household (see Bourguignon 1989; Blackorby and Donaldson 1993).

According to this principle, a replication of a situation leaves welfare, poverty, inequality, and the like unaffected.

The standards properties of the utility function do not guarantee that this property be satisfied unless one imposes additional restrictions on the utility function that still need to be identified. For instance, the quasi-linear utility function \({U}(x,G) := x + 2 \, \ln (1 + G)\) does not generate an equivalent income that decreases with household size.

This was first recognised by Glewwe (1991), who showed that a regressive transfer of income between two households might decrease the inequality of well-being when the equivalent incomes are weighted by the household sizes.

Actually, condition UDI is stronger than the principle of population to the extent that the weights are not necessarily equal to the numbers of households who have that particular income: for instance, weights may be used to improve the representativeness of the sample data.

We refer the interested reader to Ebert and Moyes (2002) for a proof of this assertion in the particular framework considered here.

Combining these two conditions, we obtain the kind of invariance property considered in the standard multidimensional inequality literature, where different scalings are used for different attributes (see, e.g., Tsui 1995).

To this extent, a between-type progressive transfer is a particular case of the more general transformation introduced by Kolm (1977) who requires that transfers take place in all attributes. It must also be stressed that Kolm (1977) imposes no restrictions on the respective positions—with respect to the different attributes—of the households involved in this generalised transfer. In particular, it is not necessary that one household be richer than another in all attributes for the transfer to make sense.

This result is reminiscent of Ebert and Moyes (2003) who obtained similar restrictions on the adjustment method but using a slightly different approach.

In the case of two variables, standard homotheticity requires that \({f({\lambda } {{\mathbf {u}}})}/{f({{\mathbf {u}}})} = {f({\lambda } {{\mathbf {v}}})}/{f({{\mathbf {v}}})}\), for all \({{\mathbf {u}}} := (u_{1},u_{2}),{{\mathbf {v}}} := (v_{1},v_{2}) \in {{\mathbb {R}}}_{++}^{2}\) and all \({\lambda } > 1\). On the other hand, (partial) homotheticity in the first variable would impose that \({f({\lambda } u_{1},u_{2})}/{f(u_{1},u_{2})} = {f({\lambda } v_{1},v_{2})}/{f(v_{1},v_{2})}\), for all \({{\mathbf {u}}},{{\mathbf {v}}} \in {{\mathbb {R}}}_{++}^{2}\) and all \({\lambda } > 1\).

Since by assumption the utility function is differentiable, so are the indirect utility function and its inverse.

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Titel: Inequality and isoelastic equivalence scales: restrictions and implications
verfasst von: Udo Ebert
Patrick Moyes
Publikationsdatum: 28.11.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 2/2017
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-016-1004-x

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