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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
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.
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.
References
Aczel J (1966) Lectures on functional equations and their applications. Academic Press, New York
Aczel J (1984) On weighted synthesis of judgements. Aequ Math 27:288–307
Apps P, Rees R (2009) Public economics and the household. Cambridge University Press, Cambridge
Atkinson AB, Bourguignon F (1982) The comparison of multidimensioned distributions of economic status. Rev Econ Stud 49:183–201
Atkinson AB, Bourguignon F (1987) Income distributions and differences in needs. In: Feiwel G (ed) Arrow and the foundations of the theory of economic policy. MacMillan, New York, pp 350–370
Atkinson AB, Rainwater L, Smeeding T (1995) Income distribution in OECD countries. OECD Social Policy Studies No. 18, Paris
Bazen S, Moyes P (2003) International comparisons of income distributions. Res Econ Inequ 9:85–111
Blackorby C, Donaldson D (1993) Adult-equivalence scales and the economic implementation of interpersonal comparisons of well-being. Soc Choice Welf 10:335–361
Blundell R, Lewbel A (1991) The information content of equivalence scales. J Econ 50:49–68
Borcherding T, Deacon R (1972) The demand for services of non-federal government. Am Econ Rev 62:842–853
Bourguignon F (1989) Family size and social utility: Income distribution dominance criteria. J Econ 42:67–80
Buhmann B, Rainwater L, Schmaus G, Smeeding TM (1988) 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
Burniaux J-M, Dang T-T, Fore D, Förster M, Mira d’Ercole M, Oxley H (1998) Distribution and poverty in selected OECD countries. OECD Economics Department Working Paper, No. 89, Paris
Chiappori P-A (1997) Collective models of household behavior: The sharing rule approach. In: Haddad L, Hoddinott J, Alderman H (eds) Intrahousehold resource allocation in developing countries: models, methods and policy. The John Hopkins University Press, London, pp 39–52
Coulter FA, Cowell FA, Jenkins SP (1992a) Differences in needs and assessment of income distributions. Bull Econ Res 44:77–124
Coulter FA, Cowell FA, Jenkins SP (1992b) Equivalence scale relativities and the extent of inequality and poverty. Econ J 102:1067–1082
Cutler D, Katz L (1992) Rising inequality? Changes in the distribution of income and consumption in the 1980s. Am Econ Rev Papers Proc 82:546–551
Dalton H (1920) The measurement of the inequality of incomes. Econ J 30:348–361
Donaldson D, Pendakur K (2004) Equivalent-expenditure functions and expenditure-dependent equivalence scales. J Public Econ 88:175–208
Ebert U (1995) Income inequality and differences in household size. Math Soc Sci 30:37–53
Ebert U (2000) Sequential generalized Lorenz dominance and transfer principles. Bull Econ Res 52:113–123
Ebert U (2010) Dominance criteria for welfare criteria: Using equivalent income to describe differences in needs. Theory Decis 69:55–67
Ebert U, Moyes P (2002) Welfare, inequality and the transformation of incomes. The case of weighted income distributions. In: Moyes P, Seidl C, Shorrocks AF (eds) Inequalities: theory, experiments and applications. Journal of Economics, Suppl 9, pp 9–50
Ebert U, Moyes P (2003) Equivalence scales reconsidered. Econometrica 71:319–343
Ebert U, Moyes P (2009) Household decision and equivalence scales. J Popul Econ 22:1039–1062
Ebert U, Moyes P (2016) Inequality of living standards and isoelastic equivalence scales. GREThA Discussion Paper 2016-27, Université de Bordeaux
Edwards JH (1990) Congestion function specification and the “publicness” of local public goods. J Urban Econ 27:80–96
Figini P (1998) Inequality measures, equivalence scales and adjustment for household size and composition. Luxembourg Income Study, Working Paper No. 185
Fleurbaey M, Gaulier G (2009) International comparisons of living standards by equivalent incomes. Scand J Econ 111:597–624
Gastwirth J (1971) A general definition of the Lorenz curve. Econometrica 39:1037–1039
Glewwe P (1991) Household equivalence scales and the measurement of inequality: transfers from the poor to the rich could decrease inequality. J Public Econ 44:211–216
Gravel N, Moyes P (2012) Ethically robust comparisons of bidimensional distributions with an ordinal attribute. J Econ Theory 147:1384–1426
Gravel N, Moyes P, Tarroux B (2009) Robust international comparisons of distributions of disposable income and regional public goods. Economica 76:432–461
Hagenaars A, de Vos K, Zaidi M (1994) Poverty statistics in the late 1980s: research based on micro-data. Office for Official Publications of the European Communities, Luxembourg
Jenkins SP, Lambert PJ (1993) Ranking income distributions when needs differ. Rev Income Wealth 39:337–356
Kapteyn A, van Praag B (1976) A new approach to the construction of family equivalence scales. Eur Econ Rev 7:313–335
Kolm S-C (1977) Multidimensional egalitarianisms. Q J Econ 91:1–13
Koulovatianos C, Schröder C, Schmidt U (2005a) On the income dependence of equivalence scales. J Public Econ 89:19–27
Koulovatianos C, Schröder C, Schmidt U (2005b) Properties of equivalence scales in different countries. J Econ 86:967–996
Lanjouw P, Ravallion M (1995) Poverty and household size. Econ J 105:1415–1434
Lewbel A (1989) Household-equivalence scales and welfare comparisons. J Public Econ 39:377–391
Maasoumi E (1999) Multidimensional approaches to welfare analysis. In: Silber J (ed) Handbook of income inequality measurement. Kluwer Academic Publishers, Boston/Dordrecht/London, pp 437–477
McElroy MB, Horney MJ (1981) Nash-bargained household decisions: towards a generalization of the theory of demand. Int Econ Rev 22:333–349
Moyes P (2012) Comparisons of heterogeneous distributions and dominance criteria. J Econ Theory 147:1351–1383
Reiter M, Weichenrieder AJ (1999) Public goods, club goods, and the measurement of crowding. J Urban Econ 46:69–79
Samuelson PA (1956) Social indifference curves. Q J Econ 70:1–22
Tsui K-Y (1995) Multidimensional generalizations of the relative and absolute inequality indices: the Atkinson–Kolm–Sen approach. J Econ Theor 67:251–265
Whiteford P (1985) A family’s needs: Equivalence scales, poverty and social security. Research Paper No. 27, Development Division, Department of Social Security
Acknowledgements
This paper forms part of the research project Heterogeneity and Well-Being Inequality (Contract No. HEWI/ANR-07-FRAL-020) of the ANR-DFG programme whose financial support is gratefully acknowledged. We are indebted to Stephen Bazen, the associate editor and two anonymous referees for very useful comments and suggestions when preparing this version. Needless to say, none of the persons mentioned above should be held responsible for remaining deficiencies.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ebert, U., Moyes, P. Inequality and isoelastic equivalence scales: restrictions and implications. Soc Choice Welf 48, 295–326 (2017). https://doi.org/10.1007/s00355-016-1004-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00355-016-1004-x