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Comparing two-dimensional distributions: a questionnaire-experimental approach

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Abstract

This paper aims to study the extent to which people’s intuitions about the distribution of two attributes within a society are consistent with the different axiomatizations proposed by economists. In particular, the objective is to compare the empirical support for two alternative principles, namely aversion to dispersion of attributes and aversion to correlation between attributes. Using a questionnaire approach, most people are found to be averse to correlation rather than averse to dispersion.

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Notes

  1. Seminal contributions are from Atkinson and Bourguignon (1982, 1987), Bourguignon (1989) and Kolm (1977). For recent contributions, see, for instance, Decancq (2012), Gravel and Moyes (2012), Moyes (2012), Müller and Trannoy (2012). Readers could refer to Weymark (2004) and Trannoy (2004) for surveys.

  2. See Schokkaert (1999), Konow (2003) or Gaertner and Schokkaert (2012) for overviews and discussion.

  3. See Kolm (1977) and Fleurbaey and Trannoy (2003) for equivalence result between utilitarian dominance and such transformation.

  4. See Gravel and Moyes (2012) for the relationship between transformations presented in the subsection and utilitarian dominance.

  5. The type is in fact defined in relation to the endowment of the other attribute.

  6. The questionnaires are presented in the online supplementary appendix.

  7. In order to study how such framing affects responses, some subjects (\(n=18\)) had to answer questions on the motivations of their choices. By showing that the main motivation of subjects is to ensure a subsistence level for villagers and that subjects take into consideration the fact that malaria is a “serious” illness and can be fatal, the analysis of this questionnaire allows us to not overplay framing problems. We would suggest that readers requiring further details on this questionnaire should refer to the online supplementary appendix.

  8. Note that attribute-specific progressive transfer and two-dimensional transformation are tested in the situation in which the two individuals involved in the transfer are differently ranked in respect with the two dimensions, i.e., \(x_h^1 < x_k^1\) and \(x_h^{2} > x_k^{2}\). The case where one of the two individuals is richer than the other in both dimensions does not allow us to distinguish aversion to dispersion from aversion to correlation.

  9. Here we assume that participants are not indecisive in numerical questions. The indecisiveness exhibited in verbal questions is explained only by the lack of information. More precisely, participants could be in favor of a transfer in some circumstances and in disfavor in some others. As these “circumstances” are not known in verbal questions, participants may be indecisive.

  10. In fact, subjects had to answer five or six questions. For the sake of proper statistical treatment, we do not exploit all of the questions but only two questions per questionnaire. However, the results are not qualitatively changed by this choice.

  11. Further econometric analysis showed that demographic and social characteristics have no significant effect on individual responses. The results of econometrics are displayed in the online supplementary appendix.

  12. Note that, in the discussion of the statistical tests, one applies a significance level of \(p\le 0.05\).

  13. Note that there are six pairs of questions for the two-dimensional transformation. The distributions of patterns of answers are compared using McNemar and Stuart-Maxell tests (see Table 10 in the appendix).

  14. Concerning the two-dimensional transformation, the proportion of inconsistent pairs of answers varies from 15% (Q3/Q4) to 38% (Q4/Q5). The subjects who are in line with such a property in both questions varies 27% (Q3/Q5) to 46% (Q3/Q4). These differences are statistically significant (see Table 10).

  15. As participants are free to choose several answers, the sum of percentages is not equal to 100%.

  16. According to the principle of income-related health transfer, transferring health from a rich person to a poor person does not lead to a reduction in social welfare provided the transfer does not change the ranking of the individuals in terms of income.

  17. Note that the income-related health transfer implies the between-type health transfer but the reverse is not true.

  18. By complex, we mean that a distribution is obtained through another one by a set of 2, 3 or 4 elementary transformations. See Bleichrodt et al. (2012), p. 89 and appendix.

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Acknowledgments

This paper has benefited from comments by Frédéric Chantreuil, Fabrice Le Lec, Marianne Lumeau, Olivier L’Haridon, David Masclet and the participants of the ESA Annual Meeting (Chicago, 2011). I would like to thank two anonymous referees and the editor Marc Fleurbaey for their valuable comments and suggestions which allow me to significantly improve the paper. I thank Elven Priour for research assistance and Eric Darmon, Gabin Langevin and Pascaline Vincent for giving me the opportunity to run the questionnaires in their classes. Finally, I would like to thank Vicki McNulty for proofreading this article in depth. Needless to say, none of the aforementioned persons should be held responsible for any deficiencies. The paper is part of the research project Conflict (Contract No. ANR-08-JCJC-0105-01) and the research project The Multiple Dimensions of Inequality (Contract No. ANR 2010 BLANC 1808) of the French National Agency for Research whose financial support is gratefully acknowledged.

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  1. University of Rennes 1 and CREM, 7, Place Hoche, 35065 , Rennes Cedex, France

    Benoît Tarroux

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Correspondence to Benoît Tarroux.

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Appendix

Appendix

See Table 10.

Table 10 Two-dimensional transformation: Stuart-Maxwell and McNemar tests for distributions of preferences for all possible pairs of questions
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Tarroux, B. Comparing two-dimensional distributions: a questionnaire-experimental approach. Soc Choice Welf 44, 87–108 (2015). https://doi.org/10.1007/s00355-014-0829-4

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