Preferences for redistribution and social structure

Abstract

We model inter-individual differences in preferences for redistribution as a function of (a) self-interest; (b) ideas about the deservingness of income differences due to luck, effort and talent; (c) subjective perceptions of the relative importance of these determinants for explaining the actual income distribution. Individuals base the latter on information obtained from their reference group. We analyse the consequences for redistributive preferences of homophilous reference group formation based on talent. Our model makes it possible to understand and integrate some of the main insights from the empirical literature. We illustrate with GSS data from 1987 how our model may help in structuring empirical work.

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Fig. 1

Notes

  1. 1.

    Corneo and Fong (2008) estimate on data from the 1998 Gallup Social Audit that the monetary value of justice for US households amounts on average to about one fifth of their disposable income.

  2. 2.

    There is a growing amount of evidence that people have wrong perceptions about inequality and its causes and that these perceptions strongly influence their ideas about whether more redistribution is needed. Recent examples include Gimpelson and Treisman (2015) and Page and Goldstein (2016).

  3. 3.

    We understand the “reference group” of an individual as the group of people that he observes and from which he derives information about the overall income distribution. Note that in a different context, a reference group is commonly understood as the group of people to which an individual compares himself in order to evaluate his relative welfare position. Corneo and Grüner (2000, 2002) analyse the determinants of redistributive taxation in such a setting in which individuals are concerned about their relative consumption.

  4. 4.

    The assumption of a linear tax scheme is very common in the literature and it has also been shown that such a scheme may be a good empirical approximation of the more complex non-linear tax schemes that are found in the real world (see, e.g., Roemer et al. 2003). Nevertheless, exploring the consequences of relaxing this restriction is an interesting topic for future research. The assumption of a quadratic income tax scheme (as in De Donder and Hindriks 2003) may be an interesting starting point for this extension.

  5. 5.

    In political economy models of the determination of tax rates, each individual voter has a zero impact on outcome. We focus on individual preferences: the optimal \(\tau \) is then determined as if the individual is a dictator.

  6. 6.

    In the welfare economic literature, the treatment of luck is far from obvious—see, e.g., Lefranc et al. (2009). We follow the terminology in which “luck” stands for circumstances.

  7. 7.

    As one reviewer notes, the empirical literature has investigated the acceptability of income differences due to effort, not those due to taste for effort. However, in most models the former is monotonically related to the latter. At a more basic level, the distinction relates to the fundamental debate in the literature on responsibility-sensitive egalitarianism on the question for which factors individuals should be held responsible: for factors under their control, or for their preferences. Within the economic model, effort choices are determined by preferences and by constraints and there is no really free choice. See, e.g., Fleurbaey (2008) for a deeper discussion of these issues.

  8. 8.

    The index i is typically used for the consumer assessing his/her preferences for redistribution, and j concerns typically consumers observed by him/her.

  9. 9.

    To save on notation, we do not use subscripts for these cultural traits, but we will derive comparative statics results with respect to them.

  10. 10.

    In an earlier version of this paper (Schokkaert and Truyts 2014), we did not impose the assumption of an homogeneous taste for effort. This complicates the derivation and the interpretation of the results, without adding useful insights. The results derived in this paper still hold in the more general model.

  11. 11.

    Note that this means that we confine ourselves to the voters’ ex ante mental exercise of determining their preferred tax rate, based on local information. As one reviewer correctly noted, in our simplified model observing the true demogrant \(\bar{m}\) would allow sophisticated voters to infer the society’s true average ability. We keep to our simple setting for the present purposes, mainly because we want to stay close to what empirical research has revealed about the informational limitations of citizens in forming their opinion about redistributive policies. Similar results as in our model can be obtained with fully Bayesian voters observing the true demogrant in combination with less lenient other information assumptions, but we leave such a significantly more complicated analysis for a later paper.

  12. 12.

    Since i has zero mass, this immediately follows from \(\left| \mu \left( i\right) \right| >0.\)

  13. 13.

    Equation (5) might suggest that individuals take a parochial attitude and are only interested in justice within their reference group. This is not our interpretation, however. We could have started from a more general society-wide measure of injustice. However, as will become clear, only the means and the variances of the different variables will enter the expressions for the preferred tax. Since we assume that these are estimated by the individuals on the basis of their own reference group, choosing a more general formulation would not change any of our results. Moreover, in our theoretical analysis, reference groups can be interpreted very broadly, e.g., they can be seen as a (probably biased) sample of the overall population.

  14. 14.

    This is similar to the egalitarian equivalent approach in the theory of responsibility-sensitive egalitarianism—see, e.g., Fleurbaey (2008).

  15. 15.

    Our setting allows for more extreme positions. At one extreme, we have the laissez-faire or libertarian conviction that considers all income differences to be justified, such that fairness warrants no redistribution at all. This can be modelled as \(\hat{c}_{j}^{\zeta }=m_{j}\) for all j. In this case, social injustice \(\Omega _{j}\) equals the average income change due to taxation and it is minimized by setting \(\tau =0.\) Note that this fairness ideal conflicts with self-interest for consumers with an income below average. At the other end of the spectrum, the pure egalitarian position corresponds to \(\hat{c}_{j}^{\zeta }=\bar{m}_{\left( i\right) }\) for all j. We will not analyse this position as such (it is rarely defended explicitly).

  16. 16.

    We assume that when deciding about the tax rate, the consumer does not yet know the realization of luck \(\varepsilon _{i}\) and therefore estimates it as the average \(\overline{\varepsilon }_{(i)}=0\).

  17. 17.

    The condition for strict concavity of the optimization problem is that \(a_{i}<2\overline{a}_{(i)}.\) We only consider individuals for which this assumption holds.

  18. 18.

    An alternative approach would take as the reference a no-tax situation, with effort equal to \(\alpha _{j}\beta \) and fair pre-tax income \(\beta \left( \zeta a_{j}+\left( 1-\zeta \right) \bar{a}_{(i)}\right) \). This would mean, however, that our “naive” consumer uses a sophisticated model to go from the observed situation to the counterfactual no-tax world. As a matter of fact, our comparative statics results remain valid under this assumption (simply set \(\tau ^{a}=0\) in all expressions).

  19. 19.

    Note that the naive idealist in our model does not take into account incentive effects in determining his/her own preferred tax rate. However, if \(\tau ^{a}\) is large, (s)he will observe a lower contribution of effort in the actual process of income creation and this has an effect on what (s)he considers to be the fair tax rate.

  20. 20.

    This mechanism is analogous to the one that is described by Alesina and Angeletos (2005) to explain the differences between the European and the US welfare states.

  21. 21.

    Alesina et al. (2012) mention that it is debatable whether or not \((1-\tau )\) should enter the definition of the “fair” wealth. An alternative is to take as reference the no-tax situation \(\tau =0.\) This alternative assumption does not change the fundamental results.

  22. 22.

    This result stands in sharp contradiction to the optimality result for linear redistributive taxation in the traditional approach with a welfarist (e.g., utilitarian) social welfare function. In fact, our sophisticated consumer is only concerned about fairness (and self-interest) and does not consider the society-wide trade-off between the global level and the distribution of income.

  23. 23.

    The (reasonably weak) conditions for strict concavity can be obtained from the authors on request.

  24. 24.

    Consumers may use other information sources, such as what they see and hear in mass media. Of course, this information is also biased. The (interesting) question of how consumers may combine different biased information sources is an open question for further research.

  25. 25.

    A discussion of reference group formation for generic distribution functions in the case where only natural talent matters is presented in Appendix.

  26. 26.

    Hence, we restrict our attention to cases where the support of \(\mu \left( i\right) \) is in the interior of \(\left[ \theta _{L},\theta _{R}\right] \times \left[ \theta _{D},\theta _{U}\right] .\)

  27. 27.

    The General Social Survey was set up by the National Opinion Research Center at the University of Chicago in 1972, and collected its 30th round in 2014. See: http://www.gss.norc.org.

  28. 28.

    The previous literature has often made use of redist1, one of the four variables that make up our factor (see, e.g., Alesina and Giuliano 2011). The results with that variable are very similar to the ones obtained with FactRedis, and can be obtained from the authors on request.

  29. 29.

    “Explanatory” is meant to refer to statistical features and does not imply causality. Indeed, one does not have to be particularly cynical to note that regressing “attitudes” on other “attitudes” is bound to lead to strong associations.

  30. 30.

    As shown in Table 13, the range of the education variable is from 0 to 20.

  31. 31.

    As can be seen from Table 12, income is just household income measured in 1986 dollars. However, the combination of rlincome with prospect can arguably be interpreted as a proxy measure of permanent income, which may better fit the income concept in our theoretical model.

  32. 32.

    Note that these data requirements w.r.t. the reference groups go well beyond the data used in the previous section. The 1987 round of the GSS contains detailed information about different memberships of organisations (sports clubs, science clubs...), but this does not allow for a characterization of the sample of society that individuals meet in such an organization. Even if individuals from all parts of society are members of sports clubs, this does not mean that they meet each other in these clubs, because the individual sports clubs can be very homogenous in terms of their members’ socio-economic and other characteristics.

References

  1. Alesina A, Angeletos G-M (2005) Fairness and redistribution. Am Econ Rev 95(4):960–980

    Article  Google Scholar 

  2. Alesina A, Cozzi G, Mantovan N (2012) The evolution of ideology, fairness and redistribution. Econ J 122:1244–1261

    Article  Google Scholar 

  3. Alesina A, Giuliano P (2011) Preferences for redistribution. In: Benhabib J, Bisin A, Jackson M (eds) Handbook of social economics, vol 1A. Elsevier, New York, pp 93–131

    Google Scholar 

  4. Benabou R, Tirole J (2006) Belief in a just world and redistributive politics. Q J Econ 121(2):699–746

    Article  Google Scholar 

  5. Cervellati M, Esteban J, Kranich L (2010) Work values, endogenous sentiments and redistribution. J Public Econ 94:612–627

    Article  Google Scholar 

  6. Corneo G, Fong C (2008) What’s the monetary value of distributive justice. J Public Econ 92:289–308

    Article  Google Scholar 

  7. Corneo G, Grüner. H-P (2000) Social limits to redistribution. Am Econ Rev 90:1491–1507

    Article  Google Scholar 

  8. Corneo G, Grüner H-P (2002) Individual preferences for political redistribution. J Public Econ 83:83–107

    Article  Google Scholar 

  9. Cruces G, Perez-Truglia R, Tetaz M (2013) Biased perceptions of income distribution and preferences for redistribution: evidence from a survey experiment. J Public Econ 98:100–112

    Article  Google Scholar 

  10. De Donder P, Hindriks J (2003) The politics of progressive income taxation with incentive effects. J Public Econ 87:2491–2505

    Article  Google Scholar 

  11. Eugster B, Lalive R, Steinhauer A, Zweimüller J (2011) The demand for social insurance: does culture matter? Econ J 121:F413–F448

    Article  Google Scholar 

  12. Fleurbaey M (2008) Fairness, responsibility and welfare. Oxford University Press, Oxford

    Google Scholar 

  13. Fong C (2001) Social preferences, self-interest, and the demand for redistribution. J Public Econ 82:225–246

    Article  Google Scholar 

  14. Gaertner W, Schokkaert E (2012) Empirical social choice. Cambridge University Press, Cambridge

    Google Scholar 

  15. Georgiadis A, Manning A (2012) Spend it like Beckham? Inequality and redistribution in the UK, 1983–2004. Public Choice 151:537–63

    Article  Google Scholar 

  16. Gimpelson V, Treisman D (2015) Misperceiving inequality. NBER: Working Paper 21174

  17. Guiso L, Sapienza P, Zingales L (2006) Does culture affect economic outcomes? J Econ Perspect 20(2):23–48

    Article  Google Scholar 

  18. Isaksson A-S, Lindskog A (2009) Preferences for redistribution—a country comparison of fairness judgements. J Econ Behav Organ 72:884–902

    Article  Google Scholar 

  19. Keely L, Tan C-M (2008) Understanding preferences for income redistribution. J Public Econ 92:944–961

    Article  Google Scholar 

  20. Konow J (2003) Which is the fairest one of all? A positive analysis of justice theories. J Econ Lit 41(4):1188–1239

    Article  Google Scholar 

  21. Lefranc A, Pistolesi N, Trannoy A (2009) Equality of opportunity and luck: definitions and testable conditions, with an application to income in France. J Public Econ 93:1189–1207

    Article  Google Scholar 

  22. Luttens R, Valfort M-A (2012) Voting for redistribution under desert-sensitive altruism. Scand J Econ 114(3):881–907

    Google Scholar 

  23. Luttmer E (2001) Group loyalty and the taste for redistribution. J Polit Econ 109(3):500–528

    Article  Google Scholar 

  24. Luttmer E, Singhal M (2011) Culture, context, and the taste for redistribution. Am Econ J Econ Policy 3:157–179

    Article  Google Scholar 

  25. McPherson M, Smith-Lovin L, Cook J (2001) Birds of a feather: homophily in social networks. Annu Rev Sociol 27:415–444

    Article  Google Scholar 

  26. Meltzer A, Richard S (1981) A rational theory of the size of government. J Polit Econ 89:914–927

    Article  Google Scholar 

  27. Page L, Goldstein D (2016) Subjective beliefs about the income distribution and preferences for redistribution. Soc Choice Welf 47:25–61

    Article  Google Scholar 

  28. Piketty T (1995) Social mobility and redistributive politics. Q J Econ 110(3):551–584

    Article  Google Scholar 

  29. Pittau M, Massari R, Zelli R (2013) Hierarchical modelling of disparities in preferences for redistribution. Oxf Bull Econ Stat 75(4):556–584

    Article  Google Scholar 

  30. Roemer J (1993) A pragmatic theory of responsibility for the egalitarian planner. Philos Public Aff 22:146–166

    Google Scholar 

  31. Roemer J (1998) Equality of opportunity. Harvard University Press, Cambridge

    Google Scholar 

  32. Roemer J, Aaberge R, Colombino U et al (2003) To what extent do fiscal regimes equalize opportunities for income acquisition among citizens? J Public Econ 87:539–565

    Article  Google Scholar 

  33. Roemer J, Trannoy A (2015) Equality of opportunity. In: Atkinson A, Bourguignon F (eds) Handbook of income distribution, vol 2A. Elsevier, Amsterdam, pp 217–300

    Google Scholar 

  34. Schokkaert E, Truyts T (2014) Preferences for redistribution and social structure. CORE Discussion Paper 2014/1

  35. Yamamura E (2012) Social capital, household income, and preferences for income redistribution. Eur J Polit Econ 28:498–511

    Article  Google Scholar 

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Correspondence to Erik Schokkaert.

Additional information

The authors thank Dirk Van de gaer, two anonymous referees and conference and seminar participants in Leuven, Marseille, Moscow, Louvain-la-Neuve, Barcelona and Brussels. Financial support by the Flemish Science Foundation (project G.0522.09N) is gratefully acknowledged. Tom Truyts gratefully acknowledges financial support from the Belgian French speaking community ARC project no 15/20-072, Social and Economic Network Formation under Limited Farsightedness: Theory and Applications, Université Saint-Louis–Bruxelles, October 2015–September 2020.

Appendices

Appendix

Data appendix: description of the variables

See Tables 6, 7, 8, 9, 10, 11, 12 and 13.

Table 6 Description of redistribution variables
Table 7 Dependent variables and factor loadings
Table 8 Description of variables: beliefs about determinants of success in life
Table 9 Description of variables: perceptions of incentives
Table 10 Summary statistics of perceptions and beliefs
Table 11 Beliefs: rotated factor loadings and uniqueness
Table 12 Description of socioeconomic variables
Table 13 Descriptive statistics independent variables

Mathematical appendix: proofs

A. Derivation of equation (10)

Implementing Eq. (5) for the naive idealist gives

$$\begin{aligned} \Omega _{(i)}^{\zeta 0} =&\frac{1}{|\mu (i)|}\int _{j\in \mu (i)}\left\{ (1-\tau )(1-\tau ^{a})(\beta a_{j}+\varepsilon _{j})+\tau (1-\tau ^{a})\beta \bar{a}_{(i)}\right. \\&\left. -\zeta (1-\tau ^{a})\beta a_{j}-(1-\zeta )(1-\tau ^{a})\beta \bar{a}_{(i)}\right\} ^{2}dj \end{aligned}$$

or

$$\begin{aligned} \Omega _{(i)}^{\zeta 0}=\frac{1}{|\mu (i)|}\int _{j\in \mu (i)}\left\{ (1-\tau )\varepsilon _{j}+\beta (1-\tau ^{a})(1-\tau -\zeta )(a_{j}-\bar{a}_{(i)})\right\} ^{2}dj \end{aligned}$$

Using condition 1, this immediately yields Eq. (10).

B. Comparative statics for sophisticated consumers

Because sgn\(\left[ \frac{\partial \tau _{i}^{*}}{\partial z}\right] =\)sgn\(\left[ \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial z}\right] ,\) and

$$\begin{aligned} \Upsilon _{i}\equiv (1-\gamma )\beta \left[ \left( 1-2\tau \right) \bar{a}_{(i)}-\left( 1-\tau \right) a_{i}\right] -\gamma \frac{\partial \Omega _{(i)}^{\zeta }}{\partial \tau }=0 \end{aligned}$$

with

$$\begin{aligned} \frac{\partial \Omega _{(i)}^{\zeta }}{\partial \tau }&=-2\left( 1-\tau \right) Var\left( \varepsilon \right) _{(i)}-2\left( 1-\zeta -\tau \right) \left( 2-\zeta -2\tau \right) \left( 1-\tau \right) \beta ^{2}Var\left( a\right) _{(i)}, \end{aligned}$$

we have for self interest

$$\begin{aligned} \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial a_{i}}&=-\left( 1-\gamma \right) \left( 1-\tau _{i}^{*}\right) \beta \le 0 \end{aligned}$$

For the cultural traits

$$\begin{aligned} \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial \zeta } =&\,(2\gamma (1-\tau _{i}^{*})\beta ^{2}Var\left( a\right) _{(i)}\left[ 2\zeta -3\left( 1-\tau _{i}^{*}\right) \right]>0\Leftrightarrow \frac{2}{3}\zeta >\left( 1-\tau _{i}^{*}\right) \\ \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial \gamma } =&-\beta \left[ \left( 1-2\tau \right) \bar{a}_{(i)}-\left( 1-\tau \right) a_{i}\right] -\frac{\partial \Omega _{(i)}^{\zeta }}{\partial \tau }\\ =&\frac{\Upsilon _{i}-\beta \left[ \left( 1-2\tau \right) \bar{a}_{(i)}-\left( 1-\tau \right) a_{i}\right] }{\gamma }\\ =&\frac{\beta [\left( 1-\tau _{i}^{*}\right) a_{i}-\left( 1-2\tau _{i}^{*}\right) \bar{a}_{(i)}]}{\gamma }. \end{aligned}$$

For the perceptions and beliefs

$$\begin{aligned} \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial Var\left( \varepsilon \right) _{(i)}}&=\left( 2\gamma \left( 1-\tau _{i}^{*}\right) \right) \ge 0\\ \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial \bar{a}_{(i)}}&=\left( 1-\gamma \right) \left( 1-2\tau _{i}^{*}\right) \beta .\\ \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial Var\left( a\right) _{(i)}}&=2\gamma \beta ^{2}\left( 1-\tau _{i}^{*}\right) (1-\zeta -\tau _{i}^{*})(2-2\tau _{i}^{*}-\zeta ). \end{aligned}$$

C. Unidimensional reference group formation based on talent

Let the joint and marginal density functions of \(\Phi \left( a,q\right) \) be denoted respectively \(\phi \left( a,q\right) ,\) \(\phi ^{a}\left( a\right) \) and \(\phi ^{q}\left( q\right) .\) If \(\delta =1\), individual i only takes into account professional talent in the formation of his reference group, so that Eq. (15) reduces to \(\mu \left( i\right) =\left\{ j|\left| a_{i}-a_{j}\right| \le \pi \right\} .\) We derive for the perceived mean and variance of a : 

$$\begin{aligned} \bar{a}_{(i)}= & {} \frac{\int _{a_{i}-\pi }^{a_{i}+\pi }s\phi ^{a}\left( s\right) ds}{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }\\ Var\left( a\right) _{(i)}= & {} \frac{\int _{a_{i}-\pi }^{a_{i}+\pi }\left( z-\bar{a}_{(i)}\right) ^{2}\phi ^{a}\left( z\right) dz}{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }. \end{aligned}$$

These expressions immediately show that the beliefs of individual i will be influenced by his position in the distribution of talents. It is obvious that \(\partial \bar{a}_{(i)}/\partial a_{i}>0\). The sign of \(\partial Var(a)_{(i)}/\partial a_{i}\) is less straightforward, as it depends on the shape of \(\phi ^{a}(a)\). If \(\phi ^{a}(a)\) is the uniform density, natural talent \(a_{i}\) has no effect on the perceived variance.

An increase in \(\pi \) (i.e., an increase in the marginal utility of social relations \(\xi \) or a decrease in the cost c) will lead to an extension of the reference group of the individual. This results in

$$\begin{aligned} \frac{\partial \bar{a}_{(i)}}{\partial \pi } =&\frac{\left( \left( a_{i}+\pi \right) \phi ^{a}\left( a_{i}+\pi \right) +\left( a_{i}-\pi \right) \phi ^{a}\left( a_{i}-\pi \right) \right) }{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }\\&-\frac{\left( \phi ^{a}\left( a_{i}+\pi \right) +\phi ^{a}\left( a_{i}-\pi \right) \right) \bar{a}_{(i)}}{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }, \end{aligned}$$

which is positive iff

$$\begin{aligned} \frac{\bar{a}_{(i)}-\left( a_{i}-\pi \right) }{\left( a_{i}+\pi \right) -\bar{a}_{(i)}}<\frac{\phi ^{a}\left( a_{i}+\pi \right) }{\phi ^{a}\left( a_{i}-\pi \right) }. \end{aligned}$$

Again, the effect of changes in \(\pi \) will depend on the shape of \(\phi ^{a}(a)\). If \(\phi ^{a}(a)\) is uniform and if \(\mu \left( i\right) \) is strictly within the support of \(\phi ^{a}(a)\), the perceived mean \(\bar{a}_{(i)}\) obviously does not change with changes in \(\pi .\)

The effect on the perceived variance is

$$\begin{aligned} \frac{\partial Var\left( a\right) _{(i)}}{\partial \pi }&=\frac{\left( \begin{array}{c} -2\int _{a_{i}-\pi }^{a_{i}+\pi }\left( z-\bar{a}_{(i)}\right) \frac{\partial \bar{a}_{(i)}}{\partial \pi }\phi ^{a}\left( z\right) dz\\ +\left( \begin{array}{c} \left( \left( a_{i}+\pi \right) -\bar{a}_{(i)}\right) ^{2}\phi ^{a}\left( a_{i}+\pi \right) \\ +\left( \left( a_{i}-\pi \right) -\bar{a}_{(i)}\right) ^{2}\phi ^{a}\left( a_{i}-\pi \right) \end{array}\right) \\ -\left( \phi ^{a}\left( a_{i}+\pi \right) +\phi ^{a}\left( a_{i}-\pi \right) \right) Var\left( a\right) _{(i)} \end{array}\right) }{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }\\&=\frac{\begin{array}{c} \left( \left( a_{i}+\pi -\bar{a}_{(i)}\right) ^{2}-Var\left( a\right) _{(i)}\right) \phi ^{a}\left( a_{i}+\pi \right) \\ +\left( a_{i}-\pi -\bar{a}_{(i)}^ {})^{2}-Var\left( a\right) _{(i)}\right) \phi ^{a}\left( a_{i}-\pi \right) \end{array}}{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }. \end{aligned}$$

This expression is positive iff

$$\begin{aligned}&((a_{i}+\pi -\overline{a}_{(i)})^{2}-Var(a)_{(i)})\phi ^{a}(a_{i}+\pi )\\&\qquad +((a_{i}-\pi -\bar{a}_{(i)}^ {})^{2}-Var\left( a\right) _{(i)})\phi ^{a}\left( a_{i}-\pi \right) >0 \end{aligned}$$

A sufficient condition for this is that \(\Phi ^{a}\) is not too skewed around \(a_{i}\), so that both

$$\begin{aligned} Var\left( a\right) _{(i)}-\left( \left( a_{i}+\pi \right) -\bar{a}_{(i)}\right) ^{2}<0 \end{aligned}$$

and

$$\begin{aligned} Var\left( a\right) _{(i)}-\left( \left( a_{i}-\pi \right) -\bar{a}_{(i)}\right) ^{2}<0. \end{aligned}$$

This condition is definitely satisfied if \(\Phi ^{a}\) is uniform.

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Schokkaert, E., Truyts, T. Preferences for redistribution and social structure. Soc Choice Welf 49, 545–576 (2017). https://doi.org/10.1007/s00355-016-0996-6

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