Abstract
Since Becker (1971), a common argument against asymmetric norms that promote minority rights over those of the majority is that such policies reduce total welfare. While this may be the case, we show that there are simple environments where aggregate sum of individual utilities is actually maximized under asymmetric norms that favor minorities. We thus maintain that without information regarding individual utilities one cannot reject or promote segregation-related policies based on utilitarian arguments.
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Notes
We assume throughout the same population density everywhere. In other words, 60 % of the population will reside in the 0.6 part of land that is allocated for the exclusive use of the majority. Hence three quarters of the majority will live in majority-only areas.
We use terminology that applies to housing and country division, but as stated before, college or association formations can be likewise analyzed. In such cases individuals will prefer to study or associate with people like them, while also preferring to have as much choice as possible.
Set \(\gamma =1\) and obtain from Eq. (6) that \(\beta = \frac{12\alpha ^2-6\alpha }{9\alpha -4}\). Observe that \(\beta \geqslant \alpha \) iff \(\alpha \geqslant \frac{2}{3}\). Substitute into the second order condition to obtain \(-2\alpha + 1\) which is negative for all \(\alpha > \frac{1}{2}\), hence \(\left( \frac{12\alpha ^2-6\alpha }{9\alpha -4},1 \right) \) is a maximum point.
For \(\beta =1\) Eq. (8) implies that \(\gamma = \frac{30\alpha - 54\alpha ^2 + 42\alpha ^3 - 12\alpha ^4 - 6}{19\alpha - 23\alpha ^2 + 9\alpha ^3 - 5}\). The second order derivative is positive for all \(\alpha >\frac{1}{2}\) (and \(\ne \frac{5}{9}\), where it is zero). In other words, when \(\beta =1\) the optimal value of \(\gamma \) is either 1 (which is covered by the previous analysis), or \(1-\alpha \). We have therefore to check the value of \(W(\beta ,\gamma )\) at two points, \((\alpha ,1-\alpha )\) and \(\left( \frac{12\alpha ^2-6\alpha }{9\alpha -4},1 \right) \). By Eq. (5), \(W \left( \frac{12\alpha ^2-6\alpha }{9\alpha -4},1 \right) - W(\alpha ,1-\alpha ) =\frac{(76\alpha - 171\alpha ^2 + 135\alpha ^3 - 12)\alpha }{72(2\alpha - 1)^2}\), whose sign is the same as that of the numerator. This function is increasing on \(\left[ \frac{1}{2},1 \right] \) and its value at \(\alpha =\frac{2}{3}\) is positive. In other words, for all \(\alpha >\frac{2}{3}\), the optimal policy is to set \(\beta = \frac{12\alpha ^2-6\alpha }{9\alpha -4}\) and \(\gamma =1\).
It may well be that maximal social welfare requires a division of the state into several mixed areas with different minority/majority ratios.
That is, for all \(0 < \lambda \leqslant \min \left\{ \frac{1}{s+t}, \frac{1}{s^{\prime }+t^{\prime }} \right\} \).
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Acknowledgments
We thank Aaron Fix and Nisha Ye, and especially Inácio Guerberoff for research assistance. We thank Eddie Dekel, Larry Epstein, Johannes Hörner, Ben Polak, John Roemer, Larry Samuelson, Joel Sobel, and a referee for their comments.
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Appendix: axioms
Appendix: axioms
In this appendix we offer axioms leading to the conclusion that the utility of each person from a policy leading to \((s,t,\pi )\) is
where \(g\) is strictly increasing, \(g(0)=0\) and \(g(1)=1\).
Consider the set \(A=\{(s,t,\pi ) \in [0,1]^3: s+t \leqslant 1\}\). From an individual point of view, the vector \((s,t,\pi )\) means that the group to which the individual belongs can reside in \(s+t\) part of the country, where in the \(s\) part only this group resides while in the \(t\) part both groups can reside, and the proportion of people of the individual’s group in this \(t\) area is \(\pi \). Let \(\succeq \) be a complete, transitive, and continuous preference relation over \(A\). There is therefore a function \(u\) representing these preferences. Assume further that \(\succeq \) is increasingly monotonic in \(s\), and for positive \(t\) and \(\pi \), it is increasing in these two variables as well.
When \(\pi =0\), there are no representatives of the individual’s type in the mixed area, so this area does not really exist. This leads to the first restriction on \(\succeq \):
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A.
For all \(t \leqslant 1-s\), \((s,t,0) \sim (s,0,0)\).
Similarly, if \(t=0\) there is no mixed area, and \(\pi \) doesn’t matter. That is,
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B.
For all \(\pi \), \((s,0,\pi ) \sim (s,0,0)\).
When \(\pi =1\), the mixed area is not mixed but is inhabited by members of the individual’s group only. Hence
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C.
\((s,t,1) \sim (s+t,0,0)\).
Next, we assume separability between the two areas. The evaluation of two possible mixed areas does not depend on the size of what is assigned to the exclusive use of the individual’s group. Formally:
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D.
\((s,t,\pi ) \succeq (s,t^{\prime },d^{\prime }) \Longleftrightarrow (s^{\prime },t,\pi ) \succeq (s^{\prime },t^{\prime },d^{\prime })\).
Define \(f(t,\pi )\) by \((0,t,\pi ) \sim (0,f(t,\pi ),1)\) and obtain by the last two assumptions that
Therefore \(\succeq \) can be represented by
Observe that by assumption C above,
Finally, we assume size-homotheticity:
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E.
\((s,t,\pi ) \succeq (s^{\prime },t^{\prime },d^{\prime })\) iff for all appropriate Footnote 5 \(\lambda \), \((\lambda s,\lambda t,\pi ) \succeq (\lambda s^{\prime }, \lambda t^{\prime },d^{\prime })\).
In particular, \((s,t,1) \sim (s,t^{\prime },\pi )\) implies \((\lambda s,\lambda t,1) \sim (\lambda s,\lambda t^{\prime },\pi )\). By Eqs. (14) and (16),
As \(f\) is homogeneous of degree 1 in its first argument, it follows that \(u(s,t,\pi )\) \(=s+tg(\pi )\) (see Eq. 15 above). Since \(u(s,t,0)=s\) it follows that \(g(0)=0\), and since \(f(t,1)=t\) it follows that \(g(1) = 1\).
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Harel, A., Segal, U. Utilitarianism and discrimination. Soc Choice Welf 42, 367–380 (2014). https://doi.org/10.1007/s00355-013-0734-2
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DOI: https://doi.org/10.1007/s00355-013-0734-2