How does the Meltzer-Richard model provide a valid justification for single peaked preferences?
It has something to do with the Envelope Theorem and showing that the middle class' preferences over tax is the median voter's preference over tax.
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If I understand your question correctly, single-peaked preferences simply reflect the fact that voters in the lower percentiles of the income distribution will prefer higher levels of redistribution, and voters in the upper percentiles lower levels. However, to establish this the Meltzer-Richard model draws from the work of Robert's (1977), who proved that if everybody has a unique most preferred tax schedule then this implies that individuals with higher incomes have most preferred schedules with lower marginal tax rates (see Lemma 1 in particular).