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Should two - band income taxes be progressive given a general income distribution? The authors provide a negative answer under utilitarian and max-min welfare functions. While this result clarifies some ambiguities in the literature, it does not rule out progressive taxes in general. If they maximize total or weighted utility of the poor, as often intended by the society, progressive taxes can be justified, especially when the 'Rich' are very rich. Under these objectives they obtain new necessary conditions for progressive taxes, which only depend on aggregate features of income distributions. The validity of these conditions is examined using plausible income distributions.
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