Abstract
Deaton (1979) showed that with an optimal linear progressive tax, commodity taxes are redundant if preferences are weakly separable between goods and labour and quasi-homothetic in goods. Hellwig (2009) later showed that any allocation with differential commodity taxes and an arbitrary linear progressive income tax is Pareto-dominated by one with uniform commodity taxes and a reformed linear progressive tax. We show that Deaton’s theorem and Hellwig’s extension apply a) with a piecewise linear income tax and tax credits, b) when some individuals earn no income either by choice or involuntarily, and c) when individual preferences for leisure differ.