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Indirect taxes such as sales taxes discriminate against the needy. The rich have a much smaller marginal propensity to consume than the needy - the rich will park most of their money in the bank or some index fund, while the poor must fork out most of their money for basic expenditure such as food and shelter. IMO, the best solution remains a negative income tax, which Philip Greenspun alluded to (it's the solution advocated by many Nobel Prize-winning economists, including Milton Friedman).
I used to be one of those who didn't really believe in redistribution of income, but now having studied a little economics, I think it makes sense. The poverty cycle is a real thing, and it can only be moderated by some redistribution of wealth. There can be fewer unfair things in this world than being unable to realise your full potential simply because you had the misfortune of being born into a family that could not afford to feed, clothe or educate you.