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You still pay taxes in those states, just not income. Most people will pay more taxes in those states compared to places like California (not the rich, of course). Texas chose a system of sales taxes (state and local), which act like flat-taxes, which put more burden on lower income people.
Hmm, I guess theoretically. I bet towns or businesses close to eachother over a state border do something to equalize prices. Or I guess the businesses in the lower taxed state would just raise their prices because they can and still get business.