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Sure a flat tax, by definition, is not regressive. However, with our current level of deductions tend to target higher wage earners. So, it is plausible to believe if you instituted a flat tax rate but left the deductions in place that you would efffectively create a regressive tax system when looking at effective tax rates rather than the nominal rates.
Deductions are necessary. Consider you sell a widget for $1 that costs you $.50 in material to make. Is your income $1 or $.50? Without deductions, your income is $1 (but it's really $.50).
The tax system is not regressive, nor would a flat tax based system be regressive.
As it stands now, the top 1% pay more taxes than the bottom 90% combined. The top 10% pay 70% of all the income taxes. If it weren't progressive, the top 10% would pay 10% of the taxes.

