Earlier quoted context omitted.
Negative income tax is certainly a very interesting idea, but it is virtually unheard of as a position in any mainstream U.S. political compass. It's generally considered that you either want to expand the current, highly convoluted system, or that you loathe welfare entirely as a matter of principle.
In the US, the Earned Income Tax Credit is similar to a negative income tax, and has bipartisan support (it was expanded under Clinton, Bush and Obama, for example). Economists love it, too.
EITC is like what a person who once heard of negative income tax but wanted to make a Rube Goldberg parody of it would create.
A negative income tax would, in its simplest form, instead of limiting AGI to a minimum of zero after deductions, let it go negative, and treat the (negative) calculated liability on that income as a negative total tax liability.
(Now, EITC helps a wider range of people -- with qualifying children, it hits about the same point for people without children -- than NIT would with the current deductions available for US taxes, but, that could be addressed by adding greater child deductions than the existing standard deduction.)