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A negative income tax would replace the current progressive income tax system used throuhgout most of the western world. This would be replaced by a set flat tax on income of say 25%, but each tax payer would also be given 10,000 dollars by the government. Thus a person earning only 4000 dollars per year would pay 1000 dollars in taxes, but overall would receive a net gain of 9,000 dollars from the government. A person making 40,000 dollars would be at the break-even point, and would neither pay taxes or recieve any benefits. A person making one million dollars per year would pay close to the full 25% tax as the 10,000 dollars would count little towards relieving their tax burden.
A negative income tax solves several problems with current systems. It would eliminate the welfare trap, it would also prevent the decrease in real wages when one moves over the border from one tax bracket to another.
A negative income tax can be but is not necessarily a guaranteed minimum income. A GMI has to provide enough money to survive on a NIT could be as low as few hundred dollars and a 2% tax rate implemented by a city government. GMI systems also often have other major reforms, such as the elimination of the minimum wage and the ending of most current social welfare schemes.
While the notion has long been popular with economists, it has never been politically feasible to be implemented. In part this is because of the very complex and entrenched nature of the current tax codes in most countries that would have to be rewritten under any NTI scheme.