Income earned above a given allowance is subject to tax by the government. Those who earn less than that allowance pay no tax. Under a negative income tax, the government would pay a fraction of the deficit to those who received less. If the percentage was 50%, the maximum subsidy was £8000, and the allowance was £16000, a person earning no income would receive £8000. If they pocket £2000 the following year, the government gives them 50% of the deficit as a subsidy: £7000. Combined with the £2000 that they’d earned, their year’s take-home would be £9000. A full £1000 more than when they were reliant on public assistance - this retains an incentive to work. More money is always better than less.
A negative income tax assures every family a minimum amount while avoiding a massive bureaucracy and a diminishment of personal responsibility. It also harms the incentives to seek employment less than the current welfare system.