Anybody who has seen Gordon Brown lose his temper does not easily forget it. The image of an angry bear springs to mind, except angry bears cannot usually get their hands on items of office equipment to fling around. The prime minister is said to be very angry at the media this weekend for creating a political storm in what he regards as a tax teacup. He is angry with the independent experts who have calculated that 5.3m families will lose out as a result of the abolition of the 10p income tax band.

Mr Brown is also angry with his own back-benchers, who he thinks are too dim-witted to have understood his government’s efforts on behalf of the poor of Britain, from his system of complex tax credits to introducing and increasing the minimum wage. There is surprise as well as anger in the prime minister’s response. He had got so used to being attacked for redistributing away from the better-off and the middle classes that he had not expected to leave himself vulnerable to the charge of deserting Labour’s core supporters.

One source of Mr Brown’s anger can be easily dismissed: the idea that his critics have got their numbers wrong. When the Institute for Fiscal Studies first produced its figure for the 5.3m families losing out, Mark Neale, a senior Treasury official, told the Commons Treasury committee that its estimates were “in the right ball-park” and consistent with Mr Brown’s own assessment that four out of five households would gain or be no worse off as a result of the change. By implication the rest, one in five, would lose.

The losers typically earn between £5,000 and £18,000, the very group the 10p tax band was intending to benefit. Damagingly for the government, they are spread far and wide by age, family type and where they live. Every MP will have losers in his or her constituency. There are 2.2m single working people who do not qualify for tax credits because they earn slightly too much or do not work enough hours; 1.2m working couples without children who are also outside the tax credit net or choose not to claim it; 700,000 two-earner couples with children, 300,000 women aged between 60 and 64, and so on.

Had Mr Brown deliberately set out to sow discontent in every corner of the land he could not have done better. The losses are not large, probably averaging £2 a week and less than £1 billion in total. But they rankle hugely. The prime minister believes the losers should take the long view and appreciate what 11 years of Labour has done for them in making them better off. The “concession” in the coming days to buy off the rebels will be a pledge to look after them in future.

The trouble is that things have moved on. When he announced the abolition of the 10p rate in March last year, the credit crisis had yet to explode and the squeeze on family incomes was in its infancy. Since then world oil prices have doubled, gas and electricity bills have shot up and there has been a double-figure increase in the cost of a week’s food shopping. There is never a good time to hit anybody with a tax rise but this is a terrible time to be hitting the low paid.

Mr Brown will adopt his best head-down approach, twisting the arms of ministerial aides who threaten to resign and promising jam tomorrow for his increasingly restive MPs. He has no alternative. Compensating the losers would be fiendishly complicated and abandoning the abolition of the 10p rate would cost the Treasury more than £7 billion a year.

The prime minister should think back to how this problem arose. It was when he was chancellor, desperate to go out on a high note from the Treasury en route to No 10 and seeking a flurry of favourable headlines. He wanted to be seen as a dynamic tax reformer by cutting the basic rate of tax to 20p, paid for by abolishing the 10p rate. Sadly for him, he did not even get the good headlines. That is where his anger should be directed.