Next Monday, the British government faces its most serious parliamentary revolt since Gordon Brown became Prime Minister.  The issue is the abolition of the 10p lower rate of income tax.  Mr Brown announced its withdrawal in his final budget a year ago and the change came into effect three weeks ago.

We have been here before.  This is the third occasion in thirty years on which a lower starting rate of tax has been introduced to help the low paid, and then abandoned.  The aim is laudable, the measure ineffectual.

The proposal reflects misunderstanding which is easily resolved by anyone who was taught calculus well at school – not very many people in Britain, unfortunately – or who paid close attention in economics 101 – even fewer people.  The rates of tax printed in ready reckoners, or announced by Finance Ministers, are marginal rates of tax.  The marginal rate is the extra tax payable on an extra pound of income. But it is not the marginal rate of tax that determines whether a tax change makes you better or worse off.  That depends on the average rate of tax – the relationship between your total tax bill and your total income.

The lesson of elementary calculus is that average and marginal rates are quite different, and can move in opposite directions.  The lesson of economics 101 is that both average and marginal rates influence behaviour, but in different ways.  The implication of these two observations together is that the people who gain most benefit from a low initial rate of tax are never the people for whom it is the marginal rate.

Last year, everyone received a tax free allowance of £5225.  After that, the first £2320 was charged at 10%, the next £32280 at 22%, and the balance at 40%.  3.8 m people paid tax at a marginal rate of 10%, 23.2m faced a marginal rate of 22%, and 3.7m paid 40% on the last slice of their income.  From April the tax free allowance is £5435 and the next £36,000 is charged at 20%.

The person who loses most from this reform is someone who earns just enough to benefit from the new, lower, basic rate of 20%.  If you are paid £8000 a year your marginal tax rate has just fallen  from 22% to 20%.  But the amount of tax you pay has risen by £220 per year.  

If you did tax at 10%, you lose less:  an individual on £6000 a year is worse off by only £40 per year.  And the biggest gainers from the changes are those who paid tax at the higher rate and still do.  Someone on £45,000 a year finds their tax bill reduced by about £15 per week.  

The outcomes are counter intuitive.  And that is why the changes received little attention when they were announced: yet  when the outcomes became visible in pay packets, they caused consternation in the Labour Party.

The Chancellor cannot now restore the 10% rate because such a move would be too costly:  the cost is £8 billion.  The reason restoring the lower tax band would be so expensive is that the measure would give over £200 to 90% of taxpayers – to all of them, in fact, except the lowest paid.   The lesson is to think before you act.

The first mistake was to introduce the 10% rate.  Ten years ago, Gordon Brown as Chancellor set that rate for the first £1500 of income.  He should instead have raised the personal allowance by £800. The cost would have been much the same, the tax structure would have been simplified, some low earners would have been taken out of tax, and the benefits would have gone to the low paid as well as the better off.

The second mistake was to abolish the 10% rate in order to cut the basic rate of tax.  The measure wrong-footed the opposition and won favourable headlines – for a few days.  The Chancellor’s advisers assured him, wrongly, that a few tweaks to the benefit system would deal with the problems that resulted.  

Do not tinker with the tax system for short term political advantage.  Tax is always more complicated than you think and the results come back to haunt you.