The Taxpayers’ Alliance did the figures for us, and put the cost of the 50p tax^{1}at £4.5bn.

I've since looked into the detail of this claim.

It seemed unreasonable to base an article on analysis of income tax by a self-described campaign for lower taxes, without providing data and methodology. The TPA's campaign director replied to my tweet, directing me to a recent book they'd published. That book did contain some more details (page 9/10), and it provided a reference to an IFS report on the 2009 budget.

The analysis is based on standard economic assumptions that if you change the tax rate then you'll change people's behaviour. As the IFS report states, people may "work less, retire earlier, emigrate, contribute more to pension or charity, convert income to capital gains, incorporate, invest in tax avoidance, ...". It also considers people's behaviour with regard to indirect taxation (e.g. spending less on consumer goods, thus paying less VAT). (This is all a reasonable application of Laffer curves, which I dislike as a general economic argument, but seems applicable when looking at a limited number of parameters.)

The direct taxation component is summarised by a parameter called "Taxable Income Elasticity" (TIE), an economic elasticity measuring change of income in relation to change in tax. The IFS have another useful report on the subject which goes into detail on the TIE.

This means that when the net-of-tax rate (that is to say, one minus the marginal tax rate) falls by 1% of its original value, then the taxable income of the very rich falls by 0.46%. For example, suppose the marginal tax rate was originally 50% and increased to 50.5%. Then the net-of-tax rate would have fallen from 50% to 49.5%, or a fall of 1%, and the elasticity of 0.46 implies that the taxable income of the very rich would decline by 0.46%.

Essentially, the income raised by increasing taxation is very sensitive to the TIE value, which for the rich (i.e. >£150,000 income) is very variable. The best data is from the 1980s (when tax rates for the rich last changed), which gives an estimate of 0.46, with a ±1 standard deviation confidence interval of 0.33 to 0.59.

The treasury has assumed a TIE for the rich of 0.4, reducing to 0.35 after the pension changes in the 2009 budget make it harder for the rich to reduce tax liability by investing in pensions. The IFS say, in the report that the TPA link to, "Government’s assumption not unreasonable" (however, they do then raise some concerns over the estimate).

The IFS show that if TIE is 0.46, then a 40% tax rate is optimal for maximising income. If TIE for the rich is 0.35, as the treasury assume, then a 50% top-rate tax will raise money. That value is on the low side, but still within 1σ of the mean. If it's closer to 0.46, then it will cost a small amount, and the higher it is, then the more it will cost.

All of which leads back to the question of where the Spectator/TPA's £4.5 billion cost of the 50% tax rate comes from. This figure is the highest of the three presented in the TPA's book, with a TIE of 1.00. That is, a reduction in income of 1% for every 1% increase in marginal tax rate. This is justified in the book by the sentence "Research by the HMRC has suggested other potential values for the TIE even higher than those used by the IFS", with a reference note saying "Response to a Freedom of Information request from HMRC and author's calculations". There's no reference to check the context for the HMRC considering a TIE of 1.00.

Even ignoring the rather extreme implications of that assumption, from the IFS data a TIE of 1.00 is over 4 standard deviations from the mean. There is a less than 0.006% chance of a value lying outside a 4σ confidence interval (given a normal distribution, which I'm assuming from the IFS's reference to a ⅓ chance of the TIE being outside a 1σ confidence interval).

In summary, it seems very unlikely from published research that the 50% tax rate will cost £4.5 billion, and it's disingenuous of the Spectator to base an article on such an extreme value. However, I've now a better appreciation of the considerations around raising money by increasing tax on the rich, there does appear to be a rather delicate balance to strike to optimise income, and relatively little hard data to make this decision on.

Of course, none of that takes into consideration any social effects of higher rate taxes for the rich. I can imagine that some people may act in an economically irrational way and support higher taxes for the rich even if it cost everybody else more. Obviously, there will be opposing views to that too, and the overall social effect of all that is rather unknowable.

While reading up on this, I found a couple of papers and a Powerpoint presentation from the IFS useful for background reading.