Peter Diamond and Emmanuel Saez have published a paper which includes a calculation of the "optimal top marginal tax rate" in the USA, on the assumption that the only criterion is to maximize revenue - there is negligible social utility in letting rich people keep their income. The calculation has attracted considerable interest from on-line commentators. Nobel laureate Paul Krugman writes in the New York Times in defence of the criterion. Brad DeLong observes that Adam Smith disagrees, partly because the rest of us take vicarious pleasure in the rich enjoying their wealth. Richard Green fears that higher taxes on high earners might cause them to pay their servants less. Kevin Drum reports with evident satisfaction that according to one number in the paper the peak of the Laffer Curve is at a (US) Federal income tax of 76%, far above the current top rate of 35%.

In the UK, the "#1 economics blogger" Richard Murphy quotes Drum at length, and concludes that we are comfortably below the peak of the Laffer Curve. Murphy is not one to concern himself with details, but he seems simply to be noting that the top UK tax rate of 50% is a lot less than the 76% he's quoted.

Meanwhile, the UK's leading libertarian scandium oligopolist, Tim Worstall, asked his readers to calculate what tax rate in the UK, including employers' and employees' National Insurance and VAT, would be directly comparable with the tax numbers used by the paper for the USA, which includes the Medicare tax and state income and sales taxes. He used the analyses they (I might say 'we') submitted to declare that Murphy is wrong (that's always Worstall's conclusion) and that the true optimum top UK income tax rate is 40%.

As you might expect, I am going to adjudicate. First, an outline of what Diamond and Saez actually did: They assume, in line with extensive empirical research, that income at the top end follows a Pareto distribution, that is it has a probability density function falling off according to a power law, p(z) = C/z^(1+a). They find empirically that the parameter 'a' in the USA is 1.5 . Then they assume, following various other authors, that taxable income is an elastic function of retained earnings (in the economic sense of 'elastic', i.e. a given change in the logarithm of the fraction of marginal income not taken in tax results in a proportional change in the logarithm of taxable income reported). This is a convenient assumption, in that it means that a change in marginal tax rate leaves the power law shape unchanged, affecting only the value of 'C'. As the tax rate increases the fraction of income retained falls, so that a given change in tax rate has a larger proportional effect on the retained fraction, and hence a larger effect on taxable income reported (e.g. a tax rate change from 0% to 1% takes away one hundredth of your net income, but a change from 90% to 91% takes away a tenth). So with the elasticity assumption there is a critical tax rate at which the reduction in taxable income when the tax rate is increased balances out the extra tax raised by the higher rate - this is the optimum rate, which turns out to be 1/(1+elasticity.a), as the mathematically inclined reader may care to prove. The difficulty now is to determine the elasticity parameter. They report a mid-range estimate from the empirical literature of 0.25, but go on to use figures from another paper, also co-authored by Saez, which finds an elasticity of taxable income for top earners of 0.57, but only 0.17 for 'broad income', which they define as "Total Income less Capital Gains [and] Social Security Benefits". The implication is that most of the elasticity is due to tax avoidance rather than reduced income.

How applicable is this to the UK? The Pareto distribution seems to hold quite generally, but the power law may be different: this paper reports a=1.37 in the USA and a=1.06 in the UK. I would expect the 'broad income' elasticity to be somewhat higher in the UK, because high-earning Britons are more likely than Americans to move overseas, if only because Americans are discouraged by the extraordinary geographical range of American tax laws. But I would expect the taxable income elasticity to be smaller in the UK, because there are fewer deductions available.

What taxes is it appropriate to include in a UK calculation? Income tax obviously, and the 2% top rate employees' national insurance contribution (it's the same for the self-employed). This corresponds to the 1.45% employees' Medicare tax included in the US calculation. Also included in the US calculation are the 1.45% employers' Medicare tax and 40% of average (state) sales taxes, which is 2.32% Analogously, we should include employers' NI of 13.8% and some fraction of the 20% VAT rate. But I'm unconvinced that this is right. The relevant taxes in the USA are quite small, so Diamond and Saez may have included them just to avoid argument. In the UK the issue is more important, and deserves some consideration. It seems to me that tax avoidance schemes are chosen by careful calculation of their benefits, but scaling of effort in response to tax changes is more emotional: I doubt that many people would think of employers' NI as a consideration there. However, the incidence of employers' NI is considered to be largely on the employee, so it may make working abroad relatively attractive financially. Regarding VAT, I doubt that much of the marginal income of high earners goes on goods subject to VAT. For the most part, a person earning well into six figures buys whatever retail goods they feel like already. And psychologically, paying tax when you buy stuff does not affect your attitude to earning money in the same way as having to hand more than half of it over to the government as you get it.

My rough numbers: a=1.25, broad income elasticity = 0.27, taxable income elasticity = 0.4, optimum combined marginal tax rate = 67%. Employer's NI contributing to elasticity effect = 2%, VAT contributing to elasticity effect = 5%, Marginal income tax rate net of 2% employees' NI to give 67% combined rate = 62%

There's a good bit of guesswork in the parameters I've used, so there's no reason why anyone else should get the same answer. But I think it's pretty hard to defend the choice of elasticities of either 0.57 in the UK (Worstall) or 0.17 in the USA (Drum).

In the interests of full disclosure I should say that I've paid tax at the 50% rate ever since it was introduced. I may not do so in the 2012-13 tax year. I can tell you that there's a psychological impact from direct taxes exceeding half one's marginal earnings: it's OK for you not to care.

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