The issue

In the UK Value Added Tax (VAT) is charged at 17·5%. government webpage. The basic rate of income tax is 22 pence in the pound. government webpage. Is it accurate to call this 22 percent? If VAT was increased to 22% would it then be at the same rate as income tax?

Tax is a wedge between what the buyer pays and the seller receives. There are three numbers in play, the wedge, the net, and the gross. They are all in proportion. The issue is whether to compute the tax rate as the ratio of the wedge to the net, or as the ratio of the wedge to the gross.

Value Added Tax

A shop keeper wishes to receive £100 for an item, so he calculates that VAT at 17·5% is seventeen pounds fifty pence and offers the item for sale at £117·50. His customer must hand over £117·50 of which £17·50 goes to the government. The customer can compute his own ratio 17·5/117·5=0.1489. He has paid a little under fifteen pence in the pound.

Income Tax

The employer has agreed to pay £100 for a days work. Deducting 22 pence in the pound, he gives £22 to the government and £78 to the worker. If the worker is familiar with Value Added Tax he may look at the £78 in his hand, at the £22 deduction on his pay slip, and calculate the percentage rate on the same basis as for VAT. He calculates 22/78 = 0.2820 and sees that he is being taxed at a little over 28%.

Easier Arithmetic

If you are simply checking my calculations with a calculator you may find that although they are correct, you cannot quite see what is going on. Try this example, which uses simpler numbers. Imposing tax of 25% raises prices by a quarter. So an item originally priced at 80p has its price raised by 80/4=20p. After the tax has been imposed it retails for 80p + 20p = £1.

Clearly twenty pence out of that pound goes to the government, so one says that a tax rate of 25% is a tax rate of 20 pence in the pound.

Political Impact

I want to rush on to describe the political impact of the choice of parameterization of tax rates, because I think it is large. However, I do have a degree in mathematics; I need to be careful not to leave my audience behind. Rather than rush on, I'm going to present some tables to bring out the practical differences between pence-in-the-pound and percentage-rate.

The first point is that the differences are unimportant for low tax rates. My first table converts from pence in the pound to percentage rate.

If one choses to describe a tax rate as a percentage rather than as a number of pence in the pound, the number would be slightly larger, for example 8.7 instead of 8.

Naturally enough it goes the other way, the other way round

If one choses to describe a tax rate as so many pence in the pound, the number would be slightly smaller, say 8.26 instead of 9.

In 1906 income tax was raised from one shilling to one shilling and two pence. In decimal money that is an increase from 5 pence in the pound to 5.833... pence in the pound. So a hundred years ago these two tables would have sufficed, and it would be difficult to see the point of my essay. Nowadays income tax rates are higher so one needs more tables covering the current range of rates.

Something dramatic is happening. As rates approach 100 pence in the pound, the percentage rate diverges to infinity.

The idea of the government taking everything, so that economic activity comes to a halt is described very differently in the two competing parameterisations.

Describing tax rates as a percentage spreads the less than total tax takes out along the whole real line from zero to infinity. Destroying the economy with total taxation doesn't have a number to describe it.

Describing tax rates as so many pence in the pound has a limited range, from zero to one hundred. It is unclear what larger numbers are supposed to denote. Usually a seller at an auction accepts the highest bid. If taxes are more than one hundred pence in the pound, may the seller avoid tax by accepting a lower bid? May sympathetic buyers spare the seller a large tax bill by declining to make high bids?

What happens when we try to create a conversion table that goes the other way, from a percentage rate to pence in the pound? We face the problem of dividing up the range from zero to infinity.

One of the problems with taxation is that it tends to discourage economic activity. Throughout history Kings and Princes have become greedy and raised taxes too high, leaving them with a shrunk economy and an empty treasury. More recently politicians have accepted that income taxes, such Income Tax, Employers National Insurance Contribution, and Employees National Insurance Contribution tend to push the economy towards higher unemployment, and have favoured indirect taxes such as VAT.

Moralists have noticed that this problem with taxation can be turned to advantage. When they cannot have a vice forbidden, they can focus on the taxes that are paid on it. Increasing the taxes suppresses the vice.

Notice that if the vice is suppressed, then the tax take falls. This leads to the notion of punitive taxation, a tax levied not to raise revenue but to suppress vice. The concept is a vague one. If the people are attached to their vices they make pay high taxes. The government may grow so dependent on the income that it loses its zeal to suppress the vice.

A complication arises when a product such a petrol for fueling motor cars is very cheap in relation to its usefulness to the consumer. In this case the government can levy a revenue raising tax limited only by the usefulness to the consumer, and many times the cost of producing the product.

Since the notion of punitive taxation is vague attempts at precise definition must be somewhat arbitrary. I find it convenient to place the dividing line between ordinary, revenue raising taxation and punitive, revenue losing taxation at 100 percent. Then I can solve the problem of choosing a range of tax rates for my table by ignoring punitive taxation and tabulating revenue raising rates of tax.

Notice how it tops out at fifty pence in the pound. The arithmetic is uncontroversial. If tax doubles the price, half of the price goes to the government in taxation. The politics is controversial. Income Tax is often raised above fifty pence in the pound in the hope of raising revenue to spend of schools and hospitals.

Let us return to the task of drawing up a conversion table from percentage-rate to pence-in-the-pound. The technical question is how to divide up the range from zero to infinity so that we are tabulating values of interest. First look at a table going the other way that zooms in on the politically contentious rates of taxation

A standard idea for solving this kind of problem is to use a logarithmic scale. Here I use stick close to taking logarithms base cube root of ten. Each row of the table shows a percentage-rate approximately 2.15 times the percentage rate in the previous row.

Political Impact One

One of the most intrigue ing aspects of the Thatcher era was the reduction of the top rate of income tax from 83 pence in the pound to 40 pence in the pound. Did this cost the Treasury a lot of money? Over time the take from the higher rate of income tax increased, and it is claimed that it actually doubled.

When ones sees this claim ones subconscious goes off on an emotionally tempting chain of illogic. First it notes that 83 is about twice 40 and it takes this two to be real and not merely an artifact of the choice of parameterization. Second it fancies that halving the tax rate must halve the tax take. Finally it looks at the claim that the tax take doubled and concludes that some-one must have got the ratio the wrong way up. The notion that halving the tax rate doubled the tax take is always tainted by the feeling that it is too glib and must be the wrong way round.

Consider on the other hand the arithmetically identical and superficially very different claim that Thatcher cut income tax from 488percent to 67percent and that this doubled the tax take. It is clear that 67% is a high rate of tax, you could take a lot of money at that rate. It is also clear that 488% is a punitive rate. People will move abroad. People will take early retirement. People will go part time. People will evade it. People will avoid it. While 67% might be self defeating, (you might get more revenue at 50%), you can hardly make that argument about 488%. It might not raise very much money, but no-one would imagine that a tax rate of 488% is intended to raise money. It is clearly intended to discourage.

Looking back I don't think that ordinary people actually realised the difference between 83% and 83 pence in the pound. 83% is only 45 pence in the pound. You can raise revenue with it. 83 pence in the pound is 488% You use it for social engineering, to drive class enemies into tax exile, to bring the era of capitalism to a long overdue close. Did the people of Britain in the 1970's really opt for the second option or did they think 83 mumble?

Political Impact Two

When the Liberal Democrats suggest increasing the top rate of tax from 40 pence in the pound to 50 pence in the pound, how are we to think about this? How do we apprehend it as a matter of quantity? How do we get it into proportion? Is it a large change or a small one?

On way is to calculate 50-40=10 and say it is a change of 10 pence in the pound. Assimilating that to 10% it seems quite small.

That is not the usual way of calculating a percentage change. Traditionally the change is expressed in proportion to the old value ¹⁰/₄₀ = ¹/₄ or 25%. That does seem like a fairer assessment of the size of the proposed change.

Notice that we are looking at the change in the parameter that describes the tax rate. The parameterization is not unique. Arguing that income tax rates are best parameterised as percentage rates, we first convert the parameterization. 40 pence in the pound is 66 ²/₃% while 50 pence in the pound is 100%. So the change is 33 ¹/₃% percent, fully half of the original value. We are considering a 50% increase in tax.

Well, which is it 25% or 50%? Imagine that technological progress is raising wages. Some-one who used to get £100 thousand is now on £120 thousand. Income tax was 40 pence in the pound = 66 ²/₃%, so he was taking home £60 thousand. The government proposes that the national interest is best served if the windfall of technological progress falls to the government so that it can distribute it fairly across the whole of the population. Thus is raises income tax to 50 pence in the pound = 100%. Our hypothetical figure sees his £120 thousand divided, £60 thousand take home pay, as in the old days, and £60 thousand tax. Notice that the government is getting 60 thousand instead of 40 thousand, a 50% increase in anybody's reckoning.

I can see the attraction for government in saying that 40 to 50 is a 25% increase in tax. It makes them look awfully clever. They have only raised taxes by 25% but have brought in 50% more revenue. Alternatively it makes them look dishonest. They are taking 50% more money but only admitting to raising taxes by 25%.

Truth?

Which parameterisation is correct?

Consider first taxes on goods. In the percentage parameterisation one relates the tax wedge to the net. In a competitive economy prices of goods tend to be driven down towards their cost of production. It makes sense to think of the net as already being as small as it can be and hence fixed. In the pence in the pound parameterisation the tax wedge is related to the gross. Thinking of the gross as fixed suggests that a tax rise will not increase the price of goods, but will instead compel the manufacturer to take a loss. That is nutty. For goods the percentage parameterisation is the only one that makes sense (except for the point that the two parameterisations are similar provided taxes are low).

Second consider taxes on income. Here the subnarrative is of aristocrats with a fixed income from a country estate which is mostly spent on balls and hunts, or as we might say today parties and adventure holidays. A change to Income Tax from one shilling in the pound to one shilling and two pence in the pound means two pence less on balls and hunts and two pence more for widows and orphans.

Three modern subnarratives.

1. Modern business is engaged in a "war for talent". When the successful manager of a medium sized business looks for promotion to managing a large business he looks for a pay rise. As he looks around the world for his next challenge he is computing net pay. A company in a country with higher tax rates must offer a higher gross in order to offer a competitive net. It makes sense to think of the net as being fixed by international competition.

2. Poor people pay Income Tax. When we ask whether a person earning £8000 a year is being taxed at 22 pence in the pound or at 28.32% we are very likely asking about some-one with fixed outgoings, on rent, on food, on gas and on electricity. If it makes sense to parameterise a tax increase against a background of a fixed gross and a variable net do we also imagine our tax payer freezing in the dark, or coming out on strike for more pay?

3. The incentive structure of society is fixed by net pay. A young person must decide whether to pursue further education and if so, for what length of time. Some important career paths, such as architect, hospital consultant, partner in law firm or accounting practice, take decades. Is it worth it? If civilisation is to thrive then the net wages of theses top position must compensate for the long, disciplined slog to attain them. If the government wants to raise income tax, it must also raise the wages of hospital consultants, or leave subsequence governments to cope, when twenty years later, too many have dropped out of the career path, leaving a shortage.

I can see that the pence in the pound parameterisation makes a lot of sense as a piece of political spin. If two different parameterisations of tax rates are regarded as legitimate and one wishes to downplay tax rates it is practical politics to use the parameterisation with the lower numerical value. The confusion between the two parameterisations means that the rate of taxation is understood in an in between sort of way. Fifty pence in the pound creates mixed emotions from being 50-mumble in a world with 2 parameterisations. It is partly being felt as 50 pence-in-the-pound = 100% and partly being felt as 50%=33¹/₃ pence-in-the-pound.

This consideration of political rhetoric is symmetrical at lower rates, but asymmetrical at higher rates. If one is opposed to raising the rate of Income Tax to 50 pence-in-the-pound one can fight it by portraying it as 100%. Notice that you cannot expect voters to react to 100-mumble as though it were 100 pence in the pound, because 100 pence in the pound doesn't make sense. 100-mumble can only mean 100% = 50 pence-in-the-pound.

Spin aside, the pence-in-the-pound parameterisation is mathematically absurd. It distorts the quantification of tax rates above 100% to fit into a limited range from 50 to 100 pence in the pound, and for what end? To create a singularity at 100 beyond which the numbers have no clear economic meaning. The parameterisation creates a trap for the unwary for no purpose beyond trapping the unwary.