Income Tax Formula

  1. Introduction

    This post is concerned purely with the mechanics of deriving the rate of income tax to be applied, and not with any discussion of the relative merits of income tax as compared with other forms of tax.

    The purpose of this post is to explore ways in which income tax rates can be derived by algebraic formulae to avoid the discontinuities and other ill-effects associated with the more usual system of tax bands. Various formulae are derived and typical results of their use are shown in graphical form. One particular formula is then selected for recommended use, and is described in the main tax discussion. I show comparisons with the existing (at the time of writing) UK tax system, not because it is typical but because it is the only one with which I am familiar and it is almost certainly as complex as any.

  2. The UK system

    Income tax consists of several separate elements in UK, only one of which actually carries that name! The others are some of the different classes of so-called National Insurance Contributions (NIC), which earn entitlements to various state benefits such as unemployment pay and pensions. The income-related classes, with which we are concerned here, are levied on employees, their employers and self-employed people. The employers’ payments affect the employee only indirectly, since they increase the cost of employing him/her, and I shall ignore them in this discussion for simplicity. Since the self-employed payments are quite different from those for employees, I shall also ignore them here. I shall, therefore, compare with my alternative formulae the result of applying the existing (at the time of writing) UK income tax plus employees’ National Insurance Contributions. I shall also ignore the complexities of different rates being applied to certain kinds of unearned income such as interest and dividends.

    The UK income tax system (as at 2006), after deducting allowances dependent on personal circumstances, applied three different rates of tax to different bands of income, 10% on the first £2,150 earned, then 22% on the next £31,150 and 40% on the remainder.

    The employee’s NIC were 0% on the first £5,035, 11% on the next £28,505 and 1% on the remainder. This is a simplified explanation, but is adequate for our purpose here.

  3. Requirements

    We require a system which applies a progressive system of tax rates and can be easily adjusted both in terms of the total tax revenue to be achieved from it and the degree to which it is progressive. It is also obviously necessary that the rate should never exceed 100%, and the formula should be capable of having set a minimum income level below which no tax should be levied. In a genuinely progressive system not only should the total tax levied increase as income increases, but so should the marginal rate.

  4. Derivation of the formula

    I looked for a formula for which both the marginal and total tax rates started low and increased asymptotically towards a maximum. Starting with the marginal rate, I came up with:


    r = A(1-C.exp(1-i/B))


    r =` marginal tax rate`
    i =` income`

    A, B and C are parameters available to make adjustments.


    From this the corresponding formula for the total tax can be obtained by integration:

    `Equation for tax rate


    T =` total tax rate`
    i =` income`

    and K is a constant chosen to make T = 0 when i = B
    Hence K = A.B + A.B.C

  5. Application of the formula

    The first point to note in using the formula is that the parameters A, B and C have specific, easily understood functions. A is the maximum tax rate to be applied to the highest incomes (marginal and total rates). B is the level of income at which tax is to be applied. C is an adjustment used to determine the overall level of tax, and hence the total revenue to be raised, without changing the other two parameters. A purely hypothetical example is used in figure 1 to illustrate the use of the parameter C, where the maximum rate of tax has been set at the high level of 90% and the lowest income to suffer tax is £5,000. The graph for the 2006 UK system is also shown for comparison.

    Figure 1

    A better comparison with the 2006 UK system is shown in Figure 2, where the maximum rate is 41% (the same as in the UK system) and with C set to 0.60 to give a comparable total tax revenue. [The UK incomes have been increased here by £5,000 so as to correspond with this as a starting level – published figures for UK rates are based on taxable income, after deducting the zero-tax amount.]

    Figure 2

    Figure 3 below shows up very clearly the illogical nature of the UK system in its effect on marginal tax rates, with not only the various steps but also the serious anomaly between £38,310 and £38,350 where the marginal rate rises to 51% before falling back to 41%.

    Figure 3

  6. Conclusion

    The income tax payable by an individual (the only tax based on income) should be calculated from the formula:

    T = A.i + A.B.C.exp(1-i/B) – A.B – A.B.C


    T = total tax payable
    i = total income (before deducting tax free allowances)
    A = maximum tax rate payable on the highest incomes
    B = tax free allowance (which can vary between individuals)
    C = a fraction in the region of 0.6.

    The total revenue can be adjusted by varying any of these last three parameters, but it should be noted that reducing C puts an increased burden on the lowest earners (and C = 0 corresponds to a flat rate tax), while increasing A increases the tax burden on all tax payers while remaining highly progressive and is therefore in general preferable. The formula proposed avoids all the anomalies resulting from fixed tax bands and multiple taxes based on income, while remaining highly flexible in all significant ways.


“Carbon” emissions

I do wish people would stop talking about “carbon emissions” when they mean “carbon dioxide emissions”. Carbon is not a greenhouse gas. It is a solid, usually black, and harmless except in a relatively minor way when emitted as black smoke. Carbon dioxide is not even mainly carbon – by weight it is more than twice as much oxygen as carbon, so it would make (very slightly) more sense to talk about “oxygen emissions”!

Mississippi Personhood Amendment

I’m passing on “as is” the original post on this subject. The idea of giving a fertilised egg (foetus) which is not viable as a separate entity any rights at all seems madness to me. It isn’t a person until it can think like one and be aware of itself and its surroundings, and is entitled to only the same rights as an animal not to be subjected to cruel treatment (e.g. unnecessary pain) before that – and that means it is sufficiently developed to have a nervous system capable of feeling pain in order to rate as highly as a rat!

Original post follows:

Originally posted by gabrielleabelle at Mississippi Personhood Amendment
Okay, so I don’t usually do this, but this is an issue near and dear to me and this is getting very little no attention in the mainstream media.

Mississippi is voting on November 8th on whether to pass Amendment 26, the “Personhood Amendment”. This amendment would grant fertilized eggs and fetuses personhood status.

Putting aside the contentious issue of abortion, this would effectively outlaw birth control and criminalize women who have miscarriages. This is not a good thing.

Jackson Women’s Health Organization is the only place women can get abortions in the entire state, and they are trying to launch a grassroots movement against this amendment. This doesn’t just apply to Mississippi, though, as Personhood USA, the group that introduced this amendment, is trying to introduce identical amendments in all 50 states.

What’s more, in Mississippi, this amendment is expected to pass. It even has Mississippi Democrats, including the Attorney General, Jim Hood, backing it.

The reason I’m posting this here is because I made a meager donation to the Jackson Women’s Health Organization this morning, and I received a personal email back hours later – on a Sunday – thanking me and noting that I’m one of the first “outside” people to contribute.

So if you sometimes pass on political action because you figure that enough other people will do something to make a difference, make an exception on this one. My RSS reader is near silent on this amendment. I only found out about it through a feminist blog. The mainstream media is not reporting on it.

If there is ever a time to donate or send a letter in protest, this would be it.

What to do?

– Read up on it. Wake Up, Mississippi is the home of the grassroots effort to fight this amendment. Daily Kos also has a thorough story on it.

– If you can afford it, you can donate at the site’s link.

– You can contact the Democratic National Committee to see why more of our representatives aren’t speaking out against this.

NHS and This Government

While no one would claim it is perfect, the NHS is generally recognised as being one of the world’s best health services, in terms of universal availability, health standards and low overall cost compared to other developed countries. By comparison, the system in the USA is one of the worst in terms of both universal availability and overall cost. Before last year’s general election Cameron promised his government, if elected, would not embark on any major reorganisation of the NHS. Now Lansley claims that the bill which comes before the House of Commons on Tuesday, will “change it beyond recognition” – and the changes would push us very close to the failed USA system. This is a far more serious breach of promise than Clegg’s let-downs over tuition fees and VAT increase, but the media are concentrating on the relatively minor side-issue of abortion.


Why do ministers want to do this? I can only see one possible reason. They can see large sums of public money being spent, with very little of it being siphoned off into the pockets of their rich friends and relatives by way pf private company profits, and that is what they are determined to change. The changes in the bill will achieve just that.


The most pernicious clause in the bill is the one which removes the duty of the Secretary of State for Health to provide a health service. This clause will totally remove all democratic accountability for the NHS. Everyone should email their MP now to demand, ideally that the whole bill be scrapped, but certainly the removal of this particular clause.


This entry is being cross-posted to my Live Journal blog.