 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 illeffects 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.
 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 socalled National Insurance Contributions (NIC), which earn entitlements to various state benefits such as unemployment pay and pensions. The incomerelated classes, with which we are concerned here, are levied on employees, their employers and selfemployed 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 selfemployed 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.
 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.
 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(1C.exp(1i/B))
`
`````````````where:` ` 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:
`
‘
`
`````````````where:` ` 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 
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.
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 zerotax amount.]
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%.

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(1i/B) – A.B – A.B.C
`
where:
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.
Income Tax Formula
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