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5Taxation model


The importance of mathematical modelling is significantly increasing as people expand their abilities in translation of mathematical equations and formulating them into concrete conclusion in addressing issues in the contemporary world. According to Dilwyn and Hamson (1996), models are essential in describing our beliefs in regard to the functioning of the world. These beliefs are translated into mathematical language through mathematical modelling. This leads to various advantages. First, mathematics is regarded as a very precise language and as a result, mathematical modelling assists in formulating ideas, as well as identifying the underlying assumptions. Secondly, mathematics is a concise language based on well defined rules required for manipulations. Thirdly, mathematics has proved better results for many years. Finally, the computers are essential in performing numerical calculations in mathematical modelling. It is also important to understand the objectives that modelling can achieve. Mathematical models are used for various reasons which include developing scientific understanding, testing the effect of change in a particular system, as well as aiding in decision making such as strategic and tactical decisions by planners and managers (Armstrong et al, 2007).

According to Voskoglou (2006), the main goal of modelling is optimization or computation of the best solution. Thus, the process of modelling involves adding value under a sequence of various steps that are common in mathematical models. These are: problem identification, identification of relevant variables, simplification, relating the variables mathematically, and solving. The important final question is whether the solution have added value.

This essay involves a discussion of a mathematical model on taxation with reference to an example of a income tax model in United Kingdom.

The efficiency of a system of revenue collection is based on a complex but careful structure and adjustment of various taxes, their rates, bases, deductions and exemptions. Optimal ratio of taxes leads to adequate sources of revenue for authorities (Creedy, 2004). Increase in tax base and general tax reform change the proportion of invest-consume in all sectors. Selective reform on taxes and tax exemptions stimulate several sectors, but damages fairness in taxes. This explains the importance of tax system analysis (Chander, &​Wilde, 1996). UK is facing challenges dealing with sovereign debt due to global financial crisis. The issue is to come up with a model that reduces the federal budget deficit and amount of debt, where taxes are the main instrument of implementing the recovery policy (Ivanitskaya, & Tregub, 2011).

Details about the income tax model

Not all the income in UK is subjected into tax. The main forms of taxable income are from unincorporated businesses, self-employment, earnings from employment, and interests from banks, property, as well as dividends on shares. However, interests from most means-tested and non-means-tested benefits are not liable to income tax. Income tax from certain savings products is not also paid. This model is aimed at providing an understanding in regard how taxes change on various factors. The model is a representation of an overview of a taxation system of UK, that the variables, recent fluctuations, as well as forecast of occurrence of future changes policy (Ivanitskaya, & Tregub, 2011).

The model will forecast the endogenous variable which in this case is UK income tax revenue. There are independent factors that provide the foundation for analysis of the relationship, also referred to as exogenous variables. They include the UK population paying taxes (total number of people levied on taxes), inflation (reflection of real money value based of various time period), the prices of oil (a variable that changes the value of all economic indices globally), and growth of GDP (represents the whole value UK economy, economic dynamics ad conditions as well as its current situation) (Ivanitskaya, & Tregub, 2011).

All these variables affect the whole economy, thus, in order to ensure optimisation in our model, estimating the relationship between taxation (endogenous variable), and the population of the country and annual level of inflation (exogenous variables) is important (Chander, &​Wilde, 1996).

Interpretation of results and conclusion

UK income tax revenue depends of four factors which are: GDP growth rate, oil prices, retail price index (component of inflation), and number of taxpayers. These factors are regressors and they influence UK income tax revenues (endogenous variable).

Mathematically, the model is as follows:

Y- Income tax revenue (UK)

— No. of taxpayers1X

— Annual PRI2X

— World oil prices3X

— GDP growth4X

The model after the definition of dependence of external factors based on scatter diagrams and correlation matrix become:

tε+3t-761,37X2t+2921,68X1t= -302970,54+16,3Xt refer the attachmenty

(35714,44) (1,39) (835,61) (169,66) (12246,48)

R² adjusted = 0,91

F = 105,84

F critical = 2,96

After testing the model, it shows that it would give an exact right result in 85% cases.

The model is adequate and trustworthy and it is applicable for forecasting. In regard to forecasting, the future values (of all exogenous variables) need to be extracted by estimating them as effective variables. By having the future values in based on independent variables, they will be used in prediction of further dependencies and fluctuations of the UK income tax revenue (endogenous variable) policy (Ivanitskaya, & Tregub, 2011).


Income tax, VAT and contributions from National Insurance are three largest sources of revenue to UK government.

). In regard to the number of taxpayer, the relationship is clear between the endogenous and exogenous variable. That is, the more the number of people paying taxes, the more revenue in tax receipts (Armstrong et al, 2007). In regard to retail price index (key inflation index), rise in inflation leads to depreciation of money, and then the average level of income taxation increases. In regard to oil prices, the dependence is subjected to other factors such as inflation. This is because the relationship between world oil prices and tax revenues can be negative in some cases. For instance, when oil prices reflect high economic activity, there will be an increase in money. Then, the increase in income tax can be an element of a policy implemented for decreasing money turnover and as well reduce a possibility of inflation. This explains the reason for a negative dependence in this case.3), and oil prices (X2), retail price index (X1The income tax revenue model assists in the analysis of external factors depended upon by the income tax revenue. The three factors affecting this revenue are: number of tax payers (X

Finally, it is worth noting that this taxation model provides an opportunity to forecast future fluctuations of income tax revenue (endogenous variable) (Voskoglou, 2012). This is very important in current economic uncertainty in the world.


Armstrong, A, Draper, N, Nibbelink, A, & Westerhout, E 2007, Fiscal prefunding in response to demographic uncertainty, CPB Discussion Paper 85, CPB Netherlands Bureau for Economic Policy Analysis.

Chander, P &​Wilde, L 1996, A general characterization of optimal income tax enforcement, National University of Singapore, Dept. of Economics and Statistics, working paper, no. 97/​98-11

Creedy, J 2004, “The excess burden of taxation”, Australian Economic Review, vol. 37, no. 4, pp. 454-464

Macmillan: London, Mathematical Modelling Skills 1996, Dilwyn E, & Hamson, M

Ivanitskaya, S, & Tregub, IV 2011, Mathematical Model of Income Tax Revenue on the UK example, accessed on May 13, <>

Voskoglou, MG 2012, “Mathematizing the Process of Analogical Reasoning”, Journal of Mathematical Modelling and Application, Vol. 1, No 7, pp. 58-69