Evolution of Risk-Statuses in One Model of Tax Control

Abstract

Nowadays information is an important part of social life and economic environment. One of the principal elements of economics is the system of taxation and therefore tax audit. However total audit is expensive, hence fiscal system should choose new instruments to force the tax collections. In current study we consider an impact of information spreading about future tax audits in a population of taxpayers. It is supposed that all taxpayers pay taxes in accordance with their income and individual risk-status. Moreover we assume that each taxpayer selects the best method of behavior, which depends on the behavior of her social neighbors. Thus if any agent receives information from her contacts that the probability of audit is high, then she might react according to her risk-status and true income. Such behavior forms a group of informed agents which propagate information further then the structure of population is changed. We formulate an evolutionary model with network structure which describes the changes in the population of taxpayers under the impact of information about future tax audit. The series of numerical simulation shows the initial and final preferences of taxpayers depends on the received information.

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Appendix

In Appendix we present additional information about probability distribution, used in this paper. Let’s recall that the density f(x) and function F(x) of the uniform distribution of the value X on the interval (b − a, b + a) are defined by the next way [13]:

$$\displaystyle \begin{aligned} f(x)= \left\{ \begin{array}{lcr} \displaystyle{\frac{1}{2a}}, & \qquad \mbox{if }\ & |x-b|\le a, \\ \vspace{-10pt}\\ 0, & \qquad \mbox{if }\ & |x-b|>a, \end{array} \right. \end{aligned} $$

$$\displaystyle \begin{aligned} F(x)= \left\{ \begin{array}{lcr} 0, &\qquad \mbox{if }\ & x<b-a, \\ \vspace{-10pt}\\ \displaystyle{\frac 1{2a}(x-b+a)}, &\qquad \mbox{if }\ &|x-b| \le a, \\ \vspace{-10pt}\\ 1, &\qquad \mbox{if }\ & x>b+a, \end{array} \right. \end{aligned} $$

The mathematical expectation MX of the uniform distribution is MX = b.

The Pareto distribution [13], which is often used in the modeling and prediction of an income, has the next density

$$\displaystyle \begin{aligned} f(x)= \left\{ \begin{array}{lcr} \displaystyle{ \frac{a b^a}{{x}^{ a+1}}}, &\qquad \mbox{if }\ & x\ge b, \\ \vspace{-10pt}\\ 0, &\qquad \mbox{if }\ &x<b, \end{array} \right. \end{aligned} $$

function

$$\displaystyle \begin{aligned} F(x)= \left\{ \begin{array}{lcr} \displaystyle{1-\left(\frac{b}{x}\right)^a}, &\qquad \mbox{if }\ &x\ge b, \\ \vspace{-10pt}\\ 0, &\qquad \mbox{if }\ &x<b, \end{array} \right. \end{aligned} $$

and the mathematical expectation $MX=\displaystyle {\frac {a}{(a-1)}} \cdot b$.

The scatter of income levels in the group with the highest income may be extremely wide. Therefore, as a value of parameter of the distribution we consider a = 2: higher or lower values significantly postpone or approximate average value to the lower limit of income.