Effectiveness of the Cutoff Audit Rule and Inequality of Income

Transcription

1 α Effectiveness of the Cutoff Audit Rule and Inequality of Income by PISSAS DIMITRIOS a and KOTSIOS STELIOS b Department of Economics, National and Kapodistrian University of Athens, Athens, Greece. dpissas@econ.uoa.gr b Department of Economics, Division of Mathematics and Computer Science, National and Kapodistrian University of Athens, Athens, Greece. skotsios@econ.uoa.gr Abstract Previous analysis in the theory of tax compliance has shown that the implementation of the cutoff audit rule produces vertical inequality in the tax system and constitutes more unequal the distribution of income in the economy. However, the ability of the cutoff audit rule to raise the government s tax revenue is affected by the distribution of income, since the cutoff audit policy targets at specific income groups. In this context, the question that arises is how the change in the inequality of income affects the effectiveness of the cutoff audit rule. On this account, the present study reveals that in a lognormal economy, the growth of the inequality of income increases the effectiveness of the cutoff audit rule, despite the fact that the extent of tax evasion in the economy increases. Consequently, the effectiveness of the cutoff audit rule is supplied by the implementation of the cutoff audit rule in the economy. Jel Classification: Keywords: Cut Off Audit Rule, Effectiveness, Gini Coefficient, Lognormal Distribution [1]

2 Introduction Tax evasion is one of the major problems facing the governments of modern states. Governments are trying through the audit policy to limit the extent of tax evasion and to reduce the negative effects created in the collection of tax revenues. The basic policy instrument for the confrontation of tax evasion is the Audit Rule. Audit rules determine the way by which the government selects the taxpayers for a tax audit and imposes penalties on the violators of tax legislation. Among the class of the audit rules, the Cutoff Rule takes the most prominent position 1. The cutoff audit rule belongs to the category of endogenous audit rules since its probability of detection depends upon the taxpayer s reported income. In the most common case, the cutoff audit rule imposes a threshold or cutoff income and a policy to audit any report below the threshold with a sufficiently high probability of detection, but to leave all reports above the threshold unaudited. According to scientific research on tax compliance 2, a cutoff audit rule is able to generate more compliance than any other audit rule since it provides incentives for taxpayers to report their incomes truthfully 3. Moreover, a cutoff audit rule produces at least as mush revenues as the random audit rule, especially in the case where the taxpayers are risk neutral 4. Finally, it creates horizontal equality by treating taxpayers who have the same income identically, both before and after the tax audits 5. On this account, the cutoff audit rule is considered to be the most effective audit rule. Consequently, the audit policy must always be based on the cutoff audit rule or, in other words, the government must select for a tax audit only low income taxpayers, that is taxpayers who report an income lower than the cutoff income in the economy. Despite its advantages, the cutoff audit rule produces vertical inequality in the tax system because dishonest high income taxpayers never get audited 6. Hence, the implementation of it reduces the proportionality of the tax system and constitutes more unequal the distribution of the total income 1 The cutoff audit rule was firstly introduced in the Theory of Tax Compliance by Reinganum & Wilde (1985). 2 See Andreoni, Erand, and Feinstein (1998) for a literature review in theory of tax compliance. 3 As Alm, Cronshaw and McKee (1993) have proven, the cutoff rule is more effective in increasing compliance than the conditional future or the conditional back or the random audit rules. Particularly, A cut off rule is able to generate compliance in excess of 80 percent, although this rule requires that a substantial number of individuals are audited. 4 As Reinganum & Wilde (1985) have showed, for any given income distribution, the cut off audit rule produces at least as much revenues as the random audit rule. Hence the cut off audit rule weakly dominates the random audit rule. 5 See Reinganum & Wilde (1985). 6 See the studies of Scotchmer (1987), Sanchez & Sobel (1993), Cremer, Marchand & Pestieau (1990), Chander & Wilde (1998). [2]

3 in the economy. However, the ability of the cutoff audit rule to raise tax revenues is affected by the distribution of the total income. This came from the fact that the tax audits targets at specific income groups and particularly at low income taxpayers who have income lower than the cutoff income. Consequently, the distribution of the total income among those who have lower and higher than the cutoff income appears to affect the effectiveness of the cutoff audit rule. Based on the above, the question whish arises is how a change in the distribution of income among the taxpayers in the economy, which may or may not be caused by the audit policy, affects the effectiveness of the cutoff audit rule. The present study intends to clarify the relation between the effectiveness of the cutoff audit rule and the degree of income inequality that characterize the economy before the implementation of the audit policy. Following Reinganum & Wilde (1985), the effectiveness of the cutoff audit rule is measured by the amount of net revenues that government expects to collect from the implementation of the optimal cutoff audit policy. Moreover, as the tradition in the literature on income distribution espouses, the degree of income inequality is captured by the value of the Gini Coefficient. Based on an economy where the income of the taxpayers is lognormally distributed, numerical analysis draws the following results. Firstly, the growth of the inequality of income does the audit policy more aggressive by increasing the value of the optimal cutoff income. However, the extent of tax evasion in the economy increases, as the number of taxpayers who honestly reports their income decreases. Despite of this, the percentage of the total income possessed by taxpayers who are not selected for a tax audit is increased and the cost of the tax audits declines. Consequently, in a lognormal economy the growth of the inequality of income increases the effectiveness of the cutoff audit rule. Combining the results of the present study with the results of previous research on tax compliance reaches the following conclusion. On the one hand, the effectiveness of the cutoff audit rule is improved by the growth of the inequality of income, despite the increase of the extent of tax evasion. On the other, the implementation of the cutoff audit rule constitutes more unequal the distribution of income. Therefore, the effectiveness of the cutoff audit rule is supplied by the implementation of the cutoff audit rule. This result should be carefully concerned, especially form those who support the cutoff audit policies. Although the continuous application of the cutoff audit [3]

4 rule would increase the government s tax revenue, it would also increase the extent of tax evasion and the inequality of income in the economy. The analysis proceeds as follows. Section (1) presents the distribution characteristic of the lognormal economy, such as the Lorenz Function and the Gini Coefficient, and justifies the reason behind the choice of it. In Section (2), the tax and audit policy of the government is introduced and the index of the effectiveness of the cutoff audit rule in the lognormal economy is developed. In Section (3), the numerical analysis is conducted and the basic results are derived. Finally, future research topics that could expand this study are provided in Section (4). Proofs of the theorems and propositions of the above sections are presented. 1. Lognormal Distribution and Gini Coefficient The majorities of the empirical studies in the literature on income distribution adopt the lognormal assumption, according to which the income is supposed to be lognormally distributed among the agents in an economy 7. The need for such an assumption comes from the fact that empirical estimation on the dispersion of income for a country requires micro-data that are generated by household surveys conducting usually by the country itself. Since these surveys take place for some counties and for some years and tend to be published very sparse summaries of their results, most studies assume that income follows the lognormal distribution 8. In this context, the present study considers an economy where the income of each taxpayer is supposed to be a random variable, independently and identically distributed according to the Lognormal Probability Density Function (1) 7 See the studies of the Harrison (1981), Cowell (1999), Bouguignon (2003), Kalwij and Verschoor (2006), Lambert (2009). See also Bandourian et al. (2003), Bresson (2009), for a contrary view regarding the lognormal assumption. 8 The Gibrat (1931) was the first who introduced the lognormal assumption by showing that if individuals experience random proportionate shocks to income, the distribution of income will converge to being lognormal in the long run. Since then, many empirical studies have verified that empirical data fit well the lognormal distribution. One of the most recent is the study of Pinkovskiy and Sala-i-Martin (2009) who by combining survey household data obtained by the World Income Inequality Database (UNU-WIDER 2008), have showed that the lognormal distribution provides superior fit in comparison with the gamma and the Weibull distributions, for the 98% of the world population. [4]

5 The is the income of the taxpayer and the and are parameters of the lognormal distribution. The Cumulative Distribution in the lognormal economy is given by the function (2) where the is the Gaussian Distribution or Error Function, and. For a given level of income, let, the value of the cumulative function represents the portion of the population of taxpayers who has income lower than or equal to, while the value of the density function represents the portion of the population of taxpayers in the economy who has income equal to. Furthermore and given the values of the parameters and, the proportionality of income in the economy is measured by the Lorenz Function, which in the case of the lognormal economy is 9 (3) The function is the inverse of the Gaussian function. The Lorenz Function gives the percentage of the total income possessed by various portions of the population, when the taxpayers are ordered at the size of their income. For example, if and, the value of the Lorenz function shows the portion of the total income which is possessed by the portion of the taxpayers with income. Finally, the degree of income s inequality is given by the Gini Coefficient that in the lognormal economy is 10 (4) 9 See proof A.1 in the appendix. 10 See proof A.2 in the appendix. [5]

6 The Gini coefficient calculates the extent in which the distribution of income among taxpayers within an economy deviates from a perfectly equal distribution. The Gini Coefficient ranges between to, that is. If, there is perfect equality of income in the economy (everyone has the same amount of income), while if, there is perfect inequality (one person has all the income). Graphically, the Gini coefficient is represented by the area between the Lorenz curve and the Line of Equality (45 o ). The calculation of the value of the Gini Coefficient requires the specification of the parameters and of the lognormal distribution. In empirical studies, these values are statistically estimated according to the collected micro-data. Nevertheless, the values of and in the present study are produced by a computational algorithm based on the assumption that the mean income is always constant and equal to a particular positive number. The set of the generated values is presented in Table 1 (see appendix B.1). 2. Lognormal Distribution and Cutoff Audit Rule Regardless of the degree of income inequality, the government asks the taxpayers to report their incomes in order to pay an income tax. The tax owed by each one is given by the tax function (5) where is the tax rate, and is the taxpayer s reported income. This tax system is called proportional with respect to the reported income because no matter the level of the reported income the tax rate holds steady. The government, in order to verify that the taxpayers report their true incomes, conducts tax audits. It is assumed that the tax audits are carried out on the basis of the cutoff audit rule, which implies that the probability a taxpayer to be selected for a tax audit is given by the function (6) [6]

7 The is the threshold or cut off income, the is the penalty rate and the is the probability of detection which ensures that the taxpayers will truthfully report their income 11. According to the function 6, each taxpayer who report an income lower than the cutoff income faces a probability of detection whereas each taxpayer who reports an income higher than or equal to the cutoff income faces a probability of detection. Since the penalty rate is always more than unity the probability is less than unity, implying that the government does not have to audit the whole population of taxpayers with income in order to ensure truthful reporting. Finally, the government imposes penalties on those who were found to report less than their true income. It is assumed that the penalties owed by the evaders are given by the penalty function (7) The is the penalty rate, and the is the evaded tax. As long as the penalty rate stays at the same level throughout the variations of the evaded tax, the penalty system is called proportional. The selection process of the cutoff audit rule forces the taxpayers to a specific reporting behavior that is described by the optimal reported function (8) According to the function 8, taxpayers with income less than the cut off income will report their true income to the government, whereas taxpayers with income greater than or equal to the 11 See proof A.3 in the appendix. [7]

8 cut off income, will report an income equal to the cut off income 12. Consequently, the expected amount of tax evasion in the first income group is zero, while the expected amount of tax evasion in the second income group is positive and equal to. Given the taxpayers reporting strategy, the task of the government is to determine the optimal cut off audit policy. This involves the maximization of the Expected Net Revenue Function with respect to the cut off income. When the tax audits are conducted according to the cutoff audit rule, the expected net revenue function is defined as follows: (9) where is the cost per tax audit and is the highest possible level of income in the economy. The first integral expresses the net revenues that government expects to collect from the taxpayer with income less than the optimal cut off income who report their true incomes, whereas the second integral gives the revenues that government expects to collect from the taxpayers with income greater than or equal to the optimal cut off income who report the cut off income. The condition for the determination of the optimal cutoff income in the lognormal economy is 13 (10) Where and. Finally, considering the value of the optimal cut off income, the government determines the optimal amount of net revenues that expects to collect, which represents the index of the 12 See proof A.4 in the appendix. 13 See proof s A.5 and A.6 in the appendix. [8]

9 effectiveness of the cutoff audit rule. In a lognormal economy the Index of the Effectiveness of the cut off audit rule is given by: (11) The function is called the complementary error function Numerical Analysis Given the values of the Gini Coefficient in Table 1, it now remains to calculate the optimal cut off income and the index of the effectiveness for each pair of. However, the optimal cut off audit policy cannot be determined by algebraic methods because the optimality condition 10 involves the cut off income in an essentially non algebraic way. Hence, calculations must be based on numerical approximations derived from numerical analysis. In order to use the method of numerical analysis, specific values for the tax rate, the penalty rate and the cost per tax audit should be adopted. The statistical values of those parameters vary from country to country and from year to year. On this account, the present study uses different set of hypothetical values in order to cover several real cases. Specifically, it is assumed that the average tax rate in the economy can be, each of which corresponds to a different level of taxation, that is, and of the reported income, respectively. Moreover, it is supposed that the penalty rate derives the values, each of which represents a different level of fine, that is, and of the evaded tax, respectively. Finally, it is considered that the cost per tax audit takes the following values, where each one corresponds to different percentage of the mean income, that is, and, respectively. 14 See proof A.7 in the appendix. [9]

10 The basic result of the numerical analysis is that in a lognormal economy, the growth of the inequality of income implies the increase of the effectiveness of the cutoff audit rule. The positive relation between the Gini coefficient and the index of the effectiveness is depicted by the data of Table 2 (see appendix B.2). It is apparent from this table that in each set of parameters values, the Gini coefficient and the index of the effectiveness are positively related. For instance, if the penalty rate is, the tax rate is and the cost per tax audit is, the rise of the Gini coefficient from to implies the increase of the index of the effectiveness from to. Similar results produced by each set of. However, the question that arises is why the effectiveness of the cutoff audit rule is positively related to the inequality of income in the economy. To understand the causes for this result, the impact of the change of the inequality in the value of the optimal cutoff income, the cumulative function and the Lorenz function should be examined. Regarding the optimal cutoff income, the numerical analysis reveals that in the lognormal economy, the growth of the inequality of income results to the rise of the level of the optimal cutoff income. According to the data of Table 3 (see appendix B.3), for each set of values of the parameters, and, there is a positive relation between the Gini coefficient and the optimal cutoff income. For example, when the penalty rate is, the tax rate is, the cost per tax audit is and the Gini coefficient rises from to, the value of the optimal cutoff income increases from to. The increase of the optimal cutoff income indicates a more aggressive audit policy from the government. Provided that taxpayers with income report their incomes truthfully (see function 8), the rise of the optimal cutoff income is expected to increase the number of taxpayers who honestly report their incomes in the economy. This policy illustrates a commonly accepted view according to which the limitation of the extent of tax evasion will increase the government s tax revenue. However, things do not work in that way in the case of the cutoff audit rule. Numerical analysis shows that in the lognormal economy, the growth of the inequality of income implies the decrease of the portion of taxpayers who honestly reports their incomes. The negative relation between the Gini coefficient and the value of the cumulative function is clearly depicted by the data of [10]

11 Table 4 (see appendix B.4). For instance, if, and, the growth of the Gini coefficient from to results to the decline of the value the cumulative function from to. This means that both the number of taxpayers who has income and the number of the tax audits that is conducted are reduced. At the same time, the number of taxpayers with incomes is increased. Therefore, despite the fact that the government is implementing a more aggressive audit policy, the extent of tax evasion in the economy rises. Hence, the positive relation between the effectiveness of the cutoff audit rule and the inequality of income cannot be explained on the ground of the extent of tax evasion. Nevertheless, the numerical analysis has to reveal a final result. Particularly, in the lognormal economy, the growth of the inequality results to the decrease of the percentage of the total income possessed by taxpayers who are selected for a tax audit. According to data of the table 5 (see appendix B.5), for each set of values of the parameters, and, the Gini coefficient and the value of the Lorenz function are negatively related. For example, when, and, the growth of the Gini coefficient from to results to the decline of the value the Lorenz function from to, which means that the percentage of the total income possessed by the taxpayers who are selected for a tax audit declines from to. At the same time, the percentage of the total income possessed by the taxpayers who are not selected for a tax audit is increased. Hence, the significance of the income of the taxpayers who honestly report their income to the determination of the expected net revenue declines. According to these results, the increase of the optimal cutoff income, the increase of the number of taxpayers who report an income equal to the cutoff income, the decrease of the audit cost and the increase of the percentage of the total income possessed by taxpayers who are selected for a tax audit, can ensure the rise of the net revenues that government expects to collect from the cutoff audit rule. The improvement of the effectiveness of the cutoff audit rule is caused despite the fact that the extent of tax evasion has increased. Consequently, what seems to be more important in the case of the cutoff audit rule is not the taxpayer who honestly report their incomes but the taxpayers who report an income equal to the cutoff income. [11]

12 4. Conclusions Previous researches on the theory of tax compliance were limited to the examination of the effect that the implementation of the cutoff audit policy has on the distribution of income. The results showed that the cutoff audit rule constitutes more unequal the distribution of income in the economy. The present study was designed to determine the inverse relation that is the effect that the change of the inequality of income has on the effectiveness of the cutoff audit rule. The results of the present study illustrate that in a lognormal economy, the growth of the inequality of income increases the effectiveness of the cutoff audit rule, despite the fact that the extent of tax evasion is increased. Consequently, the conduction of the tax audits according to the cutoff audit rule increases the effectiveness of the cutoff audit rule by constituting more unequal the distribution of income in the economy. On this account, the continuous implementation of the cutoff audit rule can cause hurt on the distribution of income and the social justice in the economy. However, this study has thrown up many questions in need of further investigation. Firstly, further work needs to be done to establish whether the above results hold for other distribution functions such, as the Weibull or the Gamma Distribution. Moreover, using the same experimental set up and real micro-data, the real effectiveness of the cutoff audit rule for a country can be estimated. It would be also interesting to compare the effectiveness of the cutoff audit rule the effectiveness of other audit rules, such as the random rule. Finally, the dynamic aspects of the correlation between effectiveness and inequality can be investigated. [12]

13 References R. Aaberge, Characterizations of Lorenz curves and income distribution, Social Choice Welfare 17 (2000), pp M. Allingham and A. Sandmo, Income tax evasion: a theoretical analysis, Journal of Public Economics 1 (1972), pp J. Alm, M.B. Cronshaw and M. McKee, Tax compliance with endogenous audit selection rules. Kyklos 46 (1993), pp J. Andreoni, B. Erand, J. Feinstein, Tax Compliance, Journal of Economic Literature, Vol. 36, No. 2 (1998), pp K.C. Border and J. Sobel, Samurai accountant: A theory of auditing and plunder. Review of Economic Studies 54 (1987), pp P. Chander and L. Wilde, A general characterization of optimal income taxation and enforcement, Rev. Econ. Stud. 65 (1998), pp J. Gastwirth, Notes and Comments: A general definition of the Lorenz curve, Econometrica, 39,6, (1971), pp Gibrat, Robert: Les inégalités économiques, Paris, Sirey, (1931). N. C. Kakwani, Applications of Lorenz Curves in Economic Analysis, Econometrica 45 (3), (1977), pp P. J. Lambert: Pro-poor growth and the lognormal income distribution, ECINEQ , September 2009 T. Pham-Gia, N. Turkkan, Determination of the Beta Distribution from its Lorenz Curve, Mathl. Comput. Modelling, 16,2 (1992), pp M. Pinkovskiy, X. Sala-i-Martin: Parametric Estimations of the World Distribution of Income, NBER Working Paper 15433, (2009). J.R. Reinganum and L.L. Wilde, Income tax compliance in a principal-agent framework. Journal of Public Economics 26 (1985), pp I. Sánchez, J. Sobel, Hierarchical design and enforcement of income tax policies, Journal of Public Economics, 50 (3), (1993), pp S. Scotchmer, The Regressive Bias in Taxation and Enforcement, Public Finance 58 (1992), S. Scotchmer, Audit Classes and Tax Enforcement Policy, American Economic Review Papers and Proceedings 77 (2), 1987, S. Yitzhaki, A note on income tax evasion: A theoretical analysis. Journal of Public Economics 3 (1974), pp [13]

14 Appendix Section A. Proofs of Functions and Optimality Conditions A.1. The Lorenz Function in the lognormal economy. According to the studies of Gastwirth (1971) and Pham-Gian & Turkkan (1992), the Lorenz Function that corresponds to any random variable with a cumulative distribution function and a finite positive mean is defined to be: (I) Where is the inverse of the function. In the case of the lognormal distribution, the mean income is, the Cumulative Distribution Function is, and the inverse of is. Substituting those function in (I) and doing the calculation, we receive that the Lorenz Function in the lognormal economy is. Q.E.D A.2. The Gini Coefficient in the lognormal economy. According to the study of Kakwani (1977), the Gini Coefficient for a random variable is defined as: (II) Where is the Lorenz Function. Recalling from function 3 that the Lorenz Function in the lognormal economy is, and substituting in the function (II), we obtain that the Gini Coefficient in the lognormal economy is. Q.E.D A.3. The probability of truthfully reporting. Consider a taxpayer who report his true income. In that case, the taxpayer is certain about his tax liabilities which are given by the function. Consider, now, a taxpayer who report less than his true income. In that case, the taxpayer is uncertain about his tax liabilities. If the taxpayer is selected for a tax audit, he will pay an income tax and a penalty. If, however, he is not selected for a tax audit, he will pay only an income tax. Provided that and are the probabilities of detection and no detection, respectively, the expected tax liabilities of the taxpayer are given by the function. The probability of truthful reporting necessitates the taxpayer to report his true income which means that the tax liabilities of the taxpayer in the case of are less than his tax [14]

15 liabilities, or. Solving the inequality we find that which means that the minimum probability that ensures truthfully reporting is Q.E.D A.4. Taxpayer s optimal reporting strategy under the cutoff audit rule. Consider a taxpayer with true income. If the taxpayer reports, he will be selected for a tax audit, and he will be punished. As a result, his after the penalty disposable income will be. If, however, reports an income, he will be again selected for a tax audit, but he will not be punished. His disposable income in that case will be. Solving the inequality, we can find that as long as the penalty rate is greater than unity, the inequality will hold. Consequently, a taxpayer with true income less than the cut off income will always prefer to report honestly. Consider a taxpayer with income. If the taxpayer reports, he will avoid the tax audit, and he will have a income equal to. If the taxpayer reports, he will avoid the tax audit, and he will have an income equal to. Since, the income will be lower than the income. Consequently, a taxpayer with true income greater than or equal to the cut off income always prefer to report an income equal t the cut off income, which is the minimum level of income ensuring no audit. A.5. Optimality condition of cutoff income, the general case. Given the taxpayer s optimal reporting strategy against the cut off audit rule, the government expected net revenue function in the general case where no specific probability density function is used, is defined as follows: (III) The first integral expresses the net revenues that government expects to collect from the taxpayer with income less than the optimal cut off income who report their true incomes, whereas the second integral gives the revenues that government expects to collect from the taxpayers with income greater than or equal to the optimal cut off income who report the cut off income. recalling that, this function can be simplified as follows: [15]

16 . So The optimal cut off audit policy is determined by the maximization of the above function with respect to the cut off income. The first derivative of with respect to is.setting the derivative equal to zero and rearranging the terms we obtain the condition. Q.E.D A.6. Optimality condition of, the lognormal case. The optimality condition in the case where no specific probability density function is used is. In the case of the lognormal distribution, the value of the probability density function in the cut off income is, and the value of the cumulative distribution function is. substituting those function in the above optimality condition, and making the necessary manipulation we receive the condition, where and. Q.E.D A.7. Index of effectiveness of the cut off audit rule in the lognormal economy. The index of the effectiveness of the cut off audit rule arises if we substitute the optimal cut off income in the expected net revenue function. This function in the general case is. However, in the lognormal economy, the probability density function is, and the expected net revenue function become replace the for, we take the index of the effectiveness of the cut off audit rule. Q.E.D. If in that function we [16]

17 B. Results of Numerical Analysis B.1: Determination of the values of the Gini Coefficient Table 1: Gini Coefficient for different values of m and s s m E(w) G 0,52 9, ,2869 0,62 9, , ,72 8, , ,82 8, , ,92 8, , ,02 8, , ,22 8, , The upper income in the economy is constant and equal to u= euro. When the income of the taxpayers is bounded from above, the mean or expected income in the lognormal economy is given by the function, which becomes when. Given the value of the upper income in the economy, we calculate the pairs of the parameters and that holds the mean income steady and equal to euros. Then, for those pairs, we compute the value of the Gini coefficient. The results are presented in Table 1. It must be noted that similar results are produced by repeating that procedure for a different level of mean income. B.2: Numerical Approximations of the Index of the Effectiveness of the cutoff audit rule. Table 2: Index of Effectiveness ER(co * ) and Inequality of Income G f t c G=0,28 G=0,33 G=0,38 G=0,43 G=0,48 G=0,52 G=0,61 1,6 0, , , , , , , , , , , , , , , ,1563 2,4 0, , , , , , , ,4547 1,6 0, , , , , , , ,1658 1,6 0, , , , , , , ,1501 1,6 0, , , , , , , ,2843 1,6 0, , , , , , , ,3621 The values of the Index of Effectiveness ER(co * ) are expressed in euros. The procedure for the determination of the value of the index of the effectiveness is the following. Firstly, given the value of the parameters,,,, and, we calculate the value of the optimal cutoff income by solving the optimality condition 10. Then, we calculate the value of the index cutoff income in function 11. by substituting the optimal level of the [17]

18 B.3: Numerical Approximations of the Optimal Cut Off Income Table 3: Optimal Cutoff Income co * and Inequality of Income G f t c G=0,28 G=0,33 G=0,38 G=0,43 G=0,48 G=0,52 G=0,61 1,6 0, , , , , , , , , , , , , , , ,242 2,4 0, , , , , , , ,857 1,6 0, , , , , , , ,883 1,6 0, , , , , , , ,840 1,6 0, , , , , , , ,852 1,6 0, , , , , , , ,965 Τhe values of the optimal cutoff income co * are expressed in euros. The optimal cut off income is numerically determined by equation 10. As mentioned above, the determination of it requires the specification of the value of the parameters,,,, and. B.4: Numerical Approximations of the Cumulative Distribution Function Table 4: Cumulative Distribution H(co * ) and Inequality of Income G f t c G=0,28 G=0,33 G=0,38 G=0,43 G=0,48 G=0,52 G=0,61 1,6 0, , , , , , , , , , , , , , , ,1117 2,4 0, , , , , , , ,1207 1,6 0, , , , , , , ,1316 1,6 0, , , , , , , ,1421 1,6 0, , , , , , , ,0831 1,6 0, , , , , , , ,0679 The values of the Cumulative Function H(co * ) are expressed as percentages. For the determination of the above values, we use the cumulative function of the lognormal distribution, and the values of the optimal cut off income which have already presented in Table 3. On this account, the term gives the percentage of the population of taxpayers who has income less than the optimal cut off income, and its complement shows the percentage of the population of taxpayers with income higher than the optimal cut off income. [18]

19 B.5: Numerical Approximations of the Lorenz Function. Table 5: Lorenz Function L[H(co * )] and Inequality of Income G f t c G=0,28 G=0,33 G=0,38 G=0,43 G=0,48 G=0,52 G=0,61 1,6 0, , , , , , , , , , , , , , , ,5020 2,4 0, , , , , , , ,5792 1,6 0, , , , , , , ,6733 1,6 0, , , , , , , ,7648 1,6 0, , , , , , , ,2594 1,6 0, , , , , , , ,1314 The values of the Lorenz Function L[H(co * )] are expressed as percentages. For the determination of the above values, we use the Lorenz Function in the lognormal economy, and the percentage of the population who has income less than the optimal cut off income. Given that, the value of the Lorenz Function give the percentage of the total income that is possessed by taxpayers with income less than the optimal cut of income, and the complements of it, that is, gives the percentage of the total income that is possessed by taxpayers with income more than the optimal cut off income. [19]