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1. [30 marks] Xuan-Thuc is a New Zealand citizen with taxable income of $110,000. According to the current personal income tax schedule, Xuan-Thuc's total tax liability on this income is $27, 220 which would leave her with $82,780 to spend on consumption. Xuan-Thuc is self- employed and can reduce her tax liability by declaring less then her full income. In effect, Xuan-Thuc can choose how much tax to pay she can choose anything from 0 to the full $27, 220. However, if she chooses to pay less than her true tax liability and her fraud is detected, she will have to pay a penalty. According to the IRD website, this penalty is to repay the evaded tax plus a fine equal to 150% of the evaded tax. For example, if Xuan-Thuc under-reports her income to reduce her tax liability to $26, 220 (i.e., $1000 less than her true liability) and is caught, she will have to pay the $1000 in evaded tax, plus a fine equal to $1500. Suppose that fraud is detected if, and only if, the IRD decides to audit a taxpayer. Let contingency 1 be the contingency in which Xuan-Thuc is audited and let contingency 2 be the contingency in which she is not audited. Therefore, if Xuan-Thuc tries to evade $z of her tax liability, her contingent consumption is 82780 - 32, 82780 + 2 2) 3 1 (i.e., if she is caught, she pays her true tax liability plus a fine of $iz, and if she is not caught, her consumption spending is $z more than it would have been if she had made an honest declaration). She must decide what z to choose and she can choose any value from 0 to $27, 220. In particular, z = 0 corresponds zero evasion (i.e., declaring her true income) and z = 27, 220 means evading her full tax liability (i.e., declaring zero income). Note that every additional dollar evaded reduces consumption in contingency 1 by $1.50 and increases consump- tion in contingency 2 by $1.00. Suppose that the IRD randomly audits taxpayers, and each taxpayer has probability p of being audited irrespective of how much income they declare Xuan-Thuc is risk averse with utility-of-money function v (c) = ln (c). (a) Draw Xuan-Thuc's budget line in a contingent consumption (CC) diagram, with consumption in the contingency in which she is audited measured on the horizontal axis. [7 marks) (b) Explain why Xuan-Thuc will make an honest declaration i.e., choose z = 0) if, and only if, p > 0.4. (Hint: Use your CC diagram. Compare the slope of Xuan-Thuc's indifference curve through the bottom point of her budget line (where z = 0) to the slope of her budget line. The Supplementary Video 1 for Week 1 illustrates this technique. [7 marks] 17 marks) (c) Suppose the IRD only audits 25% of taxpayers, so p = 1 (hence Xuan-Thuc chooses 2 > 0). (1) Find Xuan-Thuc's optimal contingent consumption bundle (i.e., the point on her budget line where the budget line is tangent to an indifference curve). [12 marks] (ii) What value of z achieves this optimal contingent consump- tion? [2 marks] (iii) Based on New Zealand's personal income tax schedule, how much income does Xuan-Thuc declare? [2 marks] This is unlikely to be true in practice, but it is a convenient assumption for our Income tax rates New Zealand has progressive or gradual tax rates. The rates increase as your income increases. From 1 April 2021 For each dollar of income Tax rate Up to $14,000 10.5% Over $14,000 and up to $48,000 17.5% Over $48,000 and up to $70,000 30% Over $70,000 and up to $180,000 33% Remaining income over $180,000 39% 1. [30 marks] Xuan-Thuc is a New Zealand citizen with taxable income of $110,000. According to the current personal income tax schedule, Xuan-Thuc's total tax liability on this income is $27, 220 which would leave her with $82,780 to spend on consumption. Xuan-Thuc is self- employed and can reduce her tax liability by declaring less then her full income. In effect, Xuan-Thuc can choose how much tax to pay she can choose anything from 0 to the full $27, 220. However, if she chooses to pay less than her true tax liability and her fraud is detected, she will have to pay a penalty. According to the IRD website, this penalty is to repay the evaded tax plus a fine equal to 150% of the evaded tax. For example, if Xuan-Thuc under-reports her income to reduce her tax liability to $26, 220 (i.e., $1000 less than her true liability) and is caught, she will have to pay the $1000 in evaded tax, plus a fine equal to $1500. Suppose that fraud is detected if, and only if, the IRD decides to audit a taxpayer. Let contingency 1 be the contingency in which Xuan-Thuc is audited and let contingency 2 be the contingency in which she is not audited. Therefore, if Xuan-Thuc tries to evade $z of her tax liability, her contingent consumption is 82780 - 32, 82780 + 2 2) 3 1 (i.e., if she is caught, she pays her true tax liability plus a fine of $iz, and if she is not caught, her consumption spending is $z more than it would have been if she had made an honest declaration). She must decide what z to choose and she can choose any value from 0 to $27, 220. In particular, z = 0 corresponds zero evasion (i.e., declaring her true income) and z = 27, 220 means evading her full tax liability (i.e., declaring zero income). Note that every additional dollar evaded reduces consumption in contingency 1 by $1.50 and increases consump- tion in contingency 2 by $1.00. Suppose that the IRD randomly audits taxpayers, and each taxpayer has probability p of being audited irrespective of how much income they declare Xuan-Thuc is risk averse with utility-of-money function v (c) = ln (c). (a) Draw Xuan-Thuc's budget line in a contingent consumption (CC) diagram, with consumption in the contingency in which she is audited measured on the horizontal axis. [7 marks) (b) Explain why Xuan-Thuc will make an honest declaration i.e., choose z = 0) if, and only if, p > 0.4. (Hint: Use your CC diagram. Compare the slope of Xuan-Thuc's indifference curve through the bottom point of her budget line (where z = 0) to the slope of her budget line. The Supplementary Video 1 for Week 1 illustrates this technique. [7 marks] 17 marks) (c) Suppose the IRD only audits 25% of taxpayers, so p = 1 (hence Xuan-Thuc chooses 2 > 0). (1) Find Xuan-Thuc's optimal contingent consumption bundle (i.e., the point on her budget line where the budget line is tangent to an indifference curve). [12 marks] (ii) What value of z achieves this optimal contingent consump- tion? [2 marks] (iii) Based on New Zealand's personal income tax schedule, how much income does Xuan-Thuc declare? [2 marks] This is unlikely to be true in practice, but it is a convenient assumption for our Income tax rates New Zealand has progressive or gradual tax rates. The rates increase as your income increases. From 1 April 2021 For each dollar of income Tax rate Up to $14,000 10.5% Over $14,000 and up to $48,000 17.5% Over $48,000 and up to $70,000 30% Over $70,000 and up to $180,000 33% Remaining income over $180,000 39%

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