Describe the behavior of the marginal tax rate under a proportional, progressive, and regressive tax system.

Let's define the three tax systems mentioned: - Proportional tax: Also known as a flat tax, it's a system where taxpayers pay a fixed percentage of their income as tax, regardless of the amount of income. - Progressive tax: In this system, the tax rate increases as the taxable income increases. The marginal tax rate rises with higher-income levels. - Regressive tax: This is a tax system where the tax rate decreases as the taxable income increases. The marginal tax rate falls with higher-income levels.

Under a proportional tax system, the marginal tax rate remains constant as an individual's income increases. Since the tax rate is fixed, the marginal tax rate behavior is straightforward: for each additional dollar earned, the taxpayer pays the same proportion of tax. In this case, the marginal tax rate is equal to the average tax rate, and it does not depend on the level of income. Mathematically, you have: \( Marginal\, Tax\, Rate = Proportional\, Tax\, Rate \)

Under a progressive tax system, the marginal tax rate increases as an individual's income increases. As income rises, taxpayers will see their marginal tax rate rise as well. In this case, the marginal tax rate is always higher than the average tax rate, and it indicates that the higher-income earners pay a larger proportion of their income as tax compared to lower-income earners. Mathematically, as income increases: \( Marginal\, Tax\, Rate > Average\, Tax\, Rate \)

Under a regressive tax system, the marginal tax rate decreases as an individual's income increases. This means that the proportion of tax paid decreases as income rises. In this case, the marginal tax rate is always lower than the average tax rate, and it suggests that more of the tax burden falls on lower-income earners compared to those with higher incomes. Mathematically, as income increases: \( Marginal\, Tax\, Rate < Average\, Tax\, Rate \) In conclusion, the marginal tax rate behaves differently under each of these tax systems: - Proportional tax system: Marginal tax rate remains constant - Progressive tax system: Marginal tax rate increases with income - Regressive tax system: Marginal tax rate decreases with income