Problem:

Suppose you live in the hypothetical country of Evergreen. Evergreen does not tax the first $10,000 of income, but imposes a tax of 10% on the next $30,000 of income, 20% on the following $60,000, and a top rate of 30% on any income above $100,000.

1. Suppose your income is $20,000. What is your marginal tax rate? What is your average tax rate?
2. Further suppose your neighbor's income is $50,000. What is her marginal tax rate? What is her average tax rate?
3. Is Evergreen's tax structure progressive, proportional, or regressive?

Answer:

1. The first $10,000 is untaxed, while incomes between $10,000 and $40,000 carry a marginal rate of 10%. Therefore, an income of $20,000 is taxed at a marginal rate of 10%. Since the 10% applies only to the part of income above $10,000, your tax due is $1,000: 0.10x($20,000 - $10,000) = $1,000. Your average tax rate is then $1,000/$20,000 = .05, or 5%.
2. The marginal rate is 20% on all incomes between $40,000 and $100,000. The total tax due on a $50,000 income is 10% of the income between $10,000 and $40,000 plus 20% of the income above $40,000. Thus, her tax due is 0.10x($40,000 – $10,000) + 0.20x($50,000 – $40,000) = $3,000 + $2,000 = $5,000. Her average tax rate is $5,000/$50,000 = 10%.
3. The tax structure is progressive: the average tax rate rises with income.