1.7 Extension - Taxes

Governments collect taxes to pay for the services they provide. In the United States, federal income taxes help fund the military, the environmental protection agency, and thousands of other programs. Property taxes help fund schools. Gasoline taxes help pay for road improvements. While very few people enjoy paying taxes, they are necessary to pay for the services we all depend upon.

Taxes can be computed in a variety of ways, but are typically computed as a percentage of a sale, of one’s income, or of one’s assets.

When taxes are not given as a fixed percentage rate, sometimes it is necessary to calculate the effective rate.

Taxes are often referred to as progressive, regressive, or flat.

Example 3

The United States federal income tax on earned wages is an example of a progressive tax. People with a higher wage income pay a higher percent tax on their income.

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For a single person in 2011, adjusted gross income (income after deductions) under $8,500 was taxed at 10%. Income over $8,500 but under $34,500 was taxed at 15%.

A person earning $10,000 would pay 10% on the portion of their income under $8,500, and 15% on the income over $8,500, so they’d pay:

$\begin{matrix} 8500 (0.10) = 850 & 10 % of 8500 \\ 1500 (0.15) = 225 & 15 % of the remaining $ 1500 of income \\ Total tax: = $ 1075 \end{matrix}$

The effective tax rate paid is $1075 / 10000 equals 10.75 %$

A person earning $30,000 would also pay 10% on the portion of their income under $8,500, and 15% on the income over $8,500, so they’d pay:

$\begin{matrix} 8500 (0.10) = 850 & 10 % of 8500 \\ 21500 (0.15) = 3225 & 15 % of the remaining $ 21500 of income \\ Total tax: = $ 4075 \end{matrix}$

The effective tax rate paid is $4075 / 30000 equals 13.58 %$ .

Notice that the effective rate has increased with income, showing this is a progressive tax.

Example 4

A gasoline tax is a flat tax when considered in terms of consumption, a tax of, say, $0.30 per gallon is proportional to the amount of gasoline purchased. Someone buying 10 gallons of gas at $4 a gallon would pay $3 in tax, which is $$ 3 / $ 40 equals 7.5 %$ . Someone buying 30 gallons of gas at $4 a gallon would pay $9 in tax, which is $$ 9 / $ 120 equals 7.5 %$ , the same effective rate.

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However, in terms of income, a gasoline tax is often considered a regressive tax. It is likely that someone earning $30,000 a year and someone earning $60,000 a year will drive about the same amount. If both pay $60 in gasoline taxes over a year, the person earning $30,000 has paid 0.2% of their income, while the person earning $60,000 has paid 0.1% of their income in gas taxes.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.