pcecon.com Class Notes
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The curve showing the relationship between the tax revenue and the tax rate is called the "Laffer Curve." It shows that raising tax rates does not always increase tax revenues to the government. In fact, if tax rates are already high, raising them will reduce tax revenues (and lowering them will increase tax revenues).

When taxing things like income, we have two ways of describing the tax rate. These are the Marginal Tax Rate and the Average Tax Rate.

Marginal Tax Rate=

^{change in taxes paid}/_{change in income}

example> if I make $2,000 more this year and pay $800 more in taxes than last year, the marginal tax rate is
=$800÷$2,000=.40= 40%

Average Tax Rate=

^{total taxes paid}/_{total income}

example> If I pay a total of $10,000 in taxes on $30,000 income I earn this year, the average tax rate is
=$10,000÷$30,000=1/3=.333333=33.33%

The two tax rates are important for different reasons.

Decisions about how much to work (or whether to work extra hours) are determined by the marginal tax rate. A person who is taxed at a high marginal rate will be less likely to try to earn extra income than she would be if she had a lower marginal tax rate.A higher marginal tax rate means a taxpayer keeps less of each additional dollar she earns.

On the other hand, the average tax rate is used to determine how taxes affect people of different income levels (overall). This is the "income incidence" of the tax.
If the average tax rate is greater for people with greater incomes, we say the tax is "progressive."
If the average tax rate is lower for people with greater incomes, we say the tax is "regressive."
If the average tax rate is exactly the same for people with low incomes as it is for people with high incomes, we say the tax is "proportional."

By the way, many people mistakenly believe that the marginal tax rate must be higher for higher income people in order for a tax to be progressive. This is not true (see the end for an example).

It is the actual proportion or percent of each person's income that he or she ends up paying that is important in making these determinations of income incidence. Most taxes, unless they are specifically based on a person's income, will end up being regressive.
For example, a "head tax" or "poll tax" is a tax in which each person is charged the same amount of money, without regard to his or her income. This tax would not affect any prices (since no good or activity is taxed, just being a person is taxed). For this reason, there is no excess burden. However, a head tax is very unpopular, since it is certainly regressive.
For example, an annual head tax of $1,000 for each person would tax a person making $10,000 a year at an average tax rate of
^$1,000/_$10,000 = 10%. A person who makes an income of $100,000 a year would only pay an average tax rate of
^$1,000/_$100,000 = 1%.
Charging everyone the same amount of money does not result in the same average tax rate for everyone. People with high incomes end up paying a smaller portion of their total income in taxes, and so this kind of tax is regressive.

Even a tax that takes more money from higher income people can be regressive in its effect. Suppose a sales tax is imposed that results in a person who earns $10,000 a year paying $500 in sales taxes.
The effective average tax rate for such a person would be ^$500/_$10,000 = 5%.
If the same sales tax rules resulted in a person who earns $100,000 paying $4,000 in sales taxes (perhaps because such a person saves some of his or her income rather than spending it), the average tax rate from this tax for such a person would be
^$4,000/_$100,000 = 4%.
Even though the person with the $100,000 income pays more money in sales taxes, the percentage of his or her income that goes to paying the sales tax is less than it is for someone earning only $10,000. Thus, even this tax is regressive (although the higher income person pays more money, the higher income person pays a smaller portion of income in taxes). Sales taxes could be proportional if a person with a $100,000 income purchased exactly ten times as many taxed goods as a person who earned only $10,000, but in practice, this probably won't happen.

Any tax that is not specifically linked to the income a person earns is quite likely to be regressive in its effects.

BONUS (not in class) STUFF
One income tax system that people propose from time to time as an alternative to our rather complex income tax structure is what is called a "flat rate" tax. Interestingly, this tax sounds proportional, but can be made progressive. Most of the versions of it that are proposed are indeed progressive. Usually, the proposals say that some amount of income can be earned without any tax being charged at all to the earner. Then, any money earned over this base amount is taxed at a flat marginal rate (which, remember, is different from the average rate).
For example, a person could earn up to $20,000 without paying any tax on the income earned. But, for each dollar earned over the $20,000, a 20% marginal tax rate is charged. This marginal tax rate stays the same, no matter how much over the $20,000 (tax free) base a person earns. While the marginal tax rate does not change, the average tax rate will, in effect, increase as a person's income increases.
If Jose earns $25,000, his total tax will be 20% of the $5,000 he earns over the $20,000 base, or 20%*$5,000=$1,000. Thus, Jose's average tax rate will be
^$1,000/_$25,000 = 4%. If Maria earns $100,000, her total tax will be 20% of the $80,000 she earns over the $20,000 base, or 20%*$80,000 = $16,000. Thus, Maria's average tax rate will be
^$16,000/_$100,000 = 16%. Maria will pay a larger portion of her income in taxes because she has a higher income, so the tax is progressive. This is true even though the marginal tax rate is the same. The marginal tax rate does not tell us what the average tax rate is, unless we do this kind of calculation. The average tax rate is what tells us that this tax is actually progressive.