Progressive taxation

Economists generally agree that the best measure of whether a tax change is progressive or regressive is the percent by which it changes the after tax-incomes of various income groups. Across-the-board rate cuts raise after-tax incomes by a larger percentage for those at the top of the income scale than for those at the middle and bottom, meaning they are regressive.  (Cutting all rates by the same number of percentage points, rather than the same percentage, is also regressive.)
To understand why cutting rates by an equal percentage produces unequal results, consider a simple tax system with two rates: 20 percent and 40 percent.  Assume that both tax rates are reduced by 50 percent, so that the new bottom rate is 10 percent and the new top rate is 20 percent.  These rate cuts enable taxpayers to keep 90 cents — rather than 80 cents — of each dollar taxed at the bottom rate, generating a 12.5 percent increase in after-tax income. [3]   But, for each dollar taxed at the top rate, taxpayers will keep 80 cents, rather than 60 cents, generating a 33 percent increase in after-tax income.[4]
Thus, even though both tax rates fall by the same percentage, the result is a regressive tax cut that benefits people in the top bracket the most, not only in dollars but in the percentage increase in their after-tax income.

• Low-income taxpayers who have all of their income taxed at the bottom rate will get a 12.5 percent increase in their after-tax income. 
• For taxpayers with incomes high enough to face the higher tax rate, the percentage increase in their after-tax income will be somewhere between 12.5 and 33 percent:  they will take home 12.5 percent more of their income taxed in the lower bracket and 33 percent more of their income taxed in the higher bracket.  The more income they have taxed at the higher rate, the closer their total increase in after-tax income will be to 33 percent.

The same principle holds for tax codes with more than two rates, like the federal income tax code.  Higher-income taxpayers, who face the top tax rates, would receive a bigger percentage boost in after-tax income from across-the-board marginal income tax rate cuts than low-and moderate-income taxpayers, who face only the lower tax rates.
Estimates from the Urban-Brookings Tax Policy Center (TPC) starkly illustrate this.  (See Figure 1.)  A 20 percent cut to all current marginal income tax and AMT rates[5] would raise average after-tax incomes by 4.4 percent among millionaires but by only 1.7 percent among taxpayers with incomes between $50,000 and $75,000.   As Table 1 shows, these percentages work out to average tax cuts of $92,400 for millionaires and $919 for middle-income taxpayers.
http://www.cbpp.org/cms/index.cfm?fa=view&id=3807

Where do federal dollars go? http://www.cbpp.org/cms/index.cfm?fa=view&id=1258

Who pays taxes: http://www.cbpp.org/cms/index.cfm?fa=view&id=3505
The fact that most people who don’t owe federal income tax in a given year do pay substantial amounts of other taxes — and also are net income taxpayers over time — belies the claim that households that do not owe income tax in a given year will form bad policy judgments because they “don’t have any skin in the game.”