Understanding the Mathematics of Personal Finance

# UNDERSTANDING PERSONAL FEDERAL INCOME TAX RATES

Every year in the United States, on April 15, our income tax filings are due. The federal government, most states, and some cities collect income taxes. In almost all cases, the federal tax is by far the highest of the personal income taxes that we must pay.

When you fill out your tax return, you list all of your earnings and then you list your deductions. When you subtract the sum of all your deductions from the sum of your earnings, you get your taxable income. The amount of tax you must pay to the Internal Revenue Service (IRS) is based on this taxable income and your filing status. Your filing status can be single, married filing jointly, and so on. Accompanying each filing status is a Tax Rate Schedule.

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

2008 Tax Rate Schedules

Schedule X—If your filing status is Single

If your taxable The tax is

income is:

But not

Over— over—

 so \$8,025 ........... 10% \$0 8,025 32,550 \$802.50 ♦ 15% 8,025 32,550 78,850 4,481.25 + 25% 32,550 78.850 164,550 16.056.25 + 28% 78.850 164,550 357,700 40,052.25 -*• 33% 164,550 357.700 103,701.75 + 35% 357,700

Schedule Y-1—If your filing status is Married filing jointly or Qualifying widow(er)

 If your taxable income is: Over— But not over— The tax is: of the amount over— \$0 \$16,050 ........... 10% \$0 16,050 65,100 \$1,605.00 + 15% 16,050 65,100 131,450 8,962.50 + 25% 65,100 131,450 200,300 25,550.00 ♦ 28% 131,450 200,300 357,700 44,828.00 -*• 33% 200,300 357,700 96,770.00 «*• 35% 357,700

Schedule Y-2—If your filing status is Married filing separately

If your taxable The tax is

income is:

But not

Over— over—

 \$0 \$8,025 ........... 10% so 8.025 32,550 \$802.60 + 16% 8,025 32,550 65.725 4,481.25 + 25% 32,550 65,725 100,150 12.775.00 + 28% 65,725 100,150 178.850 22.414.00 + 33% 100,150 178,850 48,385.00 ♦ 35% 178,850

If your taxable The tax is

 \$0 \$11.450 ........... 10% \$0 11,450 43,650 \$1,145.00 * 15% 11,450 43,650 112,650 5,975.00 + 25% 43,650 112,650 182,400 23,225.00 28% 112,650 182,400 357,700 42,755.00 + 33% 182,400 357,700 100,604.00 + 35% 357,700
 Figure 9.1 IRS 2008 Tax Tables.

Figure 9.1 shows the 2008 Tax Rate Schedules. Each schedule is a table. Each row in the table shows a range of taxable income and a formula for calculating the tax for this income. Each formula includes a percentage that is known as the tax bracket. What your tax bracket does and doesn’ t tell you about your tax and its relation to your taxable income are often misunderstood. I ’ tl point out the place where many people get it wrong as I walk through the calculations.

Table 9.1 is a repeat of Figure 9.1 for married couples filing jointly with the tax table formulas rewritten in conventional algebraic notation. As an example, if your taxable income is exactly \$100,000, then your tax is

Tax = \$8,962.50 + 0.25(100,000 - \$65,100) = 8,962.50 + 0.25(\$34,900)

= \$8,962.50 + \$8,725.00 = \$17,687.50

In Figure 9.2, I’ve plotted the tax versus the taxable income for married couples filing jointly. If you look carefully, you can see that the graph is made up of straight

Table 9.1 IRS 2008 Tax Tables for a Married Couple Filing Jointly

 Taxable income range (x) Tax \$0 < x < \$16,050 0.1x \$16,050

 Taxable income (\$) Figure 9.2 Graph of IRS 2008 Tax Tables for a married couple filing jointly.

line segments with slight corners where they join. These corners occur when the taxable income jumps from one tax bracket to the next.

The fact that the graph is continuous—there are no discontinuous jumps in the graph—is important. While it is clear that the higher your taxable income is the higher your taxes will be, there are no sudden jumps when you move from one tax bracket to another. In other words, if you earn a few dollars more, you pay a few dollars more. Somehow, the idea has snuck into the popular culture, that if you earn a few dollars more and happen to move to a higher tax bracket you will suddenly owe a huge amount more in taxes. There is simply no basis for this conclusion.

Because Figure 9.2 is not a straight line, the average tax rate is not a single number. The average tax rate on your taxable income is your tax divided by your taxable income, usually expressed as a percentage. If you remember the graphs dis­cussion in Chapter 1, this is identically the slope of the line from the lower left-hand corner of the graph to the point on the graph showing your taxable income and tax.

Figure 9.3 shows this calculation, with the average tax rate plotted versus the taxable income (for married couples filing jointly). The graph has some irregular curves in it, but it is monotonically increasing. This means that as your taxable income (the horizontal axis) increases, your average tax rate increases (the vertical axis). The details of how it increases depending on your taxable income bear some discussion, but nonetheless, it always increases.

Using Figure 9.3 , let’s consider a taxable income of \$100,000 as an example. The average tax rate on these earnings is about 18%. Your tax bill will be 18% of \$100,000, or \$18,000. Looking back at the tax table, however, a taxable income of \$100,000 puts you in the 25% bracket. If you’re only paying 18% of your taxable earnings as tax, how does this have anything to do with 25%?

 Taxable income (\$) Figure 9.3 Average tax rate for a married couple filing jointly.

 Taxable income (\$) Figure 9.4 Expansion of the lower left region of Figure 9.2.

The answer is that the term “tax bracket” refers to the incremental tax rate, not the average tax rate. Figure 9.4 shows the expanded lower left corner of Figure 9.2. The graph is actually made up of connected straight line segments, each with a different slope. These line segments connect at the break points between the tax brackets. When your taxable income is \$100,000, the first \$16,050 is taxed at a 10% rate; \$65,100 - \$16,050 = \$49,050 is taxed at a \$15% rate; and then \$100,000 - \$65,100 = \$34,900 is taxed at a 25% rate. The average tax rate comes from averaging all of these pieces together. This is called a progressive tax. The name has nothing to do with the political party or philosophy that created it. Progressive here refers to the fact that the incremental tax rate gets progressively larger as the taxable income increases.

Your average tax rate only gets close to your incremental tax rate only when you’re making an awful lot of money. Using the same tables as above, you can see that once your taxable income is more than \$357,700, you are in the top tax bracket. For a \$400,000 taxable income (the largest taxable income that I included in the graphs), your average rate is about 28% while your incremental rate is 35%. Dividing the former by the latter, we get that your average rate is 80% of your incremental rate.

If your taxable income was \$1,000,000, then your tax would be about \$322,000. Your average tax rate then is 32%, but your incremental rate is still 35%. Your average rate is now 92% of the incremental rate. At \$2,000,000 taxable income, your average rate would be 99.6% of your incremental rate, effectively the incremental rate. The average rate is said to approach the incremental rate asymptotically. This is a fancy term for “getting closer and closer but never exactly reaching it.” Finding yourself in this situation, however, is sometimes referred to as having a “very high - class problem.”

For those of you who are following the math, the line tangent to a straight line is the straight line itself. The incremental tax rate is therefore the slope of each straight line in its range of applicability. As discussed in Chapter 1, the slope of the line is also called the rate of change of the variable on the vertical axis with respect to the variable on the horizontal axis, which is referred to in this application as the “ tax bracket. ”

What the 25% tax bracket means for a taxable income of \$100,000 is that for every dollar more than \$100,000 earned, \$0.25 more is due in taxes, and also for every dollar less than \$100,000 earned, \$0.25 less is due in taxes. Remember that this last sentence applies only when the earned income is between \$65,100 and \$131,450 (the defined region for the 25% bracket).

Table 9.2 summarizes the situation for a married couple filing jointly. As the table shows, everybody is actually paying much less than their tax bracket rate would

Table 9.2 Summary of Average and Incremental Taxes for a Married Couple Filing Jointly

 Taxable income (\$) Tax (\$) Average rate (%) Incremental rate (%) 0 0 0 0 10,000 1,000 10 10 20,000 2,198 11 15 40,000 5,198 13 15 60,000 8,198 14 15 80,000 12,688 16 25 100,000 17,688 18 25 120,000 22,688 19 25 140,000 27,944 20 28 160,000 33,544 21 28 180,000 39,144 22 28 200,000 44,744 22 30 220,000 51,329 23 33 240,000 57,929 24 33 260,000 64,529 25 33 280,000 71,129 25 33 300,000 77,729 26 33 320,000 84,329 26 33 340,000 90,929 27 33 360,000 97,575 27 35 380,000 104,575 28 35 400,000 111,575 28 35 20,000,000 6,971,575 34.9 35

predict. Even with a taxable income of \$20,000,000, you’re not quite paying your tax bracket rate—but you are getting very close.

9.2 ONLINE TAX CALCULATORS

Many online calculators will actually calculate your taxable income from the infor­mation you give them. Make sure you know what’s being calculated, based on what. These links each lead to several calculators and tax information:

4. http://www. finance. cch. com/sohoApplets/index. html (This link brings you a large list of calculators. Scroll down the list to find the tax calculators.)

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