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
Copyright © 2009 John Wiley & Sons, Inc.
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 Y1—If your filing status is Married filing jointly or Qualifying widow(er)

Schedule Y2—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 
Schedule Z—If your filing status is Head of household____________
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 ($) 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 lefthand 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

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.
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:
1. http://turbotax. intuit. com/taxtools/;
2. http://www. dinkytown. net/java/Tax1040.html;
3. http://www. hrblock. com/taxes/tax_calculators/index. html;
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.)