Income tax

Summary: Mathematics is used to compute income tax returns and analyze income-tax fraud.

Albert Einstein once quipped that preparing a tax return was an activity too difficult for a mathematician and was better suited for a philosopher. Many would point to the complex and ever-changing laws regarding taxation, rather than the underlying mathematical concepts, as being the problematic part of understanding income taxes.

Numbers and their operations, along with algebra, are very useful in the calculation of the taxes owed by individuals and corporations. In addition, probability, statistics, and geometry are among the fields used by those interested in the analysis of the process and outcomes of taxation, such as tax irregularities and evasions, tax burden, and the effects of taxation on overall economic welfare.

History

In 1861, the U.S. Congress imposed a tax on personal incomes to help finance the Civil War. Prior to that time, it had depended mainly on excise taxes and customs duties. The first income tax was a proportional (or flat) tax: anyone who made an income of more than $800 per year had to pay a fixed 3% of that income in taxes. The next year, a two-tiered progressive rate structure was put into place. Taxable incomes up to $10,000 were still taxed at 3%, while higher incomes paid 5%, though people were allowed to take various deductions from their incomes before calculating the tax. Taxes were also withheld by employers for the first time. This tiered taxation method became the standard for income tax, although some countries in the early twenty-first century use a dual income tax system in which individuals and corporations are taxed at a low rate while labor income is taxed at a high rate.

Income taxes were abolished in 1872; but after a great deal of legal debate, they returned permanently with the passage of the Sixteenth Amendment in 1913. Everyone who earns income in the United States is subject to federal individual income tax and, in most cases, state income tax as well. Some municipalities also charge local income tax. Employers are required to withhold money from taxpayers’ paychecks and to remit the funds to the appropriate government agencies. Self-employed taxpayers are required to submit quarterly payments.

Calculating the Income Tax

As can be seen from the Internal Revenue Service 1040 individual income tax form, a series of calculations are required to determine the amount of income tax owed.

Total Income: All sources of taxable income must be added to calculate total income, including not just wages but also funds accrued from sources such as tips, interest earned, alimony, capital gains, retirement withdrawals, royalties, and business income.

Adjusted Gross Income: Certain types of expenses can be subtracted from the total income, including some expenses related to moving, business, education, alimony paid, self-employment, and student loans. After subtracting the allowable expenses, the result is the adjusted gross income.

Taxable Income: Additional deductions and exemptions are subtracted from the adjusted gross income to arrive at the taxable income.

• • Deductions: Taxpayers can elect to take the standard deduction, which is a set amount depending on filing status, or they can itemize their deductions to see if a tally of the allowable deductions results in more than the standard amount. People who paid mortgage interest, significant medical costs, large charitable donations, and/or business expenses will often find that itemizing produces a larger deduction than the standard.
• • Exemptions: The federal government allows taxpayers to deduct a fixed amount for each dependent in the household; in 2009 that amount was $3,650 per dependent.

Tax owed: The tax is then looked up in the tax table, reading the appropriate column depending on filing status (single, married filing jointly, married filing separately, head of household), unless the taxable income is over $100,000, in which case a tax computation worksheet is used. In general, single people pay more taxes than married couples filing jointly with the same income.

Understanding the Federal Tax Tables and the Tax Computation Worksheet

The government defines a series of tax brackets, which are percentages linked to income ranges. The income ranges for a specific tax bracket vary depending on the filing status of the taxpayer.

The federal government sets different ranges for the following categories: single, married filing jointly, married filing separately, and head of household. In 2009, for example, the tax brackets were 10%, 15%, 25%, 28%, 33%, and 35%. The range for a single tax payer in the 10% bracket was $0 to $8,350 in taxable income. For a married couple filing a joint return, the income range for the 10% bracket was $0 to $16,700.

The tax table and tax computation worksheet values do not correspond directly to the tax brackets. For example, a single person earning $62,025 in 2009 would appear to fall into the 25% tax bracket ($33,950 to $82,250). However the tax shown in the tax table is $11,694, which is less than $17,250 (25% of $69,000). The tax table value was determined by applying the tax brackets to the taxable income in stages. In 2009, the tax brackets for a single taxpayer were

• • 10% bracket: $0 to $8,350
• • 15% bracket: $8,350 to $33,950
• • 25% bracket: $33,950 to $82,250

The first $8,350 of the taxable income earned falls into the 10% bracket, yielding $835 in taxes. The next $25,600 ($33,980-$8,350) of the taxable income falls into the 15% bracket, yielding $3,840. The last $28,075 ($62,025-$33,950) falls into the 25% tax bracket, yielding $7,018.75.

The total tax is $835+$3,840+$7,018.75=$11,694 (rounded to the nearest dollar).

The tax tables are provided in $50 increments, so anyone earning between $62,000 and $62,050 would pay the same amount of tax.

The tax computation worksheet calculations work the same way. For a single person with a taxable income of $130,000 (28% tax bracket), the worksheet calculation is to multiply by 0.28 and then subtract $6,280. The $6,280 figure is subtracted to compensate for the lower taxes paid on the portions of the $130,000 income that fall into the lower tax brackets.

Other Methods of Calculating Taxes

Some groups are concerned that the federal tax code is too complicated, confusing, and unfair. There are those who advocate simplifying the tax code and leaving the graduated tax bracket structure, and others who advocate a flat tax—one percentage rate for all with no exemptions or deductions.

Most states follow the federal government’s lead and have a series of tax brackets. In 2009, Colorado, Illinois, Indiana, Massachusetts, Michigan, Pennsylvania, and Utah all had flat taxes ranging from 3 percent to 6 percent. Alaska, Florida, Nevada, South Dakota, Texas, Washington, and Wyoming did not collect any individual income tax.

Mathematical Modeling

The impact of taxation is of great personal and political concern. Income taxes, in particular, can generate a great deal of debate, and many people feel personally and directly affected by changes in these taxes. Mathematical methods are used to model a variety of phenomena related to taxes. For example, equilibrium modeling seeks to explain and predict the broad economic repercussions of different market factors, including taxes.

These complex models take into account the flow of cash, commodities, and other goods between various people and businesses, which have different motivations and constraints. Other potential variables can include prices, interest rates, and taxes. A system (like the U.S. economy) is in equilibrium when the inflows and outflows, or supply and demand, are balanced. These models are computationally intense and generally solved using numerical methods, graph theory, geometry, and stochastic simulation.

Several countries and U.S. states, as well as companies and accounting firms, use software based on Benford’s Law to check income tax returns for fraud. Benford’s Law is named for engineer and physicist Frank Benford. According to stories about Benford, he was inspired by the fact that pages of logarithm books associated with numbers starting with the digit 1 were dirtier and more worn than other pages. Thinking that it was unlikely that scientists had some special preference for these numbers, he analyzed over 20,000 sets of data from a wide variety of sources, such as baseball statistics, numbers he found in magazine articles, and atomic weights.

All of these data sets followed a similar pattern in terms of the first digits of the numbers. About 30% of the time, the first digit of the numbers was a 1. Each subsequent numeral 2 through 9 occurred less and less often as the initial digit, such that the probability of any number n from 1 through 9 being the first digit is the following:

One simple way that data can be tested is by comparing the observed first-digit counts to Benford’s Law. For example, accountant Mark Nigrini examined 169,662 IRS files and found that they follow Benford’s Law, with an allowable statistical margin of error. Former president Bill Clinton and (as of 2010) Secretary of State Hillary Rodham Clinton’s tax returns for several years were also analyzed.

Nigrini concluded that the Clintons may have used some rounded-off dollar estimates rather than exact numbers, but his test did not uncover any fraud. Generally, studies show that fraudulent data contain too few numbers starting with 1 and too many starting with 6.

Bibliography

Fu, Michael, Robert Jarrow, Ju-Yi Yen, and Robert Elliott. Advances in Mathematical Finance. Basel, Switzerland: Birkhäuser, 2007.

Nievergelt, Yves. A Graphic Introduction to Functions: The Federal Income-Tax Law. Bedford, MA: COMAP, 1989.

Radulescu, Doina Maria. CGE Models and Capital Income Tax Reforms: The Case of a Dual Income Tax for Germany (Lecture Notes in Economics and Mathematical Systems). Berlin: Springer, 2007.

Woytek, Steve. Mathematics in Living, Credit, Loans and Taxes Book IV. Boulder, CO: Pruett Publishing, 1976.