Exemptions and deductions

The tax is dependent on income, but what exactly is the income used in the tax equation?

Well, this is where the difficulties begin. Tax laws come from Congress that cannot resist its urge at pleasing specific parts of the population by introducing tweaks, normally in the form of deductions linked to specific conditions like marriage, house ownership, children, .... The base principle becomes the tax law that is then formalized in specific rules by the IRS, i.e. what is the definition of a kid.

Because the world is not as regular as can be expressed in a law, the IRS rules are complex, full of details that apply probably only to a very small percentage of cases but still must be detailed in the instructions booklets written by the IRS.

The income used in the tax equation is known as the adjusted gross income, it is essentially calculated as follows for your workpay:
* start with your nominal salary
* remove a number of special items: social security, 401k contributions, ... and you get your taxable income
* remove the exemptions, these are usually amounts that are not directly related to any real expenses
* remove deductions, these are usually amounts that correspond to specific expenses that are considered useful for the society, at least as seen by Congress
* presto, you have your adjusted gross income.

Of course, this is where the pain starts, you as a taxpayer, have to assemble all the information needed to correctly assess your taxable income, but also to substantiate your deductions. The IRS estimates are that filling a 1040 form requires 30.3 hours of which 19 hours are for record keeping. If you can take 2 hours a day after work, this requires two full weeks of work.

The IRS also estimates the average cost at $269, but this probably assumes your time is free. No surprise this is called a burden. Of course, the IRS is part of the problem as the rules are at best complex. Just an example, line 42, Exemptions has special treatment for persons displaced by Hurricane Katrina. This probably comes from a good intention, but a first obvious remark is that this is only a very small part of the population, but it impacts all taxpayers (you have to read the instructions).

Then there are the exact rules, with no less than 6 small details that you have to check if you happen to be housing a displaced person, the first one reads "The person displaced lived in your main home for a period of at least 60 consecutive days ending in 2006". The definition appears rigorous, but this is a nice example of how arbitrary the process can be, note the following:
* main home, what if you have contributed a cottage that is on the same piece of land as your main home?
* 60 consecutive days, what if the displaced person traveled for one week, say to New Orleans, is it still consecutive. If you remember that the displaced person was there in January and February, can you positively remember that he or she was there for 60 days?

This is not directly a critic of the IRS, but of the whole process of trying to carve out special cases. Every special case comes with a set of rules and all these rules will probably raise a number of secondary questions that may end up as more rules and the cycle never ends. After a few years of this, you need 30 hours to fill your tax form and everybody hates filling the form.

But this derails us from the goal, a mathematical treatment of the tax equation. Mathematically, deductions and exemptions correspond to an offset. Essentially pushing the tax curves to the right when using the taxable income for the x axis. This will be further discussed and illustrated in a next article.

Tax equation, first cut

From the definition of the statutory tax rates, it is possible to write a tax equation, giving the tax owed in function of the income, the tax brackets and the filing status.


The different variables are:
* I, income
* R, array of statutory rates, by bracket and filing status
* B, array of bracket start boundaries, by bracket and filing status
* f, filing status
* b, a bracket index
The equation uses an intermediary variable S, array of tax owed at the start of a given bracket. S is not an independent variable and can be calculated iteratively as shown above or directly expressed as a sum


The marginal rate is the ratio between the increase in tax owed and an increase in income. Using the tax equation, the marginal rate is the partial derivative of the tax equation over the variable I. As expected, the marginal rate is simply the statutory rate at this stage.


If wanted, partial derivatives over different variables provide solutions for other form of increase, e.g. if the start of the 10% bracket is changed, what is the corresponding increase in tax.

Statutory tax rates

Statutory tax rates are the most fundamental part of the tax. They give the form of the tax curve, i.e. how much tax you have to pay in function of your "income". The tax curve defined by the statutory tax rates is very simple, the complexity will come later, introduced by a large number of tweaks, in the form of deductions, exemptions, credits, phaseout, and other goodies we'll also tackle in this blog.

The US, as most countries, use progressive tax rates, that is both the tax and the tax rate increase as your income increase. The statutory tax rates are marginal tax rates, they indicate the ratio between an increase in tax and an increase in income at a given income level.

Mathematically, the marginal tax rate corresponds to the derivative of the tax curve, or equivalently the tax curve is the integral of the marginal tax rates. The statutory tax rates are dependent on the filing status and adjusted for inflation. The 2006 schedule of statutory tax rates are summarized in the table below.

	Upper limit of income bracket taxed at the indicated rate
Filing status	10%	15%	25%	28%	33%	35%
Single	$7,550	$30,650	$74,200	$154,800	$336,550	Infinity
Married filling separately	$7,550	$30,650	$61,850	$94,225	$168,275
Head of household	$10,750	$41,050	$106,000	$171,650	$336,550
Married filling jointly	$15,100	$61,300	$123,700	$188,450	$336,550

As the statutory tax rates are piecewise constant, the tax curve is piecewise linear, i.e is formed of line segments connected together as shown in the figure below. The black curve shows the tax paid in function of the income, the red curves shows each segment extrapolated, illustrating the slowly increasing rates characteristics of a progressive tax system. Mathematically, the curve appears to have a positive second derivative.


Before introducing all extra tweaks, the tax curve has some desirable properties:
* continuous
* monotonously increasing
* piecewise linear
* first derivative monotonously increasing

The first derivative however is not continuous, but piecewise constant with discontinuities at the tax bracket boundaries. This is not a big problem in practice, except that it introduces impulses in the second derivative, and generally only discussion of the first derivative makes sense.

The tax curve is directly expressed in monetary units and so in some sense is the most concrete of all, but it has the inconvenient of being continuously increasing, so that details close to the origin are lost. Two alternate ways of presenting the same information with less scale problems is to use either the first derivate (the marginal rates) or the average rate (the tax divided by the income). The next figure shows both the marginal and average rate for a single filer.


That the average rate is always lower than the marginal rate is normal for a first derivative that is monotonously increasing. The specific form of the average rate will be discussed in a next article.

Rationale

It is once more begin of January, tax forms start to trickle. And yet again, I'm reminded of the difficulty of preparing my annual tax return. I'm especially taken aback by the style of the IRS documents, especially the instruction forms.

I'm an engineer, trained to use mathematics as a tool, adept at reading technical documents where graphical elements are used to enhance comprehension. But obviously IRS instructions are cast from a different mold.

A serious peeve, from an engineering standpoint, is the wordiness of IRS documents, especially when describing what are in essence simplistic mathematical expressions, coupled with an almost total absence of explanation for the rationale behind some of the tax rules. And of course, I'm appalled to see page after page of tables that seem to be nothing more than a tabulation of simple linear equations.

This blog wants to apply mathematical concepts to give visual explanations of tax rules and to show that their form does (or sometimes doesn't) make sense. A first sequence will tackle the concept of marginal rates, with the explicit intention of explaining how the curve below, extracted from "Effective Marginal Tax Rates on Labor Income", a Congressional Budget Office document, can be recreated.

The CBO is a government office. As such the documents it produces are not copyrighted. This is confirmed by an explicit copyright policy, that however requires correct attribution. So please mention the CBO origin if you want to use the above chart elsewhere.