Visual Tax

Phasein

2007-04-19T20:47:00.000-07:00

A phasein is the opposite of a phaseout, a tax benefit is increasing with increasing income. It sounds too good to be true, but in fact the EITC works exactly like this. The EITC is a refundable credit that increases together with an increase in income, reaches a plateau and finally decreases in a phaseout range.

The EITC is such a special mechanism that it is also probably abused. Rather incredible amounts of non compliance were reported in the past by the IRS "Despite these efforts, the IRS has been unable to significantly reduce noncompliance. The most recent compliance study (of TY 1999 returns), reported that between $8.4 and $9.9 billion in EITC claims (27% to 32%) had been improperly paid. Based on “significant compliance problems” associated with the EITC, the General Accounting Office has listed the administration of the credit as a “high risk area for the federal government.”

For our purpose, a phasein results in a decrease in the marginal rate. In fact because the EITC applies to income ranges with very low statutory rates, the marginal rate may become negative. This rather strange behavior is apparent on the figure shown on the first post of this blog. Understanding how the marginal rate could ever become negative was a reason I wanted to understand in more detail the tax mechanisms in the US.

Smooth phaseout

2007-03-25T13:59:00.000-07:00

As shown before sharp phaseouts of deductions may introduce the annoying problem of your tax income decreasing even as your before tax income increases. The solution to that problem is to define a smooth instead of a sharp transition. The simplest method and the one chosen by the IRS is to linearly spread the phaseout, i.e. the deduction decreases linearly until 0 above the phaseout boundary. The previous example can be reworked, showing three different linear transition slopes.

As shown in the figure, a smooth transition makes the curve continuous, but it is still possible for the income after tax to be locally decreasing, depending on how fast the deduction is phaseout. This is illustrated by the red dashed curve in the figure.

There are two ways to define a linear transition phaseout, either fixing the slope of the phaseout, or fixing the end point of the transition. The two methods are equivalent when the deduction amount is fixed, like for the tuition fee example, but they are not equivalent when the deduction amount is a variable dependent on one's situation. In that case, fixing the slope is better because the slope is directly related to the increase in marginal rate. Fixing the slope insures that the change in marginal rate is also fixed. This is probably why this is the form universally selected by the IRS when it defines smooth phaseouts.

The IRS always uses similar text to describe the transition. The example below is taken from Publication 17 (2006): "If your itemized deductions are subject to the limit, the total of all your itemized deductions is reduced by ... some details removed at this time ... 3% of the amount by which your AGI exceeds $150,500 ($75,250 if married filing separately)."

The impact on the marginal rate can be summarized like this. If not for the phaseout, the tax close to the income of interest can be expressed by a simple linear relation, with r the marginal rate

Adding the phaseout results in a modified tax equation, with s the slope of the reduction (would be 3% for the IRS example above)

The new marginal rate can be found by differentiating the modified equation

This shows that controlling the slope as specified results in a predictable increase of the marginal rate. It is then a simple matter to specify the slope such that the phaseout doesn't introduce unwelcome effects, especially limiting the increased marginal rate in the transition region to (well) below 100%.

Sharp boundary phaseouts other problem

2007-03-04T10:15:00.000-08:00

An other problem introduced by sharp boundary phaseout is linked to an other desirable property of the tax equation that was not stated yet: the income after tax should be a monotously increasing function of the income before tax.

In simple terms, an increase in your income before taxes should always result in an increase of your income after taxes. Mathematically this means that the marginal tax rate must be smaller than 1 everywhere, and this is clearly not the case at a sharp phaseout boundary.

If income is defined as a continuous variable, the marginal rate is infinite, i.e. there is a positive impulse in the marginal rate at the position of the boundary. Income is of course quantized, and the marginal rate in that case is always finite, but significantly greater than one. In the example discussed in the previous article, the marginal rate was $500 in tax increase divided by $10 of income increase = 50 or 5,000%.

We can visualize that using a graph showing the income after tax in function of the income before tax, based on the characteristics of the fee deduction for an individual filer. The sharp phaseout boundary introduces a negative step into the curve, the highlighted areas correspond to ranges of income where an increase in income before tax can result in a decrease of income after tax as shown below.

The vertical step is simply the difference in deduction times the underlying marginal rate. The range of income affected is the vertical step divided by the slope, and the slope is 1 minus the underlying marginal rate. For the example, the vertical step is $2000 * 0.25 = $500, and the affected range $500 / ( 1 - 0.25) = $667.

Of course, the problem is only apparent when looking closely, when looking at a scale much larger than the range calculated above, the bumps in the curve become much less significant, but still present and somehow disturbing.

In the next article we'll see how to avoid these ugly steps, providing a rationale for the solution favored by the IRS.

Phaseouts: sharp boundary

2007-03-01T20:08:00.000-08:00

Phaseout is the process of decreasing some tax benefits, usually based on an increase in income. This can take the form of reducing the amount of a specific deduction, or to decrease some tax credit.

Phaseouts can take different form, the simplest form is a sharp boundary: if your income exceeds a threshold, then some deduction or credit is no more allowed. This form of phaseout is easy to describe and understand but has unwelcome consequences: it introduces a discontinuity in the tax equation. Most of the phaseouts defined by the IRS have a more elegant form that will examine in a future article, but unfortunately a few sharp boundaries do exist in the tax code.

But why is continuity of the tax curve important? Because otherwise small changes in income can create large changes in tax liability. This can be illustrated by the following scenario: John Tuition has carefully entered all the needed information in some tax preparation software, its total tax comes to X dollars. A little later, he receives an updated 1099 form, where after correction its income increases by $10. No sweat, John opens the saved tax file, updates the relevant field, and sees his tax liability jump by $500!! While unlikely, this scenario is not impossible and obviously rather disturbing

The problem is that John was deducting tuition fees. The phaseout for tuition expenses as defined in Chapter 6 of Publication 970 (2006) goes like this (in the typical IRS verbal description)

If your modified adjusted gross income (MAGI) is not more than $65,000 ($130,000 if you are married filing jointly), your maximum tuition and fees deduction is $4,000. If your MAGI is larger than $65,000 ($130,000), but is not more than $80,000 ($160,000 if you are married filing jointly), your maximum deduction is $2,000. No tuition and fees deduction is allowed if your MAGI is larger than $80,000 ($160,000).

So the phaseout has two sharp boundaries that depend on your filing status. If you cross a boundary, the deduction may decrease by $2,000, as it happened to John. The increase in tax liability is the $2,000 step multiplied by the marginal tax rate, 25% for incomes close to the boundary. So John may have reasons to curse who wrote that tax provision, but calling the customer support for the tax preparation software will be fruitless.

And things can go even worse. The increase in tax liability may cascade because this could result in John crossing an other sharp boundary. The IRS starts to assess penalties for underpayment of estimated tax based on a sharp boundary. The underpayment rules are a model of complexity that I may tackle in more detail later on, but the boundary is defined for most people like this in Chapter 4 of Publication 505 (for 2005, no change in 2005 as far as I know)

In general, you may owe a penalty for 2005 if the total of your withholding and estimated tax payments did not equal at least the smaller of:
1. 90% of your 2005 tax, or
2. 100% of your 2004 tax. (Your 2004 tax return must cover a 12-month period.)

So the $10 increase in John's income can result in an increase of about $500 in tax liability. That increase can possibly trigger the estimated tax underpayment penalty. Calculating the underpayment penalty is complex, in general you let the IRS figure that out, but it will be a positive value :-). If anybody can provide an estimate for the above scenario, that would be great.

Hopefully the bad consequences of sharp phaseout boundaries are now clear, and the tax code has a solution, tapered phaseouts, with less drastic impact on the tax equation. This will be the subject of the next article

Digression: first comment

2007-02-27T21:06:00.000-08:00

I got my first comment, but unfortunately it is rather scathing. Obviously, I missed my goal, possibly even achieving an effect contrary to the expected one: obfuscating instead of enlightening :-(

The strange thing though is that researching the subject has unearthed different papers that essentially use the approach I want to follow. I guess this is not the approach per se that is wrong, but my own take on it. I'll try to realign my presentation towards a more concise and elegant approach as suggested by the anonymous scholar.

On a side note, comments are always welcome, even negative ones. You cannot get better if you don't know your bad points. Constructive criticism is obviously preferable to a blanket repudiation, but I'm afraid I'm not the one calling the shots. But in some other ways, a blog can be seen as a form of mental masturbation, so I'll continue even if I fail in my stated goal, I can still enjoy the act of writing itself

My (restated) goal is to construct a simple mathematical expression that calculates the tax owed in function of all necessary parameters. I found in "Effective Federal Individual Income Tax Functions: an Explanatory Empirical Analysis" a citation that summarizes the approach I want to follow in a concise way: "... a tax law is a mapping from a vector whose elements are the income characteristics of the individual (wage income, dividends, capital gains, and all the other items in the income tax form) to tax liabilities. It is supposed to be a well defined function; ..." taken from Arrow, Kenneth J. "Microdata Simulation: Current Status, Problems, Prospects." in Microeconomic Simulation Models for Policy Analysis, New York: Academic Press, 1980.

I want to construct this mapping. This mapping is simple in the sense that it only uses elementary operations, the final form is piecewise linear and mostly continuous, this sounds simple to me and I want to expose this simplicity for everybody to see

Tax credits

2007-02-25T09:59:00.000-08:00

While deductions and exemptions reduce your income, credits directly reduce your tax itself. As such credits are better, their effect doesn't depend on the marginal rate. Formally, the effect of credits can be reflected in a new tax equation.

with
* C, a vector of Credits
* T₁, the tax equation defined before

The most important credit is the earned income tax credit (EITC). The EITC is a very special tax credit, it is refundable, in other words if you end up with a negative tax, the government will effectively send you back some money. Note that this is different from a refund, the EITC is only valid for low income taxpayers and correspond to a work incentive, in a certain range of income, the government chips in some extra money towards your disposable income. The EITC is one of the largest aid programs, as mentioned in "Behavioral Responses to Taxes: Lessons from the EITC and Labor Supply"

In fact, the EITC is the largest cash transfer program for lower-income families at the federal level. An unusual feature of the credit is its explicit goal to use the tax system to encourage and support those who choose to work.

The next figure is extracted from the same document (based on values provided in table 13-14, Green Book, 2004, Joint Committee on Taxation, Ways and Means Committee) and shows the evolution of the EITC. To put things in perspective, the total amount of individual tax liability is about 800 billions in 2003.

As alluded, the EITC increases with an increase in income, mathematically this means negative marginal rates, i.e. your tax decreases (in this case becomes more negative) as your income increases. The exact rates are variable and depend on the phaseout characteristics of the EITC, this will be discussed in the more general article on phaseout effects.

Deductions and equity

2007-02-17T09:35:00.000-08:00

The presence of deductions has a nasty side effects, both the average and marginal tax rates becomes dependent on the deductions one can claim for a given level of income. The new tax equation is easily derived in function of the previous equation

with
* D, a vector of deductions
* E, a vector of exemptions
* T₀, the tax equation defined before

The rates at a given income are now dependent on the deductions and exemptions. In other words, two taxpayers with the same income and filling status can be taxed wildly differently, based on which deductions and exemptions apply to them. Obviously also, people are aware of this and may adapt their lifestyle to maximize deductions, at least when possible. This is in some sense, why deductions are introduced in Congress, but sometimes with unintended side effects.

An obvious side effect is to introduce societal tensions between people able to claim specific deductions and people unable to claim them. A second side effect is the complexity of the rules needed to administer the different deductions, impacting both the taxpayer and the IRS.

An extreme example compares the federal tax and corresponding average tax rate for two couples filling jointly in the figures below. The first couple has no kid and don't itemize deductions, the second couple has three kids and itemize major deductions that scale as a percentage of the income: 7% state tax, 25% of mortgage interest and 7.5% of medical expenses (their total medical expenses is 15% of income, the first 7.5% cannot be deducted).

Because in this model the deductions are proportional to the income, the two curves diverge, asymptotically the ratio between the marginal rates is 1 - the total percentage of deductions, or about 0.6 in the example. It is a matter of discussion if such a difference makes sense, i.e. is it equitable to favor specific behaviors though differences in taxation rates?

And the difference is huge, the next figure shows the ratio between the tax paid by the couple with three kids and using aggressive deductions and the tax paid by the couple with no kid and using the standard deduction. The ratio never even reaches 50% for the range of income considered here, with the asymptote at 60% as explained before.

Deductions can so much reduce the tax paid that two different mechanisms are in place to avoid their abusive use: phaseouts and the Alternate Minimum Tax (AMT). Both of these will be discussed later on, after an article on tax credit.

Exemptions and deductions

2007-02-14T20:42:00.000-08:00

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

2007-02-11T19:47:00.001-08:00

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

2007-02-03T09:46:00.000-08:00

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

2007-02-02T20:08:00.000-08:00

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.