Sunday, March 25, 2007

Smooth phaseout

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%.

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