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.