## Saturday, February 3, 2007

### 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.

#### 1 comment:

Terry Marston said...

Is it feasible to integrate the marginal tax rate with the "deductions, exemptions, credits, phaseout, [refunds,]and other goodies" you mention to get an idea what the "effective" marginal tax rates are. For instance, you always hear that 40% us pay no taxes at all. I'd love to see you take a stab at that.