In the US, April 15th is a sort of holiday, the deadline to file Federal Income Tax returns. In the minds of some, tax brackets are an important (only?) instance of a nonlinear function. That is, since we, as a society have decided to increase the rate that income is taxed goes up as the amount increases, we have a progressive tax system. However, this leads to complications (in addition to the complications of calculating the various deductions and credits to politically favored groups, such as homeowners) when deciding how much to withhold when both spouses in a couple work and file jointly. Professional tax preparers know that when the formula for withholding is applied separately to each income, the total withheld will generally be too low when the incomes and tax liabilities are combined. This is a great example of Jensen’s inequality, in which the average of a function will be the function of the average input value only if the function is linear: <f(x)> = f(<x>) holds if f is a linear function of x, but NOT in general for some other functionality. Specifically, if f is a convex function (the second derivative is positive, <f(x)> is greater than f(<x>). In the case of the IRS, imagine that the function f is the tax liability for x adjusted gross income (after deductions, etc.). In a couple filing jointly, the total withheld will be: f(x1) + f(x2), but the actual tax liability is f(x1+x2), which is GREATER than f(x1) + f(x2), since the marginal tax rate increases for larger incomes. This means that too little was withheld and the couple will still owe money come tax day. Put another way, combining incomes puts more of the couple’s money into a higher tax bracket. For a while this meant couples actually paid more taxes – the so-called “marriage penalty” – but congress changed the tax structure to mitigate this effect. However, when calculating the withholding, each member of the couple is still considered individually.
For more, see the excellent and accessible book: The Flaw of Averages.