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Impossibility theorem of marriage tax


It is well known that there could be marriage penalty or marriage bonus. For example, in 2019, if two people each make more than $306,175, then they have to pay more tax after getting married. In the worst case, they have to pay $8,165 more. Not that bad. However, if one person make all the money, and the other has no income, then together they will always pay a smaller amount of tax.

I always thought this is because the tax code is designed to make sure in a family, there is a sole breadwinner. But recently I realized it is just mathematically impossible to have anything other than a linear tax and preserve some other nice properties.

Indeed, this was shown by Lovell [1].

Let R+\R_+ be the positive reals. Consider functions S:R+R+S:\R_+ \to \R_+ and J:R+R+J:\R_+ \to\R_+. The first is for tax paid for a single person and tax paid for a married couple file jointly. The input for married file jointly is a single number, which is the combination of the taxable income of the couple. This is called horizontal equity in marriage.

Marriage neutral is precisely when S(x)+S(y)=J(x+y)S(x)+S(y) = J(x+y). We define a few notions, it is not completely the same as the ones in Lovell’s paper [1], but it essentially demonstrate the same idea.

A tax function TT should have the following properties.

  1. Reasonable Tax: 0T(x)x0\leq T(x)\leq x.
  2. Principal of Progressiveness: there is some c>0c>0 such that T(x)x>T(y)y\frac{T(x)}{x} > \frac{T(y)}{y} for all x>y>cx>y>c.

The second one tries to tax the rich more, as in larger proportion of their money. The reasonable tax requirement makes sure S(x)=axS(x) = ax for a[0,1]a\in[0,1]. It is easy to see we cannot hope to have principal of progressiveness.

Remark

Married filing separately is always no better than them being single and file their own taxes.

I personally think there should not be a marriage penalty at any income level to encourage marriage. Of course, people might disagree and think the rich should have a marriage penalty, since it is a small amount compare to their total income so they won’t care anyways.

Anyway, consider the world where there can only be marriage bonus. That is we have the property S(x)+S(y)J(x+y)S(x)+S(y)\geq J(x+y). An easy tax function is a function that has reasonable tax property, and is a piecewise-linear convex that has at least 11 breakpoint larger than 00. This is strictly stronger than principal of progressiveness. This is satisfied by the current personal income tax function used by the IRS.

Let SS be a easy tax function, then we can obtain an easy tax function JJ that always gives a marriage bonus. Indeed, let J=infa+b=x,a,b0S(a)+S(b)J = \inf_{a+b=x,a,b\geq 0} S(a)+S(b). JJ is extreme in a way that any function greater than it at any point will cause marriage penalty. JJ is the infimal convolution of SS and itself, which would also be piecewise-linear convex. If SS is the personal income tax function for 2019, then JJ matches the 2019 IRS married file jointly function up to $612,350! It’s just for some reason the IRS decide to cut this JJ off at $612,350, and then impose a higher rate just to penalize families with two very high income earners.

References

[1] M.C. LOVELL, ON taxing marriages, National Tax Journal. 35 (1982) 507–510.

Posted by Chao Xu on .
Tags: Tax.