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 advocate sole breadwinner in a family, and the other person is stay at home husband/wife. 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 ${\mathbb{R}}_{+}$ be the positive reals. Consider functions $S:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ and $J:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$. 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 $S(x)+S(y)=J(x+y)$. 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 $T$ should have the following properties.

1. Reasonable Tax: $0\le T(x)\le x$. Indeed, one should not tax people more than their income. The taxation system does not want to give free money to low income people either.
2. Principal of Progressiveness: there is some $c>0$ such that $\frac{T(x)}{x}>\frac{T(y)}{y}$ for all $x>y>c$. Basically, the rich should pay a larger proportion of their money to taxes.

For a marriage neutral system, the reasonable tax requirement would prove that $S(x)=ax$ for $a\in [0,1]$. It is easy to see we cannot hope to have principal of progressiveness.

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 $S(x)+S(y)\ge J(x+y)$. An easy tax function is a function that has reasonable tax property, and is a piecewise-linear convex that has at least $1$ breakpoint larger than $0$. This is strictly stronger than principal of progressiveness. This is satisfied by the current personal income tax function used by the IRS.

Let $S$ be a easy tax function, then we can obtain an easy tax function $J$ that always gives a marriage bonus. Indeed, let $J={\inf }_{a+b=x,a,b\ge 0}S(a)+S(b)$. $J$ is extreme in a way that any function greater than it at any point will cause marriage penalty. $J$ is the infimal convolution of $S$ and itself, which would also be piecewise-linear convex. If $S$ is the personal income tax function for 2019, then $J$ matches the 2019 IRS married file jointly function up to $612,350! For some reason I do not know, the IRS decide to cut this $J$ off at $612,350, and then impose a higher rate just to penalize families with two very high income earners.

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