Symmetry property of fair and efficient wage taxation

2017 January 13
by Daniel Lakeland

Based on my discussion of eating your own labor one of the major points there is that the second earner in a household faces a massive tax burden that makes it generally unwise for households to have two earners unless each of them earns a very low wage. What would be a better scheme? Really we'd like two people in a household to face symmetric work incentives. This then enables households where two people have advanced degrees or other large earning potential to both do the thing that society most wants them to be doing (namely, working to provide valuable services).

This symmetry property works as follows. Suppose we have worker W1 and W2, and they earn salaries S1 and S2. We can define a function Keep(S) that describes how much of the salary a single worker keeps.

Keep(S) = \int_0^S (1-t(x))dx

Where t(x) is the marginal tax on an extra dollar that a person earning x dollars takes home.

Now, the symmetry property we want is

Keep_2(S1+S2) = Keep(S1) + Keep(S2)

That is, whether you aggregate the salaries into a single household and use the household keep function, or you consider them split out as two totally separate earners, you still get the same total amount kept by the two earners.

Now, this is something like linearity. The property of linearity would be

Keep_2 = Keep


Keep(a\times S) = a\times Keep(S)


Keep(S1 + S2) = Keep(S1) + Keep(S2)

But, our fairness/symmetry requirement imposes a weaker requirement than linearity. For example suppose Keep_2(x) = x for x < 2 and Keep_2(x) = 2 for x > 2, and Keep(x) = x for  x< 1 and Keep(x) = 1 for x > 1, (that is, you keep 100% of the first dollar each person earns and nothing after that). Then the symmetry we want holds, but of course this is a seriously pathological example.
Now let's examine a further principal. The idea is you should keep SOME of every dollar that you earn, otherwise there is no incentive to do something valuable to society for which you can get paid, since the pay would all go to the government. This principal can be translated to the following mathematics:

\forall x \ge 0, \frac{dKeep}{dx} > 0

Which says that the Keep function is always going up at least a little bit.These two principals are sufficient to show that both the Keep and the Keep_2 functions are lines with the same slope. We can do this by supposing that S1 is anything you like, and S2 is $1 which we will take to be dx, an essentially infinitesimal quantity.

Keep_2(S1+S2) = Keep_2(S1) + \left. \frac{dKeep_2}{dx}\right |_{S1} dx \\= Keep(S1) + Keep(S2) = Keep(S1) + \left. \frac{dKeep}{dx}\right |_0 dx

Which shows that Keep_2(S1) = Keep(S1) and \left. \frac{dKeep_2}{dx}\right |_{S1} = \left. \frac{dKeep}{dx}\right |_0, which is to say that the Keep and the Keep_2 function are the same, and they have the same slope at any arbitrary point S1 as they do at x=0 (so it's a line).

You can also get that the keep functions are the same from Keep_2(S1+0) = Keep(S1) + Keep(0) and Keep(0)=0.

From this, I argue, the only income/wage tax compatible with the family value of "not punishing a second earner in a married household" is a flat tax on wages (ie. a constant tax rate, you're always taxed say p% of your total earnings). In my opinion, Feminists should be arguing strongly for flat taxation as women are more likely to be second earners, and women, particularly women who are educated, have student loans, have deferred incomes to gain skills, and soforth are the ones most harmed by wage taxes with increasing rates for increasing household income.

Furthermore, society needs the skills of all the highly skilled people we can get. Highly skilled people tend to marry other highly skilled people. We can't afford to be educating anyone to get skills and then leave them unable to take advantage of those skills because they're the second earner in a high earning household. That means we either give up on highly skilled people having families (completely untenable!), or we give up the value to society they would have provided in services if their marginal tax rates were constant, and make them pay for their student loans from their spouses salaries instead of what would have been their high earning potential.

Note: those who complain about the "regressive" nature of flat taxes should defer their complaints. The amount you consume doesn't need to be equal to the wages you keep. You can also spend money you receive from the government or from non-wage sources, and I will argue for a Universal Basic Income to compliment flat taxation in a future post. Furthermore, it's entirely plausible that the harm to society from making half the potential skilled work-force stay out of work is plausibly much larger than any harm from regressive taxation.


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