No, a flat tax cannot be both revenue & distributionally-neutral.

I got into an extended twitter discussion today with economist Dr. Aaron Hedlund about whether or not it was possible to have a flat tax that maintained progressivity while also being revenue neutral. The back and forth was cordial and respectful, and this longer response will maintain that tone. But we came to a bit of a dead-end, and what I want to do here is show, using some basic assumptions and Tax Policy Center data, that it is mathematically impossible to accomplish both of those goals — revenue neutrality and distributional neutrality — with a flat tax.

But first, let me start by acknowledging that Dr. Hedlund’s initial contention is true:

He is right that, with a large enough standard deduction, people at the bottom of the income distribution could do better under a flat tax system, so long as that system replaced both the income and payroll tax. That’s because, contrary to some popular mythology about who pays taxes, even very low-income households do have positive tax liability, mostly because of payroll taxes. So, if you instead got rid of payroll taxes and exempted some large amount from the new flat income tax, a lot of those at the bottom would end up actually having zero tax liability.

A quick example. In 2015, the average household in the bottom 20 percent of the income distribution made about $13,000, and paid about $180 in combined federal income and payroll taxes. But under a hypothetical flat tax with a $13,000 deduction, that same household would pay nothing because all of their income is exempt from taxation.

Ok, so it is possible for households on the bottom of the income spectrum to be held harmless in a system like this, with the caveat that this only works if the flat tax is replacing both the income and the payroll tax.

But the discussion ranged away from this narrow question to a larger one. Namely, can you design a flat-tax system that is both revenue-neutral and more broadly distributionally-neutral? In other words, can you get roughly the same tax revenue from roughly the same income groups using just one rate (plus a large standard deduction) instead of many rates?

The answer is: no you cannot.

To see why, let’s start with some numbers. The richest 1 percent paid 27.4 percent of their income in federal income and payroll taxes under the current system in 2015. In order to maintain revenue and distributional neutrality, the new flat tax system must ensure that they pay at least the same effective rate. If they pay less, one of two things must then be true. Either, the lost revenue is made up for elsewhere in the income distribution and therefore we’ve violated our distributional requirement, or else the lost revenue is not made up for anywhere else and so we’ve violated our revenue requirement. So, the first constraint is that the top 1 percent must still pay about 27.4 percent of their income under the new system.

Ok, so where do we need to set the rate in order to ensure that the top 1 percent are paying a 27.4 percent effective rate. Why, somewhere slightly above 27.4 percent, of course. It’s that simple because, under this system, there will be some standard deduction and then all subsequent income, regardless of type or source, is taxed at one rate. With no deduction, a marginal rate of 27.4 percent ensures that every dollar earned by the top 1 percent is taxed at that rate, and therefore their effective rate is 27.4. With some deduction, the rate will need to be a little higher because some portion of their income will be taxed at zero. And notice that as the deduction goes up, the tax rate will have to go up a little bit too.

Now that we’ve got the 1 percent mostly squared away, let’s figure out where to set the deduction so that people at the bottom and in the middle don’t get screwed. Right now, the combined income and payroll tax effective rates, by income percentile are as follows:

Source: Tax Policy Center

Obviously, we can’t apply the 27.4 percent tax rate to every dollar, or even most dollars, earned by people in the bottom group, or else they end up paying a lot more than they do now and that violates our distributional requirement. In fact, we’ll need to exempt almost all of their income, so that applying a 27.4 percent tax rate leaves them paying 1.4 percent, the same as they pay now.

Figuring out what the deduction needs to be requires a little bit of simple algebra (i.e. what is the amount of taxable income needed that, with a 27.4 percent rate, yields tax payment equal to 1.4 percent of total income). The answer, for the average household in this group, is about $12,300. In other words, using the flat rate of 27.4 percent, we need to start applying the tax only after the first $12,300 so that the average household in the bottom quintile still ends up paying only 1.4 percent of their total income in taxes.

Great, so we set the deduction at $12,300, the rate at 27.4 percent. That way the bottom 20 percent pay roughly the same as they do now (on average) and same with the top 1 percent. So far so good. What about the rest of the income distribution? Here now, are the new effective tax rates (and the old ones) for the average household in each group:

Flat tax effective tax rate is for the average household in each income group.

Uh oh. Sure the top and the bottom are basically fine. But everyone else is paying much much more. Clearly, the standard deduction needs to be higher.

We can go through the same process to figure out what the deduction would need to be so that the average household in the second quintile would still pay 5.7 percent as they do now. The answer is roughly $26,000. The average household in that quintile makes about $33,000, so exempting the first $26,000 means that only their last $7,000 are subject to the 27.4 percent tax. That means their overall effective tax rate is back down to 5.7 percent. Great, but you probably already see where this is going.

Increasing the deduction to $26,000 means the second quintile is basically held harmless (again, on average), but it means the bottom quintile now pays nothing (since the richest family in the bottom 20 percent makes just under $26,000). That’s not necessarily a problem from a progressivity perspective, but it is a problem if you are trying to maintain rough distributional neutrality.

Ok, but say we don’t mind a little bit of wiggling around. And we certainly don’t mind it when the poor pay less (this is, of course, at odds with much of the rhetoric from some flat-tax proponents, but that’s a side point). Has the $26,000 deduction solved the rest of our distributional puzzle? No, not by a long shot.

Flat tax effective tax rate is for the average household in each income group.

Everyone from the middle on up to almost the very top is still paying way more, while the top 1 percent is now paying a little less because the rising deduction is eating into their effective rate.

You can see, now, the fundamental problem here. We can increase the standard deduction yet again to get the effective tax rate down for those in the middle of the income distribution. It would take a deduction of about $36,500 (and a rate of about 27.9 percent to get the top 1 percent back up to their target) to ensure that the average household in the middle quintile still pays 10.8 percent. But now the effective rate for the second quintile falls significantly below its current level, and pretty much everyone from the 60th to the 98th percentile will still be paying way more than they do now, while the top 1 percent will still paying the same.

This trade-off is inevitable when you only have one rate. Every time you increase the deduction, you lower the effective rate for those on the lower and middle part of the distribution, but unless you raise it really high, a big chunk of people in the upper-middle will a lot pay more than they do now, while the top 1 percent stays the same (by design). And you can’t raise the top 1 percent any higher without also raising the effective rates on those upper-middle income households who are already paying more, because you only have one rate!

Thus, there is no way to design this flat- tax system such that everyone pays something relatively close to what they are paying now.

BUT…

What if you relax the constraints somewhat? What if, instead of demanding strict distributional-neutrality relative to the current system, you instead only demanded general progressivity. That is, what if all we cared about was that, under the new system, the rich still paid more than the middle who paid more than the poor? Could we have flat-tax system that met that standard, while still being revenue-neutral? (And, to be fair to Dr. Hedlund, he did drive toward this construction of the question.)

The short answer is “Yes.” The longer answer is “Yes, but it would be super weird, it would involve rather large tax increases for upper-middle income people, it would mean a marginal rate increase for most middle-income households, and it would not offer any obvious benefits.”

Here’s one example. Let’s say we set the deduction at about $50,000 and the rate at about 28 percent. That would generate the following effective tax rates:

Flat tax effective tax rate is for the average household in each income group.

On it’s own terms, these effective rates are progressive. The bottom 40 percent of households pay nothing, the average household in the middle pays a little, and then the rates begin to climb. This structure also yields an annual revenue loss of about $50 billion. That’s roughly 2 percent of current income and payroll tax revenue. Not nothing, but maybe we can let it slide.

The real problem here is practical. Yes, we can make these numbers work on a spreadsheet, but imagine telling households making around $160,000 a that their tax bill is going up by 15 percent. Or a household making $250,000 a year that their tax bill is going up by 20 percent. I’m not about to let this already ridiculously long post get derailed into a discussion of who is and isn’t middle class, but from a political economy perspective, there’s just no way this is ever going to happen. Especially not when the top 1 percent get no tax increase, and the bottom 40 percent get their taxes wiped out entirely.

And truth be told, the options for improvement on this plan aren’t great. You could increase the standard deduction even further to try to limit the size of the tax increase for those 80–99 percenters. But as soon as you get too far over $50,000, the revenue losses start to pile up real quick. For example, a deduction just $5,000 higher would yield a total revenue loss of over $150 billion. You could try to mitigate the revenue loss by increasing the tax rate, but then you’re back in the same trap of massively raising taxes on the 20 million households who are just below the top 1 percent, since they feel the effects of the rate increase too (because one rate). And remember, the flat tax rate is already about 28 percent, which itself is a higher marginal rate than households who make $60,000 a year face now.

So, who, exactly, would like this new system? Conservatives who already mistakenly complain that half the population pays no federal taxes would actually be accurate under this system. Liberals who’d like to see the very rich pay more would be frustrated, since the top 1 percent’s rate would stay the same. Economists who think the marginal tax rate is important for a variety of reasons would worry that while that rate is lower for the very top few percent, and for the bottom, it’d be actually higher for a lot of middle-income people. And the 20 million upper-income, but not 1-percenter households who saw their tax bill go up by 15 or 20 percent would, of course, be utterly furious. (And I’m not going to even get into the Social Security and Medicare financing implications.)

Ultimately, I think that Dr. Hedlund’s argument that you could create a flat tax system that yielded progressive effective rates is narrowly true. On its own terms, yes, you could. I concede that point. But compared to the current system, it probably wouldn’t be as progressive, and it would have all these other serious flaws.

And what are the advantages of this system? Simplicity? Sure, but all of this plan’s simplicity comes from taxing all forms of income the same way, and having only one deduction that everyone takes with no other tax breaks, not from its single rate. It would be just as simple if, after the big deduction, there were 10 rates or 1000. Rates don’t add complexity, income calculations and special tax breaks do.

Finally, I think it’s important to point out that this type of flat tax — one with a very large standard deduction, with a high tax rate which applies to all forms of income, and which replaces both income and payroll taxes — is not one that any political candidate or officeholder that I am aware of has proposed. Dr. Carson’s flat tax, for example, does not replace the payroll tax, and exempts investment income from taxation entirely. So while it may be theoretically possible to craft a one-rate structure that is roughly revenue neutral and yields progressive effective rates (though not distributionally neutral in comparison to current law), no one should confuse any of the actual proposals out there for this theoretical approach.

Budget and econ wonk. Pittsburgh native. Father of three. Aspiring to be Leslie Knope (would settle for Ben Wyatt). Personal account. Opinions my own.

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