### Fun with Laffer curves

**Matthew Martin**10/22/2015 01:35:00 PM

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"This ignores mountains of evidence, going back for generations, showing that raising tax rates does not automatically mean raising tax revenues — and has often actually led to falling tax revenues. A fantasy expressed in numbers is still a fantasy."He softens the remark in the usual weasle words ("does not automatically mean") but gives away the game by asserting flat out that raising revenues by rasing the tax to 40 percent is "still a fantasy." Sowell claims that the Laffer curve peaks at less than 40 percent. And he's wrong.

I think the idea of 40 percent taxes makes a lot of people nervous because they implicitly think that there's some dangerous threshold at 50 percent, as if that's the threshold at which people will drop out of the labor force. But the reality is that nothing special happens if you tax more than 50 percent. First, it's important to remember that that's a 50 percent

*marginal*rate--most people in this bracket won't pay anywhere near 50 percent of their income because most of their income is in a lower tax bracket. And as for the incentive effects, the poor are actually likely to work

*more*if you tax them at a higher rate because they simply need the money--the wealth effect simply dominates. Now think about the rich. The top income bracket threshold corresponds to roughly $1,000 an hour. Take half of that and they'd still get $500 for an extra hour's worth of work per week--that incentive is still huge. For the rich, the substitution effect does dominate, just as the Laffer curve predicts, but at 50 percent that substitution effect is still pretty strongly weighted towards labor.

That's why, if you actually examine the "mountains of evidence" Sowell refers to, you'll find that economists generally estimate that the peak is somewhere in the 70 percent to 80 percent range. It's hard to know for sure, because it has been a very long time since we had tax rates anywhere near that high, and there are a lot of other taxes and wedges that affect it. But that remains the best estimate we have.

Nevertheless, out of curiosity, let's do the math. Let [$]L[$] be the number of hours per week that you work, which is a decreasing function of the tax rate [$]\tau[$]. You work for an hourly wage [$]w[$] so that [$]wL[$] is your weekly pay. Thus the taxes [$]y[$] you pay is given by [$$]y=\tau wL.[$$] In order for it to be the case that raising taxes reduces the total taxes you pay (and therefore government revenue), the derivative with respect to the tax rate, [$]\frac{\partial y}{\partial \tau}[$] must be negative:[$$]0\gt \frac{\partial y}{\partial \tau} = wL+\tau w \frac{\partial L}{\partial \tau}[$$] which implies that we must have [$$]\frac{\partial L}{\partial \tau}\lt -\frac{L}{\tau}.[$$]

Let's plug in some numbers, shall we? With a 40 hour work week and a 40 percent tax rate, that means for Sowell's claim to apply to our example, [$$]\frac{\partial L}{\partial \tau}> -\frac{40}{0.4}=-100[$$] That in turn means that a 1 percentage point increase in tax must cause you to reduce your weekly hours by at least 1 hour per week. At the median wage of $22 an hour, that's a tax hike of $8.80 per week causing you to reduce your weekly income by $22 per week in exchange for an extra hour of liesure a week.

Whether that sounds reasonable to you is opinion, I supose. But the data does not find many people making that choice.

Sidenote: there are also state and local taxes to consider, as well as other types of taxes. In addition to income tax, sales taxes also have a labor disincentive effect, albeit of different magnitude, so we need to consider those. All told though, the rest of these taxes add up to about 15 percent for a typical city in a typical state, and these tend to be regressive rather than progressive. 55 percent is still way below the peak of the laffer curve.

It's also worth noting that revenue is not necessarily the only way taxes redistribute.

Max10/22/2015 05:12:00 PM