## The Laffer Curve and the Kimel Curve

by Mike Kimel

People always talk about the Laffer curve, but have you ever seen it estimated? Have you ever wondered why you don’t? If you’re a quant guy, you know the answer to that. Because if you’re a quant guy, at some point curiosity must have gotten the best of you. That means you pulled out some data and you plugged it into whatever piece of software happened to be handy. What happened next depends on what sort of a quant guy you are. If you’re the sort that let’s the numbers do the talking, you spotted the joke and probably left it at that. If you have a strong ideological leaning in a certain direction, on the other hand, you might have tried to “fix” it. You tried a few times, failed, and kind of just left it there as something to get back to some time, but no hurry because your ideology tells you what the answer should be.

Today, by coincidence, I got two e-mails asking me about the Laffer curve. And it occurred to me… maybe someone should let non-quant people into the joke. Because the only people really discussing it are those who are driven by ideology, whereas it should be afforded the Hauser’s law treatment.

So here’s how it works. Putting numbers to the Laffer curve pretty much comes down to estimating:

(1) tax collections / GDP = A + B*tax rate + C*tax rate squared + some other stuff if desired

A, B, and C are estimated statistically using a tool such as regression analysis.

If you plug in numbers, and find that B is positive and statistically significant and C is negative and statistically significant, then it turns out that you can trace out a quadratic relationship between tax collections / GDP and tax rates. i.e., tax collections / GDP is a function of tax rates that looks like an upside down U. If you increase tax rates when tax rates are “low,” growth will increase. On the other hand, to increase growth when tax rates are “high”, you have to decrease tax rates.

When you have such a shape, you look for the top of the upside down U and there’s your maximum.

So I started with the obvious:

(2) tax collections / GDP = A + B*tax rate + C*tax rate squared

In other words, the simplest version of (1) possible. I plugged in data. That would be current federal tax receipts, line 2 from NIPA table 3.2, divided by GDP, from the BEA, and the top marginal tax rate from IRS’ historical table 23. You can go all the back to 1929—that’s when the GDP and current federal tax receipts begin.

The problem is… the data isn’t quite amenable to shoehorning into the desired shape. The fit of the model sucks, B is negative, C is positive, and neither coefficient is significant at the ten percent level.But they aren’t so far off either.

All that together means that maybe, just maybe, a slightly better specified model might do the trick. The simplest solution… find another variable that has some explanatory power and throw it in. Well, I’m supposed to be on a hiatus from blogging, so I don’t want to spend a huge amount of time at this, but it occurs to me that year is probably such a variable. There’s a good chance that over time, tax collection has become a bit more efficient.

So I reran (2) as follows:

(3) tax collections / GDP = A + B*tax rate + C*tax rate squared + D*Year

Here’s what the results look like:

B and C have the wrong sign. That means you don’t get an upside down U, you get a U. Here’s what it looks like when graphed:

The low point in tax collections happens to be about 32%. In other words… if the top marginal tax rate is below 32%, cutting it further will raise tax revenues. On the other hand, if the top marginal tax rate is above 32%, to boost revenues you have to raise tax rates.

Now this is quick and dirty, and it has boundary issues (i.e., 100% tax rate collects more than 99% tax rate – would it really? well, the model is extrapolating because its never observed tax rates of 99% or 100% in the wild). I should also throw in a few more variables to improve the fit. Worse, there’s autocorrelation. That means the error terms are correlated. The correlation between the residuals at time and the residuals at time t+1 is 75%. That in turn violates one of the assumptions of OLS regression analysis. Its fixable, but its also ignorable for our purposes since what it means is that the coefficient estimates are probably “correct” but merely less significant than they appear. Regardless, you won’t get anything that bears even a remote resemblance to what you hear from the crowd who perennially cites the Laffer curve so authoritatively.

Which brings up another piece of the joke. In the end, tax collections don’t matter. Its nobody’s goal to maximize tax collections. Taxes only matter because they pay for certain government services. They also take money out of our pockets. So there’s a tradeoff. But we’re made better off if the government services taxes pay for generate more value than they cost us. And at least to some extent, you can measure that by whether they generate more growth than they cost us.

Now, it turns out that the optimal tax rate for growth is easy to calculate. The data cooperates very nicely. There is a relationship, an easy to estimate curve which I’ve modestly called the “Kimel curve.” And the high point in the Kimel curve is somewhere around 65%. Now, the Laffer curve analysis shows us that getting to the level of taxation that produces the fastest economic growth rates would also increase our tax collections… not a bad thing at all in an era of rapidly rising national debts.

Which brings us to the biggest Laffer curve joke of them all: ain’t no way the folks who like to talk about the Laffer curve would support that.

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As always, if you want my spreadsheet drop me a line. I’m at my first name (“mike”) period my last name (“kimel” – that’s with one m only!!!) at gmail.