### Laffing Our Way To Transversal Trouble

Over in the comments section of Jim Hamilton's excellent blog Econbrowser, Barry P asked this question:

Has anybody been able to come up with a Laffer curve for the US economy with actual, reliable, meaningful numbers attached? One guy I know swears that the maximum is at about a 90% tax rate, so we *must* be to the left of the maximum, but empirically I wouldn't know one way or another.This topic happens to be related to my business (no, running CCM is just a hobby!). So I weighed in over there. Let me cross-post my comment there:

I'm pretty sure that the 90% number comes from Varian's Intermediate (undergrad) Micro text. But if you want to figure it out for yourself, just do the following:

1. Note that revenue is R(t) = tI(1-t), where t is the tax rate, 1-t is the take-home share, I is taxable income, which is a fctn of t if you believe in microeconomics (why this has come to be called "supply-side" rather than "utility-maximizing" is an interesting question for the intellectual historians). Of course, I am simplifying by assuming a flat tax rate. But I don't have time to write another dissertation right now.

2. Take logs and differentiate:

d ln R(t)/dt = (1/t) - I'(1-t)/I(1-t)

Note that the peak of the laffer curve occurs where this derivative equals zero. So we set

I'(1-t)/I(1-t) = 1/t.

Now multiply both sides by 1-t, and you get that the l curve peaks at t* such that

E = (1-t*)/t*,where E is the elasticity of taxable income (E must be positive if you believe that there is an opportunity cost to revenue-generating activity). All serious arguments about the l curve's peak come down to figuring out what the value of E is, since

t* = 1/[1+E].

Now, it's only a slight caricature to say that the luskin-bush-novak-delay-kemp types seem to think that E = infinity, so that every single tax cut raises infinity. Sigh. More respectable folks think E may be as high as 3, in which case t*=0.25. For technical reasons, I think the evidence for that claim is totally dismissable. A more plausible long-run estimate is probably something in the neighborhood of 0.4, and perhaps as high as 0.6 or as low as 0.

In the 0.4 case, t*=1/[1.4]=5/7, or over 70%.

If I remember right, the 90% estimate is based on earlier estimates of the labor supply elasticity using hours worked, rather than taxable income, as the endogenous supply-side (!) variable.

In any case, t* is a hell of a lot higher than today's tax rates.

Bet you're sorry you asked, aren't you, CHB?

## 3 Comments:

in case CHB didn't make it clear at the start, me no math so good. how about a translation, dumbed waaaaay down...or a beer. either way i'll be happy

About 25 years ago, James Tobin did a variation of this based on a labor demand = labor supply model. His parameterization of the model suggested t* = 82%. Robert Barro once claimed t* = 70%. But note my recent link to Jude Wanniski who seems to think the labor supply curve bends back. So if elasticity of labor supply is not positive, the model just don't work. Neither does Hey Jude's mind!

pgl

i think we can safely reject the null that jude is sentient....

j

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