Wednesday, July 13, 2011

A real Laffer


One more short note on John Cochrane's paper. In my last post I talked a bit about "dynamic Laffer effects," and claimed that they didn't make sense in a historical context. But I wanted to point out that they make even less sense in a theoretical context.

A "dynamic Laffer effect" is a permanent effect of tax rates on growth rates. John Cochrane illustrates the idea with this equation:


PV is tax revenues, Y is GDP, g is the long-term growth rate of GDP, t is the tax rate, and r is the interest rate.

The third term on the right hand side represents the permanent effect on growth rates of a change in tax rates. Cochrane asserts that this term is negative, and suggests a value of "only" -0.02 for the derivative dg/dlog(t).

Sounds small and insignificant, right? It makes sense that taxes have some negative effect, however small, on the rate of growth, right?

Except it doesn't make sense. If taxes change the long-term growth rate, then tiny differences in tax rates between countries will, over a long enough time scale, cause countries' incomes to diverge. A country with a 10.01% tax rate will eventually become infinitely richer than a country with a 10% tax rate.

But you don't need to invoke infinity to see how silly this is. Here's a numerical example. The United States takes in about 27% of our GDP in tax revenue. Using Cochrane's numbers, a country that took in only 6% of its GDP in tax revenue would - all else being equal - grow about 3% per year faster than us, year after year. In a mere 24 years, that country would be twice as rich as us. After a century, they would be 19 times as rich as us. That's about the same as the current disparity between us and sub-Saharan Africa.

In other words, dynamic Laffer effects can't exist, because, as a physics prof of mine liked to say, "then the Universe would explode." Tiny differences in tax rates among rich nations would eventually add up to massive differences in wealth, and we'd see exponential divergence instead of the convergence that we see in the real world. The nonexistence of dynamic Laffer effects is a stability condition of growth theory.

And yet the existence of these Universe-exploding effects is absolutely central and crucial to conservative policy ideas.

19 comments:

  1. Sounds small and insignificant, right? It makes sense that taxes have some negative effect, however small, on the rate of growth, right?

    Actually, that's wrong as well. Mike Kimel at Angry Bear has sliced and diced the real data for the U.S. every way imaginable. There is no - none - nada - zilch negative correlation between tax increases and economic growth in U.S. history, for as long as there is good data.

    No correlation = no causation.

    If anything, the correlation is that raising taxes causes higher economic growth.

    Assume a Laffer-type curve for economic growth. If this concept has any validity (and it very likely does) we are far to the low tax side of the maximum.

    Cheers!
    JzB

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  2. But that runs contrary to my ideal of perfect liberty! Therefore it must be false.

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  3. Jolly Green said...

    "But that runs contrary to my ideal of perfect liberty! Therefore it must be false."

    Which is good enough for:
    1) Joe Random on the internet,
    2) Dr. Joe 'Scholar' in a think tank, and
    3) Economics professors at really elite universities.

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  4. The only relationship between taxes and growth you can find in the data is that you can't have high growth without high taxes. That's all there is to it. All these people arguing that you can freeze water if you can only get it hot enough are just ideologues. Whenever I read a mainstream economist blathering on, I can swear I'm reading an old issue of Pravda. Drop the little pictures of Lenin's head and you wouldn't realize that the Soviet Union had collapsed.

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  5. Tax rates and growth do not correlate because higher tax rates do not correlate to higher government revenue:

    http://2.bp.blogspot.com/-9R0ESLWQfRE/TcrzO8BjpqI/AAAAAAAAE-0/4v3r1D2DjUg/s1600/Tax+Rates+vs+revenues.jpg

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  6. If a fisherman in a simple economy has 20% of his catch left over each month, and uses those fish to trade for more nets (exponentially growing his business), is it better to tax him and take those extra fish each month?

    If we do tax away his extra fish, that slows or stops the growth of his business. In order to have growth, the taxed fish must be put to good use (a better use than more nets).

    Is it logical to think that government will use the extra fish more wisely than the fisherman? If so, what would government do with those fish that would benefit the simple island economy more than an increased supply of fish?

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  7. Anonymous1:08 PM

    Q: "Is it logical to think that government will use the extra fish more wisely than the fisherman?"

    A: Yes.

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  8. If it explodes, then you are a quantum economist. If you are a Keyensian then we get an ever expanding, infinitely divisible fluid blob.

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  9. Anonymous8:06 AM

    I think this is a double fail on the critics part.

    First, Cochrane's paper is entirely correct in making the point that the short-run and present-value Laffer curves can peak at very different points. Does anyone here challenge this?

    Second, if you follow Lucas's and Cochrane's papers and thinking, they think that policies such as higher marginal tax rates move the economy to a lower (but not necessarily slower) growth path. For example: Compared to the US, Europe has bad governance at EU level and bad policies at country level. Consequently, the GDP per capita will fluctuate at around 75% of that in the US. If we move from 100 unit growth path to 75 unit growth path with progressive taxation and automatic "stabilizers," of course this will lower the present value of future primary surpluses. Given that the adjustment is between growth paths is slow, there's a big difference between short-run and present-value Laffer curves. Does anyone challenge this?

    The numerical example is just an example illustrating the difference between static and dynamic Laffer effects.

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  10. Alex Carr5:18 PM

    Who cares about the difference between short run and present value Laffer curves? Noah's point is that due to the nature of exponential growth any difference in growth due to tax rates would be extremely easy to spot and yet we see nothing. This basically debunks present value Laffer curves as well since they depend on arguments that lower tax rates increase growth which then increase government revenues. But sure, let's just keep pretending Cochrane had a useful point...

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  11. Anonymous10:29 AM

    Alex Carr, I think your email is confusing/confused.

    I don't think anyone disputes the existence of Laffer curve peak, short-term or present-value. At the limit at 100% tax rate, people will work only at gun point, which has not worked well historically. If you dispute that, I think we've agreed to disagree about a lot of things.

    If you do however take the sensible position that there is a Laffer curve peak short of the 100% limit, then we have the more interesting question of where it is / they are. "Who cares about the difference between short run and present value Laffer curves?" I do and the author of this blog do.

    All evidence points to the US left of the short-term Laffer curve peak. However, we don't know whether we are left of the present value Laffer curve point, so at minimum someone should calibrate a model. We're probably left of that as well, but also the slope of the curve matters for policy.

    I (and Lucas and Cochrane) basically agree with Noah's point that it's hard to come up with policies that change the long-term growth rate. The example in Cochrane's paper is the simplest possible numerical example that uses the simple formula for geometric series sum. It's an example, not an empirical calibration.

    Although it is hard to change the long-term growth rate, it is however not hard to come up with policies that moves us slowly to a lower growth path. Put in French or Greek labor rules and voila you've accomplished a slow transition to a lower but not slower growth path.

    Why is this relevant? If you work out the present-value math, the slow transition to a lower (not slower) growth path will cause a similar but less extreme gap between short-run and long-run Laffer curves.

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  12. @Second Anonymous (dagnab it, you people, get screen names!):

    Without an effect on the trend growth rate, the static Laffer effect (i.e. what Uhlig estimated) is the same as the present-value effect. The static Laffer effect assumes - wrongly but innocuously - that the effects of taxation all kick in instantly. In actuality, they take some time to have their full effect, but the size of the effect should be obtainable by static analysis.

    If Cochrane wants to do a numerical example, he should not do a numerical example that uses explosive solutions, especially when those explosive solutions are necessary for him to make his desired point. I think it's just bad form.

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  13. Laffer curves are very useful to Republicans making the specious claim that taxes are always too high, promising constituencies that as tax rates approach zero, revenue will approach infinity. They get their way; then during democratic administrations republicans get to complain simultaneously about deficits AND taxes being too high without appearing to be completely insane. This tactic sways voters, especially the ones that are bad at math.

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  14. Anonymous9:59 AM

    @first anonymous = @second anonymous = @this third anonymous

    Noah --

    The numerical example that you are making a mountain out of is one short paragraph in Cochrane's excellent theory paper. It's mathematically by far the easiest example to present in a paragraph, and it illustrates how extreme the present value effect is. It's not a prediction or an empirical model of the world.

    I completely disagree with you that explosive or trend growth differences are needed to create a wedge between static and present-value Laffer curves. All that is needed is slow adjustment. For example, let's say we put in European institutions (including higher taxes). Immediately, this will massively increase tax revenues. Say the GDP will not change and we'll increase the tax revenue from 20% to 30% of the GDP, or a tax revenue increase of 50%. Suppose further that over some years, say ten, this will bring the GDP to a growth path that is at 0.75 of the original growth path. With low constant interest rates, the present value of the taxes only increases 12.5%, not 50%. In this example, the trend growth didn't change at all.

    Best, ptuomov

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  15. piglet5:30 PM

    "The numerical example that you are making a mountain out of is one short paragraph in Cochrane's excellent theory paper. It's mathematically by far the easiest example to present in a paragraph, and it illustrates how extreme the present value effect is. It's not a prediction or an empirical model of the world."

    Cool. It's not a prediction or an empirical model but it is mathematically easy. You can't make this stuff up.

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  16. Anonymous7:47 AM

    Piglet --

    "Cool. It's not a prediction or an empirical model but it is mathematically easy. You can't make this stuff up."

    That's right. Have you read the paper in question, or any other recent theory papers? It's common to give simple numerical examples in theory papers nowadays. It's an extremely weak criticism of a theory paper to complain about the realism of such numerical examples.

    Noah doesn't seem to like Cochrane's politics. But I think he's making a mistake here criticizing an excellent theory paper for an unrealistic numerical example.

    Let me quote the paper from the Laffer curve section:

    "Yes, this calculation is too simple. The point is to contrast this calculation with the dynamic calculation below, not to assess realistically the U.S. tax system."

    "For a simple calculation,
    suppose growth of taxable income is steady at rate g and the interest rate is a constant r."

    "I do not digress here to the economics by which marginal tax rates lower the level or growth rate of output. ....Growth theory points to accumulation of knowledge as the main driver of long run per-capita growth rates, but I do not want to stop here to model how distorting taxes interfere with
    that process, nor tie the calculation to one particular such model."

    If the reader takes the one-paragraph numerical example as a serious prediction about how the world works after those caveats, the reader is either confused or has some other motivations than trying to understand the paper.

    ptuomov

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  17. @Ptuomov:

    A present value effect is very different from an effect on g. The coefficient g is an effect on the slope of the balanced growth path. Very different from a slow-to-occur effect on the level of the balanced growth path.

    Also, and more to the point, Cochrane is using this example to show why Uhlig's estimates of Laffer effects are too low. But Uhlig's estimates should be equal to the entire present value of the Laffer effect, not just the instantaneous response of some slow-to-adjust model. Cochrane is trying to insert something above and beyond the Laffer effect that Uhlig deals with, but such additional effects can't really exist.

    Uhlig is a smart guy, he would not make the mistake of confusing an instantaneous response with a present value.

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  18. Anonymous3:59 PM

    Noah --

    I am not critical of what Uhlig wrote (in his working paper with Trabandt). They compute the present value effects under some assumptions.

    In fact, Cochrane gives Trabant and Uhlig as an example of the proper calculation! "(Yes, this calculation is too simple. The point is to contrast this calculation with the dynamic calculation below, not to assess realistically the U.S. tax system. Trabandt and Uhlig (2009) offer a detailed Laffer calculation with fixed productivity growth and no migration, yielding the result that the U.S. is substantially below the Laffer limit.")

    I don't understand how Cochrane's Laffer curve paragraph can be seen as anything else but a numerical example, especially since he tells the readers to go read Uhlig for actual empirical work.

    Two remaining points:

    (1) In my opinion, any model or framework in which very persistent but transitory shocks and permanent shocks give completely different answers has a major weakness. In the present value formula with discount rate greater than the growth rate, very persistent growth differences and permanent growth differences are basically very close to each other. I think you're making a mountain out of molehill here if the point is that there's some deep philosophical difference between very persistent and permanent growth differences.

    (2) In practice, I think that migration can cause permanent or very persistent differences in growth. (Uhlig and Trabandt assume no migration, so that's a stated assumption in their analysis.) Yes, a country can't grow faster than a world's population in the long run, but a country can certainly grow slower than the world's population. There's no reason to think that there hasn't been for all practical purposes permanent population growth differences between India and Sahara desert.

    Best, ptuomov

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