Peter Diamond and Emanuel Saez have penned an op-ed in today’s Wall Street Journal asserting that a tax rate on top earners of between 50% and 70% will not reduce tax revenues. They argue that this is possible because, “We’re not close to the peak of the Laffer Curve.” J. Bradford DeLong, an economist alongside Mr. Saez at the University of California, Berkeley, takes their argument beyond economics and into the social justice realm:

We have a moral obligation to tax our superrich at the peak of the Laffer Curve: to tax them so heavily that we raise the most possible money from them – to the point beyond which their diversion of energy and enterprise into tax avoidance and sheltering would mean that any extra taxes would not raise but reduce revenue. [emphasis added]

All three economists rely upon the Laffer Curve to make their argument. The Laffer Curve is a much maligned and often misunderstood concept, but it is actually quite simple. Named for economist Arthur Laffer, it postulates that there exists a tax rate that maximizes tax revenues. Mr. Laffer theorized that if the tax rate was 0% the government would definitionally receive zero dollars. At the other end of the curve, a tax rate of 100%, would also yield zero tax revenues, since if citizens were to lose every dollar they made, they would have no incentive to produce any taxable income (or at least to report it). Therefore, somewhere between 0% and 100%, tax revenues are at a peak. (See Figure 1 below.)

This is uncontroversial among economists of all stripes, and in fact the Laffer Curve only begets argument because it is often mischaracterized to state that a reduction in tax rates yields an increase in tax revenues. That is only true if the current tax rate is to the right of TRmax. Diamond et al. argue that we are to the left of TRmax, and therefore, we should raise tax rates to raise tax revenues. If they are correct about where we are on the Laffer Curve, then their prescription should increase tax revenues. But even if they are correct, it will do so by severely hurting GDP as I will demonstrate below.

Caution: The boring calculus part follows.

The slope along the Laffer Curve changes along its length. We know that it is equal to zero at the peak. We also know that the slope of the curve at any point is equal to the first derivative of the curve with respect to T. We’ll call that R’. And R’ = 0 at a tax rate of T Rmax .

At that point an infinitesimally small increase in the tax rate sufficient to theoretically yield 1 extra dollar of tax revenue, actually yields 0 dollars because the extra tax revenue is exactly offset by a fall in GDP. In fact, the fall in GDP at that point is equal to the reciprocal of the tax rate. Let’s show this by zooming in on the point where the Laffer Curve peaks. (Figure 2.)

In the aggregate, tax revenues (R) are a function of GDP (G) and the average tax rate (T). So we can express the Laffer Curve as: R = G x T. The maximum Revenue (Rmax) is equal to the GDP at Rmax (G Rmax ) times the tax rate at Rmax (T Rmax ). Rmax = T Rmax x G Rmax

If we increase the tax rate by an infinitesimal amount that theoretically should have yielded one extra dollar of tax revenue–we’ll call that Delta T–then we should see some change in G, which we will call Delta G. However, since this is the peak of the curve, the addition of Delta T to the tax rate has no net effect on revenues, meaning that we can set the original amount of tax revenues equal to the new tax revenues as a result of adding Delta T ot the tax rate. We can run through some equations to see what happens at that point on the Laffer Curve.

T Rmax x G Rmax = (T Rmax + ΔT) x (G Rmax + ΔG)

If we multiply the equation through on the right side, we have this:

T Rmax x G Rmax = (T Rmax x G Rmax ) + (T Rmax x ΔG) + (ΔT x G Rmax ) + (ΔT x ΔG)

We can subtract the common element from each side to get this:

T Rmax x G Rmax = ( T Rmax x G Rmax ) + (T Rmax x ΔG) + (ΔT x G Rmax ) + (ΔT x ΔG) 0 = (T Rmax x ΔG) + (ΔT x G Rmax ) + (ΔT x ΔG)

And since we are trying to solve for Delta G, let’s isolate it to one side:

-(ΔT x G Rmax ) = (T Rmax x ΔG) + (ΔT x ΔG) -(ΔT x G Rmax ) = (T Rmax + ΔT) x ΔG -(ΔT x G Rmax ) / (T Rmax + ΔT) = ΔG

Remember, we set Delta T equal to an amount sufficient to theoretically yield one extra dollar of tax revenue, that means that:

ΔT x G Rmax = 1

By substituting that into the equation above, we get this:

-1 / (T Rmax + ΔT) = ΔG

And since Delta T is infinitesimally small relative to T Rmax , we can approximate it as zero, yielding this result:

ΔG = -1 / T Rmax

I illustrate this in Figure 3 below by showing the GDP curve as a function of the tax rate.

(If you hate calculus, you can resume reading now that most of the eye-glazing math part is over.)

In other words, when an economy is at the peak of the Laffer Curve, a decrease in the tax rate yields an increase in GDP equal to the reciprocal of the tax rate. This mathematical truism alone ought to give lie to the idea that there is a constant positive multiplier effect to money taken from taxpayers and spent instead by the government.

This is all well and good, but one of the critiques of the Laffer Curve is that no one knows where it peaks. Diamond, Saez and DeLong think that we are to the left of its peak. I disagree, but it doesn’t matter. Because by focusing on Laffer, we’re looking at the wrong curve. What we should be focusing on is not the point where tax revenues are at their maximum, but the point where GDP is maximized. (T Gmax in Figure 3 above.)

If we want to estimate where the maximum point of the GDP curve is, mathematically it is where the slope of the Laffer Curve is equal to one. (Trust me on this, unless you want me to go back into boring math mode.) (See Figure 4)

While that doesn’t seem very useful, it is. An increase in taxes that was expected to produce a statically-scored estimate of a certain amount, but instead yielded a different amount gives us an approximation of where we are on the curve. Any time that ratio of actual revenues to expected tax revenues is below one, it indicates that the economy is already to the right of the peak of the GDP curve.

There is empirical evidence to support my contention. It in fact comes from a fellow UC-Berkeley economist and former Obama economic advisor Christina Romer. She and her economist husband discovered from an analysis of tax changes between 1945 and 2007 that a one-percent increase in tax revenues results in GDP decreases of between 2.2% and 3.6%. This indicates that the economy is well to the right of the maximum of the GDP curve. In fact, this understates the severity of the deleterious effect of tax increases. That is because the Romers measure percent of change of tax revenues and percent of change of GDP. If we use the current economic conditions where government spending is roughly 40% of GDP, that implies that when measured on a dollar basis, the true discrepancy is 2.5 times as large as what the Romers’ percentage-based measures indicate. In other words, for every dollar raised in taxes, between $5.50 and $9.00 is taken from the economy. That’s huge.

Furthermore, in the post-war era, tax revenues have been remarkably consistent, hovering around 20% of GDP in spite of large changes in the marginal rates of taxation. Therefore, if we are near the peak of the Laffer Curve, we would expect the multiplier to be about 5. For every one dollar of taxes raised, GDP should fall by the reciprocal of the average tax rate of 20%. The Romers’ report strongly suggests that we are much closer to the peak of the Laffer Curve than Diamond, et al would have us believe.

What Romer and Romer tell us is that we are far to the right of maximized GDP. Average tax rates are already high enough that they significantly retard GDP growth. Additionally, while the Laffer Curve measures tax revenues at a point in time, GDP has a cumulative component. That is because GDP growth this year is transmitted logarithmically through future years. Sub-optimal GDP today is wealth that isn’t compounded far into the future. Given where the nation is in terms of the long-term debt-to-GDP rates, nearly all economists agree that to reduce it to a sustainable level, the larger portion of the difference is going to have to come from much greater growth instead of from much greater taxes. (For a great example of how there isn’t enough wealth and income available to close the gap, check out this video from a brilliant Nobel De-laureate.)

That means that anything which produces a one-year bump in tax revenues, does so at the expense of a significant and prolonged decrease in GDP. In other words, while Diamond, Saez, and DeLong attempt to use Laffer to bolster their case, what Laffer really tells us, is that they have offered a ludicrous prescription for the nation’s long-term financial health. The peak of the Laffer Curve is never the place to be.*

* NOTE: If engaged in a war of an existential nature, then the peak of the Laffer Curve is exactly where the nation needs to be.

UPDATE: Thanks to John Tamny and Real Clear Markets for the link.