The Laffer Curve is probably most notorious because of its influence on Reaganomics.

It has been reported that Arthur Laffer first drew his curve on a cocktail napkin, while demonstrating a point to a few politicians and reporters. Whether this story is true or not (though my intro macroeconomics textbook presents it as fact), it does an admirable job of giving an intuition for two things: The common sense truth the graph has at its core, and the more subtle problems that it fails to address.

Simply put, his graph measures the tax rate on one axis, and total tax revenue on the other. The curve intersects the Tax Rate axis at 0% and 100% (Thanks Oolong):


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0%                         
           Tax Rate

At a tax rate of 0%, we will of course not get any tax revenue. What makes Laffer's graph a bit more interesting is the statement that we won't get any tax revenue at a tax rate of 100% either. By this point, any and all businesses within a given state will do one of two things: Go bankrupt or start cheating on their tax forms.

No business can function when all of its profits are taken away. It cannot invest in the materials required to keep a business operating.

As a result, Laffer's curve states very simply that there must be some optimal tax rate which is less than 100% where tax revenue is maximized. Any increase in the tax rate past that point will only reduce tax revenue.

Laffer believed, and here is where he went wrong, that we were already on the right hand side of that curve. Tax rates were already so high that they were cutting into productivity, doing more harm than good.

While most experts dismissed this idea as voodoo economics, it got the attention of Ronald Reagan, primarily because Reagan had experienced the effects Laffer mentioned firsthand. David Stockman, budget director in the first Reagan administration, said that,

[Reagan] had once been on the Laffer curve himself. "I came into the Big Money making pictures during World War II," he would say. At that time the wartime income surtax hit 90%. "You could only make four pictures and then you were in the top bracket," he would continue. "So we all quit working after four pictures and went off to the country." High tax rates caused less work. Low tax rates caused more. His experience proved it.

When Reagan became president, he lowered taxes as Laffer recommended. Unfortunately, Laffer's prediction was wrong, and tax revenues plummeted. From 1980 to 1984, income tax revenues fell 9%, after adjusting for inflation, even though the average income increased by about 4%. Once the tax cuts had been enacted, they could not easily be repealed, and led to the massive deficit spending the Reagan era is known for.

What went wrong?

Laffer's primary error was the simplicity of his graph. He simply draws an upside down semicircle and so assumes that the optimal tax rate is somewhere in the middle of the graph. In reality, all his ideas regarding productivity can only tell us that there is an optimal point, but not where that point lies on the graph. This optimal point would in practice be far closer to 100% than Laffer assumed. A more accurate Laffer curve might look like the following:


                                                                                                        
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0%                         
           Tax Rate

Clarification: This new graph is not scientific in any sense and should not be taken as gospel. It is merely an approximation drawn by an economics professor of mine a few years ago. That is all. (Thanks eipi10)
And now it turns out it's not even all that accurate. The real curve should likely be convex throughout, not concave. (Thanks ariels)

Another assumption he makes is that the graph will be clear and intelligible, when this is not the case either. Many many things affect productivity and tax revenues, and sorting them out is not always easy. neo-Laffer curve has much more written on this issue.

To sum up, the Laffer curve is interesting to look at, but flawed. While it is theoretically true that lowering tax rates may increase tax revenue in some small number of cases, in reality this happens so rarely that it is silly to ever use the Laffer curve as reason for lowering the the tax rate.


Sources:
Most of this writeup uses information found in The Princinples of Macroeconomics, by N. Gregory Mankiw. Quote taken from pg. 166.

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