In my first post, I described three economic curves as a function of tax rate: a general economic curve with a maximum value at some tax rate te, the Laffer Curve, which maximizes government revenue at a tax rate tg, and the Crier Curve that maximizes the amount of money that is retained in the private economy at the tax rate tc. My assertion is that tg is always higher than te, which is always higher than tc.
Here is a simple thought experiment that proves that the peak of the Laffer Curve cannot be at a lower tax rate than the peak of the economy:
Imagine yourself standing at the peak of the Laffer Curve. To your right are higher tax rates, to your left are lower. Look to your right-toward higher tax rates. What do you see? Decreasing government revenue. Since the tax rate is increasing, but government revenue is shrinking (or even flat), the economy has to be shrinking as well. If the economy were flat, revenue would grow as the tax rate increased, and you would not be at the Laffer peak. Thus, under any conditions the economy is shrinking to the right of the peak of the Laffer Curve. This holds even if the peak of the Laffer Curve is at 100%-obviously the peak of the economy cannot be at a higher tax rate than that!
Thus, under any conditions the economy is shrinking to the right of the peak of the Laffer Curve.
The identical argument applies to the Crier Curve and the peak Economy. Imagine yourself standing at the peak of the economy curve. To your right are higher tax rates and lower economic output. In that direction, the taxpayer is keeping a smaller percentage of a smaller number, and thus the Crier Curve is shrinking. The peak of the Crier Curve cannot be at a higher tax rate than the peak of the economy.
This leads to the weak formulation of my theory:
tc is less than or equal to te is less than or equal to tg
The strong formulation drops the “or equal" formulation and depends on continuity. A simple explanation is that a function is continuous at the point in question if the slope of the function is the same from below (a lower tax rate) as it is from above (a higher tax rate). I have proven above that if you were standing on economy curve at the Laffer peak tax rate and looked right, the economy would be shrinking. Continuity means that if you look left, toward lower taxes, the economy would appear to be growing at the same rate as it is shrinking to the right. Continuity requires that the economy be bigger to the left of the the peak of the Laffer Curve. Using the same logic, it also requires that the Crier be growing to the left of the peak economy.
There may be a tax function that enables a triple point where all three peaks exist at the same tax rate. But like the triple point of of helium, this economic triple point is not going to exist on this planet outside of a laboratory.