I know that the economy cannot actually be modeled by a third order polynomial

A couple of commenters have disputed my Crier Curve theory because they assume that the theory is derived from my polynomial equation that I used. This is not the case. The theory is derived from calculus I learned in high school t(cough cough) years ago. The polynomial is simply meant to be illustrative-it's a formula that I could derive and graph easily. I gave the formula so that anyone who wanted to "try this at home" could do so.

Let's focus on just the math and treat this like a calculus test question. Forget Laffer Curves, polynomials, or any other ties to the real world. Here is how I would state the problem in purely mathematical terms:

Consider three continuous functions who relationships as a function of the variable "x" are described as follows:

f(x) = h(x) * x
q(x) = h(x) - f(x)

where x has a range between zero and one.

Question 1: What is the slope of h(x) when f(x) is at a maximum or minimum?

Answer: Assume that a local maximum or minimum occurs at and X value of "x_1". This means that f'(x_1) = 0.

f'(x) =h'(x) * x +h(x)

f'(x_1) = 0 = h'(x_1) * x_1 + h(x_1)

so

h'(x_1) = -h(x_1)/x_1 when f is at a local maximum or minimum.

Question 2: What is the slope of q(x) when h(x) is at its peak value?

Answer: Assume that h(x) is at a local maximum or minimum at a value of "x_2"

q(x) = h(x) -f(x) = h(x) - h(x) * x = h(x) * (1-x)

q'(x) = h'(x) * (1-x) + h(x) * (-1) = h'(x) * (1-x) -h(x)

q'(x_2) = -h(x_2)

Now assume that f(x), h(x), and q(x) must always be greater than zero.

This means that both h'(x_1) and q'(x_2) are less than zero.

Again, this is simply high school calculus. The only requirements I have put on this solution are as follows:

f(x), h(x), and q(x) must be continuous and positive over the range of "x".
The value of "x" must be greater than zero and less than one.

Now we can apply descriptions to f(x), g(x), and q(x):

f(x) = government revenues
h(x) = size of the economy
q(x) = money retained in the private economy (Crier Curve)
"x" = effective tax rate
x_1 = tax rate at which Laffer Curve peaks.
x_2 = tax rate at which economy peaks.

Making the substitutions above proves that for a real world economy where tax rates are between zero and one hundred percent, and absolute economic output cannot be negative, the following relationships hold:

When tax revenue is maximized, the slope of the economic output function is negative, meaning that the economy is smaller than ideal.

When the size of the economy is maximized, the slope of the Crier is negative, and the money retained in the private economy is less than optimal.

This means maximizing tax revenue occurs at a higher tax rate than the peak economic output, which occurs at a higher tax rate than the rate at which the most money is retained in the private economy.

QED