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

A triple point for Laffer/Crier/Economy curves?

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.

Hauser's Law and the Crier Curve

Hauser's Law posits that the U.S. government's revenue will always be around 19.5% of GDP, no matter what the marginal tax rate is. And, indeed, federal tax revenue as a percentage of GDP has been very flat even though marginal tax rates have varied widely. I believe that this represents the peak of the Crier curve, where the absolute amount of money retained by the private economy is maximized.


People and corporations want to optimize the money they retain. In a free and productive society like ours, the government's ability to tax is constrained. Corporations and industries will lobby to create loopholes and lower their effective rates; the people will vote out incumbents who raise taxes, especially if the economy sours. In the battle between the government trying to raise taxes, and the people trying to keep their money, the people will win as long as they control the government.

Does the Crier Curve analysis work?

In my previous post, I asserted that the economy can be taxed at three mutually exclusive rates: the lowest optimal tax rate results in the highest absolute amount of money remaining in the private economy; the next lowest optimal tax rate results in the largest possible economy; and the highest optimal tax rate results in the government collecting its highest revenue.




The key assumption to this assertion is continuity of the economic output as a function of tax rate. So let's examine source of discontinuities:




First, the economy as a whole is not subject to a single tax rate. The main sources of revenue for the government are (in order) personal income taxes, employment taxes, and corporate taxes. Each of these has a different tax structure, and so each type of tax will have its own set of economic curves.




-Personal income taxes are progressive. This means that individuals have different effective tax rates, depending on AGI. If you plot the AGI versus effective tax rate, you will see a discontinuous curve.




-Employment taxes are discontinuous because they are capped-you do not pay them above a certain amount of income.




-Corporate taxes are generally paid on profits, not revenue. Thus, GM can produce billions of dollars of product and not pay tax.




It is obvious that each of the major tax elements is discontinuous. Does that invalidate the Crier Curve analysis? I don't know. The Crier curve is a macro phenomenon, while each of these effect is a micro phenomenon. One analogy is to the Ideal Gas Law, which states that the relationship between pressure, volume, and temperature in a closed system is described by:




Pressure x Volume/Temperature = Constant




Temperature is a macro phenomenon that is an aggregation of the thermal kinetic energy of the gas molecules. Each molecule will each have its own energy function over time, and trying to describe their interactions on a molecular level is impossible. However, it does not matter because we have the macro level concept of temperature.




In the same manner, a variety of variables go into a person's economic output. These include education, intelligence, skills, luck, personal motivations, risk aversion, what company you work for, and other factors. Similarly, a lot of factors go into an individual's effective tax rate. For example, itemized deductions establish a baseline below which the tax rate is zero. Marital status affects one's tax rate.




A corporation's effective tax rate depends largely on profitability. The higher its profits, the larger its effective tax burden.

Thus, on the micro level, you have over a hundred million individuals and tens of thousands of companies, each with the possibility of contributing to the economy and the tax base. My intuition says that these aggregate to a continuous economic output as a function of the tax rate; the discontinuities cancel or are otherwise inconsequential.

Regardless, I still think that the fundamental hypothesis holds:

The aggregate tax rate of an economy can be optimized at one of three different tax rates:

-A low tax rate that maximizes the wealth retained by the private economy.

-A middle tax rate that maximizes the size of the overall economy.

-A high tax rate that maximizes government revenue.

An Engineer's Look at the Laffer Curve

The Laffer Curve is a common description of the government’s revenue as a function of tax rate. While this function is not mathematically defined, the Laffer Curve is often used in discussions about the optimal tax rate. In this post, I will use the mathematical relationships implied by the Laffer Curve to prove the following:

1) There are three different tax rates that optimize different facets of the economy:
a. tc is the tax rate at which the highest absolute amount of money remains in the private economy.
b. te is the tax rate at which the size of the overall economy is maximized.
c. tg is the tax rate at which the government collects the most revenue.

2) The tax rate at which the government optimizes its revenue (tg) always results in a smaller than optimal economy. The tax rate at which the economy is maximized (te) is always greater than the rate at which the most money remains in the private economy. In other words:

tc is less than te is less than tg

Background:

The Laffer Curve is a commonly accepted description of the revenue a government generates as a function of tax rate. At a tax rate of zero, the government’s revenue is obviously zero. At a tax rate of 100%, the government’s revenue is also zero as there is no incentive to work. A curve exists between these two endpoints with a maximum at some tax rate.

Here is a picture of a Laffer curve:

By definition, the Laffer curve is simply the size of the economy times the tax rate:

Government Revenue = Economy Size * overall tax rate

Or

G(t) =E(t) * t

where

G(t) = Government tax revenue

E(t) = Economic Output

t = tax rate and is greater than zero but less than one.

So the economic curve for the Laffer Curve shown above is:

The money that is retained by the earners is the size of the economy minus the government revenues. This relationship is called the Crier curve:

C(t) = E(t) –G(t) where "t" is between zero and one.

Here is a plot of all three curves simultaneously:

The green vertical line represents the peak of the Crier Curve-where the largest amount of money is retained in the private economy.

The pink vertical line represents where the economy has the largest value.

The dark blue vertical line is the peak of the Laffer Curve, where government revenues are the highest as a function of tax rate.

Two observations are readily apparent from the above curves:

1) The peak of the Crier curve occurs at a lower tax rate than the tax rate at which the peak economic output occurs.
2) The peak tax revenue is at a higher tax rate than the tax rate at which point economic output is maximized.

These two relationships will hold as long as the economy is continuous as a function of the tax rate.

Proof:

If it can reasonably be assumed that E(t) is a continuous function, then E(t) is differentiable. This results in:

G’(t) = E(t) + t * E’(t)

E’(t) = (G’(t) * t - G(t))/(t*t)

C’(t) = (1-t) * E’(t) – E(t)

Each of these functions will be maximized when the derivative equals zero.

First, assume a tax rate tg that maximizes tax revenue. At this point G’ = 0.

G’(tg) = E(tg) + t* E'(tg)

G’(tg) = 0

So

E’(tg) = - E(tg)/tg

Both “E” and tg are always positive. Thus, E’, the change in the size of the economy, has to be negative at the value tg. Thus, the economy is smaller than it would be at a lower tax rate than tg.

Next, assume a tax rate te that maximizes the size of the economy. At this point E’ = 0.

E’(te) = 0 = G’(te) *te – G(te)

So the rate of change of tax revenue is as follows:

G’(te) = R(te)/te = E(te)

Since both revenue and tax rate are by definition positive, G’(te) is greater than zero, and government revenue is still climbing, even though the economy is at a peak.

Finally, consider the amount of money that is left in the private economy after the government has collected its revenue:

C = E(t) * (1-t)

C’ = E’(t) * (1-t) – E

Setting C’(tc) = 0 and solving for E’:

E’(tc) = E(tc) / (1-tc)

Since both E and (1-tc) are always positive, E’(tc) is always positive as well. This means that tc is smaller than te.

Thus, the economy can be broken into four distinct phases based on the tax rate:

Tax rate of zero to tc: This is the “infrastructure” phase of taxation, where government provides services that enhance economic growth. These services include a police and justice system and national defense at the most basic level. Other government functions, such as the allocation of scarce resources (radio frequencies, water, etc.) and intellectual property protection are included in this phase. Government revenues are low during this phase but rapidly rising as the tax rate increases.

Tax rate between tc and te: This is the “specialization” phase of taxation. As the tax rate increases, it only makes sense to perform the work at which an individual is most productive. For example, consider a restaurateur who takes an hour a day to keep her own books. With a low tax rate, this may not be a burden. However, as the tax rate increases, it becomes more valuable to pay someone to keep the books and use the time spent bookkeeping to keep the restaurant open longer. The restaurant’s gross revenue increases over the lower tax rate, as do the government’s revenues. The restaurant’s net revenue is lower, and it is now split in some fashion between the owner and the bookkeeper, resulting in higher employment.

Tax rate between te and tg: This is the punitive phase of taxation, where the tax burden still results in growing revenue, but a smaller economy. Consider the restaurateur in this situation-the business is profitable, but not profitable enough to hire someone to do the books, or bus the tables.

From a tax rate tg to 100%: This is the confiscatory phase of taxation. In this case, every single measure of economic health is suboptimal: the size of the economy, the net revenue, and the tax revenue.

Summary:

The economy can be broken into four distinct ranges based on tax rate: infrastructure building, specialization and full productivity, punitive, and finally confiscatory. The ideal tax rate would maintain the economy in the “specialization and full productivity” range:

1) Economic output is near a peak.
2) Net revenue-the amount of money left over for the private economy-is near a peak.
3) Government revenue is constrained and thus is away from the range where it will damage the economy.

For anyone interested, the formula used to generate curves shown above is:

E(t) = 579.64 * t^3 - 1222.07 * t^2 + 642.43 * t

This gives a maximum economic output of 100 units at te =0.35.

This formula has no basis in reality-it's just that polynomials are easy to work with and have a nice shape.

[Update November 26, 2008: This is why I'm an engineer-my communications skills are lacking. Many people are assuming that my conclusions are derived from the arbitrary graphs. In fact, I derivd the math first and then created the chart to illstrate (not prove) my theory.]