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
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
G(t) =E(t) * t
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.
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
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.
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.]