Wednesday, November 12, 2008

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


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.]


  1. Very interesting - I am familiar with the Laffer curve, but not your conclusion. This is something worth thinking about.

  2. What if E(t) is maximized at t = 0.00? I don't think this case affects your conclusions, except inasmuch as t_c and t_e may both be equal to zero. The broader point however is that E(t) can't be derived from G(t) as you have done above. Again, I don't know if this affects anything in the case where E(t) is assumed continuous, but it might at least indicate the possibility that t_c = t_e = 0.

  3. The shape of the laffer curve varies with the time frame; i.e., in the very short term, t_g=1 maximizes government revenue and has no impact on the size of the economy, but C(t)=0. As the time frame lengthens, t_g shifts to the left, along with the corresponding values of t_e and t_c. Take a long enough time horizon, and t_g asymptotically approaches 0, since by extending the time horizon, tax revenue can always be increased more by increasing the economic base than by increasing the tax rate. Since t_e is always to the left of t_c, and shifts left as t_c approaches it, both numbers approach 0 as the time horizon approaches infinity.

  4. Varying with the time frame --> isn't this true of other phenomena as well, like being shot? Slicing an increment of time fine enough to observe no ill effect does not make a bad idea good.
    Consider the plan announced sometime in 1994 or 1995 for the government to tax everything above some amount, say $300k -- after all, 300k provides for a good life, so what if the rate on earnings above that was 100%? Some fool predicted that this would net some X-number of millions to the Treasury in the first year, and a growing rate of more millions in subsequent years. This is obviously false -- all those earners will have been crushed or chased offshore, so the earnings the next year would be very near zero, and all the other brackets would be producing less as well. This is a fair example of the point which is, again, just because I can make it look good "for an instant" doesn't make it good.

  5. Thanks to those who commented, for reading what I had to say.

    #2-Realistically, I do not think that the size of the economy can be optimized with tax rate of zero-the economy would be way too vulnerable to enemies, foreign and domestic. Establishing order and providing for defense are the first two roles of any government, and this costs money.

    # 3 and 4-I've not even begun to think about adding time as a variable.

    Thanks again,


  6. This is very interesting. I plotted many nation's tax rates vs. earnings and found a two hump 'camel' curve. Little data comes from confiscatory regimes. The space between Te and Tc seems cause average income to go down, as marginal earners are brought into the market.

  7. As the economy moves into 'punitive taxation', the average income increases, because tax accountants and lawyers tend to be among the best paid workers, though not the most productive in the tax regimes lower than Tc.

  8. There is an implicit assumption whatever the level of revenues to the government, that the government is properly spending the money it receives. This assumption is false.

    The other fallacy, as anyone who has actually run a small business, is that a higher tax rate will cause the restaurant owner to hire a bookkeeper. There is no assurance that being open extra hours will bring enough additional business to cover even the variable direct costs of labor, food, etc.

    In order to hire the bookkeeper, not only does the owner have to cover the direct variable cost, but must also cover the additional cost of the bookkeeper, so the hurdle to be open longer (providing more service to customers as an additional benefit) is now higher.

    More likely is that when the tax rates go up, the janitor gets fired, and the owner spends extra time cleaning the bathrooms himself, in order to try to maintain his after tax profitability.

    Bottom, despite the math and the numbers, no one will allocate there money, income and investment capital, as efficiently to meet their needs as the individual. At almost any rate of taxation beyond that necessary for the military, fire, police, and OK, roads, society is worse off.

  9. It's really a shell game. If the Feds cut the tax rate, the State and Local governments are there to pick up the slack. In addition, there are MANY forms of taxation such as sales tax, value added, and "fees" that are taxes in drag. Don't forget, the government also controls the dollar value of the currency that is used to pay those taxes. Perhaps the cruelest tax of all.

  10. What about the relationship between tax rates in the domestic economy versus tax rates abroad? I would guess that the curves above would end up being somewhat skewed by this factor. For example, if all foreign nations were to reduce marginal tax rates, would capital and investment not shift abroad and thus have an adverse effect on the domestic economy?

  11. Bill S. and Anonymous (4:11 PM)-

    Thanks for your comments.

    External factors (like how efficently a government spends its tax revenue, or the tax structure of competing economies) will have a huge impact on the shape and magnitude of the size of the economy. I do not dispute that at all.

    For example, a kleptocracy and a republic will have very different economic ouptuts, even if their tax rates are the same.

    If capital can flow freely between economies, then capital should seek lower tax rates, all else being equal. This obviously affects the shape of the economic curve (with higher output if you are the low tax economy, and lower output from teh high tax economy.) California's economic output curve is undoubtedly lower than it could be because companies like Intel have chosen to build new facilities outside the state rather than suffer California's tax burden.

    Thus, competition or inefficiency will affect the magnitude and shape of the economic curves.

    My assertion is that, no matter the shape of those curves, the tax rate that maximizes revenue hurts the overall economy.

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

    Could've just put that at the beginning and saved everyone a whole lot of trouble.

  13. Anonymous-

    The derivation applies to any continuous function, not just the one I chose to illustrate it with. For example, you can plot

    E(t) =200t * sin (4 * PI * t) + 200

    and get a completely different curve, but the same characteristics apply-the Crier Curve peaks first, then the economy curve, and finally the Laffer Curve.

    If you read my post entitled "A Triple Point for the Laffer/Crier/Economy Curves?" you can see how an identical conclusion can be reached without having to wade through the calculus.

  14. Marty,

    Comment #2 nailed it. You're trying to make this look like a mathematical proof but you derive your Economic Size curve out of thin air. As much as I agree with your POV, you're doing a disservice to the cause by creating fakey proofs like this. It reminds me of some of the "proofs" that ID adherents create, which only weaken their argument amongst informed people. Sorry.

  15. Anonymous-

    Please see my update to the post. I performed the calculus first, and then made the graphs to illustrate my point. The only assumptions I make are: continuity of E(t), positive values for the economic output,and that the tax rate affect economic output.

  16. Anonymous-

    Please see my latest post here:

  17. Thanks, this is interesting.

    I agree with Anonymous (November 24, 2008 3:35 PM ).
    I was expecting to see a curve with tc = 0. I agree with you that E(t) initially grows, but I did not expect that you would take this into account. I think it would be more clear to either show the reasoning with a strictly decreasing E(t) on the whole interval, or to explain why E(t) would initially grow. Not a complete explanation, but a hint, say "roads".

  18. Hi Philippe-

    Thanks for the feedback.

    E(t) can be any shape at all-it can peak at a tax rate of zero. All I claim is the following:

    1) Governemnt revenue will peak at a tax rate that causes E(t) to be smaller than it would be at lower tax rates.

    2) The money retained in the private economy peaks at a lower tax rate than the peak output of the economy.

    Practically, though, some tax rate is necessary to provide the barest services. The most fundamental services are national defense and a justice system. I cannot see how a properous nation could survive without a means to defend itself from poorer neighbors. Similarly, I do not see how the industrious members of a population could not protect itslef from the immoral members of the population without a police force and courts.

    Roads are an important service as well, but you could perhaps theorize a privatized toll road system. National defense and justice are two functions that should not be privatized.

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