January 6th

 

Bad to the Bone

And when I walk the streets
Kings and Queens step aside
Every woman I meet
They all stay satisfied
I wanna tell ya pretty baby
Well ya see I make my own
I'm here to tell ya honey
That I'm bad to the bone

You can be bad, but is being bad is not always criminal.

The purpose of a legislature is to pass laws.  Laws are passed to create penalties for behavior that the sovereign, the People of the United States, wishes to penalize.  One of the functions of a legislature is to determine if that bad behavior was, or should have been, addressed by existing laws.  If they are not, then new laws may need to be passed.  It is not merely the function of the legislature to pass laws that penalize bad behavior that is not addressed by existing laws.  It is the responsibility of that legislature to pass along any information that suggests that existing laws have been violated.

A series of actions happened on January 6^th at the US Capitol that appears to be bad behavior.  A select committee of the House has been established to examine what occurred on Jan 6^th. If there was behavior that the People wish to penalize, the committee is charged with determining if that behavior is already addressed by existing laws. If there are behaviors that are not addressed by existing laws, then they might consider laws that would address them. 

If there are bad behaviors that are not covered by existing laws, bills of attainder are prohibited by Article I of the US Constitution.  Passing laws after the fact can not penalize actions that were legal at the time the action occurred.  Laws can be enacted to address this behavior in the future.  However just as bill of attainders are NOT allowed by the legislature, if the legislature finds evidence of bad behavior that IS criminal under existing laws, it has a responsibility to pass that information along to the proper authorities.  Investigations by the legislature may discover behavior that should be prohibited in the future and laws may be proposed and enacted to address that behavior.  This is a proper legislative function. 

But if bad behavior is already covered by existing laws, no legislation is appropriate.  However any facts of the possible violations of those existing laws should be passed along to the proper authorities.  Just because those authorities did not discover that behavior by themselves, does not mean that the laws were not broken. If you are bad, and your actions were criminal, then you can be prosecuted, regardless of who disclosed that criminal behavior.  If a criminal act was committed, then who uncovered evidence of that act is irrelevant.

Sovereigns II

 

Good King Wenceslas

Bring me flesh and bring me wine
Bring me pine logs hither
Thou and I shall see him dine
When we bear them thither.
Page and monarch, forth they went
Forth they went together
Through the rude winds wild lament
And the bitter weather

What does it mean to be a Good King?

A monarch, a king, is the sovereign of his subjects.  Man, i.e. his subjects, is a tribal animal.  The sovereign is the physical embodiment of the “tribe”.  A sovereign cares for himself, his subjects, and interacts with other sovereigns and the subjects of those other sovereigns.  If a king cares more for his subjects, other sovereigns, and their subjects, than he does for himself, the sovereign may be characterized as a Good King.

The king can care more about his own subjects than other sovereigns or their subjects.  If he does not actively antagonize those other sovereign or their subjects but is a particularly effective sovereign for his own subjects, he may be characterized as a Great Sovereign,  ( e.g. Charlemagne, "Charles the Great".)

If he actively antagonizes those other sovereigns and their subjects, while  acting for his own subjects …. (he’s a son of a bitch, but he’s our son of a bitch.)…. then he is most probably beloved by his subjects but viewed poorly by others.  Vlad the Impaler, Dracula, was beloved by his own subjects but feared and hated by other sovereigns and their subjects.

A sovereign can care more for himself than for his subjects. If he cares more for himself than his subjects then he almost certainly cares more for himself than he cares for other sovereigns or their subjects.  Those Sovereigns are often given a name that indicates this condition. (e.g. Ivan the Terrible).

It is appropriate for a digression into the types of sovereigns.  In nature, sovereigns are almost always dominance sovereigns. That is the sovereign is chosen by his dominance of others in the tribe, e.g. Nation.  If after being chosen by dominance, he acts poorly for his subjects, he might personally thrive, but his tribe would be weakened.  In this case, the tribe will not thrive and Nature will chose a System, tribal, Optimal solution.  While dominance is almost always how leaders of the pack are chosen in Nature, mankind has adopted two other ways of choosing sovereigns,.  A sovereign may be elected by a set of his subjects, e.g. the Pope chosen by the college of Cardinals, or the Holy Roman Emperor was chosen by prince-electors.  It is also possible for an existing sovereign establish a dynasty and choose his own successor as sovereign.  This may be a close relative ( e.g. the first born son of the sovereign) or may be an adopted, or designated as an heir, of the sovereign  ( e.g. Augustus Caesar was the heir of his maternal great uncle Julius Caesar.)

It is possible for Dominance Sovereigns to aspire to become Dynastic Sovereigns. E.g. Kim il-Sung of North Korean was succeeded by his son, Kim Jong-il, who in turn, designated as heir, and was succeeded by, his son , Kim Jong-un.  If a Dynastic Sovereign dies without a strong heir, wars of succession may result, e.g. the War of Spanish Succession (1700-1714); the War of Austrian Succession (1740-1748). 

While a sovereign who favored other sovereigns or their subjects over his own, would not be expected by dominance or elected sovereigns, it can happen with dynastic sovereigns, who might not necessarily succeed as dominance, or elected sovereigns.  Tsarina Catherine the Great was accused of being a Francophile who valued the French more than her own Russian subjects.

The People of the United States, its subjects, are, through its Constitution, also its sovereign.  By definition, since they are the subjects, the sovereign can not be terrible.  In the view of its subjects, it is of course Great. It should aspire to be Good.

It's Relative II

 

Dead Man's Curve

(Dead Man's Curve) is no place to play
(Dead Man's Curve) you'd best keep away
(Dead Man's Curve) I can hear 'em say
Won't come back from Dead Man's Curve

Has relativity thrown us for a curve?

As anyone who has traveled long distances by air knows, the shortest distance between two points on a sphere is NOT a straight line as would be suggested by Pythagoras’ Theorem. 

c²=a²+b²

c=sqrt(a²+b² )

This theorem would suggest that for a right triangle, with sides a and side b, the shortest line between these two points would be the hypotenuse, c.  However, for large distances on the earth, this would require leaving the surface of the Earth and crashing into and travelling through the Earth.  If travel is confined to the surface, or just above the surface, traveling over the sphere suggests that the shortest distance is a Great Circle Distance.

This is an example of non-Euclidean Geometry, where Euclidean space deals with a flat surface.  The surface of the Earth is a sphere, not flat.  The Spherical Pythagorean Theorem is

cos (c/R)=cos(a/R)*cos (b/R)

As the radius of the sphere, R,  goes to infinity, the surface of that sphere becomes virtually flat, and expanding the cosines using their Taylor series, yields  

c=sqrt(a²+b²)

which is the traditional Pythagorean Theorem.  If a, b and c are very small compared to the radius, R, of the sphere, e.g. a small surface in your home, the surface is virtually flat and Pythagoras' Theorem is virtually true.

But a surface can not only be spherical, what mathematicians would call concave.  A surface can also be a hyperbolic plane, what mathematicians would call convex.  In this case the Hyperbolic Pythagorean Theorem is

cosh(c)=cosh(a)*cosh(b)

A hyperbolic cosine can also be expressed using Euler’s Number, the exponential e, as  

cosh (x)=½ *(e^x+e^-x)

This means that the Hyperbolic Pythagorean Theorem, in exponential form, is

½*(e^c+e^-c )= ½*(e^a+e^-a)* ½*(e^b+e^-b )

which becomes

c=ln(cosh(a)cosh(b)±sinh(a)sinh(b))

In most real-world applications, the traditional Pythagorean Theorem is good enough.  When the sphere is very large, the surface of the sphere is approximately flat.  When the distance between the points is large compared to the radius of the sphere, e.g. the surface of the Earth, the Spherical Pythagorean Theorem will give the Great Circle Distance on the spherical surface.

In most applications, the Hyperbolic Pythagorean Theorem will not be needed.  However, there is a notable instance where the use of the traditional, flat, Pythagorean Theorem might lead to the wrong conclusion.

Einstein showed that the velocity of an object, v, is limited by the speed of light, c.  His famous equation, E=mc², is an implication of that limit.  The Total Energy of an object is the combination of its Kinetic Energy, related to its velocity, and its Rest Energy, related to the speed of light.  Kinetic Energy is what powers cannonballs.  The equation for Kinetic Energy is ½ mv².   Rest Energy is what powers the Atomic Bomb.  The equation for Rest Energy is m₀c², where m₀ is the rest mass at a velocity of zero.  In most real-world examples, the velocity is very small compared to the speed of light and the kinetic mass, m, is approximately equal to the rest mass, m₀.  But strictly speaking the kinetic mass is a factor, γ, multiplied by the rest mass, where that factor depends on the ratio of the velocity of the mass to the speed of light. The magnitude of the combination  makes Einstein’s triangle of energy 

(γm₀ c² )²=(½ γm₀ v² )²+(m₀ c² )²

If you solve for γ using Pythagoras’ Theorem in flat space, this leads to Lorentz's factor as

γ=sqrt(1-(v/c)² )

This has the notable problem that it leads to an undefined term, 1/0, when the velocity of an object, v,  is equal to the speed of light, c. But the use of the Traditional Pythagorean Theorem also assumes that the surface is flat.  If the surface is assumed to be hyperbolic, (space-time is known to be curved, not flat.  The surface could also be Spherical in which case the Radius, R,  of the surface, i.e. universe, would need to be used. However the Radius of space is so large that it approaches infinity in which case, the surface would be equivalent to a flat surface), this relativistic factor could be:

γ=1+ln(1-(v/c)² )

This factor has a correlation with the traditional Lorentz factor, when v is less than 95% of the speed of light, of 0.986.  It also has the characteristic of being infinite, not undefined, when the velocity is equal to the speed of light. The proposed factor differs by less than 5% from the traditional Lorentz factor for speeds less than 90% of the speed of light, and differs from the traditional Lorentz factor by less than 1% for speeds less than 23% of the speed of light.

Texas Senate Bill 8

 

You’re a Mean One, Mr. Grinch

You're a vile one
You got termites in your smile
You have all the tender sweetness
Of a seasick crocodile
Mr. Grinch
You're a foul one
Friends you don't have none
I wouldn't touch you with a 39-and-a-half foot pole!

Either the Supreme Court is stupid, or it is mean.

The Supreme Court has allowed Texas Senate Bill 8 to continue.  Because of the drafting of the law, which makes citizens, not the state,  the enforcers of the law, the drafters of this law cleverly thought that they excluded state officials from enforcing the law, and thus the state had no standing and that there was no one to enjoin.  Unsurprisingly, California Governor Newsom has threatened that California should adopt the same approach targeting gun manufacturers.  SCOTUS did not have to allow the country to devolve into state-sponsored vigilantism on the issue of standing.

Standing was not relevant and should not have been considered by SCOTUS.  Texas Senate Bill 8 violates the rights enumerated in the Seventh Amendment to the US Constitution.

“In Suits at common law, where the value in controversy shall exceed twenty dollars, the right of trial by jury shall be preserved, and no fact tried by a jury, shall be otherwise re-examined in any Court of the United States, than according to the rules of the common law.”

The right to a trial by jury is NOT preserved. Texas Senate Bill 8 clearly sets the damages to be awarded in these civil suits, and that the finding be only that an abortion was supported by the defendant. The Senate bill prevents the jury from awarding legal costs to those who successfully defend themselves but requires that legal costs of the enforcers/plaintiffs should be paid. 

If Texas wants to criminalize  abortion, then they could at least try to criminalize abortion.  Texas did not do so because  they knew that would be unconstitutional.  Directing a jury in a civil case that an action is criminal, unconstitutionally limits the rights of the jury.  A civil case under common law requires damages to be found by a jury.  An Act by the legislature cannot limit this right, especially if no monetary, or pain and suffering, damages can be proven by the plaintiff.  A law saying that a plaintiff has suffered is an unconstitutional limitation on the right of the defendant to a trial by a jury. The legislature saying that a plaintiff has suffered is its opinion, not a finding by the jury.

What is next? A law saying that wearing yellow is a crime and a plaintiff can be sued for $10,000 if I support someone who wears yellow.  If that is absurd, then Texas Senate Bill 8 is also absurd.

Theft

 

Keep Your Eye On The Sparrow

Don’t do the crime
If you can’t do the time
Yeah, don’t do it.

What is the time for stealing one penny from 100 million?

Stealing a $1,000,000 from an individual is grand theft, a felony.  Society and that individual both have a vested interest in catching the thief.  The penalties for grand theft and the risk of getting caught are supposed to be sufficient to negate any possible benefits of the theft.

Stealing $100 from 10,000 individuals is petty theft in each instance, a misdemeanor.  The total theft is still $1,000,000.  Society and the individuals have a less of an interest in catching the thief.  The penalties for petty theft and the risk of getting caught are lower for each crime, but cumulatively the risk of getting caught when there are 10,000 chances to get caught are much greater.  Consequently the penalties of 10,000 instances of petty thefts are supposed to negate the possible benefits of 10,000 petty thefts of $100.

Stealing a penny, $.01, from 100 million individuals is technically also petty theft.  The cumulative theft is still $1,000,000.  However society, and those individuals, have even less of an incentive to pursue the thief, and the chances of getting caught for a single theft this small are also small.  Logistically attempting theft on this scale, of 100 million individuals, before computers, was not feasible.  It is not clear that the penalties for this crime negate the benefits.

Logistically the law and society have not kept pace with the possibility of petty theft on a very, very small scale. Mathematically there should be no difference in stealing $1 million from one person or one penny from 100,000,000 persons.  What had once been logistically infeasible, is quite feasible today.  The law and society need to catch up and make the time fit the crime.

The Law

 

I Fought The Law And The Law Won.

I'm breakin' rocks in the hot sun
I fought the law and the law won
I fought the law and the law won

The opinions of the Supreme Court are only opinions, but they are the law.

The opinions of the Supreme Court are law, but they are opinions, even when they are characterized as decisions.  There can be a difference of opinions.  However the subjects of a sovereign agree that their opinions are, doh, subject to the opinion of the sovereign. There is no determination that either opinion is correct, a decision, just that the opinion of the sovereign prevails.  If the sovereign were an individual, e.g. a monarch, we might refer to law as the King’s Justice.  Justice is always the sovereign’s, even when that phrase is not used.  When the Supreme Court renders an opinion, it is rendering the opinion of the sovereign.   There is not a determination that the opinion is correct, only that it is the opinion of the sovereign.

A jury can also render an opinion.  If the jury is unanimous, then the ruling of the jury prevails.  If the jury is not unanimous, is hung, no ruling is rendered and a mistrial is declared.  A hung decision can be because of only one member who prevented the decision from being unanimous.  A sovereign can by definition not have a hung decision, which is why a hung jury is a mistrial and NOT a ruling of not guilty.  A unanimous decision may only reflect the lowest common opinion that can be accepted by all parties.  A decision by a sovereign can not reflect the lowest common denominator.  Since that sovereign is unitary, he is his own lowest common denominator.

Much of the confusion in the United States has to do with an understanding of the sovereign.  The sovereign is The People, a collective noun.  The sovereign is not each person.  The opinions of the people are not the opinions of each individual person.  You can disagree with the opinions of the sovereign, but each person as a subject of that sovereign has agreed to abide by the opinion of the sovereign.  The sovereign is not the President.  The sovereign is not the Speaker of the House, The sovereign is not the Majority or Minority Leader  of the Senate.  The sovereign is not a majority of the Supreme Court.  The sovereign is The People and only The People.  People as individuals in the US are the  subjects.  The People as a collective, is the sovereign.  Each individual is NOT a sovereign.  If the sovereign has decided that his actions, or the actions of his predecessors, have been, for example, racist, this in no way means that each individual is racist.   There is a difference between people as collective noun, and people as individuals.  If we do not understand this, then we do not understand what a republic is, and despite Ben Franklin’s admonition, we will not keep it.  The first Republican President Abraham Lincoln thought that government “of the people, for the people, and by the people “ was worth preserving. Let’s hope that today’s Republicans also think that it is worth preserving as well.

Corporations

 

People

People,
People who need people,
Are the luckiest people in the world.

Are corporations people?

The majority of US Supreme Court rendered an opinion in Citizens United v. the FEC, that corporations are people.  That majority opinion was based on the fact that corporations are groups of people.  If individual people have free speech under the Constitution, then, in that Supreme Court’s opinion, corporations must have the same right of free speech. It is important to determine if there are any differences between individual people, groups of people, and corporations.  People have a life expectancy.  People are born, grow up, grow old and then die.  They are not immortal.  This is no different for people in groups or people as individuals.  Groups of people can outlive the individuals in that group, but only if new individuals replace the current individuals.  Corporations also can live longer than people, but again only if new owners of the corporation replace the current owners of the corporation.  The question should be whether a corporation is only a group of people.

Corporations are chartered by the sovereign.  In nations where the sovereign is an individual, that is why the phrase chartered by His/Her, Royal Majesty is used.  Corporations predate the founding of the United States.  For example, the British East India Company, originally chartered as the “Governor and Company of Merchants of London Trading into the East-Indies” was founded in 1600.  Corporations in the United States, which remember are chartered by the sovereign of the United States, its people acting through the Constitution, can take four basic forms.

·       Limited Liability Corporations, LLCs

·       Schedule C Corporations

·       Schedule S Corporations

·       Nonprofits

Each of these types of corporations have different requirements for: filing, filing fees, governance, reporting, taxation, etc.  The commonality among all of these types of corporations is the protection of the assets of the owners of the corporation, e.g. people, from liability for the debts of the corporation.  The personal property of the owners of corporations is clearly an asset of the owners. The personal property of the owner does include their personhood.  Their personhood is protected by the Constitution.  The Constitution was amended to prohibit slavery and thus no person may be owned by another.  Persons thus may not sell or transfer their personhood to their corporations.  The question that should have been the basis for the opinion is whether the rights of people listed in the Constitution are also among those protected assets. If the right to free speech was not protected, then the SCOTUS majority opinion  was correct.  If the right to free speech is a protected asset, then the dissenting opinion was correct.  If the rights of free speech are an asset of the people, and corporation shields the assets of its owners from liability, then, this is a recognition that a corporation is not merely its owners, people, and thus a corporation does not have the right to free speech.  There is no question that the owners of the corporation have continue to have free speech.  It is just that when acting as a corporation they do not have the same right to free speech that is conferred to people.  Actions that may abridge the rights of free speech of the corporation, do not abridge the rights of free speech of the owners of the corporation.  Thus the rights of the owners still exist.  Those rights can not be transferred to another, i.e. a corporation. If the right can not be transferred, and that right is still retained by the owners, then the corporations do NOT have the same right to free speech. Corporations are owned by people, but that doesn’t make them people.

Income Taxes II

 

Georgia Tech Fight Song 

I'm a Ramblin' Wreck from Georgia Tech
And a hell of an engineer 

What does an engineer think of the income tax code? 

Engineers, such as me, think Talking Barbie is correct. Math is hard.  Engineers try to come up with “Rules of Thumb” to use that are close to being correct.  Mathematicians would say is not possible to state Pi exactly because it is an irrational number. Engineers might say “Let’s just get close enough”.  If engineers said that Pi was 3, then this would result in 4.5% error in computations, which is not close enough.  If engineers said that Pi was 22/7, then the error in computations would be only 0.4%, which is probably close enough.  Similarly, non-linear equations, especially non-linear equations of an exponential since the exponential itself is another irrational number, are approximated by engineers as a series of linear equations, also known as straight lines. 

The mean income by quintile (i.e. 20% ranges) is reported annually by the US Census, which also reports the mean income of the top 5 percent. Assuming that the mean income of a quintile is the income of the midpoint of that quintile, or in plain English, if between 20% and 40% of the incomes had a mean income of $12,196 in 2019 dollars, the income at 30%, the midpoint of that quintile, is assumed to be $12,196.  With this assumption, the reported incomes by quintile can be shown to follow an exponential association with a fit of 95.3%.  The transient income, the only coefficient needed for this equation, is $53,450.26 .  This exponential association can be approximated by a series of 6 simple linear equations, with a correlation of 99.88%, as shown in the following figure.

An equitable distribution of income taxes should follow this same distribution of income.  The taxes should thus be 

Taxes = Income * Maximum Tax * Distribution of income

If the distribution of income is approximated by six linear equations, which are the equivalent to tax brackets, and the transient income which is defined by the income distribution is $53,450, then if the median rate is 20%, the Maximum tax rate should be twice that amount, or 40%.  (In a normal statistical distribution, the maximum is approximately twice the mean.)  The effective tax rate is the actual taxes due divided by the income. 

The 2020 ideal tax brackets which reflect the distribution of income are: 

Bracket	Lower Income	Upper Income	Marginal Tax Rate on Income Above the  Lower Income	Fixed tax	Effective Tax Rate Lower Income	Effective Tax Rate Upper Income
1	1	$53,450	14.7%	$-	14.7%	14.7%
2	$53,451	$106,901	30.7%	$7,865	14.7%	22.7%
3	$106,902	$160,351	36.6%	$40,682	22.7%	27.3%
4	$160,352	$213,802	38.7%	$99,335	27.3%	30.2%
5	$213,802	$267,251	39.5%	$182,164	30.2%	32.1%
6	$267,252	∞	40.0%	$287,827	32.1%	40.0%

The actual 2020 rates from the IRS according to those established in the Jobs and Tax Cut Act,  JTCA, are 

Bracket	Lower Income	Upper Income	Marginal Tax Rate on Income Above the  Lower Income	Fixed tax	Effective Tax Rate Lower Income	Effective Tax Rate Upper Income
1	1	$9,875	10.0%		10%	10%
2	$9,875	$40,125	12.0%	$988	10%	12%
3	$40,125	$85,525	22.0%	$4,618	12%	17%
4	$85,525	$163,300	24.0%	$14,606	17%	20%
5	$163,300	$207,350	32.0%	$33,272	20%	23%
6	$207,350	$311,025	35.0%	$47,368	23%	27%
7	$311,025	∞	37.0%	$83,654	27%	37%

The Marginal Tax Rates on income above the lower amount are comparable between the ideal and actual tax brackets.  The major difference between the two tables is the size of the tax brackets. They should be equal in a tax code based on income distribution.  There would be six tax brackets. The range of the tax brackets would be equal to the transient income as established by the distribution of income.  The actual ranges of income covered by the TCJA tax brackets are not equal and are much lower in the lower brackets.  Consequently the fixed amount (i.e. the taxes on the lower income in each bracket)  is lower for the upper brackets.  This results in effective rates that are far less than would be suggested by the income distribution. The upper tax brackets consequently have been paying effective tax rates that are much lower than would be suggested by the distribution of income. 

These lower than ideal effective tax rates have been in place since the Economic Recovery Tax Act of 1981.  If this was intentional, talk about a long con.  The inequality caused by that, and subsequent tax acts, may have been responsible for the growing wealth gap in the United States.  An income tax code should be based on the distribution of income.  In 1980, before that Tax Act was enacted, the transient income, T, the measure of the distribution of incomes, was only $39,622 in 2019 US Dollars .  A tax code which reflects the distribution of incomes at any T is shown below.  

Bracket	Lower Income	Upper Income	Marginal Tax Rate on Income Above the  Lower Income	Fixed tax	Effective Tax Rate Lower Income	Effective Tax Rate Upper Income
1	0	$T	14.7%		14.7%	14.7%
2	$T	$2T	30.7%	14.7%*T	14.7%	22.7%
3	$2T	$3T	36.6%	45.4%*T	22.7%	27.3%
4	$3T	$4T	38.7%	72.0%*T	27.3%	30.2%
5	$4T	$5T	39.5%	120.7%*T	30.2%	32.1%
6	$5T	∞	40.0%	160.3%*T	32.1%	40.0%

Adoption of new marginal tax rates may be a "Bridge Too Far" politically.  However adjustment of the ranges of income within each tax bracket might be undertaken which would more closely reflect the distribution of income.  The lowest tax bracket may not be necessary but it is retained here. The changes in the range of income in brackets effects the fixed amount, the amount due on income in the lower brackets, as well as beginning the highest tax rate at a lower income.  This lowering of the lower bound of income in the highest bracket actually has the effect of raising taxes for the current lowest income in the highest bracket because it increases the fixed amount due from taxes on income in lower brackets.  The proposed changes are shown below in red bold font. 

Bracket	Lower Income	Upper Income	Marginal Tax Rate on Income Above the  Lower Income	Fixed tax	Effective Tax Rate Lower Income	Effective Tax Rate Upper
1	1	$9,875	10.0%		10%	10%
2	$9,875	$53,450	12.0%	$988	10%	14%
3	$53,451	$106,901	22.0%	$7,402	14%	18%
4	$106,902	$160,351	24.0%	$19,161	18%	20%
5	$160,352	$213,802	32.0%	$31,989	20%	23%
6	$213,803	$267,251	35.0%	$49,093	23%	25%
7	$267,252	∞	37.0%	$67,800	25%	37%

The change in the tax bracket ranges, would lower the tax rate for the all incomes.  It would also lower the total amount of revenue raised by the income tax.  If the tax code is changed, it is suggested that marginal rates that more closely reflect the distribution of income should also be enacted.  As shown in the figure below, this would lower significantly the income tax rate for all those below approximately $140,000 while raising the tax rates for those with incomes above this amount.  However, these tax rates would more closely match the distribution of incomes.

The