Growth II

 

What Do I Know?

No university, no degree, but lord knows
Everybody's talking 'bout exponential growth
And the stock market crashing in their portfolios
While I'll be sitting here with a song I wrote
Sing, love could change the world in a moment
But what do I know?

You probably know more than you think!

Any exponential growth function can be expressed as a compound growth function. Compound growth functions have a constant and continuous growth. Exponential growth functions have a continuous but not constant growth. However any compound growth rate can be expressed as an exponential function.

If the growth rate, in demand, is 10% per period, then the compound growth formula would be

Future value=Present value*(100%+10%)^x

where x is the number of time periods from the present to the future.

At infinity, this will have a value of infinity. But this assumes that the time periods are also infinite. It is possible for the Future value to consume all of available supply before an infinite time unless the growth in supply is also 10% per year. If it is less than 10% then the future value in demand will exceed the future value in supply long before an infinite number of time periods have been reached. In other words, in compound growth you will approach infinity in a finite period. In exponential growth you will only approach infinity only after an infinite period.

The rich have a disparaging comment about the poor that they are poor because they live off principal instead of living off interest. Doh, if the present value, principal, is zero then it makes no difference what the interest rate is, the future value will always be zero. The ideal is to have enough principal that you CAN live off the interest. But if your living expenses are higher than the interest on your principal then your principal gets reduced and the next Future value will be smaller because the Present value in that next time period is also smaller.

An exponential growth can be made from this compound growth

Present value*(100%+10%)^x=Present value*exp(λx)

 if

λ=ln (110%)

In fact any compound growth rate, r%, can be made into an exponential growth if

λ=ln (100%+r%)

If λ is negative, less than zero, this is said to be a decay function, for example a radioactive element decay function, and λ can be converted into a half-life. Long half-lives are stable, good, and short half-lives are radioactive, unstable, bad. When λ is positive, greater than zero, this is said to be a growth function. But λ can still be converted into a time, say a doubling period. But while this will be the inverse, the qualifications are still valid. A long doubling period is good and a short doubling period is bad. Short doubling periods have lots of energy but expire soon. Long doubling periods have less energy but they last longer.

Given that, what should the best growth rate be? The question should be what is the growth rate of a market? The best is arguably an example of the Lake Wobegon effect ( “all the children are above average”) or the Yogi Bear effect (“smarter than the average bear”). If the population is normal, then the maximum, "best", of the crowd should be twice the average of the crowd ( e.g. the Z-score of 99.9% is approximately 2.)

Sustainable growth should thus be twice the growth of the market ( yes a  firm could grow by 10% if the market only grows by 2%, but eventually that firm will have 100% market share.)   The maximum sustainable growth  should not exceed twice the growth of a market. Twice is only sustainable if the average  IS the maximum ( which is only true when the standard deviation is zero, but then that is no longer normal.)

Should that firm enter other markets? Absolutely, but that growth still should not exceed 2 times the growth in its combined markets ( the original market plus the new market). If the best growth rate is compared to the revenue from the original market, the growth in  revenue will appear to be greater than the growth in the original market. But this is only because of the choice of the dominator in that growth rate.

This also defines the size of the new market. If a firm wishes a growth rate, r, that is more than twice the growth of its market, it should pursue new markets whose value, V, is larger than its current market according to.

Proposed Growth Rate=

2*(V_{Old Market}*r_{Old Market} +V_{New Market}*r_{New Market} )/(V_{Old Market)} +V_{New Market})

This also explains why you should pursue new markets.  If your old market is buggy whips and you don't go after any new markets, then the best growth rate you can expect have is twice the growth rate of the old market. Having 100% of the buggy whip market will still mean a growth rate of 0% since the buggy whip market is not growing.

Monkeypox

 

Hello Muddah, Hello Fadduh (A Letter from Camp)

Wait a minute, it stopped hailing,
Guys are swimming, gals are sailing.
Playing baseball, gee that's betta,
Mudda, Fadda, kindly disregard this letter!

Patience you must have, my young padawan.

If your time period is truly short, like the one day in this song, you can have a quite different impression than if you look at a longer time period. There is a difference between weather and climate; between revolution  and evolution; between winning a battle, tactics, and winning a war, strategy; etc. Confusing these is NOT good. There is also a difference between risk and likelihood. Risk is the product of likelihood AND consequences. If the consequences are low then the risk is probably acceptable, regardless of the likelihood. If the consequences are high, then the risk can be unacceptable even when the likelihood is low.

This came up during a conversation on the risk of monkeypox if you have already been vaccinated against smallpox. Smallpox was eliminated and the vaccination against smallpox stopped in 1972. The LIKELIHOOD  of getting monkeypox if you have been vaccinated against smallpox is 15%. The likelihood of successfully filling an inside straight in poker is about 8% so if you think that 15% is acceptable, can we play poker please???

The CONSEQUENCES of getting monkeypox are considerable, so that the risk of getting monkeypox might be unacceptable even if the likelihood of getting monkeypox is low. Don’t be confused.

Hyperspace?

 It Ain't Necessarily So

It ain't necessarily so

It ain't necessarily so

They tell all your children

The devil he's a villain

It ain't necessarily so

Believing only what you can see locally is wrong

Pythagoras’ theorem is that any number can be expressed by two smaller numbers according to the relationship c²=a²+b². This was originally solved in Euclidean geometry, space with a curvature of zero, as the hypotenuse of a triangle. The relationship is true for a triangle in any geometry, including non Euclidean. It is just that under other geometries, the formula for the hypotenuse of a triangle is different because the sum of the angles in a triangle will be different. If the sum of the angles in a triangle is greater than 180 degrees, the curvature is positive and the non-Euclidean geometry is spherical. If the sum of the angles in a triangle is equal to 180 degrees, the curvature is 0, flat, and the geometry is Euclidean. If the sum of the angles in a triangle is less than 180 degrees, the curvature is negative and the non-Euclidean geometry is hyperbolic.

In Euclidean space, with a curvature of zero, the formula for a hypotenuse is

c=sqrt(a²+b²)

But this is only the formula in Euclidean space. The Flat Earth fallacy is because while locally space is Euclidean, the Earth is a large sphere, with a radius R, has positive curvature, and the formula for a hypotenuse which is its Great Circle Distance is

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

When the radius of the sphere is exceptionally large compared to c, a, and b, this is virtually indistinguishable from the formula in Euclidean space, i.e. locally space is Euclidean but over the entire surface of the Earth the geometry is spherical.

If the curvature is negative, hyperbolic, the formula is

c=ln(cosh(a)cosh(b)±sinh(a)sinh(b))

Mabkhout[1] proposed that the universe is globally hyperbolic and locally flat. If Einstein’s equations are solved for universally flat space, then dark matter and dark energy must be invoked in order  to be consistent with cosmic inflation immediately after the Big Bang;  to solve the paradox of rotating galaxies; to solve for the apparent difference in the size and the age of the observable universe; and to result in an expanding universe. If the universe is globally hyperbolic, then all of these issues can be resolved without invoking unseen dark energy or matter. Additionally gravity may be only an apparent force and the time dilation, length contraction, relativistic mass transform would not be the Lorentz transform, but would be

γ=1+ln(cosh(v/c)±sinh(v/c))

Just because something is true locally, it is not necessarily so that it is true globally.

 

 

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[1] Mabkhout, Salah A. "The infinite distance horizon and the hyperbolic inflation in the hyperbolic universe." Phys. Essays 25.1 (2012): 112.

Marginal Tax Rate vs Effective Tax Rate

 

Slow Hand

You want a man with a slow hand
You want a lover with an easy touch
You want somebody who will spend some time
Not come and go in a heated rush

Slow and steady wins the race.

I got especially upset when I heard one of my favorite podcasts, Planet Money Indicator, repeat the falsehood that the tax rate once was as high as 91%. I know that I took macroeconomics more than 50 years ago, but it was math then and it should still be math. Saying the tax rate is 91% is confusing speed with acceleration. The marginal tax rate, second derivative or the rate of change of change for the math geeks in the crowd, might be 91%, but the effective tax rate, the first derivative, the rate of change, was no where near this. Talking Barbie might be right, Math is Hard, but just because it is hard doesn’t mean that it is not true.

If the median, what a statistician would say is normal, tax rate is 20%, then for any normal tax rate, the highest tax rate should be about twice that amount and the lowest tax rate should be about half that amount. If the median tax rate is 20%, then the highest effective tax rate should be 40% and the lowest effective tax rate should be 0%. But the highest marginal tax rate is not constrained to be 40%. It can be as high as 100% if the size of the brackets (increments) are equal and infinite. Math is hard, and it is useful to approximate a complex function such as a tax rate by a series of increments, brackets. The increments should be as small as possible for the approximation to be useful, but they still must be equal.

If the number of brackets is infinite, for a progressive tax rate, which has a first derivative of x% then the second derivative of the highest tax bracket can be 100%. The smaller the number of brackets, the lower that the second derivative appears to be.

Lowering of the marginal tax rate, should have been accompanied by dividing the intervals to highest income bracket into equal amounts.  If the size of the brackets are not equal,  the math is such that the approximation will not work.

I don’t want to know the highest marginal tax rate. I want equal tax brackets. The highest effective tax rate should be twice the median tax rate. The size of the brackets should be equal. When Maverick said in Top Gun, “I feel the need for speed,” he did not mean “I feel the need for acceleration.” The human body does not have a constraint on its maximum speed, but it does have a constraint on its maximum acceleration, G- force. Don’t confuse speed with acceleration… you want a slow hand.

Independent Legislative Authority

 

Sitting on Top of the World

"Glory Hallelujah," I just phoned the Parson
"Hey, Par get ready to call"
Just like Humpty Dumpty, I'm going to fall
I'm sitting on top of the world
Just rolling along, Just rolling along

Are you like Humpty Dumpty?

In the book Through the Looking Glass, when Alice meets Humpty Dumpty she has the following exchange:

'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it means just what I choose it to mean — neither more nor less.'

'The question is,' said Alice, 'whether you can make words mean so many different things.'

'The question is,' said Humpty Dumpty, 'which is to be master — that's all.'

The US Constitution reads ” Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector.”

The independent legislative authority theory hinges on what is meant by legislature.  If is narrowly defined as the body that makes laws in states and excludes the courts, the governor, the constitution of the state, the voters of the state, etc. then the position of independent legislative authority is correct. If the term is meant to include all of the legal bodies of the government of a state, then the independent legislative authority is wrong. And unfortunately this Supreme Court will try this case and three sitting justices — Clar­ence Thomas, Samuel Alito, and Neil Gorsuch —have already endorsed a narrow reading  and the full court gets to decide on the meaning of the word legislature. And whether by that single word whether they are masters, like Humpty Dumpty,  instead of our servants.

Local is Not Global

 

I’m Old Fashioned

I am not such a clever one
About the latest fads
I admit I was never one
Adored by local lads
Not that I ever try to be a saint
I'm the type that they classify as quaint

Maybe being adored by locals is not all it’s cracked up to be?

A phrase in the environmental movement is to “Think global, Act local.”  It acknowledges that our local actions  have a global impact. This has been true throughout history.  Recognizing that there is a difference between local and global has changed our way of thinking.

The Earth is flat.

It appears to be flat, Euclidean, to a local observer. The problem is that a local observer is very, very small compared to the size of the Earth. There was evidence that was inconsistent with a flat Earth. The shadow cast by a stick at noon became longer the father north you travelled. The mast of ships appeared over the horizon before the whole ship could be seen. The ocean did not fall off the edge of a flat earth, etc. These could all be addressed if it were recognized that the Earth was a sphere,  not flat. When you do large scale navigation you use non-Euclidean spherical  geometry to find the Great Circle Distance based on the surface as a sphere, where the sides of a triangle are very large compared to the radius of the Erath and the angles of a triangle do NOT sum to 180 degrees.

The Sun revolves around the earth.

To a local observer on Earth, it appears that the Moon and the Sun revolve around the Earth. But this geocentric model causes all sorts of problems requiring the “retrograde” movement of planets. All of these “retrograde” movements disappear if it is recognized that the Earth actually resolves around the Sun. Deep space voyages and astronomy rather than astrology are made possible by recognizing that we actually have a heliocentric earth.

My frame of refence is the correct and only frame of reference.

Is the local observer's the only valid frame of refence? Einstein showed that ALL frames of reference are possible and valid. Tying to reconcile an absolute, like the speed of light, with only one frame of reference resulted in trying to accommodate the aether though which light travelled. Einstein’s Theory of Relativity resolved this and many other seeming paradoxes such as the precession of Mercury, black holes, the apparent movement of stars during eclipses, and resulted in among things including atomic energy.

The universe is flat, Euclidean.

A local observer may assume that because his frame of refence is flat, Euclidean, that the universe is flat. However if the universe is Euclidean, then it requires dark matter and dark energy to resolve: cosmic inflation, the rotation of Galaxies, discontinuities, etc. that are either paradoxes or complications to try and resolve those paradoxes. If the universe is hyperbolic, has negative curvature as opposed to positive, spherical, curvature, these paradoxes may be resolved without the need for complications. It may also suggest that gravity is an apparent force, unlike the other three intrinsic forces. The movement that is observed in a Euclidean frame of refence might appear to be the force of gravity but it might actually be just particles minimizing their energy, which is equivalent to maximizing their entropy.

If you think globally and act locally, the Earth is spherical,  the sun resolves around the earth, and there is more than one frame of reference to an absolute. What else might recognizing that the universe might be locally flat and globally hyperbolic entail.

One Small Step

 

Blame It On The Bossa Nova.

Blame it on the bossa nova with its magic spell
Blame it on the bossa nova that he did so well
Oh, it all began with just one little dance
But soon it ended up a big romance
Blame it on the bossa nova
The dance of love

One small thing can be to blame for a whole chain of events.

If during my Freshman year at Brown University, the Kent State shootings had not occurred and the college had shut not have shut down, then I might have changed my major from Engineering to Quantum Physics.

If I had not taken an Electrical Engineering course,  then when I saw a plot of reliability, I would not have said that it looks like the response of  Resistor-Capacity, RC, circuit..

If the Atlanta Regional Commission, ARC, had not asked  Cambridge Systematics, the firm at which I work, to comment on its Volume Delay Functions, VDF, that varied by Time of Day, TOD, and the response had not been that vehicles should have the same VDF all day, then it might not have occurred to me that maybe the VDF was the reliability of the drivers, which could vary by TOD,  AND the time of the vehicles. Then both parties could be correct.

If COVID-19  had not confined me to home, then I might not have time to research an alternative to the speed-volume curve for vehicles that could be combined with reliability .

If my engineering courses had not included Fluid Mechanics, then it might not have occurred to me that the speed-volume vehicle flow in traffic looked like fluid flow in pipes.

If that proposal had not been reviewed and rejected, and one of the rejections said snidely that people in vehicles don’t behave like water, then I might not have been gotten mad enough to try and prove them wrong.

If I had not gotten mad, then I might not have started looking at other instances that looked like flow in a pipe, with separate stable and chaotic domains, which led me to cosmology and the proposal that space, the universe, was hyperbolic.

If I had not tried to understand  the articles on a hyperbolic universe, then I would not have learned about Minkowski Space. I had learned about  Lorentz Transforms before, in Quantum Mechanics, but it had been to flat Euclidean Space not hyperbolic non-Euclidean Space.

If I had not watched YouTube videos on Minkowski space, then I would not have leaned the concepts of invariant spacetime intervals, or that the  transition from Minkowski to Euclidean space was a Lorentz Transform, and I would not have proposed an alternative transform from Minkowski Space to hyperbolic space.

If I had not watched those YouTube videos and learned that the worldline of an event in Minkowski space which was the transform of a trajectory in the universe,  then I would not have thought about the quantum entanglement dilemma. While events with the same trajectory are the same in space, events with the same worldline are the same in Minkowski space. Thus spooky action at a distance, quantum entanglement when particles are separated by a distance, might be nothing more than the same particles having the same worldline, even when they are separated by a spatial distance.

If I had not had a career in traffic engineering, then I would not have known about Entropy Maximization and proposed that Entropy Maximization in hyperbolic space might be responsible for what appeared to be gravity in a flat Euclidean  space.

So if I am correct that... space is hyperbolic, the transform between Minkowski and hyperbolic space is NOT the Lorentz Transform, and gravity is apparent not fundamental, …blame it on the shootings at Kent State.