Exponential Curve 1
Arithmetic, Population and Energy, Part 1
For the love of the human
race.
Friday, November 14, 2014
Our Thesis
We are greatly indebted to Dr. Albert Allen Bartlett (1923-2013),
former emeritus professor of physics at the University of Colorado, Boulder.[1] These are Dr. Bartlett’s ideas, we are merely
reporting them. We have performed a
lengthy analysis of Dr. Bartlett’s “arithmetic” elsewhere.[2]
One cannot investigate either energy policy or energy theory
without a thorough understanding of the exponential curve.[3]
Arithmetic, Population and Energy, Part 1
The basic exponential equation is, y = a * bt: Where, a, is the initial value
of y or y0, the intercept where y
crosses the y axis, the initial value when the horizontal, or x
value is zero; b, is the exponential constant; and, t,
is the time, distance, or other factor plotted on the horizontal, or x
axis. If b is expressed as
1 plus some rate of increase, r, b = 1 + r,
then we can apply the basic exponential equation to the study of growth, where
growth is r.
“The
greatest shortcoming of the human race is our inability to understand the
Exponential Function.”
In exploring this line of thought Dr. Bartlett first
emphasizes the incredible size reached by the exponential curve in a very short
distance or time. A simple graph of the
exponential curve reveals that it approaches infinity without hardly trying. Since all exponential curves approach
infinity; in real life, growth requires infinite energy. The reader should realize that this is
impossible, so it is impossible to sustain growth for any length of time.
Continuous growth is impossible for the same reason that
continuous acceleration of vehicles is impossible. Either a limit is reached where the engine no
longer has enough power to cause acceleration, or the vehicle reaches a
velocity where it is no longer stable, runs out of control, and crashes.
Dr. Bartlett dramatizes this great size by reminding us of
the infamous chess board bet. A chess
board is divided into 8 x 8, or 64 squares.
A single insignificant object (usually a grain of rice or a penny) is
placed on the first square. That
quantity is doubled with each succeeding square. What does that add up to? It cannot be a very large number can it? I’ll be that it’s not more than a few
thousand; certainly not more than one million.
So we take the bet which requires that we provide the insignificant
objects, and loose them all if it exceeds our maximum estimate of one million. You can do the math, but the last square has
9.223 E+18 items plus change. That’s
9.223 E+12 million items; or 9.223 E+9 billion items; or 9.223 E+6 trillion
items. The number of items on all the
squares is twice that size. In terms of
dollars, that’s enough money to bankrupt the United States; maybe enough to
bankrupt all the nations on earth.
Exponential curves result in seriously big numbers, in very
few steps.
Doubling time, or ordinary steady growth.
Since we doubled the quantity on each square, the question
naturally arises, is there a relationship between growth and time? If we have a percentage of growth in mind can
we calculate the length of time it will take to double? Yes, it’s easy.
Since, y (t) = a * bt, we want to
find out how long it will take for to
reach 2 * a. Applying a
little simple algebra we get 2 = bt. Then with the help of logarithms we find that
ln (2) = t * ln (b).
So, t (doubling time) ≡ ln (2) /
ln (b). The ln (2)
is 0.693, which is real close to 0.7.
The ln (b) = ln (1+r), which is real close to r. So all we need to do to get the approximate time
is divide 0.7 by r, or if r is in percent we divide 70
by r. This handy dandy
rule of thumb is called the Rule of Seventy.
So, if we put a dollar in the bank at 1% interest per year,
it will be worth two dollars in seventy years (except for the fact that
inflation and taxes will steal all of it.)
The population of the United States is about 325 million people and
growing at a little more than 1%. In
seventy years the population of the United States will be roughly 650 million
people unless something changes. If you
need to know how accurate this Rule of Seventy is, just cook up a spreadsheet
table and compare the actual ln (2) / ln (1+r), using the decimal
for r here, with 0.7 / r (the decimal) or 70 / r
(%).
Size Matters
The size is likewise easy to calculate, not in the head, but
using a scientific pocket calculator, or a computer spread sheet.
The number of grains on any square is y = 1 * 2(n-1). The number of grains on the sixty-fourth square
is y = 1 * 263.
The sum of grains on the board at any square is y = 1
* 2n - 1. The number
of grains on the board at the sixty-fourth square is y = 1 * 264
- 1. The number of grains on any
square is 1 more than the total amount on all the previous squares combined.
When a percentage of growth is being considered the doubling
time = 70 / r. The number
of times it doubles = r.
The amount of the object being doubled in 70 years, approximately one
human lifetime is 2r.
If growth is at 5%, the object will double in amount every 14 years, and
be 32 times the original amount after 70 years.
The Standard Social Model
The exponential growth model is, was, and continues to be
the dominant social model chosen by our leadership. Last year’s federal budget discussions[4] were arguing the merits of
a roughly 2.5% growth plan and a 4.5% growth plan. Neither side is willing to discuss a zero
or negative growth plan.
We
should have paid closer attention to President Carter’s speech on energy in
1977. That was the last honest
presidential appraisal of the energy crisis: we have been living in a state of
denial ever since. Carter said “in each
of these decades (the 1950’s and 1960’s) more oil was consumed than in all of
mankind’s previous history.” That is a
profound observation.
Conclusion
Steady growth is in fact an aggressive, uncontrollable,
vicious monstrosity that eventually destroys the culture in which it is allowed
to exist. Growth must be
restrained. If we are to take this
“arithmetic” seriously, as we must; we must convince; nay, we must compel, we
must demand that our leaders develop steady negative growth plans. A 5% growth plan will double our
consumption in fourteen years. A
5% reduction plan will halve our consumption in fourteen years. The growth model must be put to death, before
it puts us to death. Nobody can live
with the 64th square.
The only growths that are tolerable to civilization are the
growth of conservation and the growth of food.
If we work at these we may, someday, once again have clean water, plenty
of oxygen, and rich soil.
[1] http://en.wikipedia.org/wiki/Albert_Allen_Bartlett
[2] http://swantec-ep.blogspot.com/2014/11/energy-policy-analysis-1-ra.html
[3] http://www.albartlett.org/presentations/arithmetic_population_energy_video1.html
[4]
Fall 2013
[5] If you have been blessed or helped by any of these meditations,
please repost, share, or use any of them as you wish. No rights are reserved. They are designed and intended for your free
participation. They were freely
received, and are freely given. No other
permission is required for their use.
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