Please remember that a [short scale !!] trillion is a one with twelve zeros after it. Also: The volume of the known universe, where the unit of volume is the volume of a hydrogen atom, is roughly a one with one hundred and twelve zeros. That's a googol times a trillion.
If we try to grow one economic (or any other kind of) unit at an AVERAGE rate of one percent, or at an AVERAGE rate of ten basis points, or at an AVERAGE rate of 1 basis point, or at an AVERAGE rate of two tenths of a basis point, what will we get? Let the compounding period be one year. For how many years? Consider please ...
I would like to define the economy by the continuous use of tools by homosapiens and its ancestor species. Then, for the sake of argument, I think we can assume that our economy is two million years old. Furthermore, it is right to demand that our economy should be of such a nature that it has a realistic prospect to survive ANOTHER two million years [else we today have less intelligence and less survivability than did our stone age ancestors]. Then ...
Let x0 be the initial amount, let p be the AVERAGE growth rate in percentage points, let f be the compounding period as a fraction of one year, and let n be the number of compounding periods. Then the final amount, x, is given by the next cell. Please note that the 'cells' are numbered and usually, but not always, have both an input and an output part.
If we want the AVERAGE growth rate that will produce a given final amount, xn, then we use the expressions shown in the next three cells.
Now let the initial amount be one. It could be an economic unit. But let us consider the economy in a form that is appropriate for very long periods of time. Out of respect for those who will say that the spiritual part of the economy can grow in an unlimited way, let us look only at that part of the economy which is a combination of matter and energy. This is some proper subset of all the matter and energy in the universe. We can use Einstein's E=mc^2 to express the size of this economy as either all matter or all energy. Then the economic unit could be a unit of mass or energy and "growth" is just the change with time, and the units are not affected by drastic changes that may come about because of technological change over long times. That is, for example, it might be argued that the dollar or any other currency is irrelevant for millions of years, but the units of energy and mass are not subject to this restriction. If we are to continue to live continuously for millions of years, then our use of tools must also continue, and the size of the resulting economy is some number of mass or energy units.
Let the initial amount be one:
And let the number of compounding periods be two million:
And let the compounding period be one year - so that f, which is the compounding period as a fraction of a year, is equal to one.
Then, if we try, for two million years, to maintain growth at an AVERAGE rate of one percent, which is the second element between the square brackets in the input part of the next cell, our one unit will grow into a number which is bigger than a one with eight thousand zeros.
At an AVERAGE rate of ten basis points, we still have more than eight hundred zeros.
Given the size of the universe, the numbers above are obviously unphysically large, and we can conclude that sustained growth at the AVERAGE rate of one percent or ten basis points is utterly impossible. Even one basis point seems obviously to be out of the question:
Sustained growth at an AVERAGE rate of two tenths of a basis point produces ...
To find a realistic AVERAGE growth rate, I arbitrarily assigned a value of one economic unit to the use of stone tools two million years ago. I estimated what we have now to be one thousand trillion economic units. That is a one with fifteen zeros. In the 28-29 March 2009 edition of Australia's national newspaper, the global economy was estimated to have a value of sixty trillion US dollars. That would make the stone tool economic starter unit equal to about six US cents in March 2009 value. Then an AVERAGE growth rate of 0.1727 basis points produces ...
And now let's calculate the AVERAGE growth rate that will produce exactly one thousand trillion units from one unit in two million years:
Here is a twenty significant digit approximation to p4OneQuadrillion:
And this is an AVERAGE growth rate between one tenth and two tenths of a basis point.
Finally, I would like to explain why I have repeatedly used the word average in capital letters: The growth rate from year to year can fluctuate very much. It might be more than ten or even twenty percent in some years. But, after the two million years of such fluctuating growth, we can always calculate the AVERAGE rate that would produce the same growth. The fact that we must end up with something that fits in the universe imposes limits on how big the average can be. Survival for a long time seems to force the average to zero. It seems to me that
Below are 'initialization' cells - which are evaluated first when the notebook is evaluated. They are placed at the end because they are extraneous to the reading of the notebook. I did not delete them from this web page version of the notebook because I wanted the reader to be able to see why the numbering starts at four at the beginning of the presentation. The 'Interrupt' cell prevents the initialization cells from being evaluated twice.