My Example Growth Page (2012)

Following on from 30 March

I was watching CNBC from San Francisco USA, Friday 30 March 2012: Apple's share price had grown 1600% since Sony's CEO started with Sony. Internet: That was in June 2005. Call it seven (no eight) years. That is from 2005 through 2012 inclusive.

In the special case chart on the left below, the initial amount is   100,   and the growth rate is   1%   per year. In the general chart on the right, the initial amount is   x0,   the growth rate is   p   percentage points, the compounding period,   CP,   can be any length of time, and   c = y = 100   are used to keep the binomial numbers conspicuous. As we have seen in the special case, the columns beyond the second start small but eventually dominate. By subtracting the sum of the first two columns from all the rest, we get a quantity which (when plotted) shows how all the rest catch up to the first two.

Special Case: x0=100, p=1, with annual compounding
General Case: x0, p, and the compounding period are arbitrary
Each power of ten is multiplied by its subsequent column entries.
The constants   c = y = 100   keep the binomial numbers conspicuous.

The Exponential Divergence:

Remember that the sum of all the columns is the growth,
x0(1+p/100)^n - x0,
where
x0  is the initial amount,  n  is the number compoundings, and  p  is the growth rate in percentage points.

The original growth was taken to be a  16  fold increase in  8  years: 2005 to 2012 inclusive. So one year's increase would be the  8th  root of  16  which is the square root of two, which is approximately equal to  1.414.   Then, to calculate the growth factor in each case, raise  1.414  to a power equal to the number of years. The zero of time on the horizontal axis in these plots is roughly the beginning of 2005.

Below (see the derivation here) are plots of

(sumOfColumnsGreaterThan2 minus sumOfFirstTwoColumns)/theInitialAmount
where the growth rate is 41.4% compounded annually for 80 years:


---------- ---------- ----------

(sumOfColumnsGreaterThan2 minus sumOfFirstTwoColumns)/theInitialAmount
where the growth rate is 41.4% compounded annually for 60 years:


---------- ---------- ----------

(sumOfColumnsGreaterThan2 minus sumOfFirstTwoColumns)/theInitialAmount
where the growth rate is 41.4% compounded annually for 40 years:


---------- ---------- ----------

(sumOfColumnsGreaterThan2 minus sumOfFirstTwoColumns)/theInitialAmount
where the growth rate is 41.4% compounded annually for 20 years:


---------- ---------- ----------

(sumOfColumnsGreaterThan2 minus sumOfFirstTwoColumns)/theInitialAmount
where the growth rate is 41.4% compounded annually for 10 years:


In all the plots above except for the last, the first part of the curve corresponding to the first few years is negligible compared with the last years. Only in the last curve are the first two columns of growth significant. In all the other cases it seems to me that the exponential divergence has become dominant. It seems to me that this exponential divergence should be removed from the fundamental mathematical model for our economies. It seems that a sinusoidal model would be better for the long term evolution of our species. Given that the divergence disrupts the growth, it seems that the sinusoidal model would also be very beneficial for our immediate prosperity.

Conclusion: Although I have suggested that the inflection point seems optimal for the point at which expansion could be reversed, the entire negative region of the plot seems also good - as does the beginning of the positive portion of the plots. The growth of Apple is clearly possible since it happened. What will get us into trouble is trying to extend growth into the portion of the curve which might be called the exponential divergence or explosion. When we try to turn the one episode into ten, then we fail. The expansion should be reversed before failure, and we might describe the oscillation as work-relax, work-relax, work-relax, ad infinitum. Believers in perpetual growth may rightly point out that at lower growth rates (like 7 percent?) we may get by for enough decades for one lifetime. But with a sinusoidal model, it seems we could keep the high growth rates as well as the low ones because the growth would be cyclic - like a spring being loaded and unloaded, or like healthy lungs breathing. Greater prosperity could result - with the pain removed from the economic contraction. Now please see the exponential divergence a little earlier here.

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Appendix:

My original caluclations for this page are below. I wanted to look at the growth in a sequence of cases approaching continuous compounding - which I think is approximated by the daily and shorter compoundings. It is easy to get confused if we think instinctively that we are looking at the growth factor in these plots. Instead, it must be remembered that these plots are about the extent to which the columns beyond the second are dominant. For a small number of compoundings the first two columns dominate.

[Graphics:Images/index_gr_1.gif]
[Graphics:Images/index_gr_2.gif]

The "colon equals", : =, in the next cell indicates definition without evaluation. The underscores on the left indicate that, when evaluation occures, whatever is between the square brackets on the left will be used in place of its corresponding name, from this definition, in the expression on the right.

[Graphics:Images/index_gr_3.gif]
[Graphics:Images/index_gr_4.gif]

The square root of  2  is approximately  1.414  so that the third element, which is the annual compounding rate, is approximately  41.4%.

[Graphics:Images/index_gr_5.gif]

What is plotted on the vertical axes in the plots below is, as a multiple of the initial amount, the excess by which the sum of all the columns beyond the second exceeds the sum of the first two columns.

[Graphics:Images/index_gr_6.gif]

[Graphics:Images/index_gr_7.gif]

[Graphics:Images/index_gr_8.gif]

[Graphics:Images/index_gr_9.gif]

[Graphics:Images/index_gr_10.gif]

[Graphics:Images/index_gr_11.gif]

[Graphics:Images/index_gr_12.gif]

[Graphics:Images/index_gr_13.gif]

[Graphics:Images/index_gr_14.gif]

I call the eight year period considered here "one episode".

[Graphics:Images/index_gr_15.gif]
[Graphics:Images/index_gr_16.gif]
[Graphics:Images/index_gr_17.gif]

Ten "episodes" are then  80  years, and, if we imagine the growth continuing for what amounts to the working life time of today's newborn infants, then the American trillion (German billion) fold growth amount may be problematic for those generations which are subsequent to us.

[Graphics:Images/index_gr_18.gif]
[Graphics:Images/index_gr_19.gif]

I note that the slope angles tend to be smaller for the shorter compounding periods because the horizontal scales are very stretched compared to the vertical scale for the short compounding periods. As the period gets very short, the horizontal scale becomes VERY stretched out.

Initializations

[Graphics:Images/index_gr_20.gif]
[Graphics:Images/index_gr_21.gif]
[Graphics:Images/index_gr_22.gif]
[Graphics:Images/index_gr_23.gif]
[Graphics:Images/index_gr_24.gif]


Converted by Mathematica      March 31, 2013