(c) 2013 Barry Davies
Hypothesis: The fundamental mathematical model for our economies should not contain an exponential divergence.
Abstract: The mathematical properties of rolled over growth, (1+p/100)^n, are examined by tabulating the growth so that each row corresponds to one compounding, and each column entry contains growth from one compounding on elements from the previous column in previous compoundings. The growth is then partitioned into three groups: "Column1", "Column2", and "All Other Columns". The former two are subtracted from the latter and this difference is ploted as a function of time where the unit of time is the compounding period. The zeros, local minima, and inflection points in these plots seem to be points of interest for avoiding the exponential divergence which all rolled over growth ultimately produces. In particlar, the inflection points seem to occur very uniformly near the first doubling for growth rates from 1% up to 40%, (27 Feb 2012: 100%), (28 Feb 2012: 200%). Please have an intuitive look at the
and then click here to go to the results.
Conclusion: Therefore, the first doubling seems to be a very safe and optimal point at which to reverse the expansion. I imagine that there could be millions or billions of different economic elements - perhaps even one for each person on the planet plus many more for other entities - oscillating with different growth rates and compounding periods. Consequently, a steady state might result - without the pain of recession. Now please, see the exponential divergence here.
Results 1 - Inflection Points
It seems remarkable to me that the first doubling, d1, and the inflection point, z2, occur so close to one another uniformly as shown in the last column of the table, MatrixForm[presData], immediately below. It seems that, for growth rates from 1% to 200%, the inflection point occurs just over one compounding beyond the first doubling. Therefore, the first doubling seems to be a very safe and optimal point at which to reverse the expansion in a healthy lung analogy. We would reverse the expansion even before the curve changes concavity! Again, the headings z2, z1, z0 stand for the number of compoundings required to reach the zeros respectively of the functions ddfn, dfn, fn. See the definitions below at the cell tag 3functionDefinitions.
Please Note (RED TEXT): If you got here after scrolling after clicking on the link in the abstract, then this is the place to stop reading and click the back button.
cell tag: 3functionDefinitions
If you got here by clicking on the "3functionDefinitions" link, then scroll please to the bottom of the page and then click on the back button twice to read the conclusion and go on to see the exponential divergence.