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%). If you are one who is short of time, then please have an intuitive look at the

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and then click here to go directly to the result and do not click on any other links before you reach the RED TEXT.

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 and compatible with life on earth.

Grow 100 at 1%  per year

Introduction: This page presents the original calculation in full. I have tried to make the presentation as readable as possible without altering the calculation - which would decrease the reliability of the results.

The blocks of information that are called cells in my version of Mathematica often show both input and output, with blue text as label, and the cell's sequence number appears in square brackets. Although the input and output are each a cell, I refer to them together as a unit. I do this even when the input contains several statements - each ending in a semi-colon - which makes the sequence number of the output different from the sequence number of the input. Initializations are placed at the bottom of the page because they generally interfere with the presentation. Please, if you did not click on the "Initializations" link, do so now and scroll on to see the message in red. Note, in the next cell, the initialized value of the symbol  nYears:

nYears = 8.  But this is a text cell, and the actual assignment of the value was made in the initialization section at the bottom of the page. The  handDrawnChartReplica  is described below the next cell.

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As described in the abstract, the growth is tabulated 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 years. The very small numbers in the last few columns seem utterly negligible, but we will see that the exponential divergence occurs beyond the second column. Furthermore, I would like to point out that, if we use a powerful financial calculator to do future time value of money calculations, the calculator uses a formula which is mathematically equivalent to the process which the table illustrates. The calculator gives a very accurate approximation to the exact result which the chart represents. The powers of ten are only to be multiplied by the chart elements which are below the row in which the powers of ten appear. For clarity, the original small numbers are retained in the rows which are above the row in which the powers of ten appear.

At this point, readers short on time may wish to skip ahead to the "Results 1 - Inflection Points" section below. Scroll down quickly to see the plots in passing or click here.

With  bT  standing for binomial table in the next cell, the symbol  bT[nYears]  is defined at the beginning of the initialization section.

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From previous experience and the expression  (1+p/100)^n,  I recognized the binomial coefficients in my hand drawn chart. In the pyramid of numbers, the first two columns of the hand drawn chart become the left hand diagonal line of  1's  and the left hand diagonal line of consecutive integers. The column numbers are in the right diagonal line of consecutive integers, but these numbers correspond to the columns in such a way that the right diagonal sequence stops at one less than the number of columns in the handDrawnChartReplica. The powers of ten have been completely removed from the pyramid but they are essential in the hand drawn chart. And, for clarity, it is essential that the powers of ten appear in the middle row of the hand drawn chart.

Also please note that what is considered here is the growth of  100  as a pure number and not as a currency amount or as any other particular quantity that may grow in a rolled over way - rolled over meaning of course that the growth from any compounding becomes part of the starting amount for the next compounding. And the compounding period need not be a year, but can be any interval. Also, in the plots that appear below,  p  and  n  will be considered to be continuous variables.

In the next cell, the modules  sum1Row  and  sum1RowAfterCol2,  which were defined in the initialization section, compute the indicated amounts and the first few numbers in each list can be verified by inspection of the hand drawn chart replica, which is reproduced after the next cell. The module  sum1RowAfterCol2  returns zero if the row has less than three elements. See the note in the initialization section at cell tag "sum1RowAfterCol2".

cell tag: growth each year and after column 2

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cell tag: avoid the easy to make mistake

Note well that the third element of the first line of the matrix immediately above is the sum of the third row of the hand drawn chart replica, while the third element of the second line is the last element in the third row of the chart. The first two elements in the first line are the sums respectively of the first and second rows of the chart, and these numbers do not pertain to the third row of the chart. So we should avoid the easy to make mistake of thinking that the third element in the second line is related to the first two elements in the first line. I find this mistake very easy to make anytime I have not looked at this work for a while!

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The next cell checks the result from the list above (cell tagged "growth each year and after column 2") obtained with the module  sum1row  by comparing it to the result obtained from the general formula for the final amount. Please note again that what is considered here is the rolled over growth of  100  as a pure number. Consequently, I use the terms x0, growth, and final amount instead of initial deposit, interest, and final balance. As noted at the beginning of the initialization section, symbols such as  x0Is[100]  and  percent[1]  are just labelled representations of the initial amount and growth rate values  100  and  1%  respectively. Mathematica's built in function symbol  Tr  is short for trace, and it computes here the sum of a simple one dimensional list of numbers.

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Also checked here is the result obtained from the initialization section defined module,  getFinalAmountFromBT,  where  BT  stands for the binomial table and where the module  sum1Row  is used. We can easily check that the number obtained is the same as shown in the previous cell.

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Now, as another check, the results for 20 compoundings are computed.

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And provision has been made to vary the growth rate, because it will be desired to look at growth rates higher than 1%. However, unless a module has  x0  as an explicit argument, the initial amount is assumed to be 100.

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It should be noted again (see the cell tag "avoid the easy to make mistake" above) that the growth after column two presented immediately above can't be readily deduced from the preceding line which shows output from the symbol growthForEachYear, which is defined right here and is not a module. The first two elements of the first list are for the first and second rows respectively, and the first two elements of the third row are not shown.

The first line above is reproduced in the next cell and checked against the general expression for the growth table. The growth table was derived in earlier work and is here saved in the symbol growthTable[8] - using the function  growthTableForYearN,  which is defined in the initialization section. Note that output is generally suppressed by the semicolon at the end of each line, unless the line is a  Print  statement as in the second and third lines below - where  growthTable[8][[3,3]]  extracts the third element of the third row, and  growthForEachYear[nYears,percent[2]]  was calculated in the previous cell. Also: The "slash period"  /.  is Mathematica's  replacement operator with variables and values connected by the arrows in the list which follows the replacement operator.  It will be shown below that the first nonzero term in the second line of the previous cell is equal to the last term in the third row of  growthTable[8].

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The general expression for the last term of the third row above is printed in the previous cell in the line before the growth number list. In the general expression it was necessary to replace instances of the number 100  by  c  and  y  so that the binomial coefficients remain evident. Note that  y occurs in every term whereas  c  does not occur in the first column which is the left edge of the pyramid. If,  x0  and  y  are both equal to  100  as in the present problem, then they would cancel leaving only  p's  in the first column of the general chart. In this first column, the power of ten in the "factor" is zero, which explains why  c  does not appear in the general table in the first column.

To get a general expression for the growth after column two, it is necessary to set to zero the first two diagonal lines at the left side of the pyramid of numbers. This is done below with a pitfall noted.

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The pitfall is shown in the output of the cell immediately above: It is the fact that the element  {1,1}  of  tempGT,  i.e.  tempGT[[1,1]],  is not the same as the elements  {i,1}  when  i  is greater than one. But it does not matter in what follows because these elements are replaced by zero in the next cell. Note that  ReplacePart  is a built in Mathematica function, and  tempGT=ReplacePart[tempGT,0,{i,1}]  sets  tempGT  equal to what we get if we replace the element at position  {i,1}  by    0.

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However, now the pitfall is that an error message will result if we try to do the replacements starting at the first element of  tempGT  with the assumption that the first element is at position  {1,1}.  But, since the first element is already zero, we just do the replacements starting from the second element - which is why the iteration list in the  Do  structure is  {i,2,8}.

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Note finally that sums and lists behave the same when we reference their parts. Below is shown the fourth row of the output immediately above. The whole row is at position  {4},  while the two terms in the sum are at positions  {4,1}  and  {4,2}  respectively. The built in  Extract  function uses the list form of the position, while the curly braces are omitted in direct references with the double square brackets.

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Now the specific and general expressions yield the same results as shown next.

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The modules shown below calculate the sum of the first two columns of the growth table where  x0  is assumed to be  100,  but where the growth rate is variable.  Looking at  growthTable[8],  we see that the first column is just  p  times the number of years, whereas the second column is given by the following cell:

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Since the calculation is exact, it is not necessary to show the numbers since their difference is exactly zero.

Many Compoundings

Below, the calculation is performed for  100,000  compoundings, and this makes it clear that a divergence occurs beyond the 2nd column.

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It is shown in the appendices  _1_  and  _2_  just before the initialization section, that it is not feasible for large numbers of compoundings to compute the growth using the binomial table. Therefore, I calculate below the sum of all columns beyond the second by subtracting the first two column sums from the general formula minus the initail amount.

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Next, results are presented for higher growth rates. It seems clear that there is what might be called an explosion in the "AllOtherColumns" column. I am tempted to call it a detonation when the number of compoundings is such that "AllOtherColumns" and the first two are equal.

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Plots and Tables

The purpose of the next module is to look more closely at various growth rates to see the number of compoundings required for "AllOtherColumns" to catch up to the sum of the first two columns.

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cell tag: 3functionDescriptions

The function  fn  below is  (
     (the sum of all columns beyond the second)
  minus  (the sum of the first two columns)
)/x0
,
where  x0  is the initial amount. The function  dfn  is the derivative with respect to  n  of  fn, and this is of interest for calculating the local minima in the plots below. Also of interest will be the second derivative,  ddfn,  for calculating the inflection point in each curve. These three functions are defined under "3functionDefinitions" in the initialization section. The function  fn  was derived twice - once on paper and separately with Mathematica. The two derivations are checked against each other in the next cell. The differentiations were performed by Mathematica, and  Log  is the natural logarithm function. These three functions give us three points of interest when considering mathematical criteria for making our economies breathe like healthy lungs. It seems to me from what follows that the inflection points should determine when expansion reverses.

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If you got here by clicking on the "3functionDescriptions" link, then now is the time to click on the back button.

We can visually inspect the tables below to see where the "AllOtherColumns" column catches up to the sum of the first two columns. Since the point at whith AllOtherColumns exceed the sum of the first two seems to be the onset of what I call the explosion, I refer to this event in the output below as the detonation: True means the event has occured, and False means that it has not occured.

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There seems to be nothing in any of the above material which prevents us from changing the compounding period from years to months. If the following module is interpreted in this way, then the horizontal unit in the plot is a month rather than a year. And ...

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... therefore we have a curve with a local minimum at about six months - half a year. If we have a reformed economic element functioning like a pair of healthy lungs, then this half-year point might be the time at which "inhalation" stops and a half-year period of "exhalation" would begin, and it would be of such a nature that, at the end of the year, the economic element would be in the same state that it was in at the beginning of the year. Perhaps it would be six months of work followed by six months of relaxation.

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First Doubling

A function which returns the number of compoundings to the first doubling is defined at the cell tag "firstDoublingInitialized" near the end of the initialization section. The derivation is shown in the next cell.  Exp is the exponential function, and  Log  is the natural logarithm function.

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Please Note: If you got here after scrolling after clicking on the "first doubling" link, then now is the time to click on the back button.

Zeros

To begin, we are unable analytically to solve for the number of compoundings at which the zeros occur for the function  fn. In what follows,  Log  is the natural logarithm function.

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Local Minima

Below is the calculation for the zeros of the first derivative. An analytical expression was obtained for the zeros of the first derivative, but the solution we need was not found.

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Inflection Points

Below is the calculation for the zeros of the second derivative. As we go to higher derivatives the polynomial part of the function falls away and the resulting functions are easier to differentiate.

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Below are the zeros we need, and they were found manually in another notebook. The successful function calls were pasted into this notebook. See the definitions near the end of the initialization section. The column headings  z0, z1, z2  respectively stand for  zero0[p],  zero1[p],  zero2[p],  where  p  is the growth rate in percentage points, are the "number of compoundings" to the zeros respectively of the functions  fn,  dfn,  ddfn.

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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 descriptions above at the cell tag "3functionDescriptions and see the definitions below at the cell tag 3functionDefinitions").

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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.

And, furthurmore,  z2,  keeps us near the point at which the exponential function is equal to the number  E.  See the  firstDoubleTextString  above and see the definition of the list argsE below. The  example  shows the exponential expression for the growth factor,  x/x0, with the argument of the exponential function. See the first three numbers in the first row of the table. The table above, combined with the one below, make me think that the inflection points should be used to define the point at which expansion reverses.

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Results 2 - Two Column Truncation

It seems that truncating the growth to the first two columns radically increases sustainability, as shown by the numbers below for the ultra long times.

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Results 3 - Equivalent Untruncated Rates

Here are the calculations for the equivalent untruncated growth rates for 7% truncated to the first two columns for times from  100  to  100,000  years.  The function  eqUTGR  is derived and defined in the initialization section.

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And here is the calculation for the equivalent untruncated growth rates for  20%  truncated over the decades from  10  to  100  years.

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This last table makes me think that people with today's short term outlook would not even notice the truncation.

Appendix - Other Columns

The material below was not needed, but it is interesting and seems to provide a good check of the binomial table calculations presented above.

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Column 3

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In the  Module  below, the powers of  p  and  10  can be seen to be correct for column three in the general 'pyramid' for the growth chart which appears near the top of this page in the expression  growthTable[8]  and is reproduced just before the section on column four below.  Looking at column three in the  handDrawnChartReplica, we see below that the function  c[3,n]  gives the correct sums.

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Column 4

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Column 5

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General Column: m

We can see from the patterns above that the general column is given by the recursive definition shown and tested immediately below.

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The growth with recursion is defined and tested below.

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The patterns above enabled me to guess and test the following:

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The growth is calculated as the sum of the columns for the case of  1000  compoundings using the product function which was derived above.

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The calculation immediately above took less than ten seconds on my Dell Vostro mini-laptop - which has two processors at 1.6GHz and 1GByte of RAM.

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The number shown immediately above was obtained in the expression  tenTo100000  in the first part of this page above.

Appendix - Large Binomial Tables 1

The cells below lack the blue input and output number labels in these appendices on large binomial tables because the execution times were long. Therefore, once done, these cells were made not evaluatable.

Shown here: (1) The time required to read from the hard drive the binomial table in the case of  1000  compoundings. The file requires more than  74,000 KBytes  of storage memory. (2) The time required to calculate the growth from the binomial table. (3) Verification of the result. These results make it very clear that it is not feasible to use the binomial table for calculations involving very large numbers of compoundings.

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Appendix - Large Binomial Tables 2

Here the calculation of the binomial table for the case of  1000  compoundings is divided into sections to show the times required. The next two cells were used as a guide to write the subsequent Module.

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The triple equals sign tests that the objects on its two sides are identical. If they are, then  True  is returned; otherwise  False  is returned.

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Initializations

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Before beginning, annoying spelling warnings are turned off, and the following symbols are defined so that the calls to Modules will be more readable. These calls can seem quite cryptic when the arguments are just numbers. Also: The  Interrupt  above prevents the initialization cells from being evaluated twice. Note that, whereas computer programmers talk about running or executing programs, in my version of Mathematica the term evaluation is used.

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Initialize: Grow 100 at 1%  per year

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Please Note: If you got here after scrolling after clicking on the "Initializations" link, then now is the time to click on the back button. But if you are reading from the beginning, please continue by scrolling down, and ignore text in red if you have not just clicked on a link.

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The next two modules are ONLY for the case where the initial amount is 100. However, the growth rate  p  has been included because it will be desired to look at growth rates other than 1%.

cell tag: "sum1RowAfterCol2"

Note in module  sum1RowAfterCol2  that the  Do  structure stops executing if the iteration list,  {i,3,Length[row]}  is not of the form  {i, imin, imax}, presumably with  imin  less than  imax  because the default index step is positive one. Consequently, if there are less then three elements in the row - as is the case in the first two rows, then the value of  sum  is not changed and zero is returned.

[Graphics:Images/index_gr_189.gif]
[Graphics:Images/index_gr_190.gif]

From earlier work, here is what seems to be the general expression for the growth table. To show the binomial numbers, it was necessary to replace certain instances of the number  100  with  y  and  c. The initial amount is  x0.

[Graphics:Images/index_gr_191.gif]

Initialize: Plots

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cell tag: 3functionDefinitions

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If you got here by clicking on the "3functionDefinitions" link, then now is the time to click on the back button.

Initialize: First Doubling

[Graphics:Images/index_gr_194.gif]

cell tag: firstDoublingInitialized

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Scroll up a few clicks ONLY to see the definition of the symbol  iList. I say "ONLY" because you should not want to read the definition of the symbol  firstDoubleTextString  because that should be read in the main body of the page. All the initializations have been included for completeness, but the "text string" initializations should not be read in the initialization section.

Please Note: If you got here after clicking on the "firstDoublingInitialized" link, then now is the time to click on the back button.

Initialize: Zeros, Local Minima, & Inflection Points

The zeros below were found using Mathematica's  FindRoot  function, where the initial values were estimated from the plots. Once found, the values are tested to verify that they are good approximations to the required zeros.

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[Graphics:Images/index_gr_197.gif]

Initialize: Results 1 - Inflection Points

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[Graphics:Images/index_gr_199.gif]
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Initialize: Results 2 - Two Column Truncation

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Initialize: Results 3 - Equivalent Untruncated Rates

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Initialize: Appendix - Other Columns

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[Graphics:Images/index_gr_212.gif]
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