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The Scales Of 70

Introduction

This article makes extensive use of the Rule Of 70 in preference to the Rule Of 72 to explain how population doubling and population halving works in the real world. Even though the Exponentialist focus is on population growth modelling, the same growth model applies to investments and loans.

For a comparison of these two popular competing rules of thumb, see my article The Rule Of 70 and The Rule Of 72 Compared

The Rule Of 70 is nearly always used for just constant positive rates of compound interest (exponential growth). The Scales Of 70 extend this rule of thumb to cover variable rates of compound interest at both positive growth and negative growth rates (Couttsian growth).

In the interests of accuracy, the Rule Of 70 is best restricted to growth rates up to 5%. Usually it is possible to adjust the time period used for measuring doubling and halving periods to ensure that growth rates fall within 5%. That being said, the Rule Of 70 can still be used (but with less accuracy) up to growth rates of 10%. These same restrictions apply to the Scales Of 70.

Crude Examples

Example 1: Crude Population Doubling (rate is positive 1% per annum, population doubles every 70 years):

70 / 1 = 70

Example 2: Crude Population Doubling (rate is positive 2% per annum, population doubles every 35 years):

70 / 2 = 35

In fact, you can also derive the crude population halving period the same way (if the rate is negative then divide the rate into 70 to get the halving period).

Example 3: Crude Population Halving (rate is negative 2% per annum, population halves every 35 years):

70 / -2 = -35

Or, using the absolute value of growth rate:

70 / 2 = 35

Given that sustained negative population growth always results in population halving, perhaps it would be easier to take the absolute value of the growth rate when performing the calculation. This would turn the calculation in Example 3 into the same calculation as Example 2. The difference is that Example 2 calculates the crude population doubling period caused by sustained positive population growth, and Example 3 (using absolute value) calculates the crude population halving period caused by sustained negative population growth.

The Rule Of 70 is thus usually explained through division, but of course it can also be explained through multiplication. Hence, for positive population growth, if you multiply the crude population doubling period by the rate then you get 70. This seems obvious, but here is an example to make it even more obvious:

Example 4: Crude Population Doubling (rate is positive 2% per annum, population doubles every 35 years):

35 * 2 = 70

To make things even simpler, multiplication can always be explained through a series of additions. Thus, if you add 35 two's together it is the same as multiplying 35 by 2. Either way, through multiplication or addition, the result is 70.

How does this work for negative population growth? Clearly,  if you multiply the crude population doubling period by the absolute value of rate then you get 70. The calculation would therefore be identical to Example 4, though remember that it is the crude population halving period which is now being calculated.

Mixing Positive and Negative Growth

Things start to get more interesting if you then allow for the mixture of negative and positive rates. Even when the absolute value of the rate is constant, some hidden subtleties of exponential growth are revealed. For instance, add a negative 1 somewhere in the middle of 70 positive 1's then you only have a total of 69:

 Rates for years of negative growth Rates for years of positive growth 1 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 and 1 Red Time : 1 year @ 1% = Red Rate Total : 1 Black Time: 70 years @ 1% = Black Rate Total : 70

Example 5: A single year of negative growth cancels a single year of positive growth, preventing the population from doubling. The sum of the growth rates can be calculated as 70 - 1 = 69 (Black Rate Total - Red Rate Total = 69).  The elapsed time can be calculated as 70 + 1  = 71 (Black Time + Red Time = 71 years).

Although the sum of the growth rates is 69, 71 years have elapsed (70 years of positive growth and 1 year of negative growth). 71 years exceeds the 70 years required by the Rule Of 70. Has the population doubled? No! It is the sum of the growth rates which must add up to 70, not the elapsed time. So, due to just 1 year of negative growth, it would now require a further year of positive growth to bring the sum of the growth rates up to 70 (and the elapsed time would now be 72 years). Thus, in this extension of Example 5, a single year of negative growth adds 2 years to crude population doubling period.

The following general rule is now proposed:

"When the difference between the sum of the positive rates and the sum of the negative rates reaches an value of 70, then the larger value determines whether the population has doubled or halved.

It is easier to picture this as a weighing scale, with the rates for years of positive growth on one side, and the rates for years of negative growth on the other:

Figure 1. The Scales Of 70. If the totalled positive growth rates exceed ( "outweigh") the totalled negative growth rates by 70 then the population under consideration will double. If the totalled negative growth rates exceed ("outweigh") the totalled positive growth rates by 70 then the population under consideration will halve.

Regardless of the growth rate used, the Scales Of 70 apply. Thus, if we use a constant growth rate of  2% then the Scales Of 70 apply. For instance, add a negative 2 somewhere in the middle of 35 positive 2's then you only have 68. Has the population doubled? No! Again (just like the previous example) it means that it will now take a further year of positive growth to result in the population doubling:

 Rates for years of negative growth Rates for years of positive growth 2 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 and 2 Red Time : 1 year @ 2% Red Rate Total : 2 Black Time : 36 years @ 2% Black Rate Total : 72

Example 6a: This population will double. Using the Scales Of 70 to measure the doubling time for mixed positive and negative growth rate of 2%. The sum of the growth rates can be calculated as 72 - 2 = 70 (Black Rate Total - Red Rate Total = 70).  The elapsed time can be calculated as 36 + 1 = 37 (Black Time + Red Time = 37 years).

Thus, the general law is that a single year of negative growth at a constant rate always cancels a single year of positive growth at that rate, requiring a further year of positive population growth at that rate to achieve a population doubling. The converse situation is also true, specifically that a single year of positive growth at a constant rate always cancels a single year of negative growth at that rate, requiring a further year of negative population growth at that rate to achieve a population halving

 Rates for years of negative growth Rates for years of positive growth 14 x 5 28 x 5 Red Time : 14 years @ 5% Red Rate Total : 70 Black Time : 28 years @ 5% Black Rate Total : 140

Example 6b: This population will double. Using the Scales Of 70 to measure the doubling time for mixed positive and negative growth rate of 5%. The sum of the growth rates can be calculated as 140 - 70 = 70 (Black Rate Total - Red Rate Total = 70).  The elapsed time can be calculated as 28 + 14 = 42 years (Black Time + Red Time = 42 years).

Applying the Rule Of 70 to Example 6b, a population growing at 5% doubles in 14 years and a population shrinking at 5% halves in 14 years. Thus, effectively, this population has doubled twice and halved once. That is 2 x 2 x 1/2 = 2. On balance then, it has doubled once. For example: 1 doubles twice to 4, then halves once to 2.  Hence, on balance, 1 doubles once in this example.

Example 6b demonstrates how the Scales Of 70 can be used to "weigh" the results of multiple applications of the Rule Of 70

 Number of 14 year halving periods Number of 14 year doubling periods 1 2

Thus, it becomes obvious that any population that halved once and doubled twice in the given time periods would experience a single net doubling after 42 years. Here's one more:

 Rates for years of negative growth Rates for years of positive growth 70 x 1 140 x 1 Red Time : 70 years @ 1% Red Rate Total : 70 Black Time : 140 years @ 1% Black Rate Total : 140

Example 6c: This population will double. Using the Scales Of 70 to measure the doubling time for mixed positive and negative growth rate of 1%. The sum of the growth rates can be calculated as 140 - 70 = 70 (Black Rate Total - Red Rate Total = 70).  The elapsed time can be calculated as 140 + 70 = 210 years (Black Time + Red Time = 210 years).

Again, this population has doubled twice and halved once. So, it will on balance double once in 210 years:

 Number of 70 year halving periods Number of 70 year doubling periods 1 2

Variable Growth Rates

It should be entirely logical to suppose that any mixture of positive population growth rates which add up to 70 will always result in population doubling, and any mixture of negative population growth rates which add up to negative 70 will always result in population halving. Of course, using the Rule Of 70, these are approximate calculations.

Before we explore a mixture of positive and negative variable growth rates, let's examine an example of variable positive growth rates on their own:

Example 7: Variable positive population growth rates:

1 + 2 + 1 + 3 + 4 + 2 + 1 + 3 + 4 + 5 + 3 + 2 + 1 + 2 + 3 + 4 + 2 + 1 + 2 + 4 + 2 + 3 + 3 + 2 + 5 + 5  = 70

How many years did it take the population to double? Rather than adding the rates together, simply count how many rates there are - the answer is 26. In fact, it doesn't even matter which order these growth rates appear in, the result is always the same for the same set of growth rates.

Take a look at the Total Midyear World Population for 1950 to 2050 AD from the U.S. Census Bureau, or look at an analysis of the CIA World Fact Book page for Populations Ranking. You can see a similar situation with the variable growth rates between 1960 and 1999, which resulted in our global population doubling in 39 years. This is a genuine example of variable compound interest in action (try it for yourself - the annual growth rates are provided). This example is also useful in demonstrating that growth rates are rarely whole numbers.

Take another look at Example 7. If all of these rates were all negative (rather than all positive) then it would take 26 years for a population halve (rather than double). Again, it doesn't even matter which order these growth rates appear in, the result is always the same for the same set of numbers.

Putting it all together

If there is a mixture of positive and negative rates, together with variable rates for both, then this is as complex as it can possibly get. The solution is revert back to the Scales Of 70, with negatives on one side, and positives on the other. As soon as one side "weighs" 70 more than the other side then that side "wins". If the winning side is negative, then the population in question is halved.  If the winning side is positive, then the population in question is doubled.

 Rates for years of negative growth Rates for years of positive growth 1, 3, 2, 5, 1 and 1 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 5, 5, 4, 3, 3, 3, 3, 5, 5 and 3 Red Time : 6 years @ variable rates  Red Rate Total : 13 Black Time : 23 years @ variable rates   Black Rate Total : 83

Example 8: This population will double. Using the Scales Of 70 measure the doubling time for mixed positive and negative variable growth rates. The sum of the growth rates can be calculated as 83 - 13 = 70 (Black Rate Total - Red Rate Total = 70).  The elapsed time can be calculated as 23 + 6 = 29 (Black Time + Red Time = 29 years).

The Scales Of 70 provide a simple way to calculate rough population doubling periods and population halving periods for any replicator population, regardless of any apparent complexity caused by a mixture of positive and negative growth rates and regardless of whether or not the growth rates are variable or constant.

It is my contention that Example 8 represents the most common real-life example showing the mathematical nature of how replicator populations actually halve or double. That is to say that replicator populations regularly experience variable growth rates, and a mixture of positive and negative growth rates, and yet still experience exponential growth or exponential shrinkage.

Balancing The Scales For Negative Growth

As examined in my articles Positive and Negative Growth Compared, The Rule Of 70 and The Rule Of 72 Compared, and Rules Of Population, there is an minor imbalance between positive and negative growth at any given growth rate. By a wonderful coincidence, for a population halving and doubling, this difference is always very close to Ln 2 (= 0.693147) which is where we get the Rule Of 70 from in the first place!

This imbalance is addressed quite simply by adjusting the doubling or halving period as follows:

• whenever a population doubles, total the negative rates and multiply by 1/100. Subtract this amount from the doubling period.
• whenever a population halves, that means the halving period should be calculated as normal, minus 70/100ths.
 Rates for years of negative growth Rates for years of positive growth 1, 3, 2, 5, 1 and 1 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 5, 5, 4, 3, 3, 3, 3, 5, 5 and 3 Red Time : 6 years @ variable rates Red Rate Total : 13 Black Time : 23 @ variable rates  Black Rate Total : 83

Example 9: Taking Example 8, the Scales Of 70 would be balanced for negative growth by subtracting 13/100ths from the calculated doubling period of 29 years.  Hence, doubling would occur after 28.87 years.

Here's another one:

 Rates for years of negative growth Rates for years of positive growth 70 x 1 140 x 1 Red Time : 70 years @ 1% Red Rate Total : 70 Black Time : 140 years @ 1% Black Rate Total : 140

Example 10: Taking example 6c,  the Scales Of 70 would be balanced for negative growth by subtracting 70/100ths from doubling period of 210 years.  Hence, doubling would occur after 209.3 years.

Given that you have the actual growth rates - and could calculate the exact doubling period anyway - it's up to you whether you feel it worthwhile to apply this minor refinement to the Scales Of 70.

What is interesting is the fact that natural logarithms apply equally to growth based on fixed rate compound interest (exponential growth) and growth based on variable rate compound interest (Couttsian growth).

Handling Zero Population Growth

One last point is that a year of growth at 0% simply adds a year to the time it takes a population to double or halve. You might think you can place it with the negative rates or the positive rates - that it makes no difference. Wrong! This would lead to an unnecessary adjustment for negative growth. Thus, it is good practice to include years of growth at 0% in the positive half of the Scales Of 70.

Arithmetic Mean and Geometric Mean

Professor Roper (below) comments that the Scales Of 70  "...is a way to explain doubling times when there is compound growth in discrete units of time." Another way of putting it that variable rate compound interest is equivalent to discrete consecutive periods of fixed rate compound interest:

Period 1 - 70 years @ 1%

Period 2 - 35 years @ 2%

Both the Rule Of 70 and the Scales Of 70 predict that any population growing at these rates for the given periods would double twice.

The arithmetic mean for this 105 year period would be 1.33 recurring, or 1 and a 1/3. Both the Rule Of 70 and the Scales Of 70 predict that any population growing at this rate for 105 years would not quite double twice (it doubles every 52.63 years, and 2 * 52.63 = 105.26). However, when averaging growth rates, the geometric mean should be used.

The geometric mean for this 105 year period would be 1.26. Both the Rule Of 70 and the Scales Of 70 predict that any population growing at this rate for 105 years would not quite double twice (it doubles every 55.55 years, and 2 * 55.55 = 111.1).

However, the Rule Of 70 does not work due to either arithmetic means or geometric means. It is based on the Natural Logarithm of 2, which is approximately 0.693. In short, the Period 1 growth can be added to the Period 2 growth as follows:

e (LN(1.01) * 70) * e (LN(1.02) * 35) = e(0.009950331 * 70) * e(0.019803 * 35) = e(0.69652316 + 0.693092) = e1.38961516 = 4.013305

Hence, any population with a starting value of P multiplied by e1.38961516 will (slightly more than) double twice (P * 4.013305).

This is the exponential factor. It can be calculated for any combination of variable positive and negative rates.

As you can see, population doubling (and halving) in the end all comes down to e

The Scales Of 70 works (as an approximation) for the same reasons as The Rule Of 70. It works because,  effectively, we're adding positive and negative Natural Logarithms until we get the Natural Logarithm of 2. Refer to my article the Scales Of e for a more detailed explanation.

This confirms the Exponentialist view that exponential growth and Couttsian growth are intrinsically linked. The world of finance has effectively accepted this view by calling the former fixed rate compound growth and the latter variable rate compound interest. Perhaps it's time we also recognised them constant rate exponential growth and variable rate exponential growth

At the heart of it all lies the irrational, transcendental number e, the black jewel of the calculus. See my article Population Growth Models for more.

What Is Exponential, Anyway?

Period 1 growth can be described as growing exponentially. Period 2 growth can be described as growing exponentially. Yet put them together into a 105 year period and, strictly speaking, they cannot be described as growing exponentially. Daft, isn't it? See my article What Is Exponential? for more.

In finance, Period 1 growth can be described as growing via fixed rate compound interest. Period 2 growth can be described as growing via fixed rate compound interest. In finance, put them together and they are described as growing via variable rate compound interest. Clever, isn't it?

So, given that fixed rate compound interest is equivalent to exponential growth (and the Rule Of 70 applies to both equally), and given the very close (and obvious) connection between variable rate compound interest and fixed rate compound interest, why can't scientists and mathematicians (in addition to Prof. Roper) recognise that variable rate compound interest is equivalent to discrete consecutive periods of exponential growth (fixed rate compound interest)?

The fact that the Scales Of 70 works just as well for variable rate compound interest as it does for fixed rate compound interest demonstrates the unexpected but deep connection between variable rate compound interest and exponential growth.

The Scales Of 70 - Conclusion

Although the Scales Of 70 are only a crude measure of population doubling and halving, the mathematical arguments involved do provide irrefutable proof that:

• sustained but variable positive growth rates always result in population doubling
• sustained but variable negative growth rates always result in population halving

In fact, as proven above, even mixed negative and positive variable growth rates can also result in population doubling or population halving. Remember, if growth rates are measured in years, the rules of thumb for population doubling and halving are:

• if the totalled positive growth rates exceed the totalled negative growth rates by 70 then the population under consideration will double in a number of years equal to the total number of growth rates measured. For each year of negative growth at rate r, reduce the growth / shrinkage period by r/100s.
• if the totalled negative growth rates exceed the totalled positive growth rates by 70 then the population under consideration will halve  in a number of years equal to the total number of growth rates measured. For each year of negative growth at rate r, reduce the growth / shrinkage period by r/100s.

If growth rates are measured in minutes, hours, days, weeks or months then substitute the appropriate unit of measure in the statements above. As noted by Malthus, the initial size of the population does not matter, as the same set of rates will double or halve a population of any size.

Approximate though they may be, the Scales Of 70 provide a simple but powerful visualisation of how growth works in the real world. The fact that the Scales Of 70 applies to fixed rate compound growth (exponential growth) and variable rate compound growth (Couttsian Growth) is a strong clue to the link between the two. The Exponentialist view is that, in fact, exponential growth is merely an obvious special case of Couttsian growth.

It is no longer sufficient to think of population growth adhering to an approximate law of exponential growth - namely the Exponential Law.  Population growth adheres to a universal law of nature - namely variable rate compound growth. This is not just "growth". It is, effectively, variable rate exponential growth. Or, if you prefer, discrete consecutive periods of exponential growth at different rates.

I believe this is what Malthus (1830, A Summary View) meant by the following:

"It may be safely asserted, therefore, that population, when unchecked, increases in a geometrical progression of such nature as to double itself every twenty-five years. This statement, of course, refers to the general result, and not to each intermediate step of the progress. Practically, it would sometimes be slower, and sometimes faster."

That is, geometric (exponential) progression based on population doubling (or halving) at variable doubling (or halving) periods.

It's been staring us in the face for centuries.

What you're doing, as it see it, is to offer an explanation of discrete compounding formulas. In finance they talk about, say, daily interest rates versus, say, an annual rate and they are related by the equation

(1+i)^365 = (1+i_1)(1+i_2)... (1+i_365) which

means that (1+i) is the geometric mean of (1+i_1)(1+i_2)... (1+i_365)

i.e., 1+i = [(1+i_1)(1+i_2)... (1+i_365)]^1/365

Your 'scale of seventy' is, it seems to me, is a way to explain doubling times when there is compound growth in discrete units of time."

More comments by Professor Roper on the Exponentialist homepage.