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Linear Growth Versus Exponential Growth
Logistic Growth Versus Exponential Growth
Compound Growth Versus Exponential Growth
Exponential Brownian Motion
The Mechanism Of Population Doubling
Generations and Population Doublings
What Is Exponential?

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What Is Exponential?

Introduction

The usual explanation relates to a variable raised to the power of something. For example, 10 to the power of 2 is 100 (102 = 10 * 10 = 100). The 2 in this case in the exponent, hence the word exponential. Exponential functions of the nature F(x) = xn produce the classic exponential curve when plotted on a graph.

Another common, less mathematical explanation involves explosive growth (Kelly, 2002):

"Most people have an intuitive understanding of what it means to have exponential growth. Basically, it means that things are increasing in an out-of-control way, like a virus in a horror movie."

This is the sort of growth that, it is generally inferred, far surpasses any other kind of growth. Is it a necessary condition, however, that the growth rate must be constant? From the endnotes for Chapter 1 of "The Population Explosion" (Ehrlich, 1990):

"Exponential growth occurs when the increase in population size in a given period is a constant percentage of the size at the beginning of the period. Thus a population growing at 2 percent annually or a bank account growing at 6 percent annually will be growing exponentially."

In this article I will be challenging the assumption that exponential growth requires a constant rate.

Powers Of 2

Taking "powers of 2" as our example, here's what happens when 1 A-Pop keeps on doubling:

A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024
B-Pops 1 2 4 8 16 32 64 128 256 512 1024
C-Pops 1 2 4 8 16 32 64 128 256 512 1024

Table 1 - Exponential Growth Outside Of Time

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops, 1 C-pop = 1024 B-pops etc. 

From 1 to 1,073,741,824 - it's pretty explosive growth, isn't it? The problem with this view is that, from a real-world perspective, the growth occurs outside of time. How long did it take to get from 1 A-Pop to 1,024 C-Pops? Without factoring in time, such representations of exponential growth have no real-world value.

The solution is simple - add a constant doubling period and it's easy to calculate. If the doubling period is 25 years, each row represents 250 years of growth. Hence, 3 rows worth of growth equals 750 years of growth.

Doubling Period   25 25 25 25 25 25 25 25 25 25
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024
Doubling Period   25 25 25 25 25 25 25 25 25 25
B-Pops 1 2 4 8 16 32 64 128 256 512 1024
Doubling Period   25 25 25 25 25 25 25 25 25 25
C-Pops 1 2 4 8 16 32 64 128 256 512 1024

Table 2 - Exponential Growth With Constant Doubling Period (requiring a constant growth rate of 2.8%)

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops, 1 C-pop = 1024 B-pops etc. 

This is classical exponential growth. Given that it took 750 years in this case, would you call that explosive growth? 

The Rule Of 70

Using the Rule Of 70, it is commonly accepted that a population that grows at 1% will double every 70 years (approx) whereas a population that grows at 2% will double every 35 years (approx). Giving each population 700 years to grow, it would look like this:

Doubling Period   35 35 35 35 35 35 35 35 35 35
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024
Doubling Period   35 35 35 35 35 35 35 35 35 35
B-Pops 1 2 4 8 16 32 64 128 256 512 1024

Table 3 -Population Doubles Every 35 years for 700 years. 

It's exponential growth, but is this explosive growth? The following example is even slower.

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops

Doubling Period   70 70 70 70 70 70 70 70 70 70
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024

Table 4 -Population Doubles Every 70 years for 700 years

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops

Exponential Growth But Not Always Explosive Growth

Clearly, the smaller the constant rate, the less "explosive" exponential growth becomes. Perhaps it's worth quoting Darwin at this point (Darwin, 1859, from the chapter The Struggle For Existence):

"There is no exception to the rule that every organic being increases at so high a rate, that if not destroyed, the earth would soon be covered by the progeny of a single pair."

So, the point to remember about the popular perception of "explosive" out-of-control exponential growth is that all species of living creatures are capable of it. However, where the constant growth rate is kept close to zero, it should be no surprise that exponential growth can produce the illusion of almost stable growth. For example, a human population growing exponentially at 0.1%  would take approximately 700 years to double. Hardly explosive growth in human terms! However, even this humble growth rate is theoretically capable of producing a population that outweighs the Earth itself, in time. This is, in fact, a major argument against the "simple exponential" growth model (Kelly, 2002):

"Truth be told, there are not a lot of natural cases in which exponential growth is exhibited. An exponential growth model assumes that there is an infinite amount of resources from which to draw."

So Does Exponential Growth Require A Constant Rate Of Growth?

However, the interest rate supposedly needs to remain constant for the growth to be classified as exponential. Really?

Try these examples:

Doubling Period   35 35 35 35 35 70 70 70 70 70
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024
 

Exponential growth for 175 years

Exponential growth for 350 years

 

Exponential growth?

Table 5 -Population doubles at a constant rate of 2% for 175 years, and then  grows at a constant rate of 1% for the next 350 years. 

Note:  1A-pop = 1024 pops

So, if we describe the growth from 1 to 32 billion it can be said to be exponential. If we examine the growth from 64 to 1024 billion it can also be said to be exponential. What about the overall growth from 1 to 1,024 billion? What single word in the English language describes such growth? Why not exponential? If not exponential, then at which point was the growth no longer exponential? After all, both periods of growth are described in isolation as exponential. OK, then let's assume for the moment that we can define such growth - as seen in Table 5 - as exponential. However, such definitions lead to this...

Doubling Period   35 35 70 70 35 35 70 70 35 35
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024

Table 6 -Population doubling alternates every couple of periods between 1% growth and 2% growth. Is this exponential?

Note:  1A-pop = 1024 pops

Or this...

Doubling Period   35 70 35 70 35 70 35 70 35 70
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024

Table 7 -Population doubling alternates between 1% growth for 70 years and 2% growth for 35 years. 

Is this exponential? Certainly, each period of growth is, in itself, exponential.

Note:  1A-pop = 1024 pops

Refer to US Census Bureau  - Incorrect Use of the Exponential Method for a discussion on actual variable rate compound interest at work (and the failure of the US Census Bureau to understand the growth model behind variable rate compound interest). Note the global population doubling between 1960 and 1999 from 3 billion to 6 billion - this was the result of population growth rates that varied between 1.2% and 2.09%.

To illustrate the fact that variable rates of exponential growth are just as powerful as constant rates of exponential growth consider the following example for 3 starting populations of 1,000 each. One is growing at a constant rate of 1% per annum, one at a constant rate of 2% per annum, and one is growing at a 1% and 2% on alternate years.

Exponential Growth

Graph A - Note that any  variation of the growth rate between 1% and 2% over time will always fall between the Constant 1% line and the Constant 2% line. Refer to The Scales Of 70 for a intuitive approximate explanation of how this sort of growth works in practice, and The Scales of e for an exact explanation using natural logarithms.

Tightly Bounded Exponential Growth

Or we could try closer and closer rates of growth, as shown in Table 8 and Table 9:

Doubling Period   69 69 69 69 69 69 69 69 69 69
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024

Table 8 -690 years of constant, exponential growth

Note:  1A-pop = 1024 pops

and 

Doubling Period   71 71 71 71 71 71 71 71 71 71
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024

Table 9 -710 years of constant, exponential growth

Note:  1A-pop = 1024 pops

Now throw them together....

Doubling Period   69 71 69 71 69 71 69 71 69 71
A-Pops
(billions)
1 2 4 8 16 32 64 128 256 512 1024

Table 10 -700 years of inconstant, non-exponential growth. Hang on! Isn't this very close to Table 4 and constant exponential growth? Why is one exponential and the other not exponential? Isn't the result the same over the same period, tightly bounded by a narrow but variable rate of growth?

Note:  1A-pop = 1024 pops

Here's another graphed example:

Graph B - The closer the upper and lower constant rates get, the more tightly bounded the line for variable rate growth gets.

Clearly, the more tightly bounded the variable growth is, the more and more obviously like exponential growth (at a constant rate) it gets. Nonetheless, the growth rate can vary how it likes between 1.1% and 1.2% and the line for variable growth rate will always fall between the two lines for constant rate growth. It is now absurd to argue that the term exponential growth must only be applied to populations that grow at a constant rate. 

Critics of the simple exponential growth model (growth at a constant rate of exponential growth) are quite right to dismiss it as having no real-life meaning. Logically, it cannot exist in nature as a universal law of nature because it is impossible for a population to keep growing forever without hitting a limit to growth

Compound Interest

Yet all the critics of simple exponential growth seem to miss the next most obvious candidate - variable rate compound interest - and most leap immediately to another failed growth model, the Logistic Growth Model (see Logistic Growth Vs Exponential Growth for more). 

Simple exponential growth (at a constant rate) is equivalent to fixed rate compound interest. So what sort of exponential growth is equivalent to variable rate exponential growth? See my article Compound Growth Versus Exponential Growth for more. Just as the following statement is true:

Constant rate exponential growth = Fixed rate compound interest

So too is the following statement true:

Variable rate exponential growth  = Variable rate compound interest

In fact, fixed rate compound interest (leading to constant exponential growth and constant doubling periods) is merely a special case of variable compound interest (leading to variable exponential growth and variable doubling periods). The fact is that no population can grow forever at any constant positive rate of growth. Hence, for real-world growth, simple exponential growth can only ever be used to describe growth for a limited period of time and never for all time.  

            

Fixed Rate Compound Interest is a special temporary case of Variable Rate Compound Interest.

A Constant Doubling Period is a special temporary case of Variable Doubling Periods.

Exponential growth (at a constant rate) is typically used in theoretical models to describe fixed rate compound interest resulting in constant doubling periods, but for real-life populations it is variable rate compound interest resulting in variable doubling periods that explain actual population growth (and variable negative compound interest resulting in variable population halving periods).

Simple exponential growth can only ever be a building brick. For humans for example, our annual growth rates are all simple exponential growth building bricks. However, variable exponential growth leads to a population growth model built from those bricks. 

Fixed rate compound interest resulting in constant doubling periods is described by the Malthusian Growth Model. This is also commonly known as exponential growth, or the simple exponential growth model. It copes with zero population growth, or a positive rate of growth, or a negative rate of growth. It does not cope with any combination of these 3 possible types of growth rate. Think of it as an illustrative theoretical model, rather than a real-life growth model.

Variable rate compound interest resulting in variable doubling periods is described by the Couttsian Growth Model. It copes with zero population growth, and any number of positive rates of growth, and any number of negative rates of growth. It easily copes with any combination of these 3 possible types of growth rate. This sort of growth doesn't have an agreed, recognisable name, though to me it is fundamentally a form of exponential growth. Think of it as variable exponential growth, and not complex exponential growth. It's really not that complex! It's also fairly commonly used, for example in variable rate mortgage loans. It's really quite a familiar concept, though oddly it has never been recognised by the scientific community as a perfect replacement for the routinely dismissed Malthusian Growth Model.

Variable exponential growth can easily match the out-of-control reputation of simple exponential growth, or produce an even better illusion of stable growth than we saw with simple exponential growth. This is because a population growing within a the Couttsian Growth Model is not compelled towards any limit to growth because the growth rate can vary, and is even allowed to go negative. The Couttsian Growth Model describes in simple mathematical terms a universal law of nature that applies to all populations of all species anywhere in the universe for all time. 

Annual Percentage Yield, Compound Interest and Exponential Growth

(Thanks to Paul Russo for inspiring me to write this section)

It is important to note the distinction between Annual Percentage Yield (APY) and exponential growth.  Here's an example of a starting amount of 400000 with 10% exponential growth for a year compared with calculating the APY using compounding monthly for a 10% annual growth rate:

  APY for 10% annual growth rate
compounded
monthly
10% Exponential
Growth
for a Year
Monthly
growth
rate=>
1.0083333333 1.0079741404
 Period    
0 400000.00 400000.00
1 403333.33 403189.66
2 406694.44 406404.75
3 410083.56 409645.48
4 413500.93 412912.05
5 416946.77 416204.66
6 420421.33 419523.54
7 423924.84 422868.88
8 427457.54 426240.89
9 431019.69 429639.80
10 434611.52 433065.81
11 438233.28 436519.14
12 441885.23 440000.00

For exponential growth, the growth for the year is effectively capped at 10% for the year, so the trick is finding a constant monthly exponential growth rate that will result in 10% for the year. This is the formula used:

M = (1 + p)C

M = Monthly Growth Rate
C = number of compounding peri
ods
p = percentage (e.g. 10% = 10/100)

So, we have:

M = (1 + 10/100))1/12 = 1.0079741404289

Growth for 12 months = (400000 * 1.0079741404289)12 = 440000.00

For APY even though an annual growth rate of 10% is the starting point, the aim is to increase annual growth beyond 10% by increasing the frequency of compounding period. The formula to calculate the monthly APY figure is:

M = (1 + p) / C

M = Monthly Growth Rate
C = number of compounding peri
ods
p = percentage (e.g. 10% = 10/100)

So, we have:

M = (1 + (10/100)) / 12 = 1.00833333333333

Growth for 12 months = (400000 * 1.00833333333333)12 = 441885.23

So in both cases the result is in fact exponential growth at a constant rate, and the distinction (in this case) is whether or not you are interested calculating a monthly growth rate that will cap exponential growth at 10% for the year or whether you are interested in exceeding 10% growth for the year by increasing the frequency of compounding periods to monthly. In both cases, fixed rate compound interest is the same as constant rate exponential growth.

For any combination of Nominal Interest Rate and compounding periods, the limit to the exponential factor is represented by e to the power of that Nominal Interest Rate. For a more detailed treatment of  APY and continuous compounding, see my article Calculating APY and Continuous Compounding.

Conclusion

Sustained exponential growth at a constant rate is impossible. All supposed examples of such exponential growth in fact turn out to be temporarily sustained. There is no loan, and no investment, that could ever sustain a constant rate of growth. There is not one single documented example of any population of any species ever managing to sustain indefinite exponential growth at a constant rate. 

Take human populations. An annual population growth rate represents a temporary period (365 days, 52 weeks or 12 months) at a single averaged rate of growth. For example, if the target is an annual growth rate of 1% then the daily growth rate (ignoring leap years) is  1.011/365 for a limited period of 365 days. Surely 365 time periods at exactly the same growth rate constitutes "exponential growth"! Yet when we say 1% for 1 year people question whether this is exponential growth!

Each year's population growth is, at best, described as temporary exponential growth. Hence, populations grow via consecutive periods of temporary exponential growth, with a different constant each period. This is a very messy and confusing way of describing how populations actually grow. 

Thankfully some demographers (Preston, Heaveline & Guillot, 2001, p.11) - although somewhat tentatively - appear to be ready to "raise questions" over the use of the term "exponential growth".  Describing the population growth rate as a continuously varying within the exponential function over time, and given that positive, negative and even zero growth rates all obey the same exponential function, the authors of one recent demographic tome argue (Preston, Heaveline & Guillot, 2001, p.11):

"In this sense the term 'exponential growth' is a redundancy; all growth is exponential by our measure of growth as a proportionate rate of change in population size. When people use the term 'exponential growth' they are often (but not invariably) referring to a sequence produced by a constant positive growth rate within some interval."

The authors suggests reverting to Malthus' term "geometric growth" or perhaps "constant rate growth" for such growth. I note that this would therefore leave the term "exponential growth" available for variable rate exponential growth. For a different approach to explaining the same thing, read my articles the Scales of 70 and the Scales of e.

Another way of saying that populations grow via consecutive temporary periods of exponential growth (at a constant) is simply to admit that populations grow via variable rate compound interest. 

If scientists insist that exponential growth requires a constant rate then they have nothing to use it for as no real-world examples exist. If scientists then resort to describing temporary periods of exponential growth (at a constant rate) then they are actually describing variable rate exponential growth. However, to spare them the embarrassment, just explain to them that what they are actually describing is variable rate compound interest.

References

Darwin, Charles. Origin Of Species The Illustrated Edition*. Sterling Publishing Co. 1859, 2008*.

Ehrlich, Paul R.&  Anne H, The Population Explosion. Touchstone, Simon & Schuster. 1990.

Kelly , W. Michael, The Complete Idiot's Guide To Calculus. Alpha (A Pearson Education Company). 2002.

Preston, Heaveline & Guillot, Demography: Measuring & Modelling Population Processes. 2001.