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?

External Links:
Malthusian Growth Model - by Steve McKelvey.
Mathematical Modelling in a Real and Complex World - by the Connected Curriculum Project
Exponential Growth and The Rule Of 70
- by EcoFuture 

Mathematical Modelling - Malthusian Growth and Population by SDSU & Joseph M. Mahaffy

An Exploration in Exponential Growth and Compound Interest - T Barron, Georgia University

Compound Growth versus Exponential Growth (and Couttsian Growth)

"The most powerful force in the universe is compound interest" Albert Einstein (possibly apocryphal)


In this article I will examine two very similar models for population growth - compound growth and exponential growth. What are they? What are the distinguishing features of each?

Let us take a starting figure of 1000, and apply an annual positive 1% exponential growth rate to Pop B:

Years   10 40 80 160 320 360 400 500 600 700
Pop A
Compound Growth
1 2 4 8 16 32 64 128 256 512 1024
Years   70 140 210 280 350 420 490 560 630 700
Pop B
Exponential Growth
1 2 4 8 16 32 64 128 256 512 1024

Table A. Compound Growth and Exponential Growth compared. Pop A and Pop B both reach 1,024,000 in 700 years. What does this mean?

As you can see, using the Rule Of 70, the population experiencing 1% Exponential Growth reached 1,024,000 in just 700 years by doubling every 70 years. Very impressive. Yet the population which experienced Compound Growth also reached the same figure in the same time, by doubling at irregular periods. Before I explain what this all means, it is worth a closer look at each growth model on its own. 

Exponential Growth - Constant Rate

In the following article I explore the exponential growth model using a constant rate of growth:

Positive Growth - Constant Rate

For a starting population of 100 at a 1% exponential growth rate, doubling every 70 years, 980 years of growth would look like this:

Exponential Growth

100 200 400 800 1600 3200 6400 12800 25600 51200 102400 204800 409600 819200 1638400
Years 0 70 140 210 280 350 420 490 560 630 700 770 840 910 980

Table B: Starting population of 100 at 1% exponential growth rate, doubling every 70 years

For a starting population of 100 at a 2% exponential growth rate, doubling every 35 years, a 490 years of growth would look like this:

Exponential Growth

100 200 400 800 1600 3200 6400 12800 25600 51200 102400 204800 409600 819200 1638400
Years 0 35 70 105 140 175 210 245 280 315 350 385 420 455 490

Table C: Starting population of 100 at 2% exponential growth rate, doubling every 35 years

This growth model is clearly in trouble. It is in trouble because, as Malthus himself helped to point out, any population which sustains positive population growth could fairly quickly cover the Earth - clearly an absurdity! However, if the upper growth limit is reached, simply invoke a negative rate of growth to reduce the population below the limit to growth. The negative rate of growth could be a constant rate, even the same absolute value as the previous positive rate used to reach the limit to growth. 

Still, it is also in trouble because, as many authors are fond of pointing out, it only rarely seems to apply in reality (perhaps for bacterial growth in controlled conditions). 

Perhaps the most famous feature of exponential growth is the famous exponential curve, with slow growth at the start turning into faster and faster growth reaching an almost vertical spike.

It is also easy to see that exponential growth at a constant rate is exponential because each number in the resulting series can be expressed as the original population raised to an ever increasing exponent:

Standard notation: 1 2 4 8 16 32 64 128 256 512 1024
Scientific notation: 20 21    22    23    24    25    26    27    28  29 210

The same cannot be said when the rate of growth varies from period to period.

Compound Growth - Variable Rate

Yet the true nature of the population growth does not reveal itself until variable rates of compound growth are considered. 

Consider the following example (which makes use of The Rule Of 70). For a starting population of 100 at variable growth rates between 1% and 2%, 700 years of growth could look legitimately like this:

Compound Growth

100 200 400 800 1600 3200 6400 12800 25600 51200 102400 204800 409600 819200 1638400
Years 0 70 105 150 200 250 300 370 405 440 500 570 605 665 700

Table D: Starting population of 100 at variable rates of compound growth between 1% and 2% inclusive

Logic alone should be enough to show that, if a constant 1% growth rate doubles a population in 70 years, and a constant 2% growth rate doubles a population in 35 years, then a population which experiences variable growth rates falling from 2% to 1% will double somewhere between 70 and 35 years. This is what's happening in Table D. 

As you will see, this explains how our global population of 3 billion in 1960 doubled to 6 billion by 1999 (doubling in 39 years). See Total Midyear World Population for 1950 to 2050 AD from the US Census Bureau for a real-world example. This is clearly Compound Growth at a variable rate.  

If you look carefully at Table D, you will see doubling periods of exactly 35 years and 70 years throughout the 700 years. Plus there are doubling periods which fall between the two other periods (e.g.. 50 years). What word, phrase or term is used to describe this sort of growth? Certainly, we could call it Compound Growth. But check your dictionary and you won't find it defined there. 

Now suppose during the first period as the population doubles in 70 years that the interest is a constant 1%. Then suppose that during the second period as the population doubles in 35 years and that the interest is a constant 2%. Now we're talking about successive periods of constant rate exponential growth, at different rates. Doesn't that sound a little odd? After all, either the rate is constant or variable, isn't it? 

What, in fact, constitutes constant rate growth? How long must the rate stay constant to qualify? If a population grows at 1% for 70 years it clearly qualifies as exponential growth (at a constant rate). So, if we take any 10 of those 70 years, the population in question is still growing exponentially, isn't it? So, if we take any one year of that 10 years, then the population is growing exponentially for that whole year, isn't it? That's 12 months of exponential growth, if that sounds better.

Well, at the end of that year, suppose it's the last year of our original 70 years, the rate changes to 2% and stays there for 35 years of constant exponential growth. 

How do you now describe the two year period when the rate changes from 1% to 2%? Remember, that's 12 months at 1% and 12 months at 2%. Two successive periods of constant rate exponential growth, at different rates. 

Doesn't that sound a little odd? After all, either the rate is constant or variable, isn't it? 

The Big Lie

OK, if you've read any of the other articles on this web site then you're probably confused - where'd this term Compound Growth come from? I haven't helped any by failing to provide my customary links to my Glossary in this article until now. This was because I know my Exponentialist definitions differ to the commonly perceived definitions, and I wanted the reader to read this far on the basis of their own understanding. 

Until 11th March, 2003 (see What's New), throughout the Exponentialist web site I have consistently referred to Exponential Growth and included both constant rate and variable rates  in that term. Now I've introduced the term Compound Growth. To understand why you should be confused, ask yourself these questions:

So what's the difference between compound interest at a fixed rate and Exponential Growth at a constant rate?

Answer: None

Doesn't that mean that Exponential Growth (at a constant rate) could also be called Compound Growth (at a constant rate)?

Answer: Yes

So what's the difference between Compound Growth and Exponential Growth at a variable rate? 

Answer: None

Doesn't that mean that Compound Growth is really Exponential Growth?

Answer: Yes

Compound Growth = Exponential Growth 

The bottom line is that all exponential growth is inherently compound growth by nature, and that it is clearly absurd to state that exponential growth requires a constant rate. It is time to update your dictionaries, and replace the definition of exponential growth you find there with the following definition:

Term traditionally used for positive  population growth at a constant rate per period (graphically depicted by the exponential curve).  The defining feature of exponential growth is compound interest. Thus a population which experiences  positive population growth at variable rates of compound interest can also be said to be growing exponentially. For negative population growth refer Exponential Shrinkage.

I owe my own understanding of exponential growth to Malthus (who, writing between 1798 and 1830, used the term geometrical progression). The fact is, Malthus made his case plain enough for variable rates back in 1830 (A Summary View):

"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."

Malthus was referring, of course, to the well documented population doubling of that recently created nation, the United States Of America. See Reverend Thomas Robert Malthus - An Exponentialist View for more. The key thing to note is that, practically, he expected rates of growth to vary - as indeed it does! However, I wonder whether people can be made to understand poor old Malthus. In the words of Darwin (From a letter to Sir Charles Lyell, 6th June, 1860):

By the way what a discouraging example Malthus is, to show during what long years the plainest case may be misrepresented and misunderstood.

As Plain As The Nose On Your Face

Indeed, Malthus' case is as plain as the nose on your face, and yet few - even Darwin - fully appreciated Malthus' mathematical argument, though at least Darwin understood the importance of mathematics to our understanding of the world as this reflection upon his time at Cambridge shows (from Darwin's autobiography):

"I attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense."

Malthus possessed this extra sense for he was trained as a mathematician, and indeed did rather well at it during his time at Cambridge before Darwin.

So if Malthus' case was so obvious, why has nobody else drawn the obvious conclusion that population growth works on the principle of compound growth? Well, the answer is that at least one other scientist did come to this rather obvious conclusion, and that scientist and atheist was Sir Julian Huxley (1887 - 1975), grandson of the famous Darwin's Bulldog - Thomas H. Huxley (1825 - 1895).

In his Evolutionary Humanism (1964), in an essay entitled Crowded World, the more recent Huxley writes:

"The Neo-Malthusians, supported by progressive opinion in the Western World and by leading figures in most Asian countries, produces volumes of alarming statistics about the world population explosion and the urgent need for birth control, while the anti-Malthusians, supported by the two ideological blocs of Catholicism and Communism, produce equal volumes of hopeful statistics, or perhaps one should say of wishful estimates, purporting to show how the problem can be solved by science, by the exploitation of the Amazon or the Arctic, by better distribution, or even by shipping our surplus population to other planets."

Having firmly established his Humanist credentials, Huxley then explicitly refers to compound interest rates in relation to the burgeoning human population:

"...as a result of the great explorations during and after the Renaissance, and still more of the rise of natural science and technology at the end of the seventeenth century, the process was stepped up, so that by the beginning of the present century world population stood at about 1 1/2 billion, and its compound interest rate of increase had increased from under 1/2 of 1 per cent in 1650 to nearly 1 per cent (and we all know what big results can flow from even a small increase in compound interest rates).

But the real explosion is a twentieth-century phenomenon, due primarily to the spectacular developments in medicine and hygiene, which have drastically cut down death-rates without any corresponding reduction in birth-rates - death control without birth-control. The compound interest rate of increase meanwhile crept, or rather leapt, up and up, from under 1 per cent in 1900 to 1 1/2 per cent at mid-century, and nearly 1 3/4 per cent today; and it will certainly go on increasing for some decades more. This means that the rate of human doubling itself has doubled within the past 80 years. World population has more than doubled since 1900 to reach about 2 3/4 billion today; and it will certainly reach well over 5 1/2 billion, probably 6 billion, and possibly nearly 7 billion by the year 2000."

Apologies for the rather long quote but, in all my years of searching, I have never seen another writer express the population growth model (and its consequences in terms of human numbers) so well. His prediction of "probably 6 billion" by the year 2000 was excellent, as the UN Population Fund marked 12th October, 1999 as the Day Of Six Billion..

The main drawback with Huxley's essay, as far as I can see, was that he was not actually proposing a population growth model. Perhaps to him it was just too obvious. Hence, even though he articulated it rather well, he wasn't writing the essay to explain the population growth model but was more concerned (as was Malthus) with the human implications of such high compound growth rates. Also, Huxley does not explicitly explain how compound growth and population doubling (or halving for that matter) are linked. For more on this topic, read my article The Scales Of 70

Couttsian Growth

If you're confused, my advice is to read the article Population Growth Models, and follow my new naming convention introduced there.

To picture Couttsian Growth (or Huxlian Growth if you prefer), think of human population growth. Each year - be it for a town, a county, a state, a nation or the whole Earth - an averaged figure is used to describe the growth or shrinkage of that population. This is an average of the figures for each month, or each week. Hence, the average population growth rate is an average of variable rates of interest across the year.

As per my article The Scales Of 70, these annual averaged rates accumulate over time until - typically - either the population in question halves or doubles. 

e and the law of compound interest

In e: The Story Of A Number by Eli Maor (1994), Maor points out the close relationship between fixed rate compound interest and exponential growth. Indeed, Maor dates the obscure origin of this number (only later christened e by Leonhard Euler in his two volume 1748 work Introductio in analysin infinitorum) to the early 17th century and the world of trade and finance - roughly when John Napier, in 1614,  invented his logarithms.

Much attention was focussed on what Maor refers to as "the law of compound interest" by various financial institutions of the time. This soon lead to the discovery of the following infinite series:

e = 2 + 1/2! + 1/3! + 1/4!...1/n!

Other mathematical approaches have confirmed the value and nature of e,  but it is fitting to end this article by noting that an understanding of e began with an understanding of compound interest and that such an understanding has led me to the Couttsian Growth Model which in practice works through variable rate compound interest (though it also accommodates fixed rate compound interest). 

In his book Maor (Princeton University Press, 1994) notes:

"The growth of population follows an approximate exponential law."

Oddly, Maor does not mention the discoverer of this "approximate Exponential Law" - namely Malthus. No explanation is given of why Malthus' law is considered only approximate. However, it's a commonly held view (read through my Famous Exponentialists articles for a glimpse of such views). I've also confirmed via Cassell's Laws of Nature by James S. Trefil (2002) that this must be the "official" scientific view. 

I repeat, the way to turn this approximate law of nature (based on fixed rate compound interest) into a true universal law is through the Couttsian Growth Model, and variable rate compound interest.

All it takes is a single word change, from fixed to variable.

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Copyright 2003 David A. Coutts
Last modified: 10 August, 2012