Asimov - An Exponentialist View
Bartlett - An Exponentialist View
Darwin - An Exponentialist View
Dawkins - An Exponentialist View
Drexler - An Exponentialist View
Ehrlich - An Exponentialist View
Malthus - An Exponentialist View
Sagan - The Secrets of the Universe
Savage - An Exponentialist View
Turchin - An Exponentialist View
Wallace - An Exponentialist View
Witting - An Exponentialist View

The Rule of 70 and the Rule of 72 Compared

External Links:

Professor talks at an exponential rate - Energy Bulletin

Arithmetic, Population and Energy  - Dr. Albert Bartlett

The Massive Movement to Marginalise the Modern Malthusian Message by Professor Albert Bartlett

The New Flat Earth Society - Professor Albert Bartlett

Exponential function - Wikipedia

Total Midyear World Population for 1950 to 2050 AD - US Census Bureau

Albert Bartlett - Wikipedia

Albert Bartlett - An Exponentialist View

Cornucopians vs.  Malthusians, and why both sides are wrong


Bartlett is strong proponent of the Malthusian Message, as he terms it, and is famous for stating (Bartlett, 1998):

"The greatest shortcoming of the human race is our inability to understand the exponential function."

He has given his Arithmetic, Population and Energy presentation over 1,500 times. He is opposed to cornucopian, techno-optimist views, and his message is one of a sustainable future for humanity on Earth.

In his presentation Bartlett encourages feedback on any errors (Bartlett, 1998):

"Now, don't take what I've said blindly or uncritically, because of the rhetoric, or for any other reason. Please, you check the facts; please check my arithmetic; if you find any errors let me know. If you don't find any errors, then I hope you'll take this very very seriously."

Whilst I greatly admire his passion for the Malthusian Message, and in promoting sustainable living, I believe his interpretation of the mathematics involved is flawed, and thus detracts from the effectiveness of his message. Note that I do not challenge his mathematics as such, but I do challenge his understanding of the applicability of the exponential function to human population growth. I also challenge Bartlett's (1998) assertion that Zero Population Growth is "going to happen whether we like it or not" and that (Bartlett, 1998):

"...today's high birth rates will drop; today's low death rate will rise till they have the same numerical value."

I will point out that I have contacted Professor Bartlett to try to discuss my concerns. Initially I contacted him in early 2003. His response was very polite, and he graciously commented that he found my Exponentialist website "very impressive." He forwarded me a number of his articles, which I've read. 

However, when I tried to draw him into a specific discussion on the nature of exponential growth he was too busy to provide much of a response. I have tried again since (as late as 2006), but have received no further response to date.

Bartlett's view is the traditional scientific view that exponential growth requires a constant rate of growth to be considered exponential. It is this view which lies at the heart of the concerns I have with Bartlett's conclusions. 

I offer this Exponentialist article as a way to improve Bartlett's argument, and thus his conclusions on sustainability. Only then should people be able to take Bartlett's presentation "very, very seriously."

The Rule of 70

Bartlett's use of the heuristic tool The Rule of 70 in his presentation is typical of scientists who use it to illustrate exponential growth. He uses it to explain doubling periods based on a constant rate of exponential growth (Bartlett, 1998, my emphasis):

"If you had something that was growing at 5% per year, you'd write the exponential function to show how large that growing quantity was growing year after year. And so we are talking about a situation where the requirements required for the growing quantity to increase by a fixed fraction is a constant 5% per year. The 5% is a fixed fraction, the three years is a fixed length of time. So that what we want to talk about. It's just ordinary steady growth.

Well if it takes a fixed length of time to grow 5%, it follows that it takes a longer fixed length of time to grow 100%. That longer time is called the doubling time and we need to know how to calculate that doubling time. It's easy."

Bartlett then explains that all you do is divide the growth rate into 70 to get the doubling time. Thus using the Rule of 70, a population that grows at 1% will double in 70 years, and a population that grows at 2% will double in 35 years. 

One of my main concerns with Bartlett's argument is that he uses this explanation as the basis of how populations grow. Well, it's simply just not true. There is no documented case of any population of any species ever sustaining a constant rate of growth! "Ordinary steady growth" is not normal, and not ordinary. All real-world examples of exponential growth (at a constant rate) in fact turn out to be temporary periods of exponential growth at a given rate. 

As a thought experiment, I provided Bartlett with an example of a hypothetical population that grows at variable rates between 1% and 2%. Bartlett insisted that if the growth rate varies then what you get is "just growth, but not exponential growth."  

In fact, the Rule of 70 works just as well for variable rates as it does for constant rates. Thus, it is possible to predict that my hypothetical population would double somewhere between 35 and 70 years. This is true regardless of what rates you use, so long as the rates always fall between 1% and 2%. 

So how do you calculate exactly how long a population growing at variable positive rates of growth would take to double? It's startlingly simple. All you do is add up your variable positive rates on a year by year basis. When you hit 70, your starting population will have doubled. The same mixed bag of rates will double any sized starting population in the same number of elapsed years. See my Scales of 70 for a fuller explanation, and have a go yourself.

It is thus no surprise to me that our human population doubled from 3 billion in 1960 to 6 billion in 1999. How does Bartlett's explanation explain that? It is not sufficient to call it " just growth". 

The smallest growth rate for this 39 year doubling time was 1.26% and the largest was 2.19%. The Rule of 70 shows that the doubling time for a rate of 1.25% is 55.55 years, and for 2.19% it is 33.34 years. The 39 year doubling time falls within the 33.34 year doubling time and the 55.55 year doubling time. 

Of course it does! Each year of growth represents temporary exponential growth (for a given fixed rate, for a given fixed period). Another way of describing consecutive temporary periods of constant rate exponential growth (e.g.. 1% for 2 years, 3% for 1 year, 2% for 1 year etc) is to call such growth variable rate exponential growth. Sadly, scientists resist such terminology. This is entirely illogical thinking, as a population that is said to be growing exponentially at 1% for 70 years is in fact growing exponentially for each individual year of that 70 year period. Hence, a population that grows at 1% for 1 year is also growing exponentially, albeit for a fixed period of 1 year (or 12 months, 52 weeks, or 365 days, depending upon your preference). For the daily rate (ignoring leap years) the rate is 1.011/365 percent for a limited period of 365 days.

Rather than worry about definitions of "exponential growth", I suggest Bartlett uses the phrase "fixed rate compound interest" when referring to purely theoretical exponential growth (at a constant rate), and "variable rate compound interest"  when referring to actual population growth. You see, this is how all populations of all species actually grow, via variable rate compound interest. Also, the world of finance regularly uses such terminology without batting an eyelid. For example, mortgages can be fixed rate or variable rate. The rate is typically explained in terms of compound interest.

Bartlett can still use the Rule of 70, but with the commonly accepted concept of variable rate compound interest. He can then point to our global population doubling between 1960 and 1999 as an actual documented example of real-world population growth that follows a variable rate compound interest rate growth model.

Does Population Grow Exponentially, or Not?

If Bartlett insists that exponential growth requires a constant rate and that populations grow exponentially, then his argument is weakened by comments by equally-well qualified scientists such as Professor of Population Joel E. Cohen who argues (Cohen, 1995):

"Surprisingly, in spite of the abundant data to the contrary, many people believe that human population grows exponentially. It probably never has and probably never will."

Another highly qualified scientific critic of Malthus is John Maddox who attacked arguments made by Paul R Ehrlich (similar to those of Bartlett) regarding the dangers of exponential growth at a constant rate (Maddox, 1972):

"The real absurdity, however, is not the estimate of how many people there would be in 900 years but that demography can ever be as simple as this. The rate of increase is the balance between the birth rate and the death rate, both of which are continually changing, differently in different places....

... If demography is a numbers game, it is much more intricate than Dr Ehrlich implies."

Yet another highly qualified critic of the Malthusian argument is self-styled environmental skeptic Bjorn Lomborg (Lomborg, 2001):

"Malthus' theory is so simple and attractive that many reputable scientists have fallen for it. But the evidence does not seem to support the theory. The population rarely grows exponentially, as we saw in the introductory section (Figure 11). Likewise, the quantity of food seldom grows linearly...."

I take issue with the comments made by Cohen, Lomborg and Maddox in my article Paul R. Ehrlich and the prophets of doom - An Exponentialist View. You see, Bartlett is closer to the truth (just as Malthus was), in arguing that populations grow exponentially. 

Cornucopians Versus Malthusians

Ronald Bailey, science correspondent for Reason Online and strongly anti-Malthusian, regularly writes on the Cornucopian vs. Malthusian debate. Bailey (2001a) notes the following classical cornucopian claim made by Julian Simon (Simon, 1995):

"We now have in our hands - in our libraries really - the technology to feed, clothe and supply energy to an ever-growing population for the next 7 billion years."

A-pops 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
D-pops 1 2 4 8 16 32 64 128 256 512 1024
E-pops 1 2 4 8 16 32 64 128 256 512 1024
F-pops 1 2 4 8 16 32 64 128 256 512 1024
G-pops 1 2 4 8 16 32 64 128 256 512 1024
H-pops 1 2 4 8 16 32 64 128 256 512 1024
I-pops 1 2 4 8 16 32 64 128 256 512 1024
J-pops 1 2 4 8 16 32 64 128 256 512 1024
K-pops 1 2 4 8 16 32 64 128 256 512 1024
L-pops 1 2 4 8 16 32 64 128 256 512 1024
M-pops 1 2 4 8 16 32 64 128 256 512 1024

Table A - Bartlett's refutation of Julian's Simon's assertion that we can sustain an ever-growing population for 7 billion years.

The doubling period (assuming 1994 rates) is 43 years. Therefore, at 10 doublings per row, each row represents 430 years of growth. Bartlett calculates that after 6,000 years the resultant population would equal the weight of the universe. Using my New Malthusian Scale, that's somewhere towards the end of the M-Pops row, as 6,000 divided by 430 is not quite 14.

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

Garret Harding (1999, pp.33-35 ) also uses Bartlett's analysis to refute Simon's assertion, and notes that Simon says he actually meant that we could sustain an ever-growing population for 7 million years (not 7 billion years). Garrett explains Bartlett's response:

"Let us, Professor Bartlett said, assume that 7 million is what Simon had in mind. Assuming the present world population of six billion and the recent rate of population growth of 1 percent per year, how long would it take for the human population to equal all the atoms of the universe. The answer is shocking: just 17,000 years."

In both cases, exponential growth at a constant rate (and hence a fixed doubling period) is assumed. Bailey (2001a) criticises Bartlett and Earth First! co-founder David Foreman for attempting to de-bunk Simon's assertion using false premises:

"The trouble is, Foreman and Bartlett are reflexively assuming exponential population growth."

Bailey then supports his argument by explaining that growth rates are dropping and (quoting Nature, 2001) that the global population may even drop by the end of the century (for my views on predictions on human population growth see Human Replicators - An Exponentialist View). Bailey is basing his criticism on the fact that populations do not grow at a constant rate. This is true - populations do not grow at a constant rate. However, the key point here is that exponential growth at variable rates of growth is just as powerful as exponential growth at constant rates of growth (see What Is Exponential? for more). Even though constant rate exponential growth is purely theoretical and never encountered in the real world, it is still a valid illustration of the equivalent power of population growth at variable rates of growth. Variable positive rates of population growth are just as unsustainable as constant positive rates of population growth.

Bailey (2000) has claimed that humanity knows how to "keep the Malthusian trap from closing" - through inventiveness and innovation thanks to the "exponential growth of knowledge." Here Bailey appears to be guilty of the fallacy of  equivocation in his use of the word "exponential". Although I share Bailey's enthusiasm for human inventiveness, I doubt that any analysis of the growth of knowledge shows that knowledge grows at a constant rate of exponential growth. So when applied to its usage by Foreman and Bartlett, the term "exponential" clearly means exponential growth at a constant rate. However, when Bailey uses this term in relation to the growth of knowledge he appears to be more relaxed in his own usage of the word. I would accept that knowledge grows exponentially at variable rates of growth. I also agree with Bailey's suggestions that this will mean that humanity can produce more with less, and that technological progress will extend the carrying capacity of the Earth - but not indefinitely, not for 7 billion years, not even for 7 million years.

Not even Bailey's heroically termed "Post Scarcity Prophet" - economist Paul Romer (co-creator of the New Growth Model) - expects to keep the Malthusian trap from closing. In an interview conducted by Bailey, Romer argues that although the Malthusian Ehrlich got the timing wrong, growth must cease due to limits to growth (Bailey, 2001b):

"If by growth you mean population, more people, then Ehrlich is actually right. There are physical limits on how many people you can have on Earth."

Bailey (2001a) stresses that humans are not subject to "the strategy of the deer" in that we don't get better off just by consuming more resources. Humans are cleverer, and can get more from less (Bailey, 2001a):

"As we become cleverer about re-arranging material, the more goods and services we can get from relatively less stuff."

However, not even the visionary of molecular nanotechnology - K. Eric Drexler - believes that cleverly re-arranging material (and let's face it, there's nothing cleverer in the re-arrangement of material than theoretical molecular nanotechnology!) means we will not face inevitable limits to growth on Earth. Drexler (1986) asks whether Malthus was wrong, and answers as follows (For more see Drexler -An Exponentialist View):

"Not fundamentally: he was wrong chiefly about timing and details. Growth on Earth does face limits, since Earth has limited room, whether for farming or anything else. Malthus failed to predict when limits would pinch us chiefly because he failed to anticipate breakthroughs in farm equipment, crop genetics, and fertilizers."

Bailey might also be surprised to find that Malthus was equally aware of this difference between humans and animals in our human ability to push back the limits to growth (Malthus, 1798):

"The main peculiarity which distinguishes man from other animals, is the means of his support, is the power which he possesses of very greatly increasing these means."

Bailey (2005) elsewhere criticises Jared Diamond's adherence to the "orthodox Malthusian claims" that population grows exponentially, whereas food grows linearly. Looking at the history of the 20th century, Bailey claims (Bailey, 2005):

"...it is very clear that increases in food production have been exponential, too;""

Again Bailey appears to be guilty of the fallacy of  equivocation in his use of the word "exponential." Also, the use of the word "too" is odd, as this now implies that population is also growing exponentially. If that's not what Bailey means, then what word does he use to describe human population growth? And what does Bailey mean that food is growing exponentially? Does he literally mean that food increases at a constant rate of exponential growth? But nothing in the real-world grows exponentially at a constant rate - it is impossible. The exponentialist view is that all populations of all species grow via variable (positive and negative) rates of exponential growth all of the time - this includes food (all of which grows in populations). Sure enough, a look through the statistical appendix for UN's Crop Prospects and Food Situation for July 2008 reveals that the growth rate for crop production has indeed varied in recent years (and I have no doubt it has always varied):

  Average growth rate 1998-2007 2004 2005 2006 2007 2008
Changes in world cereal production 1.2 9.3 -1.1 -2.0 5.4 2.8

Growth rates for world cereal production - UN's Crop Prospects and Food Situation for July 2008

This just highlights that the word "exponential" is misused and misunderstood by all parties to the debate. However, rather than get to the bottom of this confusion, the Cornucopian vs. Malthusian debate is instead full of ad hominem attacks. Harding (1999) refers to Simon as a "Population Ostrich." Bartlett and Foreman prefer the derogatory term "Flat Earther". Amongst Bailey's other anti-Malthusian articles for Reason Online are "Malthusian Twaddle" (2003) where Malthusians are accused of being "simple-minded" and prone to "dogma", and "The Lingering Stench of Malthus" (2006) in which Jeremy Rifkin is accused of being a "Malthusian boob." So the fallacy of ad hominem attacks is prevalent amongst Cornucopians, too.

Rather than make personal attacks, I believe it would be more productive if Cornucopians and Malthusians (or Neo-Malthusians) upgraded their thinking and revised their understanding of exponential growth. It is strange that both sides of the argument, the Cornucopians and the Malthusians, fail to understand the simple but deep significance of actual exponential growth - that is consecutive limited periods of exponential growth at a different constant rate per period. Hence, a population growing at 1% for 70 years is growing exponentially, for each individual year of that period. If that population then grows at 2% for 35 years then it is growing exponentially, for each individual year of that period. Thus, using logic, it must be true that a population that grows at 1% one year and 2% the next year is also growing exponentially! In other words, no matter what the rate per year (assuming positive rates), the population is growing exponentially! Also, despite traditional views on the nature of exponential growth (that is requires a constant rate), a population that grows at 1% for 70 years then 2% for 35 years is growing exponentially every year for 105 years! 

However, a less contentious assertion (avoiding arguments over what constitutes exponential growth) is to say that populations grow via variable rate compound interest. Such growth is comparable in power to exponential growth (at a constant rate), and the Rule of 70 still applies as a simple heuristic to help get the message across. 

Exponential growth (at a constant rate) is not a useful explanation for real world population growth (for the reasons given by Cohen, Lomborg and Maddox), but variable rate compound interest (which I have termed Couttsian Growth) is an excellent explanation of real-world population growth. 

To Professor Cohen I say that variable rate compound interest is comparable in power to theoretical exponential growth, and is a simple model that can be easily demonstrated via the Rule of 70 and my own Scales of 70. It is also a simple model that does describe human population history.

To Professor Bartlett I say that exponential growth (at a sustained constant rate) is "just theoretical growth, but not actual real-world growth". For that you need variable rate compound interest

Zero Population Growth

Ignoring any arguments in favour of or against ZPG, I argue that ZPG can never be sustained. Populations interact with their environment, and they interact with each other. The terms "population dynamics" and "dynamic equilibrium" are significant due to the use of the word "dynamic". The only thing constant about populations is change!

Equilibrium is maintained in see-saw fashion, and can usually be explained via population halvings and population doublings. This is why my Scales of 70 is a better explanation of actual population growth than the Rule of 70. However, for a completely accurate model of population growth and shrinkage via variable rate compound interest see my article The Scales of e.

The evidence is overwhelmingly against ZPG achieved via a sustained growth rate of zero percent, as argued by Bartlett. It is a fictional state of growth that can only ever be temporarily maintained. Take a look at the growth rates of populations around the world, and you might find one or two examples of ZPG for the current year. Now look at the demographic history of those countries. I guarantee that no population of any country has ever sustained ZPG via a zero percent growth rate. 

Thus, Bartlett's argument would be more effective if he demonstrated a realistic grasp of population growth and not a purely theoretical or hypothetical grasp.  

First Law of Sustainability

However, Bartlett is still correct when he states the first law of sustainability:

"Here it is - population growth and/or growth in the rates of consumption of resources cannot be sustained."

Bartlett is right for two reasons. Firstly, as I argued previously, exponential growth (at a constant rate) cannot be sustained and hence is impossible. This is irrefutable. However, this is not what Bartlett means. Bartlett means that neither population growth nor growth in consumption can be sustained regardless of the rate, and regardless of whether or not that rate varies - and he's right. This is due (second reason) to limits to growth. Bartlett refers to those who believe in unlimited growth as The New Flat Earth Society (Bartlett, 1996). I agree with Bartlett that nothing can grow forever because otherwise it would consume the Earth, or the Solar system, or the Milky Way and ultimately the Universe. Growth has limits. 

 Hence, what he should state is that, positive rates of compound interest cannot be sustained, even if they are allowed to vary. My own version of the law of sustainability is therefore:

Anything that grows via positive variable rate compound interest will be inevitably forced to shrink via negative variable rate compound interest. 


Despite the flaws in Bartlett's arguments (though not his mathematical calculations, per se), Bartlett's presentation is enduringly popular. I believe that his arguments will only be improved through the use of the term variable rate compound interest in lieu of exponential growth (at a constant rate). I further argue that a more realistic view of the dynamic nature of population growth will strengthen his deeply significant argument in favour of sustainability.

Albert Einstein is widely reported (perhaps apocryphally) as stating that compound interest is the most powerful force in the universe. True or not, I agree. Sadly, it is also the least well understood. 

Rather than assert that the greatest failing of humanity is its failure to understand the exponential function, Bartlett's argument would be better if he argued that:

The greatest shortcoming of the human race is our inability to understand variable rate compound interest.


Bailey, Ronald, David Foreman Vs. the Cornucopians. Reason online. 2001a

Bailey, Ronald, Earth Report 2000. 2000

Bailey, Ronald, Post-Scarcity Prophet (interview with Paul Romer), 2001b

Bailey, Ronald, Under The Spell of Malthus. Reason online. 2005

Bartlett, Albert, Arithmetic, Population, and Energy. 1998

Bartlett, Albert, The New Flat Earth Society. Web site accessed 7th July, 2009

Cohen, Joel E., How Many People Can The Earth Support? 1995

Drexler,  K. Eric, Engines Of Creation . (see chapter on Limits to Growth). 1986

Hardin, Garrett, The Ostrich Factor: Our Population Myopia. 1999

Lomborg, Bjorn, The Skeptical Environmentalist. 2001

Maddox, John,  The Doomsday Syndrome - John Maddox Launches An Attack On Pessimism.  McGraw-Hill.1972

Simon, Julian, The State of Humanity: Steadily Improving, Cato Policy Report, 1995 (based on the introduction to his 1995 book, The State Of Humanity)

United Nations Food and Agriculture Organisation, Statistical appendix for Crop Prospects and Food Situation No.3, July 2008.

Back to Top

Send email to exponentialist@optusnet.com.au with questions or comments about this web site.
Copyright 2006 David A. Coutts
Last modified: 07 August, 2012