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

Simple and Compound Interest - Answers and Explanations, Mathematics Network, University Of Toronto, Canada

Linear Growth versus Exponential Growth (and Couttsian Growth)


In this article I will examine two fundamental models for growth - linear growth and exponential growth. What are they? What are the distinguishing features of each?

Let us take a starting figure of 100, and apply an annual positive 1% growth rate for both models:

  Start Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7

Linear Growth

100 101 102 103 104 105 106 107
100 101 102.1 103.0301 104.060401 105.10100501 106.1520150601 107.213535210701

Table A: Positive Linear Growth and Positive Exponential Growth compared

To be honest, the difference appears to be that exponential growth is messy but not all that dissimilar to linear growth (Year 1 is even the same!). However, as the rest of this article will show, this "reality" is an illusion.

Linear Growth

It is clear from Table A that the 1% interest rate for linear growth has only been applied to the original starting sum of 100 for each year of growth. This is simple interest. Continuing the example started in Table A, it would take 100 years for starting figure of 100 to double to 200 at a 1% growth rate. Increase the starting figure to 1000, it would still take 100 years for the starting figure to double at this rate. Double the growth rate and the time it takes to double the starting figure would halve (at a 2% linear growth rate a starting figure of 100 would double in 50 years).

Wait a minute, doesn't population doubling imply exponential growth? Well, not exactly. The key element missing from the linear growth model is compound interest. Put simply, compound interest means that the interest rate applies not only to the starting sum but also to the previously accumulated interest, for each successive period in which it is applied.

In fact, with linear growth it takes just as long to get through the next hundred as it did through the first. For a starting population of 100 at a 1% linear growth rate, a 1,000 years of growth would look like this:

Linear Growth

100 200 300 400 500 600 700 800 900 1000 1100
Years 0 100 200 300 400 500 600 700 800 900 1000

Table B: Starting population of 100 at 1% linear growth rate

For a starting population of 100 at a 2% linear growth rate, 500 years of growth would look like this:

Linear Growth

100 200 300 400 500 600 700 800 900 1000 1100
Years 0 50 100 150 200 250 300 350 400 450 500

Table C: Starting population of 100 at 2% linear growth rate

I have deliberately bolded the figures for 100, 200, 400 and 800 to demonstrate population doubling using a linear growth model. However, population doubling from linear growth does not lead to exponential growth. Why not?

Exponential Growth - Constant Rate

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

Positive Growth - Constant Rate

In this article I stated that population doubling leads to exponential growth. So why not say that population doubling leads to linear growth?

For a starting population of 100 at a 1% exponential growth rate, 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 D: Starting population of 100 at 1% exponential growth rate

For a starting population of 100 at a 2% exponential growth rate, 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 E: Starting population of 100 at 2% exponential growth rate

Compare the results of exponential growth with those for linear growth. The differences are startling. I have taken the liberty of using the Rule Of 70 to quickly extrapolate my results for exponential growth (Note: The Rule of 70 is only useful for growth rates between negative 7% and positive 7%). Approximate as this method is, the sheer power of exponential growth over linear growth is revealed. Don't forget, I have taken the same starting population and applied the same two rates of growth (1% and 2 %) for each growth model (linear and exponential). It was a fair contest, though the result was never in doubt.

With exponential growth at a constant rate note that, for each successive period of time, the absolute increase constantly doubles.

The same does not apply to the linear growth model. That is why I feel justified in stating that population doubling "leads to" exponential growth. However, perhaps a better way of expressing this is to say that the nature of exponential growth is more accurately represented by the population doubling series. The nature of linear growth is more accurately represented by a linear series

e is the key

A key difference between linear growth (simple interest) and exponential growth (compound interest) relates to the number e defined as a limit, which is explored in brief in Simple and Compound Interest and The General Situation (from Answers and Explanations, Mathematics Network, University Of Toronto, Canada (1997). 

As described in the Academic Press (Daily Insight) in "Math Buffs Find an Easier e" (1998):

"For both bankers and bugs, e describes a basic limit to exponential growth. For example, if you invested $1 at 100% interest, compounded monthly you would have $2.61 at year's end. If the interest is compounded every 30 seconds, you would end up with about a dime more. No matter how frequently you earned interest, you would never take home more than e multiplied by the number of dollars you first deposited."

In short, e effectively defines the difference between linear growth and exponential growth.

The constant of proportionality

One last feature of exponential growth at a constant rate was eloquently summed up by Malthus himself:

"A thousand millions are just as easily doubled every twenty-five years by the power of population as a thousand."

Trefil (2002) called this effect the "constant of proportionality".

In fact, the same feature applies to variable rates of compound growth (but not to linear growth). Apply variable rates of growth to amount A until it doubles. Now apply the same variable rates to amount B and you will double the original amount in the same doubling period. This leads to the following universal laws defined by the Couttsian Growth Model (for both constant and variable rates of growth):

My view is that population growth is the most pressing problem that humanity faces, yet most people cannot easily distinguish between linear growth and exponential growth. Many confuse a growth model with a series (and I have probably confused a series with a sequence!). Furthermore, few people appreciate how variable rates of growth lead to exponential growth.

Exponential Growth - Variable Rate

The true nature of the exponential growth of populations does not reveal itself until variable rates of growth are considered. Modern demographers often seem to get confused over this point, as discussed in Ehrlich And The Prophets Of Doom - An Exponentialist View.

I have explored the real-life nature of population doubling (and halving) in the articles The Scales Of 70 and The Scales of e. I find that population doubling and halving are adequate to explain almost all cases of actual exponential growth. The Couttsian Growth Model refinement of the Malthusian Growth Model itself is universal in its application across all time for all replicator populations. Should you require a greater degree of granularity than is provided through the use of the New Malthusian Scale (and its reliance on population halving and population doubling) then I recommend the use of the Scales of e

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

Perhaps it is worth mentioning at this point that Malthus, the founder of modern demography, did not use the term exponential and preferred to use the term geometric. See Malthus - An Exponentialist View for more. Perhaps the use of the term exponential implies to some people that each number is the resulting series of an exponential growth model must be represented by the original sum raised to an exponent. The term geometric would not infer the same association. However, both terms suffer from the fact that they each imply that a constant rate of growth is required in order for growth to be considered exponential or geometric. 

For those that doubt the assertion that variable rates of growth result in exponential growth, consider the following example. For a starting population of 100 at exponential growth rates between 1% and 2%, 700 years of growth could look legitimately like this:

Exponential 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 F: Starting population of 100 at variable rates of exponential growth between 1% and 2%

Remember, using the Rule Of 70 the population doubling period at 1% is 70 years and at 2% it is 35 years. You will note that the population growing at variable rates between 1% and 2% (inclusive) always doubles somewhere between 35 years and 70 years. This example leads to the following universal laws:

Note that for any set of growth rates (ignoring a zero growth rate), and for any starting value, exponential growth at a variable rate would easily outstrip linear growth at a variable rate.

Malthus is wrong

It was Malthus (An Essay On The Principle of Population, 1798) who most dramatically drew world attention to the disparity between geometric (exponential) and arithmetic (linear) growth:

Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will shew the immensity of the first power in comparison of the second.”

Malthus' essay describes a bleak and constant state of affairs for humanity, caused by this disparity:

"The perpetual tendency of the race of man to increase beyond the means of subsistence is one of the general laws of animated nature, which we can have no reason to expect to change."  

Initially Malthus uses the populations and food supply of Great Britain and the United States of America to make his point. Malthus then rightly attempts to avoid the confusion of national immigration and emigration figures on natural increase. To do so, he proposes to model population growth and food supply for the whole Earth. First, in a roundabout way, Malthus introduces the concept of the limits to growth imposed by Earth itself:

"But to make the argument more general and less interrupted by the partial views of emigration, let us take the whole earth, instead of one spot, and suppose that the restraints to population were universally removed. If the subsistence for man that the earth affords was to be increased every twenty-five years by a quantity equal to what the whole world at present produces, this would allow the power of production in the earth to be absolutely unlimited, and its ratio of increase much grater than we can conceive that any possible exertions of mankind could make it."

Malthus is saying that, taking the whole Earth, it is unrealistic to assume a continued increase in food supply to support an exponentially growing population. He continues:

"Taking the population of the world at any number, a thousand millions for instance, the human species would increase in the ratio of - 1,  2,  4,  8,  16, 32, 64, 128, 256, 512, etc. and subsistence as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 etc. In two centuries and a quarter, the population would be to the means of subsistence as 512 to 10: in three centuries as 4096 to 13, and in two thousand years the difference would be almost incalculable, though the produce in that time would have increased to an immense extent."

Graph A. Malthus' comparison of arithmetic (linear) growth and geometric (exponential) growth

It is generally accepted that population does not grow exponentially (at a constant rate), nor does food supply grow linearly. At best, the simple exponential growth model (at a constant rate) can be said to be an approximation of actual population growth. This is sometimes known as the Malthusian Growth Model. Thus, on the surface, Malthus appears to be wrong.

Oddly, Malthus also insisted that the geometric growth model applied to all populations of all species for all time. Now, if this were true, then logic dictates that food supply (which is comprised of populations of living plants and animals) must also be geometric in nature. 

Let's assume for the moment that all populations of all species grow exponentially all of the time, and see if we can make Malthus' argument work. After all, food supply does generally seem to keep up with population growth.

Malthus is right

Here is Malthus' argument for exponential population growth (A Summary View - 1830):

"The immediate cause of the increase of population is the excess of the births above deaths; and the rate of increase, or the period of doubling, depends upon the proportion which the excess of the births above the deaths bears to the population."

The inverse case (not provided by Malthus) can be used to provide an argument for exponential population shrinkage:

"The immediate cause of the decrease of population is the excess of the deaths above births; and the rate of decrease, or the period of halving, depends upon the proportion which the excess of the deaths above the births bears to the population."

And here Malthus makes it plain that doubling periods will vary (because of the variability of the checks on population growth):

"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."

Note that the 25 year doubling period is specific only to human populations.

Together, these statements of Malthus' lay the foundations of the Couttsian Growth Model which alone can explain the results shown in Table F. Perhaps it would be simpler all round if we called such real-world growth Malthusian. 

For a real-world example, see Total Midyear World Population for 1950 to 2050 AD from the US Census Bureau. For those that continue to doubt Malthus on this point you must now explain the growth model which results in the figures displayed in Table F, and the real-world example from the US Census Bureau. Unfortunately for the US Census Bureau, even they fail to adequately explain such growth. Even worse, their use of the exponential method is flawed, as proven in my article US Census Bureau - Incorrect use of the Exponential Method

Exponential growth (constant rate) is equivalent to fixed rate compound interest. Exponential growth (variable rate) - which is not supposed to exist, according the definition of exponential growth - is the equivalent to variable rate compound interest. 

Variable rate compound interest is how all populations of all species grow, all of the time. 

Variable rate compound interest leads to variable population doubling and population halving periods, as described in my articles The Scales of 70 and The Scales of e. The sorts of timeframes we are talking about here are equivalent to those experienced in classical exponential growth (at a constant rate).

Malthus is so clearly right that I wonder why it is still necessary to try and persuade people (including many professional demographers and evolutionists) that variable growth rates result in growth that is equivalent in power to exponential growth. Still, after so long, people continue to refute Malthus' claim. 

The find the truth behind Malthus' work, it is necessary to correct his own self contradiction and dismiss the arithmetic (linear) growth model from his argument. Then, it is simply a matter of understanding that Malthus proposed the geometric (exponential) growth model as an approximation (that is "the general result") of actual population doubling. As Malthus said,

"Practically, it would sometimes be slower, and sometimes faster."

The greatest challenge, it seems to me, is to understand that variable rate compound interest is the equivalent of variable rate exponential growth. See What is Exponential? for more.

Gigantic Inevitable Famine

Given Malthus' dire predictions of famine due to the imbalance of linear food supply and exponential population increase does that mean we no longer need to worry about famine? After all, exponential food supply can keep up with exponential population increase. I believe Malthus might have even understood why:

"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."

We humans are capable of harnessing exponential forces when we encourage the growth of populations of plants and animals for our own consumption.

However, although exponential food supply can keep pace with exponential population increase that doesn't mean it is always guaranteed to do so. See my article Gigantic Inevitable Famine for more.

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Copyright 2001 David A. Coutts
Last modified: 26 June, 2012