Basics of Exponential Growth

Understanding Exponential Growth

Exponential growth drives resource usage for a very simple reason:

Human population increases exponentially:

Currently the world's population is growing at the rate of 1.25% per year, down significantly from the growth rate 30 years ago. But, small differences in growth rates make big differences in the ultimate population that appears

At the moment there seems to be very little that can be done about this and hence, this represents our fundamental problem. That is, you can't really stop it, only slow it down and plan accordingly.

Intelligent planning for exponential resource usage is directly related to environmental problem solving as follows:

Now, of course, the problem is made worse by the perception that we are all afraid of math and that "formulas" don't apply to real life.

"Math, formulas and other things that don't apply to real life"

(--anonymous comment from student evaluation as the part they liked least about the course)

So we live in a society that is afraid of and doesn't understand numbers.

This again is a recipe for disaster as it means the public can be sold most anything

An example:

A survey of Boulder Colorado residents about the optimal size for growth returned a result that most residents thought that a growth in population at the rate of 10% per year was desirable.

Well 10% a year may not seem innocuous but let's see how these numbers would add up?

So in 7 years (year 2--7) the population has doubled and by then 10,000 new residents per year are moving to Boulder!

If one had asked the question on the survey: Is it desireable for the population of Boulder to double in 7 years, there would have been an overwhelming NO. Clearly, the general population can not equate 10% growth rate with 7 year doubling time.

Clearly, Exponential growth, in general, is not understood by the lay public. If exponential use of a resource is not accounted for in planning - disaster can happen.

The difference between linear growth and exponential growth is astonishing.

In this animation, one can clearly see that no matter what the growth rate is, exponential growth stars out being in a period of slow growth and then quickly changes over to rapid growth with a characteristic doubling time of

70/n years; n =% growth rate


Its important to recognize that even in the slow growth period, the use of the resource is exponential. If you fail to realize that, you will run out of the resource pretty fast:

Material Rate Exhaustion Timescale
Aluminum 6.4% 2007 -- 2023
Coal 4.1% 2092 -- 2106
Cooper 4.6% 2001 -- 2020
Petroleum 3.9% 1997 -- 2017
Silver 2.7% 1989 -- 1997

The above estimates include recycling.

Exponential growth means that some quantity grows by a fixed percentage rate from one year to the next. A handy formula for calculating the doubling time for exponential growth is:

Doubling Time = 70/n years

The math behind this

where n is the percentage growth rate. Thus, if the growth rate is say 5%, the doubling time would be 14 years.

Try the Salmon Sustainability Applet

In exponential growth, the rate of growth may well change, but the growth is still exponential!

Note: There is some disagreement about the form of the population growth for the world. In population dynamics, most all species are subject to something called the logistic growth curve . It is unclear if that is the destined growth of human beings or not, as long as we are on the r-selected part of the logistic growth curve, population growth is exponential, characterized by a specific doubling time for that growth rate. There is no evidence in the data, yet, that we are on the flat portion of the logistic growth curve. Therefore it is most scientifically accurate to state that , in the year 2003, human population growth is exponential in nature, with a current growth rate of 1.35% per year.

When considering growth over a period of years, it is important to note that taking the natural logarithm of the ratio of the final value to the initial value and dividing by the time period in years gives the average annual growth rate.


Number of Building Permits in Pasco WA

Growth Rate:

What kinds of things grow exponentially?

Clearly exponential rates of growth are an integral part of the planning process. Different aspects of a growing population have different exponential growth rates and these need to be considered.

For instance, suppose your urban area is growing at the rate of 5% a year. How does this translate into the following:

Exponential Growth and Trend Extrapolation

Whenever schools get crowded, freeways get jammed, airline hubs get crowded, oil gets used up, there are no more available phone numbers, the federal debt goes beyond recovery, etc, etc is an indication of poor planning and trend extrapolation.

In a nutshell: there is no reason that we should ever be surprised at the rate of resource utilization. If we simply pay attention to past history, in general, its a fairly good guide for future resource use.

The difference between linear growth (constant number of units growth per year) vs exponential growth (constant percentage increase) is difficult to see initially, if the exponential growth rate is small.

A good example of this is provided in the case of the population data for California .

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