Exponential Growth Doubling time

All resource usage and/or pollution will grow exponentially but with different couplings to the population growth this is extremely relevant to greenhouse gas emissions which we will discuss later. For instance, the the previous "lecture" we learned that electricity generation scales as the population growth to the 3.5 power!

Returning to the theme of the very first "lecture in this class" regarding objectivity and science:

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

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. Exponential growth means that some quantity grows by a fixed percentage rate from one year to the next.

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.

A handy formula for calculating the doubling time for exponential growth is:

Doubling Time = 70/n years

The math behind this

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. The following table shows some resource exhaustion timescales done in the year 2000 given the growth rate at that time. Since 2000 some, but not all, of these rates have come down. Note that the exhaustion timescales do include recycling.

Material Rate Exhaustion Timescale
Aluminum 6.4% 2007 -- 2023
Coal 4.1% 2092 -- 2106
Cooper 4.6% 2001 -- 2020
Petroleum 3.9% 2011 -- 2023

A good example of poor public understanding.

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 almost 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.

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.

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: