Toward a stewardship of the Global Commons:

engaging “my neighbor” in the issue of sustainability

By members of the Critical Issues Committee, Geological Society of America

Part III


Doubling time: 
it works for ANY rate of change

A. R. Palmer, Institute of Cambrian Studies, Boulder, CO


A parable: “When was the pond half full?”I lived by a large pond with a thriving community of fish, so fishing was good.  One day not too long ago some algae began to grow in the pond.  Their population was doubling every minute.  Yesterday morning I went fishing and everything was fine.  Yesterday noon when I looked out at the pond, it was suddenly filled with green algal scum and the fish were dying from lack of oxygen.  Why didn’t I see the disaster coming and do something?  When was the pond half full [11:59]?  One-quarter full [11:58]?  One eighth full [11:57]?  Suppose, instead of my pond, we were considering an island, or a continent, or Spaceship Earth?


At the heart of the concept of doubling (or halving) is the exponential function familiar to many from mathematics, science and engineering.  Geologists are perhaps most familiar with this in its backward-running version, i.e. the description of the rates of decay of radioactive isotopes.  Most of us, however, learned about exponential growth as compound interest in the context of a personal savings account.  If we put our money in a bank and let the interest accumulate, our annual income grows as the capital increases. Even though the interest rate remains constant - our capital grows at an exponential rate.  Recognition of doubling (or halving) time for ANY rate of change was not always emphasized.

Because we are discussing a “doubling” time, i.e. the time for a quantity in question to double (or halve), the exponential function has a user-friendly aspect that is very helpful.  It turns out that any finite rate of change expressed as a percent (e.g. 5% per year) can be converted, to a good approximation, to a doubling time simply by dividing it into 70.  For example, if the rate is 2%, you might expect to have twice the number, i.e. 100% more stuff, in 100/2=50 years, but because of the compounding effect the correct answer is 70/2 = 35 years.  The proof of this statement makes a good exercise for a math class at the appropriate grade level.  Rates of loss (for instance, depletion of resources) have a halving time that is calculated the same way.  For anything diminishing at 2% per year there will only be half as much in 35 years.


This simple way to calculate doubling time (or halving time) should be an essential part of everyone’s education.  When the mayor is proud because the city has a healthy growth rate of 3% per year, that means the city will double in about 23 years if that rate continues, and double again in another 23 years, and double yet again in another 23 years, thus octupling from its original size in 69 years.  One might ask if a city with 8 times the present population is viable.  Garbage also has a rate of growth, as does traffic, pollution, schools and housing.


A city is not a closed container, thus its limits to growth will be determined by cultural factors, but Earth IS a closed container for all practical purposes (more on this in Part IV ).  Growth in a closed container will ultimately fill it up.  Thus, we should look carefully at anything in our culture with a growth rate, calculate its doubling (quadrupling and octupling) time and make a judgment about whether we think this is healthy for our future.  We should do similar calculations and make similar judgments about those aspects of our culture where, as a result of our consumption habits, a resource is diminishing at a measurable rate.  Thus, calculations of doubling (or halving) time are critical components of the issue of sustainability – by which we mean the indefinite continuation of the entire human enterprise within some steady-state limits imposed by space and resource availability (see Part X in this series).


Global population growth rates may be diminishing, but they are still positive.  Thus population is still growing and it has a doubling time.  Currently the rate is about 1% per year (thus doubling in 70 years).  Consumption of resources (Part IV) increases with population size, even if individual rates of consumption do not increase.


If we could slow the present population growth rate to 0.1% per year, it would still be double its present size of about six billion in 700 years, quadruple in 1,400 years, and octuple in 2,100 years – equivalent only to the time represented by the Christian era.  Forty-eight billion people may make things a bit crowded.  Such a population, with its attendant FOOTPRINTS (the areas of productive land necessary to support each one of us, see the upcoming June essay), may not be sustainable.  It is not even clear that we can handle one more doubling with a reasonable quality of life for all.


Such doubling-time scenarios should make us wonder if there is such a thing as the politically popular “smart growth” – perhaps it’s a euphemism for “predictable and voluntary disaster”.  Every time you see a headline or magazine article mentioning rates of change (either increase or decrease), do the quick mental math to calculate the doubling (or halving) time.  It is a very revealing exercise.



DEMONSTRATION 1 -   Challenge the computer skills of the students by having them write a program that can check the statement that any rate of growth or loss, divided into 70 will give you its doubling or halving time.


DEMONSTRATION 2 -   Have the class check the newspapers and news magazines for mentions of rates of change (these could include population size, waste generation, growth of GNP, inflation, depletion of forested land, or demands on natural waters) and then do the calculations of doubling or halving time.  Discuss the implications of the calculations.

Return to Introduction
Guidelines to Sustainability Literacy
Part I: Stewardship of the Commons
Part II: Understanding Deep Time
Part III: Doubling Time

Part IV: Sustainability and Resources
Part V: The Connectedness of Everything
Part VI: Ecological Footprint and Carrying Capacity
Part VII: Spaceship Earth: There's No Place Left to Go
Part VIII: Part of the Global Ecosystem

Part IX: We Live in a World of Change
Part X: What Do We Mean by Sustainable World?
Part XI: Cultural Context of Sustainability
Part XII: We Have The Option of Choice

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