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Special series: Ecological economics, part 3
NewsNotes, May-June 2009

In part two of this series we examined the concept of uneconomic growth, or “illth,” that takes place when the costs of using up natural resources are greater than the benefits from the new product built with those resources. In this article, we look specifically at the little understood, yet extremely important, concept of exponential growth, which is when something grows at a constant rate over time, for example a bank account that receives fixed interest or world population. Few people truly grasp the profound importance of this concept when we talk about the economy today. As Dr. Albert Bartlett, University of Colorado physics professor, said, “The greatest shortcoming of the human race us our inability to understand the exponential function.”

Growth is usually expressed by a percentage of increase per year, for example, a bank account that grows at two percent annually, etc. Yet a steady growth rate is misleading because, unlike the rate of change, that usually stays more or less constant, the amount of growth at a given rate of change per unit of time is not constant at all, but increases more and more with time. If you were to graph anything that grows at a constant rate, it would look like a hockey stick lying on the ground, with a long horizontal line of apparently slow growth that at a certain point turns upward rapidly into an almost vertical line where the amount of increase grows incredibly quickly in a short period of time. A few examples will help to understand this “speeding up” factor:

One way to think of exponential growth is to consider that the interval of time needed for the item which you are measuring to increase by a certain amount shrinks as time passes, even though the rate of change remains constant. Consider world population, which has been growing at about a one percent annual rate: Around the year 1804, Earth’s population hit one billion for the first time. That number doubled in 123 years (around the year 1927). By 1959, only 32 years later, the population had grown another billion, and 15 years later (around 1974), the population hit four billion. By 1987, another billion was added, and 12 years later (1999), the population was six billion. The Census Bureau estimates Earth’s population will reach seven billion in February 2012.

Another example is monetary growth in the United States. It took over 300 years (from 1492-1973) for the U.S. (including the colonial period) to create its first $1 trillion in wealth. (It’s unclear if this number includes the buying, selling and use of slaves.) The value of everything that was ever created in the U.S. and the colonies that preceded it – every road, building, automobile, etc. – before 1973 added up to $1 trillion. According to Chris Martenson of chrismartenson.com, in the fall of 2008, the most recent $1 trillion that the U.S. made took only 18 weeks to create. Where will this end? When $1 trillion is created in 18 days? 18 hours? 18 minutes?

Knowing that this “wealth” creation also represents the consumption of natural resources, it is clear that this exponential growth cannot continue for long.

Another way to think about exponential growth is to think of the amount that is added growing larger with each new time period. An example would be the legendary story of the king who wanted to reward his mathematician for inventing the game of chess. The mathematician said, “My needs are modest. Please take my new chess board and on the first square, place one grain of wheat. On the next square, double the one to make two. On the next square, double the two to make four. Just keep doubling till you’ve doubled for every square. That will be an adequate payment.” We can guess the king thought, “This foolish man. I was ready to give him a real reward; all he asked for was just a few grains of wheat.” Yet this simple doubling of grains 64 times over would result in an amount of grain larger than the entire world could produce. Just in the last square alone the king would have to place 184,467,440,737,095,000 grains. And this is equal to the sum of all the previous 63 squares added together.

One final example that may more clearly show how rapidly things increase on the vertical end of an exponential graph comes from Martenson’s Crash Course. Imagine that Fenway Park in Boston is sealed off to be able to hold water. You start to drop water onto the pitcher’s mound. In the first minute, you place one drop. Double that each minute, so in two minutes, you’d place two drops, in three minutes placing four drops, then eight drops, etc. Now imagine you are on the highest bleacher chained to the fence. If they started to place water drops at noon, at what time would the park still be 93 percent empty? It would be only seven percent filled at 12:44 p.m. At what time would the park be full of water? An hour later? A day? A week? No, it would be full at 12:49 p.m. In fact, only one minute before you are drown by the water, the park is still only 50 percent full! While watching the water level slowly rise to only halfway, you would probably not panic, but that would leave you only one minute to escape before being drowned.

This is the power of compounding, of exponential growth. See the graph below, which shows the dramatic growth in population, carbon dioxide concentration, water use, species extinction and other areas. How much longer can these lines continue to rise before we reach the limits of Earth? We will have to rapidly decrease our use of resources before we run out of them completely. And clearly this expectation that growth will solve the problem of poverty is unrealistic. Rather, we need to focus on better distribution. Keep these facts in mind as politicians throughout the world strive to get the economy growing as fast as possible once again.

Good websites that treat exponential growth in interesting ways:

Otherwise.com's Exponential growth

Raju Varghese's website

Chris Martenson's Crash Course, chapters 3 and 4

The rule of 70

An easy math trick called “the rule of 70” uses the growth rate to estimate how quickly something will double in size. By dividing 70 by the growth rate you estimate the doubling time. For example, world population, which has grown at a rate of about one percent per year, would double in size in 70 years (70 divided by one). Global use of oil has been growing at a three percent rate, meaning we double the amount of oil we use every 23 years (70 divided by three). Next time you are listening to the news and hear about something growing at a certain rate, use this formula in order to have a better idea of how soon it will double in size. While China’s recent economic growth rate of 10 percent doesn’t mean much to most, knowing that this means the Chinese economy will double in size in only seven years is much more meaningful.

exponential growth

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