|The arithmetic of compound growth|
|Written by Jeremy Wakeford|
|Friday, 11 May 2012|
Economic growth is the favourite mantra and apparent cure-all for the majority of politicians and economists in South Africa, and, indeed, in the world at large. But how many people really understand the nature and implications of compound – or exponential – growth?
Albert Bartlett, an American professor of physics, famously declared that “the greatest shortcoming of the human race is our inability to understand the exponential function”. A simple explanation of how compound growth works and what it would mean for some key variables in South Africa will show why.
Exponential growth refers to a quantity that increases by a certain percentage each unit of time. For example, suppose you have R100 on January 1, 2012. Then, if it grows by 10% a year, you will have R110 on January 1, 2013. In another year’s time, you will have R121, as the 10% growth rate is applied to (compounded on) R110 – not your original R100. After a decade of such growth, you will have R259. Nice work if you can get it.
A continuous process of exponential growth at a constant rate has a couple of very interesting properties.
First, it results in a doubling of the starting value after a certain number of time periods. It is actually very easy to calculate an approximation of the doubling time for an annual growth process: a quantity growing at a fixed rate of x% a year will double every 70 years divided by x (70/x) years. So, if the growth rate is 7%, the doubling time is ten years (70/7). If the rate is 10%, doubling takes place after just seven years (70/10), and so on. In our earlier example, your money will double in seven years, and will double again (to R400) after about 14 years.
The second surprising feature of continuous exponential growth is that the quantity added in the last doubling cycle is greater than the cumulative sum of all previous cycles. Take the simplest example of doublings: 1, 2, 4, 8, 16. The last doubling cycle added up to 16, while the previous cumulative sum was 15 (1 + 2 + 4 + 8).
Now let us apply our new arithmetic skills to some pertinent aspects of collective South African life: the economy, mining and coal production.
Suppose our economy – as measured by real gross domestic product (GDP) – were to grow by 7% a year, which is the rate our Finance Minister has said is necessary to reduce unemployment, then the GDP will double every ten years, which sounds very nice.
But consumption of goods also produces a stream of waste. So, assuming the structure of our economy remained more or less the same, as did our consumption habits, we would double the annual volume of garbage produced every ten years. If the rate of growth continued at 7% for a few decades, then each new doubling cycle would produce more waste than was generated in our entire history. You can easily imagine that our cities’ landfills would very quickly be overflowing.
Next, let us assume mining production also grew merrily at 7%. Then, after a decade, we would have doubled the annual extraction of minerals, which would bring in a lot of foreign exchange, profits to the mining companies and taxes for the State.
But, if we maintained this rate of growth for several doubling cycles, then, in each new decade, we would mine nearly as much ore as we mined in our entire previous history! (Note that the actual historical growth rate in mining output was mostly considerably less than 7%, which means that we cannot say the very next doubling cycle would produce more than our actual cumulative production at this point in time.)
Not only that – our mine dumps would grow prodigiously, and the amount of new acid mine drainage released into our precious rivers would, before too long, exceed all that went before it.
But just as significant, our remaining – finite – mineral reserves would be depleted at an accelerating rate.
David Rutledge, a professor at the California Institute of Technology, has estimated that our country has about ten-billion tons of mineable coal remaining. At the 2010 rate of production of 25-million tons, we would have just less than 40 years of coal remaining. But, if production grew at 4% a year, then the reserves would run out after just 24 years.
This coal example, by the way, is not at all realistic. Coal output cannot keep steady – or keep increasing – for a number of years and then suddenly collapse to zero. It will reach a peak – possibly around 2020 – and then gradually decline year after year.
So, next time you hear a politician or economist talk about the merits of continuous growth, think about the downsides of the arithmetic as well. Your children are counting on you.
Published in Engineering News , 11 May 2012
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