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In the abstract realm of philosophy, Achilles can never overtake the tortoise; in the real world, it is not so. (Screengrab: OpenEdu video)

Paul Krugman and The Tortoise: Why the Limits to Growth Are Real

Ugo Bardi

One of the best known of Zeno's paradoxes says that Achilles will never be able to overtake the tortoise in front of him, no matter how fast he runs. Why? Because Achilles must first reach the point from where the tortoise started, which means the tortoise is always ahead. It is a nice illustration of how easy it is to build abstract theories that hold little relation with reality.

Paul Krugman seems to have created some kind of a similar paradox with a recent New York Times article (“Slow Steaming and the Limits to Growth”) where he sets out to demonstrate that the world's gross domestic product (GDP) can continue growing even while reducing energy production. He does that by means of the example of “slow steaming.” Noting that the energy consumption of ships grows more than linearly as a function of speed, Krugman states that you can always save energy by slowing ships down and increasing their number. If the relation of speed to energy consumed is, let's say, quadratic, then by doubling the number of ships and halving their speed you reduce the energy consumed to a half. So, it is possible to maintain a constant economic throughput while reducing the input of energy to the system. Q.E.D.? Well, unfortunately things are not so simple.

The story of these slow steaming ships looks suspiciously like one of Zeno's paradoxes in the sense that you could keep forever doubling the ships and forever reducing the energy consumed – just as Achilles can keep running forever without ever reaching the turtle. It just doesn't sound right for the real world and, indeed, if we look carefully into into Krugman's argument, we see that there is something important that he has overlooked. The trick works if – and only if – the cost of ships remains constant while you increase their number. That's far from being obvious. 

A freight ship is not just an empty hull that stores containers; it needs metals, ceramics, and semiconductors for its electronics, its engines, its control system, and more. Making more ships means that more minerals have to be produced and utilized. But minerals are limited resources, in the sense that the high-grade ores from which we mine them exist in limited amounts. An extensive attempt to substitute energy with capital in the world's economy would result in a severe strain on the mining industry that would be forced to produce at high costs from low-grade ores. The consequence is that the prices of mineral commodities would go up (this is not just theory, it is exactly what's happening in the real world, owing to the gradual depletion of high grade ores). But, if prices go up, demand is destroyed and, as a result, the GDP goes down, not up, as Krugman maintains.  

So, Krugman's argument may to be indeed a paradox in the sense that the attempt to increase the GDP by saving energy may well backfire; creating the opposite effect. But why is Krugman (and many others) so concerned about energy and so cavalier about mineral resources? It is because it is generally understood that today energy is mainly obtained from fossil fuels and it is also commonly understood that fossil fuels exist in limited and non-replaceable amounts that we are gradually consuming. So, it is generally understood that we have a depletion problem with fossil fuels. What is not so commonly understood is that the situation is the same for all mineral commodities. So, switching from a limited mineral resource (fossil fuels) to other limited mineral resources (metals and others) merely amounts to shifting the problem to one sector of the economy to another. Energy saving is a good thing, but we don't have to take it as the miracle that keeps GDP growing forever.

Krugman's article is not merely about juggling resources from one place to another, but it takes a broad sweep at the very concept that there exist limits to the growth of the GDP (see also a previous article of his), for instance asserting that Bill Nordhaus (his old mentor) had effectively demolished forty years ago the book on limits by Jay Forrester that had preceded the more famous 1972 study titled “The Limits to Growth.” Alas, I am afraid that this is not the case. It is true that, in 1973, William Nordhaus claimed to have found fundamental flaws in Forrester's model. However, Forrester could show in his rebuttal that Nordhaus had simply made a mistake in interpreting the equations of the model. Nordhaus himself appeared to backtrack on his previous interpretation in an article on the same subject that he published in 1992; since he never mentioned these supposed flaws again. The story of this debate is told in some detail in my book The Limits to Growth Revisited (Springer 2012). All this doesn't mean that the “Limits to Growth” does not have limits in its approach, but simply that it cannot be dismissed so easily, especially considering that it is becoming clearer and clearer that it has been able to correctly describe reality, up to now.

In the abstract realm of philosophy, Achilles can never overtake the tortoise; in the real world, it is not so. In the abstract realm of economics, the GDP can grow without natural resources; in the real world, it is not so. In the end, we need to be wary of abstract theories and remember that the limits to growth are real.


Our work is licensed under Creative Commons (CC BY-NC-ND 3.0). Feel free to republish and share widely.

Ugo Bardi

Ugo Bardi teaches physical chemistry at the University of Florence, Italy. He is the author of The Limits to Growth Revisited (Springer 2012) and Extracted: How the Quest for Mineral Wealth is Plundering the Planet (Chelsea Green 2014).

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