Finite Resources and Infinite Growth

Today’s Globe and Mail featured a column by Gary Mason on a world without oil.  ”If you believe that the economy is structured in such a way that it needs to grow continually in order to survive,” it states, “then it will take an endless supply of energy to feed it. ” The article then raises the question,  ”How does an economy grow exponentially forever if the one element it needs more than anything to flourish is contracting with time?” This is a common refrain from environmentalists such as David Suzuki (here, here, here and likely a thousand other places): “it’s absurd to rely on economies based on constant growth on a finite planet.” But, is it? I’ll have more on this at Macleans in a couple of days, but this will serve as a technical primer.

Intuitively, it sounds simple – if I use up a certain amount of a finite quantity each year, it will eventually run out. But that tells you that you can’t have constant or increasing resource extraction from a finite resource, it doesn’t tell you anything about what you do with the resources you extract, how productive they are, or whether or not they enable continued economic growth. It’s certainly possible to sustain exponential growth infinitely with finite resources, as long as productivity improves.

Let me take you through an example (this is a really basic model, but I’ve fit it with some reasonable numbers so its intuitive).  Suppose that gross world product (real, including all environmental costs) is given by 1450*R*X, where R is resource productivity and X is extraction.  If you use oil extraction as a proxy for resources, and we extract about 31.4 billion barrels of oil per year, and let R equal 1, you’ll get a gross world product of $45,515 billion, about the same as the CIA World Factbook estimate of 2012 gross world product. Let’s also suppose, for the sake of this argument, that the 1.8 trillion barrels of oil in current global reserves represents the sum total of all the oil which will ever be extracted – a finite resource.

With those numbers, the myopic approach to maintaining constant growth with no change in productivity would lead to all oil resources being exhausted in 55 years, and then instant economic collapse.

hartwick1

Myopic resource extraction

Of course, this would not actually happen, since prices would adjust even if there were no productivity changes.  To understand what would happen, go to the last period before the collapse – a period in which the world extracts 35 billion barrels of oil out of a remaining stock of about 40 billion barrels.  Knowing what was going to happen if you stuck with that plan, you’d likely decide that it makes sense to carry some extra oil through to the following year, to stave off collapse and/or to profit from absurdly high prices. In doing so, you’d raise prices in that year. Of course, people would have seen this coming too, leading to conservation of oil from previous years as well.  This is a clumsy explanation of what Harold Hotelling wrote down almost 100 years ago – that since oil is like a capital asset, owners will act to maximize returns and this will smooth price and extraction decisions over time.   If you imposed a Hotelling solution – one which maximized the value of oil over time, you’d end up with something which looks something like this:

hartwick2

Smoothed resource extraction

However, Hotelling doesn’t get you to economic growth with finite resources – production is still decreasing over time, and tends asymptotically to zero – it’s just that there is no collapse and oil is distributed over time such that there are no gains in net present value to be achieved by shifting production forward or back in time. (In the graph above, I approximated a 400 year solution – I didn’t solve the full optimal control problem).

If you want to get to increasing economic growth with a finite resource, you need an increase in productivity.  Suppose that you still have the same finite resource stock, but that you become 3% more productive each year in your use of resources – you generate 3% more total product from each unit of resource extraction. The growth in productivity allows you to use fewer resources each year, while still increasing production.  Resource stocks still decline, and approach zero asymptotically, but it’s like going half the distance to the goal line in football – you’ll get closer every time but you’ll never score.

Resource extraction with increasing productivity

Resource extraction with increasing productivity

So, how do you increase productivity?  Energy is used in our economy as a complement to labour and capital, so if you want to increase the productivity of your finite resource then increase energy efficiency, decrease the resource-intensity of energy, increase labour productivity, or increase the quality of your human and physical capital.  This is what Queen’s University economist John Hartwick had in mind when he wrote down the Hartwick rule – the mathematical proof of what I’ve just tried to do in words: as long as you invest sufficiently in improvements in productivity, and manage resources optimally, its possible to sustain infinite growth from a finite resource.  Of course, the Hartwick rule is not a law – it doesn’t guarantee that this will always be achieved, and it certainly doesn’t say that it can be accomplished with any level of investment – it just tells you that its mathematically possible.

Saying that it’s impossible to achieve exponential growth infinitely with finite resources does nothing to advance our discussions of resource management and ignores plenty of evidence to the contrary in the economics literature.  What we should be discussing instead is how to make sure we follow Hartwick’s rule, but that’s another story for another day.

27 responses to “Finite Resources and Infinite Growth”

  1. Blair King

    As a Chemist I would ask a question about your argument. As scarcity becomes an issue and petroleum prices rise we will realize that using petroleum products for their energy content is not an optimum use. The last drop of petroleum refined will not be used to power the future’s version of a Honda,it will be cracked and/or combined into critical petrochemicals with a portion being used to produce a then ultra-expensive drug/chemical and the remainder turned into inputs for a future 3D printer to create some ultra-valuable widget/valve (since it will be the last one created using the technology).

    In the current resource scenario (with ample relatively cheap oil) the limited volume, high-value uses of oil do not really make a dent in the market, but as supply dwindles it will become a much more significant part of the equation (and input cost into the final products).

    Does the presence of a competing, alternative, high-value use for the resource simply change the rate of the curve, move the curve to the left or do something altogether different at some critical price-point?

    Thanks

    Blair

  2. Rolf Muertter

    Absurd. If my paycheck is cut in half every month, I can still increase my standard of living forever? I’m sure you can make the math work out, since we know from calculus 1A that Infinity * 0 = 7. The problem lies in the premise of eternal exponential growth of productivity. How is that possible, especially while the resource base declines exponentially? You can’t make more and more out of basically nothing. Anyone can demand an increase in productivity, but that doesn’t make it happen. This sounds like armchair philosophy to me. Yes, the GWP continues to increase, but so does the human ecological footprint. Since nature tends to be nonlinear, you could have short-term growth in productivity even with a decline in resources, but infinite growth is just a Utopian fantasy. The only kind of “infinite” growth I can imagine is if the resource supply is constant in time, i.e. renewable resources like solar energy, and the GWP approaches a maximum possible value asymptotically. Or, in the language of ecology, the human ecological footprint approaches Earth’s carrying capacity for humans asymptotically.

  3. Roundup: Reorganizing Elections Canada? | Routine Proceedings

    […] economist Andrew Leach looks at the mathematics of infinite growth versus finite resources as a technical primer to a discussion on why we’re not […]

  4. Sherwin Arnott

    “Resource stocks still decline, and approach zero asymptotically, but it’s like going half the distance to the goal line in football – you’ll get closer every time but you’ll never score.”

    This is a Zeno’s paradox redux. One tacit premise of this is that the units are divisible ad infinitum. In other words, this proof works in a math that uses real numbers, but its application to the world we live in is actually a reductio ad absurdum argument against the idea that you can “divide the distance in half” infinitely.

    Hey, where did you go on Twitter? Can’t seem to find you.

  5. Mick Womersley

    I saw this on Revkin’s FaceBook page and thought it (and the MacLeans article) an interesting and misleading oversimplification, so I wrote a correction (or debate) and emailed it to you. You didn’t respond. perhaps you haven’t seen it yet. Here is the link.

    http://ucsustainability.blogspot.com/2014/02/the-wrong-way-to-talk-about-it.html

  6. Mick Womersley

    OK, like I said in my response, we can probably agree on the application of Hotelling. (Hartwick is an addition to Hotelling, so I will continue to refer to the original. It’s the asymptotic Hotelling trajectory that does the intellectual “work” you want done. Apologies to Canadian pride.) I would just put more emphasis on the details of the thinking, engineering, and finance (including subsidy) that goes into advancing beyond fossil energy.

    I think you gloss over my other, possibly more important point, which is the effect to which your attenuated explanation has on the politics and public discourse. You published an article in Macleans blog which, if misread (and it almost certainly will be), essentially justifies oil sands exploitation on the grounds of Hotelling’s rule. Not a word about externalities.

    Not a note about Alberta’s intellectually crippling conflict of interest. I think this is a fairly clear case of denial.

    I’m going to copy over the comments to my own blog, so my students can follow this.

  7. Mick Womersley

    Quick follow up: Isn’t the reason you had to stop your Twitter feed because of the positive and negative blowback from your posts? Demonstrating that you’ve been misread?

  8. Can we have infinite, exponential growth with finite resources? Andrew Leach says Yes. | Sherwin Arnott

    […] the thing about Andrew Leach is that he’s a smart guy and a PhD and he’s really into energy policy and economics and […]

  9. Mick Womersley

    Andrew, it’s not me that’s misreading the piece. I’m fully familiar with the theory, possibly more so than you since I was made to learn the ecological economics criticism too.

    It’s the people making comments on the Twittersphere and the comments on Maclean’s blog that are misreading your piece. This should be obvious, and I find it funny that you can’t see this.

    The reason, which is clear to me, if not to you, is that your explanation is attenuated, if not entirely misplaced in the context in which it is published, especially in Maclean’s, and has political impact that you didn’t imagine would occur.

    Your mystification over “misreading” is perhaps excusable naïveté. What would perhaps be a greater fault would be if you teach this material without balancing it with the various alternate viewpoints.

Leave a Reply