Why growth is falling in all developed countries (as a long term trend)

In this post I argue that economic growth is not the normal state of affairs, it is a blip caused by the discovery of a one-off resource of non-renewable energy.

The logistic equation
$$
\frac{dy}{dt} = y(1-y)
$$

Here's a simple model.  Humans discover an exploitable but limited resource and start consuming it.  The amount consumed, $y(t)$, is a function of time.  The general form of the equation is $\frac{dy}{dt} = \alpha y (\beta - y)$, but if you choose the right units $\alpha$ and $\beta$ both become $1$.

Why should this work, in principle?

Early on, the factor $1-y$ is approximately $1$ and can be ignored.  So the model states that annual consumption $dy/dt$ starts off proportional to $y$.  In other words, $y$ grows exponentially at first.  This could happen if exploiting the resource enables further exploitation of the resource.  For example, suppose a few humans are shipwrecked on an island with 1000 trees: they take ages to cut down the first tree as they are using their bare hands, but later trees are cut down with tools made from wood, and iron smelted with wood.

The consumption rate is also proportional to the total remaining resource $1-y$, which eventually comes to dominate: the islanders are spending far more effort for each ton of wood, as the last remaining trees are smaller and harder to access.

Is there any evidence this works in practice?

Surprisingly, yes.  Let's rearrange the equation a bit

$$
\frac{dy/dt}{y} = 1-y
$$

This says that yearly consumption as a fraction of cumulative consumption (the LHS) is a linear function of cumulative consumption, $y$.  This is something we can check: simply choose a resource $y$ whose yearly consumption has been recorded, plot the LHS against $y$ and check to see if you get a straight line.  This is exactly what Thomas Murphy, a physics professor at San Diego, has done for a number of fossil fuel datasets.  Here's the result for UK coal

 

Credit: Thomas Murphy
https://dothemath.ucsd.edu/2011/11/peak-oil-perspective
When the data is displayed in this way the reason the UK government closed down the mines becomes obvious: there was no coal left!

 

The beautiful thing about this method is that, not only does it validate the model, it also predicts the total size of the resource before it's exhausted (it's where the line crosses the horizontal axis).  Repeating the same trick with the other datasets, including global oil production, Murphy finds the same result: all the points, except perhaps the very earliest, fall near a straight line.

 

How does this relate to GDP growth?

The solution to the logistic equation is the logistic function

$$
y = \frac{1}{1+e^{-t}}
$$

(Where $t$ is time since an arbitrarily chosen origin.)  But $y$ corresponds to cumulative production, which is not what we're looking for.  To get a contribution to GDP we need to look at $\frac{dy}{dt}$, and to get a contribution to growth we need to look at $\frac{d^2y}{dt^2}/\frac{dy}{dt}$.  Let's see what these three graphs look like

 

If cumulative production is a logistic function then annual production peaks when half the resource is exploited.  At that point, growth switches from positive to negative.

So, in this crude model, growth appears steady until a little before the peak annual consumption, at which point it starts falling noticeably, eventually becoming negative.  This is approximately what the modellers in the Limits to Growth team found in the early 1970s.  The only way to avoid the crash was to update the model with a change to human values, in particular to eliminate the desire for growth. 

 

Is there any evidence of a slowdown in real life?

Yes. Here's a graph produced by the Australian Treasury showing what has happened to economic growth over the last few decades in developed countries

 

https://treasury.gov.au/publication/economic-roundup-issue-2-2013-2/economic-roundup-issue-2-2013/slowing-productivity-growth-a-developed-economy-comparison

 

The smoothed data shows an obvious downward trend, which has continued in the 10 years since this data was produced.  The other interesting thing about this graph is how the timescales relate to fossil fuel resource extraction.

 

Credit: Thomas Murphy
https://dothemath.ucsd.edu/2011/11/peak-oil-perspective/

 

Oil production grew exponentially until the early 1970s and then started to plateau.  This is exactly the time that growth started falling.  One could argue that the plateau in production was a symptom of falling growth rather than a cause.  However, oil discovery had already peaked and by 1980 we were discovering less each year than we were extracting.  This means that even if demand stayed high, eventually supply would fall short, and throttle growth.  The data for other fossil fuel resources follows a similar pattern.

 

The upshot

Growth has been seen as an economic fact for decades and is taken by politicians (and most economists) to be the normal state of affairs.  If an economy is not growing it is labelled "stagnant" and if it is shrinking we call it a "recession".  Both of these are seen as aberrant situations which can be remedied, either by borrowing to invest or by slashing taxes, depending on the leanings of the politician.  The uncomfortable truth pointed to by the data is that growth was a phase the human race stumbled into when it discovered how to exploit a bunch of non-renewable resources.  Those resources are not completely exhausted yet, but growth starts to fall before even the mid point in resource exploitation, and we are way past that point now.

The solution is not - as Liz Truss, Rishi Sunak, Kier Starmer, and virtually every politician in the world claims - to try and kick start growth.  Growth must end, and it will soon.  The real solution is to fix our institutions so that they no longer depend on growth $(\ddot{y} > 0)$.  Once we've done that we can begin to address our dependency on extraction $(\dot{y} > 0)$.  That will involve learning how to reuse and repair, and how to survive on renewable resources alone.