# On Exhaustible Resources

Yesterday, George Monbiot wrote in the Guardian that the survival of capitalism relies on persistent economic growth and persistent economic growth is impossible in the long-run because there are finite resources in the world. In response, I made the following popular, but sarcastic tweet.

The tweet was meant to be funny. The format itself is a meme. Nonetheless, it does drive home the point that the source of economic growth is finding more efficient uses of resources. With this being the internet, however, I started receiving replies telling me that I was an idiot who doesn’t understand exhaustible resources and even had one person recommend that I read up on resource economics. As it turns out, I know a little bit about resource economics — and wouldn’t you know it, resource economics actually supports my position. So I thought it was worth a blog post.

Let’s imagine that we have an exhaustible resource. Suppose that the quantity of the exhaustible resource at time $t$ is given by $R(t)$, where $R(0) = R_0 > 0$. Now let’s suppose that $R(t)$ follows a geometric Brownian motion:

$dR = -cR dt + \sigma R dz$

where $c$ is the rate of resource extraction, $\sigma$ is the standard deviation, and $dz$ is an increment of a Wiener process. The intuition of this assumption is as follows. First, zero is an absorbing barrier here. What I mean is that once $R(t) = 0$, it is permanently there. This is the exhaustible resource part. Second, on average the amount of the resource that is available is declining by the consumption of the resource. Third, there is some uncertainty about the quantity of the resource that is actually available. For example, one might observe positive or negative shocks to the supply of the resource. In other words, there are times when new supplies of the resource are discovered. There are other times in which there is less supply than had been estimated. In addition, one could also include “technology shocks” as a source of positive movement in the supply of resources in the sense that better production processes tend to economize on the use of resources, which is basically the same thing as a discovery new amounts of the resource. In short, what we have here is a reasonable representation of how the supply of an exhaustible resource is changing over time.

Now suppose that the consumption of the resource gives us some utility, $u(cR)$ where utility has the usual properties. The objective is to maximize utility over an infinite horizon (with finite resources). Given the process followed by the resources, I can write the Bellman equation for a benevolent social planner as:

$rv(R) = \max\limits_{c} u(cR) - cR v'(R) + \frac{1}{2} \sigma^2 R^2 v''(R)$

where $r$ is the rate of time preference (or the risk-free interest rate). The first-order condition is given as

$u'(cR) = v'(R)$

Intuitively, what this says is that the marginal utility of the consumption of the resource is equal to the marginal value of the resource. Or that marginal benefit equals marginal cost. In fact, this implies that $v'(R)$ is the shadow price of the resource, or the spot price (more on this below).

Now, for simplicity, let’s suppose that consumers have the following utility function:

$u(cR) = \frac{(cR)^{1-\gamma}}{1 - \gamma}$

It is straightforward to show (after A LOT of algebra) that

$c = \frac{r}{\gamma} + \frac{1}{2}\sigma^2 (1 - \gamma)$

So the rate of resource extraction is constant and a function of the parameters of the model. Or, if we assume that there is log-utility, we can simplify this to $c = r.$ Let’s make this further simplification to economize on notation.

So we can re-write our geometric Brownian motion under log utility as

$dR = -rR dt + \sigma R dz$

So now we have the evolution of resources in terms of exogenous parameters. We might be interested in the quantity of resources in existence at any particular point in time, say time $t$. Fortunately, our stochastic differential equation has a solution of the form:

$R(t) = R_0 e^{-[r + (\sigma^2/2)]t + \sigma z(t)}$

Since exponential functions are always positive and $R_0 > 0$, it must be the case that $R(t) > 0, \forall t$.

So what does this mean in English?

What it means is that given the choice about how much to consume of a finite resource over an infinite horizon, the rate of resource exhaustion is chosen to maximize utility. Given the choice of consumption over time, the total supply of the resource will decline on average over time with the rate of resource exhaustion. However, the quantity of the resource will always be positive.

How is this possible?

$u'(cR) = v'(R)$

Recall that I defined $v'(R)$ as the marginal value of the resource, or the shadow price of the resource. Note that as time goes by, $R$ is declining on average. Since $c$ is constant, when $R$ declines, the marginal utility of consumption rises because total consumption $cR$ is declining. It must therefore be the case that shadow price of the resource increases as well. But the problem I described is a planner’s problem (i.e., how a benevolent social planner would allocate the resource given the preferences for society). Nonetheless, a perfectly competitive market for the resource would replicate the planner’s problem. What this means is that as the resource becomes more scarce, the spot price of the resource will rise so that people economize on the use of the resource. Consumption of the resource declines over time such that the resource is never completely exhausted.

Thus, and somewhat ironically given Monbiot’s point, it would be a deviation from competitive markets for the resource or poorly-defined property rights that might lead us to depart from this outcome. So it’s the markets that save us, not the people who want to save us from the markets.

### 25 responses to “On Exhaustible Resources”

1. Iskander

Good to see you make a post after almost a year! Monbiot has been making the same silly claims for years so its nice to see a detailed exposition of why he is wrong.

2. Paul

An effective response to ecological calamity cannot be algebra, Josh. Monbiot’s point is that, even if the quantity of the resource available falls towards zero aymptotically, eventually systematic reform, or otherwise collapse, will be required.

3. normansdog

pure, total and utter nonsence. Ask the Eater islanders how it went with the trees (oh you can’t, they are all dead), maybe try white rhinos or the cod in the newfoundland cod fisheries.
What a tool

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5. roscopcoldchain

Your post has been deemed “the dumbest post I’ve ever read” over at nakedcapitalism.com. Congrats! Dummy.

6. Amos

Congratulations, you discovered Zeno’s paradox, you nitwit.

7. Matt

Respectfully, Josh, this isn’t about (exceptionally smart economic algebraic) equations. This is about understanding the planet we inhabit and the others we share it with, and the way the natural systems (that we depend on) function.

8. m sam

Reality: from the burning Amazon to mass die offs our world is buckling under resource depletion. If you want proof that limitless growth on a planet of finite resources, simply open your eyes.

You: Look! markets are saving us! Here’s some econometrics.. and so it must be true! I love my ivory tower (the floods will never reach me here), and the warmth of keeping my head under a pillow (all I can see here are my flimsy figures that barely mask the corrupt ideology I serve).

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10. Lord Koos

So as food crops become depleted due to climate change, food will get really expensive, and presto, problem solved! Let them eat dirt.

11. This is why ordinary human beings hate economists.
chuck roast, BA Econ. ’71

12. Harvey

Of course, there are some resources that still might exist approaching zero infinitely, but everyone is dead because an infinitely small amount of the resource is not enough to sustain life. Oxygen! Water! Food!
So even though the elegant math equations hold, there is no-one left to read them because they are all dead. Economists 1 Humanity 0

13. Ander

It tickles me that you think markets provide us with maximum utility over a infinite period, or anything close to it. Markets provide the capitalists who control them with maximum profit, they have nothing to do with societal good or ‘utility’, unless utility is just another word for consumer spending. Even then, companies operate at the behest of shareholders who have no interest in maximizing ‘utility’ over infinity, they have an interest in making a profit within their lifespans.

I understand enjoying this sort of mental play. But I can’t understand how someone could feel that it’s remotely relevant to real world resource consumption, or the ecological implications of that resource consumption.

14. MarkMoon

I wonder how much less asshole-like various commenters would be without anonymity?