# Monthly Archives: September 2019

## On Exhaustible Resources, Part 2

Yesterday’s post on exhaustive resources has drawn a lot of ire from critics. Some have argued that I didn’t address the problem of economic growth. In short, the argument is that there are two sources of economic growth. The first is that increased efficiency of resources allows us to produce more stuff with the same amount of resources. The second is that because resources are more productive we tend to use more of them. Others have argued that algebra is irrelevant to the problem.

I’d like to address both of these concerns because they are wrong. First, let’s address the algebra issue. The model I presented in my previous post is an example of using formal economic theory to make a point that is apparently not obvious to people. If society has exhaustible resources, will markets completely deplete those resources and leave us with nothing? What the model shows is that this will not happen. It doesn’t happen because as the resource is depleted, the price of the resource rises thereby encouraging people to use less of it. (Correspondingly, if resources are near the point of depletion shouldn’t energy prices be a lot higher?) So attacking me for using algebra will get applause from a certain type of audience and “algebra doesn’t solve environmental calamity” makes a really good bumper sticker, but it is not a valid critique. The model is an exercise in maintaining consistent logic.

Now to the more substantive critique. This is the critique that growth not only comes from changes in productivity but that these changes in productivity lead to greater resource use. So let’s tackle this problem head-on using a modified version of the Solow Model.

Before going through the model let’s recall the crux of the debate:

• George Monbiot claimed that perpetual growth is not possible in a world of finite resources.
• I replied that perpetual growth comes from finding more efficient ways to use resources (the ability to produce the same amount of stuff with fewer resources).

Let’s imagine that there is an aggregate production function that is given as

$Y = (AR)^{\alpha}K^{1 - \alpha}$

where $Y$ is output, $R$ is the quantity of exhaustible resources, $K$ is capital, $\alpha \in (0,1)$ is a parameter, $A$ is the productivity of energy use. So $AR$ has the interpretation of “effective units of resources.” Now let’s assume that

$dR = -cRdt$

where $c$ is the rate of resource extraction. Note here that I am assume no uncertainty. The amount of resources are known and declining with use.

Also, I will assume that

$dA = gAdt$

where $g$ is the growth rate of the productivity of energy use.

Finally, the law of motion of the capital stock is given as

$dK = (sY - \delta K)dt$

where $s \in (0,1)$ is the savings rate and $\delta \in (0,1)$ is the depreciation rate on capital.

Define $e = AR$ as effective units of resources and $k = K/e$ as capital per effective unit of resources. The corresponding law of motion for capital per effective unit of resources is given as

$dk = [sk^{1 - \alpha} - (\delta + g - c)k]dt$

From this equation, there is a stable and unique steady state equilibrium for $k$ if $\delta + g - c > 0$. A sufficient condition for this to hold is $g - c > 0$.

Now, let $y = Y/e = k^{1 - \alpha}$. Note that this implies that in the steady state, $dy = dk = 0$. Thus, output per effective unit of resources should be constant in the steady state. This implies that the growth rate of output itself satisfies

$\frac{dY}{Y} = (g - c)dt$

It follows that in the steady state equilibrium, we can experience perpetual economic growth so long as the productivity of energy use rises by more than enough to offset the rate of resource extraction. Put differently, we can experience long-run economic growth even in a world of finite resources as long as we continue to use those resources more efficiently. Recall that Monbiot argued that it is impossible. I, on the other hand, argued that this is incorrect because growth is the result of being able to produce the same amount of stuff with fewer resources. This is precisely what I meant.

Of course, we might wonder if this is actually going on in reality. So let’s go to the data. We can measure the productivity of resource use by plotting GDP relative to energy consumption. The following figure is from the World Bank.

As one can see from the graph, there has been a considerable productivity increase in the use of energy over the last few decades. This is not the whole story since this graph only measure $g$. One would need to compare this to $c$ to determine whether we are currently on a sustainable path. Nonetheless, the claim made by Monbiot was that perpetual growth is not possible in a world of finite resources. What I have shown is that this is wrong as a logical statement. Furthermore, my basic model in this post actually understates our ability for perpetual growth since I assumed that it is not possible to substitute from the exhaustible resource to either another exhaustible resource or to a renewable resource.

## 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.