Exhaustible Resource Use in Continuous Time: Hotelling’s Rule and Beyond

When you look at a finite resource, like a seam of coal, a deposit of lithium, or a reservoir of oil, what determines how fast we use it? It might seem like a chaotic rush to extract, but economists have found a surprisingly elegant logic hiding beneath the surface. We’re not just digging randomly; we’re solving a complex puzzle over time. How much should we extract today versus saving for our grandchildren? This question is at the heart of resource economics, and its foundational answer was first proposed by economist Harold Hotelling in 1931.

His insight, now known as Hotelling’s Rule, provides a framework for understanding why the prices of exhaustible resources *should*, in a perfect world, follow a very specific path. Let’s dig into this idea, starting with the simplest case and gradually adding real-world complexity.

The core idea: A mine owner’s big decision 🔗

Imagine you own a small mine containing a finite amount of, say, copper. To keep things simple, let’s make a big assumption: extracting this copper is completely costless. (This is a thought experiment, like a physicist imagining a frictionless surface.)

You face one central dilemma every single day: Should I extract a ton of copper and sell it today, or should I leave it in the ground to sell next year?

To answer this, you need to compare two things. If you sell it today, you get cash, which you can put in a bank (or invest) to earn interest. Let’s call this interest rate, or more broadly, the discount rate (δ). This rate represents the opportunity cost of money; a dollar today is worth more than a dollar tomorrow because today’s dollar can be invested and grow.

If you leave the copper in the ground, it’s like an investment. Its value will (hopefully) increase as the resource becomes scarcer. Your decision hinges on which investment pays better: the bank or the “ground.”

Enter Hotelling’s Rule (The ‘Golden Rule’ of extraction) 🔗

Harold Hotelling’s genius was in defining the equilibrium point. He argued that in a competitive market, an equilibrium is reached only when the mine owner is *indifferent* between selling today and selling tomorrow.

When would you be indifferent? You’d be indifferent if the value of the copper in the ground is appreciating at *exactly the same rate* as the money you’d get from selling it.

This gives us the famous Hotelling’s Rule: The price of the resource must rise at a rate equal to the discount rate. In mathematical terms: (dp/dt) / p = δ

• If the price of copper was rising *faster* than the discount rate (e.g., 10% price appreciation vs. 5% interest), no sane mine owner would sell today. Why take 5% from the bank when your “ground investment” is yielding 10%? Everyone would hoard, supply would vanish, and today’s price would be bid up until the expected future appreciation fell back to 5%.
• If the price was rising *slower* than the discount rate (e.g., 2% price appreciation vs. 5% interest), everyone would sell *immediately*. Why keep a 2% investment when the bank offers 5%? This glut of supply would crash today’s price, which in turn would make the *future* price rise seem much larger, pushing the appreciation rate back up toward 5%.

The rule ensures that the present value of the resource is constant, no matter when you extract it. It creates a smooth, predictable path for extraction over time.

When does the mining stop? The terminal conditions 🔗

A path is great, but we also need a destination. We know the resource is finite, so extraction *must* end at some “terminal” time (let’s call it Time T). Economic models use special conditions, sometimes called transversality conditions, to make sure the ending isn’t absurd (like, for example, a model where you decide to save the resource forever, even when it has no value).

The performance test 🔗

This is the most intuitive condition. It simply states that at the terminal time (Time T), the reason you stop extracting is that it’s no longer profitable. In our simple costless extraction model, this is straightforward: you stop at Time T because you have run out of the resource. There is physically nothing left to extract.

As we’ll see later, if extraction *has a cost*, this test means you stop when the market price is no longer high enough to cover the cost of pulling that last ton out of the ground.

The asset test 🔗

This is a more formal mathematical condition, but it has a clear economic meaning. It requires that at the terminal time T, the product of the “shadow price” (the marginal value of the resource in the ground) and the remaining stock of the resource must be zero.

(Shadow Price at Time T) x (Stock at Time T) = 0

This equation tells us that one of two things must be true at the end of the line:

1. Stock at Time T = 0: This is the most common case. The resource is physically exhausted. We used it all up.
2. Shadow Price at Time T = 0: This is the economic alternative. It means that even if there is some resource left in the ground (Stock > 0), it has no *economic* value. Why? Perhaps the cost of extracting it is higher than anyone is willing to pay. The mine is economically exhausted, even if not physically depleted.

These two tests work together to ensure the model has a sensible and economically rational endpoint.

From a single mine to the whole industry 🔗

Now, let’s zoom out from one mine owner to the entire competitive mining industry. All owners are price-takers, and all are following Hotelling’s Rule, so the market price for everyone rises at the discount rate (δ).

But how is the starting price (P0) determined? This is where a new concept comes in: the choke price.

Finding the ‘choke price’ 🔗

The choke price is the price at which demand for the resource falls to zero. This price is usually determined by the cost of a perfect substitute. For example, the choke price for gasoline might be the price at which it becomes cheaper for everyone to buy and operate an electric vehicle. The choke price for coal in power generation might be the price at which building new solar or nuclear plants becomes universally cheaper.

The entire industry’s extraction path is now a beautiful (though theoretical) puzzle:

• We know the total amount of the resource available (Q_total).
• We know the choke price (P_choke), which sets the *final* price.
• We know the price must rise at the discount rate (δ).

The market must find a starting price (P0) that is just right, so that when the price rises at rate δ, it hits the choke price (P_choke) at the *exact moment* the last ton of the resource (Q_total) is extracted. If P0 is too high, the resource will run out while the price is still below the choke price. If P0 is too low, the price will hit the choke price (and demand will drop to zero) while there is still plenty of valuable resource left in the ground.

What if one company controls everything? The monopolist’s game 🔗

The competitive story assumes no single miner can influence the price. But what happens if a single firm-a monopolist-controls the *entire* world’s supply of this resource?

A monopolist doesn’t just take the price; they *choose* the price (or more accurately, the quantity to sell) to maximize their total profit. Your intuition is probably correct: a monopolist will restrict supply to keep prices high. This means they will be more “conservative” than a competitive industry.

The monopolist’s (different) rule 🔗

A monopolist’s calculation is different. They don’t maximize profit by letting the *price* rise at the discount rate. They maximize profit by ensuring their *marginal revenue (MR)* rises at the discount rate.

Remember from basic economics: for a monopolist, marginal revenue is *always* less than the price (MR < P). To sell one more unit, they must lower the price on *all* units, so the revenue from that extra unit is less than its sticker price.

If the monopolist makes their MR rise at rate δ, and we know `P > MR`, this mathematical property means the *price* path will be different. For a standard linear demand curve, a monopolist will:

1. Start with a higher initial price (by extracting less).
2. Exhaust the resource more slowly.

This leads to a fascinating, and somewhat ironic, conclusion: a profit-hungry monopolist ends up “conserving” the resource for longer than a perfectly competitive market would. They aren’t doing it to be green; they are doing it to maximize their own long-term profits.

Getting real: What about extraction costs? 🔗

The “costless extraction” assumption was useful, but it’s time to drop it. Mining is expensive. How does this change the rule?

When costs are constant (The Hotelling Rent) 🔗

Let’s add a simple, constant marginal cost (C) for every ton extracted. Say it always costs $50 to extract one ton of copper.

Now, the “value” of the resource in the ground isn’t its market price (P), but its net price, which is also called the resource rent or Hotelling rent. This is the profit you make on it: `(P – C)`.

The logic from before now applies to this *net price*. The mine owner is indifferent only if the *net price* rises at the discount rate.

The modified rule is: d(P-C)/dt / (P-C) = δ

This changes the price path. The *net price* (P-C) grows exponentially at rate δ. The *actual market price* (P) is just this net price plus the constant cost C. This means the price path for P *parallels* the net price path. Crucially, the *growth rate* of the market price (P) will now be *slower* than the discount rate. This is because the constant cost C makes the base (P) larger than the base (P-C), so the same absolute increase is a smaller percentage growth.

Complicating it further: The ‘stock effect’ 🔗

Let’s add one more layer of reality. Extraction costs are rarely constant. It’s almost always cheaper to get the easy-to-reach stuff first. As the resource stock (S) depletes, the cost of extraction (C) rises. Think of deep-sea oil drilling versus an old-fashioned Texas gusher. This is the stock effect: `C` is a function of `S`.

How the rule changes again 🔗

This fundamentally alters the mine owner’s decision. Now, holding onto the resource (not extracting) gives the owner a “return” from two different sources:

1. Capital Gain: The net price `(P-C)` is still expected to rise. This is the standard Hotelling effect.
2. Future Cost Savings: This is the new, tricky part. By extracting *less* today, you leave a larger stock (S) in the ground. Since a larger stock means *lower* future extraction costs, you are actively lowering your future cost profile by waiting. This “cost-saving” benefit is an additional return on your “ground investment.”

Because the owner is getting this *extra* benefit from waiting (the cost-saving component), the net price `(P-C)` no longer needs to rise as fast as the discount rate (δ) to make them indifferent. The rule becomes more complex, incorporating this cost-saving term.

This can also lead to a shutdown point where extraction stops *before* the resource is physically exhausted, simply because the cost of extracting the last dregs from a depleted stock becomes prohibitively high, exceeding any price the market is willing to pay.

The final twist: Not all resources are created equal 🔗

Our model has one last flawed assumption: that all “copper” is just “copper.” In reality, deposits vary wildly in quality, or “grade.”

The Ricardian insight 🔗

This idea goes back even further than Hotelling, to the economist David Ricardo. The Ricardian insight is simple: you pick the low-hanging fruit first. In mining, this means you extract the highest-grade, lowest-cost deposits first.

No one is going to spend a fortune on a difficult, low-grade mine when a rich, easy-to-access deposit is available. As the market price for the resource rises (following its Hotelling-inspired path), it eventually becomes high enough to cover the expenses of the *next-best* deposit. At this point, that new, lower-grade mine comes online.

This creates a real-world extraction path defined by a sequence. We start with the best ores, and as prices rise, we gradually move to extracting lower-grade, more expensive ores. This continues until the price hits the choke price, at which point even the lowest-grade ores are abandoned in favor of substitutes.

What do you think? Real-world commodity prices (like oil or copper) are famously volatile-they crash and spike. Why do you think they don’t follow the smooth, upward-sloping path that Hotelling’s Rule predicts? And, given that a monopolist slows down extraction, could a resource cartel (like OPEC) be considered an accidental force for sustainability, even if its motives are purely profit-driven?

References

1. https://www.lse.ac.uk/granthaminstitute/explainers/what-is-hotellings-rule/
2. https://www.sciencedirect.com/topics/earth-and-planetary-sciences/hotellings-rule
3. https://econ.ucsb.edu/~tedb/Courses/230A/L18.pdf