The Hotelling Rule
Why the Price of a Finite Resource Must Rise at the Rate of Interest
Before You Start
You should know
That extracting a finite resource today means it cannot be extracted later, and that interest rates create an opportunity cost for waiting.
You will learn
How the Hotelling rule is derived, what scarcity rent means, and why high-cost and low-cost producers experience the same price path differently.
Why this matters
Resource extraction timing is one of the core economic decisions behind royalties, reserves, and long-run energy strategy.
If this gets hard, focus on…
The basic arbitrage question: should a producer sell now or wait one more year?
In 1931, Harold Hotelling published a paper that remains one of the most elegant results in resource economics. The question he posed seems almost trivially simple: when should you extract a finite resource? Yet the answer — that the price must rise at exactly the rate of interest — has profound implications for how we understand oil, minerals, groundwater, and every other exhaustible resource on Earth.
Alberta sits at the heart of this problem. The Athabasca oil sands hold roughly 165 billion barrels of proved reserves — the third-largest oil reserve in the world. Every decision about how quickly to extract that bitumen, what royalty to charge, and what to do with the revenues is, implicitly or explicitly, a decision shaped by Hotelling’s logic.
The Fundamental Arbitrage
The core insight is an arbitrage argument. Suppose you own an oil well. You have two options:
- Extract today, sell the oil at price P_0, invest the proceeds at interest rate r.
- Wait one year, extract then, sell at price P_1.
For you to be indifferent between these options — for there to be no profitable arbitrage — the payoff must be equal:
P_0(1 + r) = P_1
Or in continuous time, the Hotelling rule:
\frac{dP}{dt} = r \cdot P
This is the fundamental no-arbitrage condition for exhaustible resources. If price rises faster than r, everyone waits — supply collapses, price rises even faster. If price rises slower than r, everyone extracts now — supply floods the market, price falls. The only stable equilibrium has price rising at exactly r.
The solution is exponential growth from an initial price P_0:
P(t) = P_0 \cdot e^{rt}
Sell Now Or Wait? Hotelling Says The Owner Must Be Indifferent
The chapter becomes much easier once the rule is seen as a timing comparison. If selling later beats selling now and investing, owners delay extraction. If not, they accelerate production. Equilibrium requires the two options to match.
Extract today
Sell at P₀, invest proceeds at interest rate r, end next year with P₀(1+r).
Wait one year
Leave the barrel in the ground and sell later at P₁.
No-arbitrage condition: if P₁ > P₀(1+r), wait. If P₁ < P₀(1+r), extract now. Equilibrium requires P₁ = P₀(1+r).
Deriving the Rule Formally
The formal derivation uses optimal control theory. A resource owner has a finite stock S_0 and chooses an extraction path q(t) to maximise the net present value of profits:
\max_{q(t)} \int_0^T \left[ P(t) - c \right] q(t) \, e^{-rt} \, dt
subject to:
\dot{S}(t) = -q(t), \quad S(t) \geq 0, \quad q(t) \geq 0
where c is the (constant) marginal extraction cost and S(t) is the remaining stock at time t.
The Hamiltonian for this problem is:
\mathcal{H} = \left[ P(t) - c \right] q(t) e^{-rt} + \lambda(t) \cdot (-q(t))
where \lambda(t) is the costate variable — the shadow price of the resource stock, measured in present-value terms. The optimality conditions require:
\frac{\partial \mathcal{H}}{\partial q} = 0 \implies (P - c)e^{-rt} = \lambda
and the costate equation:
\dot{\lambda} = 0 \implies \lambda = \text{constant}
This means (P - c)e^{-rt} must be constant, so the net price (price minus extraction cost) must rise at rate r:
\frac{d(P - c)}{dt} = r(P - c)
Two Components of the Resource Price
This derivation reveals something crucial. The full price of a resource has two components:
P(t) = c + \rho(t)
where: - c = marginal extraction cost (the cost to lift one more barrel) - \rho(t) = scarcity rent (also called the royalty or Hotelling rent)
The scarcity rent is the shadow price of the finite stock — it compensates the owner for permanently depleting a non-renewable asset. It rises at rate r over time. The extraction cost (if constant) does not rise.
Alberta illustration: The oil sands have a marginal extraction cost of roughly $25-$35 per barrel for SAGD operations, compared to roughly $5-$10 for conventional Alberta oil and $2-$5 for Middle Eastern conventional. This means:
- At $80 WTI, an Alberta SAGD producer earns a scarcity rent of roughly $45-$55/bbl
- A Saudi producer earns a scarcity rent of roughly $70-$75/bbl — much larger
- High extraction costs compress the scarcity rent, which is why conventional producers have more to lose from depletion
The royalty charged by the Alberta government is supposed to capture some portion of this scarcity rent — the return that belongs, in principle, to the resource’s public owners.
The Backstop Technology
Extraction stops when the price reaches the backstop technology price — the cost of a substitute that can perfectly replace the resource. If oil can be replaced by synthetic fuel at $150/bbl, then no one will pay more than $150 for conventional oil, regardless of scarcity.
The backstop price \bar{P} sets the terminal condition for the Hotelling path. The initial price P_0 is determined by:
P_0 = \bar{P} \cdot e^{-rT}
where T is the time to exhaustion. This means the initial price and the depletion horizon are jointly determined — you cannot set one without the other.
For Alberta’s oil sands, the backstop is arguably set by the carbon-adjusted cost of electric vehicles, hydrogen, and synthetic aviation fuel. As these costs fall, \bar{P} falls, and the entire Hotelling price path shifts down. This is one reason why “stranded asset” discussions have intensified since 2015.
Interest Rates and Extraction Speed
One of the most practically important implications of the Hotelling rule is the relationship between interest rates and extraction speed.
When r is high, the price must rise steeply — which means the initial price is low (to fit the exponential path under the backstop constraint). Low initial prices mean high early demand, fast extraction, rapid depletion.
When r is low, the required price growth is shallow — the initial price can be higher, demand lower, extraction slower.
This explains why the decade of near-zero interest rates from 2010 to 2021 was, paradoxically, associated with lower incentives to rapidly extract — and why rising rates after 2022 changed extraction economics. Lower r means patience is less costly; the Hotelling logic says wait.
The chart shows three stylised Hotelling paths with different interest rates, all terminating at the same backstop price. High interest rates (blue) produce fast price growth from a low initial level — rapid early extraction. Low interest rates (green) produce slow price growth from a high initial level — more patient depletion.
Does Oil Actually Follow the Hotelling Rule?
The empirical track record is mixed. Observed oil prices are far more volatile than any Hotelling path would predict — they whipsaw with geopolitical events, OPEC decisions, demand shocks, and financial crises. The 2014 price collapse from $110 to $45 in six months has no counterpart in Hotelling’s smooth exponential.
What the rule gets right is the long-run framework. Over decades, resource prices do tend to reflect scarcity — they have risen in real terms for most finite commodities over the past century, broadly consistent with Hotelling’s logic. The rule is best understood as a description of the gravity well that market prices orbit around, not a precise forecast of day-to-day prices.
Price Decomposition: Oil Sands vs Conventional
The two-component price structure looks very different across resource types. When WTI is at $60, $80, or $100, the split between extraction cost and scarcity rent is not the same for all producers.
The chart illustrates why conventional producers (Middle East, conventional Alberta) have much larger scarcity rents per barrel than oil sands or shale — and therefore more to lose from depletion. It also shows why at $60 WTI, high-cost oil sands producers are near their break-even: the “scarcity rent” in that column is really almost entirely a return of capital, not economic rent.
What This Means for Resource Policy
The Hotelling rule has direct policy implications:
Royalty design: If the scarcity rent is the portion that should accrue to Albertans as public resource owners, then royalties should track the net price (P - c), not the gross price P. Flat percentage royalties on gross revenue systematically under-capture rent when prices are high and over-burden producers when prices are low.
Depletion rate: The optimal depletion rate depends on the interest rate, which is itself partially a policy variable. Alberta’s low-royalty, fast-extraction approach implicitly accepts a high effective discount rate — prioritising current revenue over future resource wealth.
Energy transition timing: If the backstop price is falling — as electric vehicle costs decline and carbon prices rise — the Hotelling path shifts. Optimal extraction accelerates now, before the backstop arrives, not slows. This creates a political economy tension: the resource curse logic says go slow, the Hotelling logic says go fast before the window closes.
References
Alberta Energy Regulator. 2025. Alberta Energy Outlook (ST98): Crude Bitumen Supply Costs. https://www.aer.ca/data-and-performance-reports/statistical-reports/alberta-energy-outlook-st98/crude-bitumen/crude-bitumen-supply-costs
Alberta Energy Regulator. 2025. Alberta Energy Outlook (ST98): Reserves. https://www.aer.ca/data-and-performance-reports/statistical-reports/alberta-energy-outlook-st98/reserves
Government of Alberta. 2025. Oil Sands Facts and Statistics. https://www.alberta.ca/oil-sands-facts-and-statistics
Government of Alberta. 2025. Oil Prices and Value. https://www.alberta.ca/oil-prices-and-value
Hotelling, Harold. 1931. “The Economics of Exhaustible Resources.” Journal of Political Economy 39 (2): 137–175. https://doi.org/10.1086/254195
International Energy Agency. 2021. Net Zero by 2050: A Roadmap for the Global Energy Sector. Paris: IEA. https://www.iea.org/reports/net-zero-by-2050
Norges Bank Investment Management. 2025. The Fund. https://www.nbim.no/en/
Pindyck, Robert S. 1978. “The Optimal Exploration and Production of Nonrenewable Resources.” Journal of Political Economy 86 (5): 841–861. https://doi.org/10.1086/260711