How to Calculate Scarcity Rent (Hotelling's Rule)
Scarcity rent is the price of a nonrenewable resource minus its marginal extraction cost, and in competitive equilibrium that rent grows at the interest rate every year.
The Scarcity Rent formula
Calculator
Enter the resource price, the extraction cost and the interest rate to get the scarcity rent and its path over time.
What a unit of the resource sells for right now.
The cost of getting one more unit out of the ground, with no allowance for the resource itself.
What the owner could earn by selling now and investing the proceeds.
How far into the future you want the rent and the price.
Growing at the interest rate the whole way, the rent reaches $32.58 a unit by the year you entered.
- Scarcity rent today
- $20
- Scarcity rent next year
- $21
- Price after the wait
- $72.58
- Annual growth rate of the price
- 1.92%
- Rent share of the price after the wait
- 44.9%
Price minus extraction cost leaves $20 a unit, which pays for the resource itself rather than for the work of lifting it.
An owner leaves a unit in the ground only if the rent reaches $21 next year, which is what selling now and investing the proceeds would deliver.
Adding extraction cost back on top of the grown rent puts the price at $72.58 a unit.
The price climbs 1.92% a year, slower than the rent, because the extraction cost underneath it does not grow at all.
The rent moves from 33.3% of the price today to 44.9% of it later, so scarcity takes over more of what buyers pay.
How to calculate Scarcity Rent, step by step
- 1Strip out the extraction cost. Subtract the marginal extraction cost from the current price. What is left is the scarcity rent, the payment for the unit of resource itself rather than for the work of lifting it.
- 2Use the interest rate as the growth rate. An owner leaves a unit in the ground only if waiting pays as well as selling now and investing the money, so in equilibrium the rent has to grow at the interest rate r.
- 3Grow the rent forward. Multiply today's rent by (1 + r) for one year, or by (1 + r)^t for t years. This is the compounding step, so the rent rises faster the longer the wait.
- 4Add the extraction cost back. Price in year t equals marginal extraction cost plus the grown rent. Extraction cost is assumed steady here, so it does not compound.
- 5Compare the two growth rates. Work out the annual growth rate of the price itself. It comes in below r whenever extraction cost is above zero, and that gap is the result students most often miss.
Worked example: Scarcity Rent
Oil sells for $60 a barrel and lifting the next barrel costs $40, so the scarcity rent is 60 − 40 = $20. Take an interest rate of 5 percent. An owner holds the barrel back only if next year's rent reaches 20 × 1.05 = $21, which means a price of 40 + 21 = $61. Ten years out the rent has grown to 20 × 1.05^10 = $32.58 and the price to 40 + 32.58 = $72.58. That price rose only 1.92 percent a year, far below the 5 percent the rent grew at, because the $40 of extraction cost underneath it never compounds. Rent has gone from 33.3 percent of the price to 44.9 percent of it.
Scarcity Rent questions
Why does scarcity rent grow at the interest rate?
Because an owner has two ways to hold the same wealth. Selling a unit now and investing the proceeds earns the interest rate, so leaving the unit in the ground has to earn the same return through a rising rent. If the rent were expected to grow more slowly, owners would extract faster today, and the extra supply would push today's price down until the two returns matched again.
Does the price of a nonrenewable resource rise at the interest rate too?
No, only the rent portion does. Price equals extraction cost plus rent, and the extraction cost part does not compound, so the price grows more slowly than r. In the example above the rent grew at 5 percent a year while the price grew at 1.92 percent. The larger extraction cost is relative to price, the wider that gap.
What is scarcity rent also called?
It goes by user cost, royalty and Hotelling rent as well. All four names point at the same thing: the opportunity cost of selling a unit today instead of keeping it for a future sale. It is the reason a competitive resource price sits above the cost of bringing the next unit out of the ground, which is not true of an ordinary manufactured good.
What happens if new reserves are discovered?
A discovery adds to the stock, so the opportunity cost of selling a unit today falls and the scarcity rent drops with it. Price falls toward extraction cost even though nothing about the cost of drilling changed. The same logic runs in reverse when a substitute technology arrives, since a cheaper backstop caps the price the resource can ever reach and pulls the rent down today.
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