Financial & Accounting
Hotelling’s Rule Formula: Exhaustible Resource Scarcity Rent & Price Trajectory
Project competitive mine, lithium, oil, and mineral equilibrium prices P_t = (P_0 - c) * (1 + r)^t + c using Harold Hotelling’s 1931 arbitrage law that in-situ net scarcity rent (P_t - c) rises at the real interest rate r.
Why a mineral owner is indifferent between extracting today and leaving ore in the ground only when net rent grows at r!
Harold Hotelling (1931, Journal of Political Economy — "The Economics of Exhaustible Resources") proved that non-renewable mineral reserves in the ground are capital assets! If constant unit extraction cost is c and spot price is P_0 > c, extracting 1 ton today yields immediate net scarcity (royalty) rent R_0 = P_0 - c that can be invested in financial markets at real interest rate r to become (P_0 - c) * (1 + r)^t at year t!
Therefore, intertemporal no-arbitrage requires in-situ scarcity rent R_t = P_t - c to compound at rate r: P_t = (P_0 - c) * (1 + r)^t + c! Notice that gross market price P_t grows SLOWER than r whenever extraction cost c > 0! For spot price P_0 = $85/bbl, marginal lifting cost c = $35/bbl (initial scarcity rent R_0 = $50/bbl), and real discount rate r = 6.0% over t = 10 years, equilibrium price reaches P_10 = 50 * 1.06^10 + 35 = $124.54/bbl (climbing to $142.95/bbl at r = 8.0%)!
| Resource Valuation Scenario (P_0=$85, Cost c=$35) | Initial Scarcity Rent R_0 | Year-10 Rent R_10 = R_0*(1+r)^10 | Hotelling Price P_10 = R_10 + c |
|---|---|---|---|
| Real Discount Rate r = 6.0%, t = 10 yr | $50.00 / unit | $89.54 / unit | $124.54 / unit |
| Higher Capital Cost r = 8.0%, t = 10 yr | $50.00 / unit | $107.95 / unit | $142.95 / unit |
| Uneconomic Deposit P_0 <= Cost c | Non-Positive Rent | Deferred Extraction | "Requires Spot Price Above Extraction Cost" |
Hotelling Equilibrium Exhaustible Resource Price P_t = (P_0 - c) * (1 + r)^t + c
Projects competitive resource price at horizon t (B5) from initial price P_0 (B2), constant unit extraction cost c (B3 < P_0), and real discount rate r (B4 > 0).
=IF(OR(B2 <= B3, B3 < 0, B4 <= 0, B5 < 0), "Requires Spot Price Above Extraction Cost", ROUND((B2 - B3) * ((1 + B4) ^ B5) + B3, 2))Initial In-Situ Net Scarcity Rent (Royalty Value) R_0 = P_0 - c
Computes the unit resource shadow price / user cost in the ground at t = 0.
=IF(OR(B2 <= B3, B3 < 0, B4 <= 0, B5 < 0), "Requires Spot Price Above Extraction Cost", ROUND(B2 - B3, 2))Try the example
Use the original values or change them to test your understanding.