Suppose the rectangle below represents the total quantity of some non-renewable natural resource. The various stocks of this resource are sorted left to right by degree of certainty, and top to bottom by cost of extraction. Since we aren't necessarily certain about the total quantity that would be available at very high prices or that would be found after every part of the Earth is thoroughly explored, the right and bottom boundaries of this rectangle are fuzzy.

The USGS defines "reserves" as the portion of the natural resource base that is currently identified and economical to extract.  As consumption reduces reserves, the price of the resource increases, converting some sub-economic resources into reserves by definition.  The price increase also motivates exploration (shifting the reserves boundary to the right), R&D into lower-cost extraction technologies (shifting the reserves boundary downward), demand-side conservation and substitution of alternative resources.

The static reserve index is calculated by dividing current reserves by annual consumption to estimate the number of years remaining until current reserves are depleted.  This is very misleading, since it confuses reserves with the overall resource base, and it doesn't account for price responses (motivating conservation, substitution, R&D and new exploration) to increasing reserve scarcity.

Static efficiency means resources are allocated optimally within a single time period. Dynamic efficiency means resources are allocated optimally over multiple time periods.

We assume the market for an exhaustible resource is competitive and each individual resource owner is trying to maximize his or her profits (rents) from the resource through time.   Rational owners try to anticipate future market prices (P_t) extraction costs (MC_t) and marginal resource rents (MRR_t = P_t - MC_t). Selling a unit of the resource now (t=0) yields MRR₀, but also involves an opportunity cost which is the largest foregone MRR_t that unit could be expected to earn if sold in any other time period t.

In comparing current versus expected future rents, the owner discounts the future rents at some discount rate r, and only sells now if MRR₀ >= MRR_t/(1+r)^t for all future time periods.

The amount all resource owners collectively sell in any time period determines the market price in that time period, so each individual owner watches the market for the best opportunities to sell.   Owners are indifferent between selling in different time periods if their expected rents are rising at the rate of discount through time:

This step rule characterizes the rent-maximizing trajectory of MRR.   It was originally formalized by Harold Hotelling, and is sometimes referred to as "Hotelling's Rule".

Assume the total stock X = 30 units, WTP_t = 10 - Q_t, MC = $2, and r = 0.10.
Determine the optimal MRR, price and extraction schedules, and the optimal time to depletion.

These problems are best solved backwards: We don't know how many years off the depletion point (year T) will be, but we do know that at the time of depletion Q_T = 0, WTP_T = $10 and therefore MRR_T = $8.

If MRR_T = $8, Hotelling's Rule implies that the marginal resource rent in the previous year MRR_T-1 = MRR_T/(1+r) = $7.27.

Since MRR_t = P_t - MC, P_T-1 = MRR_T-1 + $2 = $9.27.

And since WTP_t = 10­Q_t, Q_T­1 = 10­P_T­1 = 0.73 units, and the total stock remaining would be 0.73 units.

In the prior year T-2, R_T­2 = R_T/(1+r)² = $6.61, P_T­2 = $8.61, Q_T-2 = 1.39 units, and the total stock remaining would be 0.73 + 1.39 = 2.12 units.

In year T-3, R_T-3 = R_T/(1+r)³ = $6.01, P_T-3 = $8.01, Q_T-3 = 1.99 units, and the total stock remaining would be 0.73 + 1.29 + 1.99 = 4.11 units.

You can solve for R, P and Q in each prior year until the total stock remaining matches your current stock level at some period T-N, which is today. So depletion will occur in N years. The full schedule is easily solved for using a microcomputer spreadsheet program:

                      Model 1: r = 0.10; X=30

             Time     Rt    MCt     Pt     Qt     Xt

     (future)  10   $8.00  $2.00 $10.00   0.00   0.00
         ^      9   $7.27  $2.00  $9.27   0.73   0.73
         |      8   $6.61  $2.00  $8.61   1.39   2.12
         |      7   $6.01  $2.00  $8.01   1.99   4.11
         |      6   $5.46  $2.00  $7.46   2.54   6.64
         |      5   $4.97  $2.00  $6.97   3.03   9.67
         |      4   $4.52  $2.00  $6.52   3.48  13.16
         |      3   $4.11  $2.00  $6.11   3.89  17.05
         |      2   $3.73  $2.00  $5.73   4.27  21.32
         |      1   $3.39  $2.00  $5.39   4.61  25.93
     (present)  0   $3.08  $2.00  $5.08   4.92  30.84

The time to depletion is approximately T = 10 years. Notice that resource use Q declines gradually as P approaches the choke price of $10. The competitive market allocates the resource so that it is used just as P reaches the choke price. It would not be rational for resource owners to hold the reserves any longer, since rents would stop rising at all after time T.

While this basic model describes a stable optimal MRR trajectory with perfect certainty about future prices, extraction costs and reserves, it also explains why changing expectations about the future will affect current markets.

In reality, since expectations about future prices, extraction costs and reserves are constantly changing, competitive resource markets constantly adjust their extraction and MRR trajectories to a moving depletion target point.

For example, if rents are expected to rise faster than the rate of discount, rational owners will withhold the resource from the current market, increasing current prices to a level from which expected rents do rise at the rate of discount.   If rents are expected to rise more slowly than the rate of discount, rational owners will sell off more in the current market, reducing current prices to a level from which expected rents do rise at the rate of discount.  Changing expectations about the future will shift the entire extraction, price and MRR trajectories upward or downward.