The USGS defines reserves as resources that are currently identified and economical to extract. The magnitude of the overall resource base is unknown (fuzzy boundaries). 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 the reserves are depleted. This is very misleading, since it doesn't distinguish between reserves and the overall resource base, and 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 (Pt) extraction costs (MCt) and rents (Rt=PtMCt). Selling a unit of the resource now (t=0) yields profit R0, but also involves an opportunity cost (Tietenberg calls it "marginal user cost"), which is the largest foregone rent that unit could be expected to earn if sold at any other time. In comparing current versus expected future rents, the owner discounts the future rents at some discount rate r, and only sells now if R0 >= Rt/(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:
R0 = R1/(1+r) = R2/(1+r)2 = ... = RT/(1+r)T
(This condition is known as Hotelling's Rule.) Competitive markets adjust prices to follow this rent trajectory. 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.
Note that changing expectations about the future shift the entire price and rent trajectories upward or downward.
Example: Constant Marginal Extraction Cost Depletion Model:
Assume the total stock X=30 units, WTPt=10-Qt, MC=$2, and r=0.10. Determine the optimal rent, price and extraction schedules, and the optimal time to depletion.
These problems are best solved backwards (I don't know why Tietenberg doesn't use this approach): We don't know how many years off the depletion point (year T) will be, but we do know that QT=0, WTPT=$10 and therefore RT=$8. If RT=$8, Hotelling's Rule implies that rents in the previous year RT1=RT/(1+r)=$7.27. Since Rt=Pt-MC, PT-1=RT-1+$2=$9.27. And since WTPt=10Qt, QT1=10PT1=0.73 units, and the total stock remaining would be 0.73 units. In the prior year T-2, RT2=RT/(1+r)2=$6.61, PT2=$8.61, QT-2=1.39 units, and the total stock remaining would be 0.73+1.39=2.12 units. In year T-3, RT-3=RT/(1+r)3=$6.01, PT-3=$8.01, QT-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 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.