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.
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: 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.
Effects of Unanticipated New Discoveries
Unexpected discoveries of additional reserves trigger immediate market adjustments. The time horizon to depletion is extended, current rents and prices (and the entire rent and price schedules) are shifted downward, and current consumption levels are increased.
Assume (as before) WTPt=10-Qt, MC=$2, and r=0.10, but suppose unanticipated discoveries suddenly increase X from 30 to 50 units. Determine the new optimal rent, price and extraction schedules, and the optimal time to depletion.
To solve, simply extend Model 1 for 4 additional time periods until Xt reaches 50 units, and redefine the time horizon. Now T=14. Note that periods 4 through 14 are identical to periods 0 through 10 in Model 1.
Model 2: r=0.10; X=50Time Rt MCt Pt Qt Xt 14 $8.00 $2.00 $10.00 0.00 0.00 13 $7.27 $2.00 $9.27 0.73 0.73 12 $6.61 $2.00 $8.61 1.39 2.12 11 $6.01 $2.00 $8.01 1.99 4.11 10 $5.46 $2.00 $7.46 2.54 6.64 9 $4.97 $2.00 $6.97 3.03 9.67 8 $4.52 $2.00 $6.52 3.48 13.16 7 $4.11 $2.00 $6.11 3.89 17.05 6 $3.73 $2.00 $5.73 4.27 21.32 5 $3.39 $2.00 $5.39 4.61 25.93 4 $3.08 $2.00 $5.08 4.92 30.84 3 $2.80 $2.00 $4.80 5.20 36.04 2 $2.55 $2.00 $4.55 5.45 41.49 1 $2.32 $2.00 $4.32 5.68 47.17 0 $2.11 $2.00 $4.11 5.89 53.07
Effects of a Change in r
A decline in r implies a slower rate of growth in rents and prices than before. This in turn implies that initial rents and prices will be higher and the time to depletion will be longer than under the higher discount rate. An increase in r implies a shortened depletion schedule and lower initial prices and rents.
The following example is the same as Model 1, but with r=0.05.
Model 3: r = 0.05; X=30Time Rt MCt Pt Qt Xt 14 $8.00 $2.00 $10.00 0.00 0.00 13 $7.62 $2.00 $9.62 0.38 0.38 12 $7.26 $2.00 $9.26 0.74 1.12 11 $6.91 $2.00 $8.91 1.09 2.21 10 $6.58 $2.00 $8.58 1.42 3.63 9 $6.27 $2.00 $8.27 1.73 5.36 8 $5.97 $2.00 $7.97 2.03 7.39 7 $5.69 $2.00 $7.69 2.31 9.71 6 $5.41 $2.00 $7.41 2.59 12.29 5 $5.16 $2.00 $7.16 2.84 15.14 4 $4.91 $2.00 $6.91 3.09 18.23 3 $4.68 $2.00 $6.68 3.32 21.55 2 $4.45 $2.00 $6.45 3.55 25.09 1 $4.24 $2.00 $6.24 3.76 28.85 0 $4.04 $2.00 $6.04 3.96 32.81
Backstop (Substitute) Resource
The existence of a backstop or substitute resource has an effect similar to a reduced demand choke price. For example, solar power represents an alternative energy source which may not be economical at the present time, but becomes economical as current exhaustible energy resources (oil and gas) are depleted and their prices rise. If the cost of the backstop technology is less than the demand choke price, it sets the upper limit on the price of the exhaustible resource.
The following example is the same as Model 1, but includes a backstop technology costing $6.
If the maximum feasible price for the exhaustible resource is $6, the maximum possible rent to be earned at the point of depletion (time T) will be RT=Pmax-MC=$4. So we can calculate the rent, price and consumption trajectories backward through time:
Model 4: r=0.10; X=30; Backstop = $6Comparing the solutions from Models 1, 2, 3 and 4, new discoveries, an increase in the discount rate or the emergence of a backstop technology all shift the rent and price trajectories downward, increasing current consumption levels.Time Rt MCt Pt Qt Xt 6 $4.00 $2.00 $6.00 4.00 4.00 5 $3.64 $2.00 $5.64 4.36 8.36 4 $3.31 $2.00 $5.31 4.69 13.06 3 $3.01 $2.00 $5.01 4.99 18.05 2 $2.73 $2.00 $4.73 5.27 23.32 1 $2.48 $2.00 $4.48 5.52 28.84 0 $2.26 $2.00 $4.26 5.74 34.58
Differential Extraction Costs
Note that the "resource stock" in these problems is equivalent to "potential reserves" in the USGS resource taxonomy, not "current reserves." Over time, rising prices effectively convert marginally sub-economic resources to reserves, and market expectations account for anticipated future discoveries of new reserves.
What if there are two types or sources of the resource with different extraction costs MC(A) < MC(B) (which are both constant through time) but the same price P? The lower-cost resource will be extracted first. The cost differential implies that rents from the lower cost source R(A)=PMC(A) will be larger than rents from the higher cost source R(B)=P-MC(B). Therefore, under any plausible price trajectory, the rate of increase in R(A) will be less than the rate of increase in R(B).
Assume there are two stocks of a resource, X(A)=20 units with MC(A)=$2 and X(B)=10 units with MC(B)=$5, and WTPt=10-Qt for either stock. Determine the competitive rent, price and consumption trajectories for the two stocks.
Model 5: 2 Stocks with Different Extraction CostsTime R(A) R(B) MC(A) MC(B) Pt Q(A) Q(B) X(A) X(B) 13 $8.00 $5.00 $2.00 $5.00 $10.00 0.00 0.00 0.00 0.00 12 $7.55 $4.55 $2.00 $5.00 $9.55 0.00 0.45 0.00 0.45 11 $7.13 $4.13 $2.00 $5.00 $9.13 0.00 0.87 0.00 1.32 10 $6.76 $3.76 $2.00 $5.00 $8.76 0.00 1.24 0.00 2.57 9 $6.42 $3.42 $2.00 $5.00 $8.42 0.00 1.58 0.00 4.15 8 $6.10 $3.10 $2.00 $5.00 $8.10 0.00 1.90 0.00 6.05 7 $5.82 $2.82 $2.00 $5.00 $7.82 0.00 2.18 0.00 8.22 6 $5.57 $2.57 $2.00 $5.00 $7.57 0.00 2.43 0.00 10.66 5 $5.06 $2.06 $2.00 $5.00 $7.06 2.94 0.00 2.94 10.66 4 $4.60 $1.60 $2.00 $5.00 $6.60 3.40 0.00 6.33 10.66 3 $4.18 $1.18 $2.00 $5.00 $6.18 3.82 0.00 10.15 10.66 2 $3.80 $0.80 $2.00 $5.00 $5.80 4.20 0.00 14.34 10.66 1 $3.46 $0.46 $2.00 $5.00 $5.46 4.54 0.00 18.89 10.66 0 $3.14 $0.14 $2.00 $5.00 $5.14 4.86 0.00 23.74 10.66
Technology Reduces MC Over Time What happens if exogenous improvements in extraction technologies cause MC to fall over time? If rents are to rise at the rate of discount, market expectations of future reductions in MC will cause resource owners to defer some extraction, so that Pt starts at a higher level with lower initial consumption rates, and rises more slowly than under the constant MC case. Same as Model 1, but assume MC falls 25 percent each year (MCt+1 = 0.75MCt).
Model 6: Exogenous Decline in MC
Time Rt MCt Pt Qt Xt
10 $9.89 $0.11 $10.00 0.00 0.00
9 $8.99 $0.15 $9.14 0.86 0.86
8 $8.17 $0.20 $8.37 1.63 2.49
7 $7.43 $0.27 $7.70 2.30 4.79
6 $6.75 $0.36 $7.11 2.89 7.69
5 $6.14 $0.47 $6.61 3.39 11.07
4 $5.58 $0.63 $6.21 3.79 14.86
3 $5.07 $0.84 $5.92 4.08 18.94
2 $4.61 $1.13 $5.74 4.26 23.20
1 $4.19 $1.50 $5.69 4.31 27.51
0 $3.81 $2.00 $5.81 4.19 31.70
This model shows that declining MC may imply a phase in the depletion schedule where rents rise at the rate of discount while prices actually decline and consumption increases (periods 0 through 2).
Extraction Costs Increase as Stock is Depleted
Suppose the marginal cost of extraction MC is a function of the stock remaining X, so that MC = MC(X); and suppose MC(X) increases as X declines, so that dMC/dX < 0. (For example, as you dig deeper and deeper to extract additional coal deposits, your marginal cost of extraction increases.)
Since extraction itself causes the increase in MC, this adds another element to the opportunity cost of consumption: each unit extracted directly increases the costs of each subsequent unit extracted. This model is not easily solved in the same manner as the previous models, but you may be able to intuit (and it can be proved mathematically) that Pt has to rise more slowly than in the constant MC case. Note that once MC reaches the demand choke price, any remaining resource will never be extracted, so the effective stock level is implicitly defined by the function MC(X). Given the same economically viable total stock, this case involves higher initial prices and lower initial consumption levels than the constant MC case.
Severance Taxes
Governments often impose severance taxes on resource extraction. A per-unit severance tax has the same effect as an increase in marginal extraction costs: it increases initial prices and reduces initial consumption, reduces rents, reduces the rate of price increase through time, and extends the depletion horizon.
A lump-sum severance tax per time period (regardless of quantity extracted) has no effect on price or consumption trajectories, but is very difficult to implement.
Under fixed or rising marginal extraction costs MC, an ad valorem severance tax (a% of the market price) takes an increasing percentage bite out of rents (Rt = Pt - MC) as time goes on. It forces pre-tax rents to rise faster than the rate of discount, implying lower initial prices and higher initial consumption rates, and faster rates of price increase.