This lecture will discuss several variations on the basic exhaustible resource allocation model discussed in the previous lecture.
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=50
Time 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=30
Time 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 = $6
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
Comparing 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.
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 Costs
Time 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.
(Tietenberg says that in this case "marginal user cost" declines over time, going to zero at the time of depletion. This isn't the same as opportunity cost, however, since it excludes the increases in MC caused by prior extraction.)
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.