EWU – ECON 457 - Natural Resource Economics
The 2-period model
The 2-period model. Static Efficiency.
Assume that we have a fixed reserve of a depletable resource to allocate between two periods.
The marginal willingness-to-pay in period 1 is given by the following equation: P = 8 - 0.4 Q,
and the marginal cost is constant at $2 per unit.
1. In the space below, plot period 1 demand curve (marginal social benefit curve) and
supply curve (or marginal cost curve) for the resource.
2. Determine the efficient level of production of the resource in period 1. Get the exact
equilibrium quantity through calculation. Identify this output on your diagram as Q*.
3. Assume that the demand and marginal cost are unchanged in period 2. Deduce the
efficient level of production of the resource in period 2. How much of the resource is
produced (and consumed) over the two periods?
4. Further assume that the reserve of depletable resource is fixed at 20 units. How does this
change your answer to questions 1-3?
The 2-period model. Dynamic Efficiency -- Present value of net benefits for both years.
Assume that the reserve of the depletable resource is fixed at 20 units. The objective of dynamic
efficiency is to balance present and future uses of a resource by maximizing the present value of
net benefits derived from the use of the resource. The present value of the net benefits for both
years is simply the sum of the present values in each of two years.
5. In the space below, re-plot period 1 demand curve (marginal social benefit curve) and
supply curve (or marginal cost curve) for the resource, as in question 1.
6. Consider the following levels of production of the depletable resource in period 1, and for
each, determine the total net benefits:
a. Q1a= 5 units,
b. Q1b=15 units.
7. Assume that the demand and marginal cost are unchanged in period 2. Further assume a
discount rate of 10%. Determine the present value of net benefits for each of the
following levels of production of the depletable resource in period 2:
a. Q2a = 5 units,
b. Q2b =15 units.
8. Which of the following allocations (Q1a, Q2b) and (Q1b, Q2a) yields the greatest total
present value of net benefits? Does this allocation correspond to the dynamically efficient
allocation?
The 2-period model. Dynamic Efficiency -- Determining the efficient allocation.
The present value of the net benefits from both years is maximized when the resource is
allocated between the two periods at the point where the present value of the marginal net benefit
from period 1 is equal to the present value of the marginal net benefit from period 2. We will
show this graphically below.
9. Using the marginal benefit curve equation (P = 8 - 0.4 Q) and marginal cost curve
equation (P=2) algebraically determine the marginal net benefit curve for period 1.
Assuming at 10% discount rate, deduce the marginal net benefit curve for period 2.
10. In the space below, plot the marginal net benefit curve for period 1, using origin O1 as the
reference point. Plot the marginal net benefit curve for period 2, using origin O2.
marginal
net benefits
in period 1
(dollar per unit)
marginal
net benefits
in period 2
(dollar per unit)
O1
O2
Quantity in period 1
20 units
20 units
Quantity in period 2
11. Re-consider the allocation (Q1b, Q2a) and illustrate on the graph above the total present
value of net benefits associated with it. Can you deduce graphically which allocation
would maximize the present value of net benefits for both years? Compute the exact
dynamically efficient allocation and its associated total present value of net benefits. How
does the efficient level of consumption/extraction of the resource change over the two
periods?
The 2-period model. Dynamic Efficiency -- Marginal User Cost.
Intemporal scarcity imposes an opportunity cost refer to as the marginal user cost. When
resources are scarce, greater current use diminishes future opportunities of using the resource.
The marginal user cost is measured by the present value of the marginal net benefits at the level
of the efficient dynamic allocation. In other words, if one additional unit of the resource was
available, the additional marginal net benefits to society from extracting or consuming that
additional unit of the resource is its user cost. (Note, the discount rate measures the opportunity
cost of using resources at different points in time, while the marginal user cost measures the
opportunity cost of the scarcity of the resource.)
12. Assume a total reserve of depletable resources of 30 units. In the space below, proceed as
in question 10 and determine the efficient allocation of the resource. What is period 1
level of consumption? What is period 2 level of consumption? What is the marginal user
cost of the resource? Does the value of marginal user cost make sense in this case?
13. Assume a total reserve of depletable resources of 20 units. In the space below, re-plot
period 1 marginal benefits and marginal costs curves, as in questions 1 and 5. Show that
an efficient market would have to consider not only the marginal cost of extraction of the
resource, but the marginal user cost of extracting the resource as well. HINT: Determine
what the market price of the resource ought to be for the efficient amount of the resources
to be consumed in period 1. Deduce that this market price is equal to the marginal cost of
extracting the resource + its marginal user cost.
14. Repeat the exercise from question 13 for period 2. How does the market price of the
resources change over time? How does the marginal user cost change over the two
periods? What is the present value of the marginal user cost for each period?