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:
- Q1a= 5 units,
- 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:
- Q2a = 5 units,
- 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.
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?