4. Suppose that South Africa is trying to find the efficient amount of coal to e

Question

4. Suppose that South Africa is trying to find the efficient amount of coal to extract, which is a depletable resource. There are two periods and the inverse demand function is P = 15 - 0.5QD and the marginal cost of extraction is 5.

(a.) Use the static efficiency model to find the efficient allocation for the first time period. Also, compute the price of coal at that efficient allocation.

(b.) Now suppose that the total available quantity of coal is Q = 40. Is the static efficiency criteria sufficient? If yes, what is the efficient allocation of coal over the two periods? If no, why not?  

(c.) Now suppose that the total available quantity of coal is Q = 35. Is the static efficiency criteria sufficient? If yes, what is the efficient allocation of coal over the two periods? If no, why not?

Explanation / Answer

a.) The inverse demand function is P = 15 - 0.5QD and the marginal cost of extraction is 5. Static efficiency occurs when we maximize the total benefits. This occurs when marginal benefits (MB) equals marginal costs (MC) or when the marginal net benefits (MNB) equals zero. Thus, for the first time period, MB1=MB2 or 15 – 0.5QD = 5; QD = 20. Thus, the efficient allocation is 20 units in the first period and the price of this allocation can be found by substituting the QD in the inverse demand equation, we get, 5 units.

b.) Since the efficient allocation is 20 units in the first period, it will be 20 additional units in the second period, so the total efficient allocation of coal over the two periods is 40 units which are equal to the total available quantity. So the static efficiency criteria are sufficient.

c.) Since the efficient allocation of the resource over the two periods is 40 units, the given total quantity of 35 units is insufficient. This is because the marginal net benefit for the two periods will be zero if the efficient allocations are 20 units per period.

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