Question

Consider a two-period model of mineral extraction with a 3%
discount rate, in which the total supply of minerals is fixed at 100. A social planner is trying to
decide how much we should consume in each period (i.e. what the efficient consumption
pattern would be). Q1 is our consumption in period 1; Q2 is our consumption in period 2.
Prices (P1), marginal benefits (MB1), and marginal costs of extraction (MCE1) in period 1 are:
P1 = MB1 = 80 – Q1
MCE1 = 5
In period 2, demand decreases because a substitute is discovered. As a result, prices (P2),
marginal benefits (MB2), and marginal costs of extraction (MCE2) in period 2 are:
P2 = MB2 = 40 – Q2
MCE2 = 5
11a What are the efficient quantities to extract in each period (Q1, Q2) and what would the
resulting prices in each period (P1, P2) be? (4 pts)
11b In answering 11a) above, we are implicitly assuming that the social planner knew in
period 1 that the substitute would be found in period 2. Now, suppose that wasn’t the
case (i.e. suppose the planner thought MB2 and MCE2 would have the same equations as
MB1 and MCE1). Would the planner in this situation have allocated more or less
consumption in period 1 than the amount Q1 you found in 11a) above? Explain your
answer either verbally or mathematically. (1 pt)
11c. BONUS: Now suppose that the social planner decides they don’t care about period 2
(i.e. an infinite discount rate), but now they also don’t know the demand curve (i.e. they
don’t know the equation for MB1). You ask them how they are going to set the efficient
consumption in period 1 (Q1) without knowing demand. They respond by saying they
don’t need to know demand in this circumstance. They say they have enough
information set the optimal price (P1), and consumers will then demand the optimal
quantity. Are they right? If not, why not? If so, how will they figure out the optimal (i.e.
efficient) price, P1? (2 pts)

Answer 1