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

QUESTIONS

11) What are the efficient quantities to extract in each period (Q1, Q2) and what would the resulting prices in each period (P1, P2) be?

12) In answering the question 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 11) above? Explain your answer either verbally or mathematically.

Answer 1

SOLUTION:-

11) The marginal net benefit of extraction:

MB1 = P1 - MCE 1 = 75 -Q1

MB2 = P2 - MCE 2 = 35 -Q2

The social plannex maximize benefit by setting marginal net benefit present value in both period equal. That in

PV (MNB1) = PV (MNB2)

  or,    

or,   

or, (75-Q1) X 1.03 = Q1_65

or, 77.25-1.03 Q1 = Q1-65

or, 2.03 Q1 = 142.25   

Resulting price: P1 = 9.93, P2= 10.07

12) In that case social planners problem

PV(MNB1) = PV(MNB2)

or,

or, 75-25-1.03 Q1 = Q1-25

or, 2.03 Q1 = 102.25

   Q1 = 50.4 < 70.07 (in 11a)

The social plannex would have allocated half the supply in period 1 because both period have same demand  unlike the case with known perios 2 demand where demand falls in period 2.

There, in first case, the social planner allocates more in period 1 than in 2.

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