Question

Assume a two-time-period model for a depletable resource, with identical demand functions in each of the tow time periods (i.e. P = 10 - 0.5q). The marginal cost of extraction is also constant for each of the two time periods (i.e. MC = $3). Calculate the dynamically efficient allocations of the depletable resource across the two time period when: (i) there are 20 units of the resouce available for use in total, and (ii) there are 30 units available. Also calculate the marginal user cost in each case. ( Assume discount rate of 10%)

Answer 1

A) For dynamic efficiency, the present value of marginal net benefit should be same across both the periods

Marginal net benefit in period 1 is P1 – MC1 = 10 – 0.5Q1 – 3 = 7 -0.5Q1

Marginal net benefit in period 2 is 7 -0.5Q2

PV of MNB in period 1 is same as no discounting is done. PV of MNB in period 2 after discounting is (7 -0.5Q2)/1.1

Now the present value of marginal net benefit should be same across both the periods so we have

(7 -0.5Q1)1.1 = 7 -0.5Q2

7.7 - 7 = 0.55Q1 – 0.5Q2

0.7 = 0.55Q1 – 0.5Q2

And

Q1 + Q2 = 20

Use Q2 = 20 - Q1

0.7 = 0.55Q1 - 10 + 0.5Q1

1.05Q1 = 10.7

Solving this will give Q1 = 10.19 and Q2 = 9.81. The prices are P1 = 4.90 and P2 = 5.10

Marginal user cost is MUC 1 = P1 - MC = 4.90 - 3 = 1.90 and MUC 2= 5.10 - 3 = 2.10

B) For Q1 + Q2 = 30, we use Q2 = 30 - Q1

0.7 = 0.55Q1 - 15 + 0.5Q1

1.05Q1 = 15.7

Solving this will give Q1 = 14.95 and Q2 = 15.05. The prices are P1 = 2.52 and P2 = 2.48

Marginal user cost is MUC 1 = P1 - MC = 2.52 - 3 = -0.48 and MUC 2= 2.48 - 3 = -0.52