Problem one is the same problem from the lecture with different mnumbers. See th

Question

Problem one is the same problem from the lecture with different mnumbers. See the lecture notes on blackboard for a worked example. . . Problem two is optional and worth extra credit. Problem 1 Consider two time periods, t1,2. Coal is a depletable natural resource that must be allocated between the two periods. The resource constraint is Q1+Q2 = 15 where Q is the quantity of coal consumed in period one, and Q2 is the quantity of coal consumed in period two. The marginal benefit functions, or inverse demand functions, are given in each period by MB(Q)-12-30 MB(Q) -12-3Q And the marginal extraction cost functions in each period are Suppose the discount factor is 9, Find the dynamically efficient allocations and prices of coal for the two time periods. Show your work. Problem 2 (extra credit) a) Previously, we let 9 Solve for the dynamically ecient allocation in period two, Q , as a function of . That is, keep as a variable in your equations. Your expression for Q will have in it. b) With your expression for Q; in terms of , give values for Q: when -3.5,7. Is Q; increasing or decreasing in ?

Explanation / Answer

1.

Two-period model for efficient allocation:

Given, Q1 + Q2 = 15 ………… (1)

In case of (t = 1), the net marginal benefit (NMB1) is (MB1 – MC1)

Therefore, NMB1 = (12 – 0.3Q1) – 3 = 9 – 0.3Q1

In case of (t = 2), the net marginal benefit (NMB2) is (MB2 – MC2)

Therefore, NMB2 = (12 – 0.3Q2) – 3 = 9 – 0.3Q2

It would be an efficient allocation, if (NMB1 = PV of NMB2)

NMB1 = PV of NMB2

9 – 0.3Q1 = (9 – 0.3Q2) / discount factor

9 – 0.3Q1 = (9 – 0.3Q2) / 0.9

9 – 0.3Q1 = 10 – 0.33333Q2

9 – 0.3 (15 – Q2) = 10 – 0.33333Q2

9 – 4.5 + 0.3Q2 = 10 – 0.33333Q2

0.63333Q2 = 5.5

Q2 = 8.68 = 9 units (rounded)

Now by putting this value in (1), Q1 = 15 – 9 = 6 units.

Marginal benefit would be the price for respective periods.

MB1 = 12 – 0.3 × 6 = 10.20

MB2 = 12 – 0.3 × 9 = 9.30

Answer: Q1 = 6 units, Price = 10.20; Q2 = 9 units, price = 9.30

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