Consider a 2-period model of oil extraction by a perfectly competitive industry.

Question

Consider a 2-period model of oil extraction by a perfectly competitive industry. There are Q=40 barrels of oil in the ground. In each period, the inverse demand curve is given by P=10-0.2q . Marginal costs in each period are MC=4 . The interest rate is r=0.05 .

1. Solve for static efficiency. How much is extracted in each of the two periods?

2. Why is a dynamic model needed? What is wrong with static efficiency?

3. What are the dynamically efficient values for q1 and q2?

4. What are the dynamically efficient values for P1 and P2 ?

5. What are the dynamically efficient values for the shadow prices m1 and m2 ?

6. What is the interpretation of the shadow price m ?

7. What is the Hotelling rule?

Explanation / Answer

Static efficiency occurs in each period when we maximize the total net benefits. As we

saw in class, this occurs when marginal benefits (MB) equals marginal cost (MC) (or

equivalently, when MNB = 0). Mathematically, using the above equation, we need:MB1 = MC1

or

10-0.2q1 = 4

Or

q1 = 6/0.2 = 30

the equation is same so for q2 = 30

so the both period combined wants 60 units but we are available with 40 units, we have scarcity problem so that dynamic model needed.

We have two pieces of information available to us in solving this problem.

First, we know we will want to use up all of the water, since a shortage of water exists for

both periods. Mathematically, this can be written:

q1 + q2 = 40

or

q2 = 40 - q1 .

The second piece of information we have is that we want the allocation to be dynamically

efficient. As we saw in class, this occurs when the present value of the marginal net

benefits in the two periods are equal. From the above equations, we have:

MNB1 = MB1 - MC1

= 10 -0. 2q1 - 4

= 6 – 0.2q1.

Similarly,

MNB2 = 6 – 0.2q2. .

The present value of the marginal benefits in period 1 is just the marginal net benefits in

period 1 (i.e., we do not discount the present):

PV[MNB1] = 6 – 0.2q1. .

However, we have to discount the marginal net benefits in period 2, because we do not

receive those benefits until one period from the present. Thus,

PV[MNB2] = (6 – 0.2q1.)/(1 + .05)

= 5.71 - .4q2

Since dynamic efficiency requires that the present value of the marginal net benefits be

equal in the two periods, we have:

6 – 0.2q1 = 5.71 - .4q2

Substituting in our resource constraint (i.e., q2 = 40 - q1) gives us

6 – 0.2q1 = 5.71 - .4(40 - q1)

= 5.71 – 16-.4 q1

or

q1 = 27.1429

and

q2 = 40 - q1 = 12.8571

P1= 10-0.2q1 = 4.5714

P2 = 10-0.2q2 = 7.4286

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