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1. Dynamic allocation of depletable resources.

The N-Period, Constant-Cost, No-Substitute Case:
The inverse demand curve in year t can be written as:
p_t=8-2q_t-3〖q_t〗^2
The marginal cost of extracting that resource is constant c=2. Total amount of resource Q=40, r=0.01.
Write down the maximization problem for the dynamic allocation of the resource over n years. (5 pts)
Write down the equations describing the allocation that maximizes the present value of net benefits. (You do not need to solve it) (5 pts)

(2) The N-Period, Constant-Cost, with an Abundant Renewable Substitute case:
The inverse demand curve in year t can be written as:
p_t=8-2q_t-3q_t2
Let q_t be the amount of a constant-marginal-cost depletable resource extracted in year t and q_st be the amount used of another constant-marginal-cost resource. The marginal cost of extracting the depletable resource is constant c=2. The marginal cost of the substitute is assumed to be d=3. Total amount of depletable resource Q=40, r=0.01.
Write down the maximization problem for the dynamic allocation of the resource over n years. (5 pts)
Write down the equations describing the allocation that maximizes the present value of net benefits. (You do not need to solve it) (5 pts)

Discuss the topic of natural monopolies and utility regulation giving rise to inefficient pricing. (10 pts)

Illustrates how inefficiencies can lead to deforestation, and identify the sources. (10 pts)

Suppose that the marginal damages to society from air pollution are MD = e – 25, where e is the level of air pollution. Suppose also that the marginal cost of reducing the air pollution on the part of firms is MC = 200 – 2e.
Find the optimal level of pollution. Illustrate graphically. (5 pts)
Find the net gain to society, assuming firms were initially not controlling emissions at all.
What level of tax would achieve the optimal level of pollution? (5 pts)
Suppose instead of a tax a standard is set so that the level of pollution is 10% below the efficient level. Calculate the cost to society. (5 pts)
Suppose that the tax is set so that it is 10% too low. Calculate the cost to society. (5 pts)