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Environmental Economics
A1A2 Term, 2021
Homework 2 –Resource extraction

All answers must be written in English; any answers written in other language will not be graded.
Your answer can be typed (single-space, 12 fonts) or handwritten, unless otherwise noted.
Illegible answers will not be graded. Your homework should be no more than 5 pages. The
homework is due on December 7th at 13:00 by email (no exceptions).

Question 1 (40 points)
Consider the extraction of nonrenewable resource stock Q with constant marginal extraction cost
c across periods 1 and 2. The marginal benefits (not marginal net benefits) in both periods are
given by b – q, where q is the level of extraction and b is a positive constant which is greater than
c.
A) (10 points) Suppose that the resource stock is scarce and it is optimal to extract positive
amounts at both periods. The discount rate is r which is constant and positive. Graphically
show the optimal extraction levels at periods 1 and 2 (q1 and q2). Graphically show the
marginal user costs at periods 1 and 2 (1 and 2). Discuss how these two marginal user costs
are related to each other. Interpret the results.
B) (10 points) Discuss how an increase in discount rate r affects q1, q2, 1, and 2. Interpret the
results. Graphically show how the present value of net benefits at each period change as the
discount rate increases. Graphically show how the present value of total net benefits over two
periods change.
C) (10 points) Suppose that Q < b – c. Discuss what values q1 and q2 ultimately reach as we
further increase the discount rate. Interpret the results. What is the value of the discount rate
at or above which q1 and q2 take such values? Now, let us decrease the discount rate. What
are q1 and q2 when r = 0? Why?
D) (10 points) Now, suppose that the possible range of Q is from 0 to 3(b - c). Graphically show
how 1, 2, q1 and q2 change over Q. In another word, plot 1, 2, q1 and q2 together using Q
as a horizontal axis. Interpret the results. Notes: You can solve this question either
graphically or mathematically. The mathematical solution gives you closed-form solutions as
a function of Q. If you solve graphically, you just need to change the value of Q and trace the
optimal solutions carefully. In either case, be careful about the corner solutions.

Question 2 (40 points)
Let Rt denote the remaining reserve of ore at time t. Consider a competitive mining industry
facing a linear inverse demand function, pt = a – bqt, where pt is a market price at time t, qt is
aggregate output at time t, and a and b are positive constants. Suppose that exhaustion occurs at
time T. For all firms, the marginal extraction cost is fixed at c and the discount rate is fixed at r
(> 0). Let us consider social planner’s problem of efficient resource extraction.

A) (15 points) Obtain pt and qt as a function of T. Obtain an equation from which the optimal T
is derived.

Notes: You need to show each step to drive your answers. If your final equation is wrong, you
cannot correctly answer the remaining questions. If you want, you can ask your TA whether your
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equation is correct. All the TA can tell you is whether your equation is correct; the TA does not
give you the correct answer. After you were told that your equation is correct, you cannot tell the
answer to other students.

B) (10 points) Suppose that a = 1, b = 0.2, c = 0.08, r = 0.05, and R0 = 50. Find the optimal T.
Graphically depict the extraction path over time. Graphically depict the price path over time.

C) (10 points) Consider a higher discount rate (r), 0.08, holding all other parameter values
given in B. Find the optimal T. Graphically depict the extraction and price paths over time
on the same graphs you made in B. Compare these paths with those you obtained in B.
Interpret the results. Is your finding consistent with your answer in Question 1B? Discuss.
Now let us increase the discount rate further holding all other parameter values given in B.
Obtain the level of the discount rate such that the optimal T is 16.

D) (5 points) Consider higher initial reserve of ore (R0), 60, holding all other parameter values
given in B. Find the optimal T. Graphically depict the extraction and price paths over time
on the same graphs you made in B. Compare these paths with those you obtained in B.
Interpret the results.

Notes: You need to use software to accurately generate extraction and price paths in B-E. To
obtain optimal T, you can use spreadsheet software like Excel or write a code in any computer
language. In the former case, you just need to find an integer T for which the equality in the
equation you obtained in A holds most closely. In the latter case, you must show your code. In
either case, considering 30 periods is sufficient.

Question 3 (20 points)
Consider sustained yield of open-access fishing. The total benefit (TB) is a concave function of
fishing effort (E). The marginal cost of effort is constant at c (> 0). Solve the following problems
at the steady state. You can assume that the social planner’s discount rate is 0, that is, you can
consider static sustained yield.
A) (10 points) Graphically depict the total benefit and total cost of fishing over effort E.
Graphically depict the marginal benefit (MB), marginal cost (MC), average benefit (AB), and
average cost (AC) of fishing over effort E. On these two graphs, show the effort EM that
attains the maximum sustainable yield (MSY), the efficient fishing effort E** for the society,
and the equilibrium effort E* under the open-access regime. Compare EM, E**, and E*, and
corresponding fish stock. Interpret the results.
B) (10 points) Suppose that technological progress occurs in fishery: c is reduced to c’. On the
two graphs you generated in A, show how the technological progress alters EM, E**, and E*.
Discuss how differently EM, E**, and E* change. Which is affected more, E** or E*? Why?