Economics 5250/6250 Dr. LozadaFall 2013 Final Exam

This exam has 67 points. There are seven questions on theexam; you should work all of them. Each question 9 points each,except the first two questions, which are worth 11 points each.

Put your answers to the exam in a blue book or on blanksheets of paper.

Answer the questions using as much precision and detail as thetime allows. Correct answers which are unsupported by explana-tions will not be awarded points. Therefore, even if you thinksomething is “obvious,” do not omit it. If you omit anything,you will not get credit for it. You get credit for nothing whichdoes not explicitly appear in your answer. If you have questionsabout the adequacy of an explanation of yours during the exam,ask me.

Answer all of the following questions.

1. [11 points] Below is a long excerpt from a handout we used in fish-eries economics. Explain how each of its equations (1)–(11) wouldchange if we instead were trying to derive the Hotelling Rule. If anequation would not change, say that in your answer.

The profit of each firm is

Π(Ht, Xt) = TRt(Ht) − TC(Ht, Xt) (1)

where H is the harvest, X is [definition omitted on this Final Examso as not to give away the answer to Question 2(a)(i)], TR is the totalrevenue, and TC is the total cost, all at time t. The objective of thefirm is to

max

∞∑t=0

Πt

(1 + δ)ts.t. (2)

Xt+1 −Xt = F (Xt) −Ht (3)

where F is the natural excess of births over deaths. (3) represents aninfinite number of constraints on (2). Using k1, k2, . . . , to denote theLagrange multipliers, the Lagrangian is

L = Π0 + · · · +Π6(H6, X6)

(1 + δ)6+Π7(H7, X7)

(1 + δ)7+Π8(H8, X8)

(1 + δ)8(4)

+Π9(H9, X9)

(1 + δ)9+Π10(H10, X10)

(1 + δ)10+ · · ·

+ k1(X1 −X0 − F (X0) +H0) + · · · + k6(X6 −X5 − F (X5) +H5)

+ k7(X7 −X6 − F (X6) +H6) + k8(X8 −X7 − F (X7) +H7)

+ k9(X9 −X8 − F (X8) +H8) + k10(X10 −X9 − F (X9) +H9) + · · · .

We wish to maximize this with respect to Xt and Ht for all t. Forexample,

∂L

∂X8= 0 =

∂Π8/∂X8

(1 + δ)8+ k8 + k9(−1 − F ′(X8)) (5)

∂L

∂H8= 0 =

∂Π8/∂H8

(1 + δ)8+ k9 (6)

∂L

∂H7= 0 =

∂Π7/∂H7

(1 + δ)7+ k8. (7)

(6) and (7) can easily be solved for k9 and k8. Substituting these valuesinto (5) yields

0 =∂Π8/∂X8

(1 + δ)8− ∂Π7/∂H7

(1 + δ)7+∂Π8/∂H8

(1 + δ)8[1 + F ′(X8)] , (8)

1

so

0 =∂Π8

∂X8− (1 + δ)

∂Π7

∂H7+ [1 + F ′(X8)]

∂Π8

∂H8. (9)

From (1), Π8 = TRt(H8)−TC(H8, X8), so ∂Π8∂X8

= −∂ TC∂X8

; call this −CX8

for short. By definition, ∂Π7∂H7

= MΠ7 and ∂Π8∂H8

= MΠ8. Also, let F ′(X8)

be abbreviated by F ′8. Then substituting these results into (9) yields

0 = −CX8 − (1 + δ)MΠ7 + [1 + F ′8]MΠ8 (10)

which can be rewritten as

(1 + δ)MΠ7 = [1 + F ′8]MΠ8 − CX8 (11)

or as

(1 + δ)MΠ7 = [1 + F ′8]MΠ8 +

∂Π8

∂X8. (12)

2. [11 points] We have spent significant time in class discussing conse-quences of the equation

X = rX (1 − X

K) . (13)

(a) In this equation, what is:

i. X?

ii. X?

iii. X/X? (This does not appear explicitly in the equation, butwhat would its interpretation be?)

iv. K?

v. r?

Especially for the last four, do not merely give the name of thevariable: explain why the name is appropriate. For the last one(r), explain the relationship between r and X/X.

(b) Draw a graph of X versus X for (13).

(c) Draw a graph of X versus time t for (13). You need not explainits derivation, but give some idea of why your answer makes sense.

(d) Give a realistic example of a fisheries growth function F (X) whichdoes not obey (13). Explain why the difference between the F (X)you gave and (13) is important.

(e) What is “logistic growth”?

2

3. [9 points] Concerning exhaustible resources, contrast the viewpointsof the schools of thought which your textbook calls the “Malthusian’and the “Ricardian” school.

4. [9 points]

(a) Draw a simple graph (with only two curves) showing how a socialplanner would compute the optimal Pigouvian tax on the outputof a good whose production causes pollution. Explain, as always.

(b) Draw a simple graph (with only two curves) showing how a socialplanner would compute the optimal Pigouvian tax on pollution.Explain, as always.

5. [9 points] Discuss “hot spots” in the context of tradeable pollutionpermits. (Don’t merely define “hot spots.” Thoroughly discuss them.)

6. [9 points] Discuss “starting point bias.”

7. [9 points] Pollution laws have sometimes mandated the use of “BestAvailable Control Technology.” What is that?

3

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