nr_exams.pdf

Peter J. Wilcoxen Department of Economics
s
Ec 384N, Spring 1992 The University of Texa
Exam 1
x
w
Suppose the stock of water in a reservoir changes according to the following equation: s
.
=a −
here s is the stock of water, a is an exogenous constant equal to the rate at which water
a
flows into the reservoir from rivers, and x is the rate at which water is extracted for use. In
ddition, suppose that x units of water can be sold for p .x dollars and that operating the reser-
voir costs c .x +d /s dollars, where p , c and d are exogenous constants.
2
(a) Suppose the owner of the reservoir wants to maximize the present value of the profits she
b
earns. If the interest rate is r , show that the first order conditions for the optimum can
e written:
λ = p − 2.c .x
h
hh
d
λ
.
= r λ −
s2
(
s
.
= a − x
b) Now find the steady state values of λ, x and s . How would these be affected by an
(
increase in d ?
c) Solve for the system’s equations of motion in terms of x and s .
(
(d) Evaluate the stability properties of the system in part (c).
e) Draw a phase diagram for the model in terms of x and s . Show the steady state, the iso-
l
a
clines, the directions of motion and the stable path, if applicable. Put x on the vertica
xis and label everything carefully.
(f) Based on your results, describe how the owner of the reservoir should set x over time.
(
Be sure to discuss how your recommendations depend on the initial stock of water.
g) Extra Credit. Discuss what would happen to the pattern of optimal extraction if d sud-
s
denly and permanently increased. If it helps, assume the system is initially at the steady
tate. Be specific and use an appropriate phase diagram.

Ec 384N, Spring 1992 The University of Texas
Optional Take Home Exam
You may take as long as you want to complete this exam but you must do it by yourself
and may not use any books or notes.
Imagine you have been contacted by the owner of an artesian well which produces mineral water.
The owner would like your advice on how to maximize the present value his profits and gives you
the following facts: the well produces water at rate x at zero cost; all water produced can be sold
at a fixed price p; the well’s flow changes over time according to the relation ẋ = y − a ⋅ x, where
y represents maintenance carried out by the owner and a is an exogenous rate of deterioration;
and y units of maintenance costs c ⋅ y2
dollars. Thus, the owner would like to solve the following
problem:
max
∞
0
∫ ( p ⋅ x − c ⋅ y2
)e−rt
dt
subject to ẋ = y − a ⋅ x
(The structure of this problem is somewhat unusual. Before you go on be sure you understand
which variable is the state variable and which is the control.)
(a) Show that the first order conditions for this problem can be written as follows:
2 ⋅ c ⋅ y = λ
˙
λ = (r + a)λ − p
ẋ = y − a ⋅ x
(b) Now find the steady state values of λ, y and x. How do these respond to an increase in p?
(c) Find the system’s equations of motion in terms of x and y.
(d) Evaluate the stability properties of the system in part (c).
(e) Draw a phase diagram for the model in terms of y and x. Show the steady state, the iso-
clines, the directions of motion and the stable path, if applicable. Put y on the vertical axis
and label everything.
(f) Based on your results, describe how the owner of the well should set y over time. Be sure
to discuss how your recommendations depend on the initial flow of the well.

-2-
(g) Draw a phase diagram showing what the owner should do if p suddenly and permanently
increased. Show what happens to the steady state, the isoclines and the stable path. You
may assume the system is initially at the steady state.
(h) Based on your results for part (g), discuss what the owner should do if p were to increase by
10%. Illustrate your analysis with appropriate phase diagrams and integral curves. If y
were initially 10, what would it be after the price increase?
(i) Extra Credit. Draw a phase diagram to illustrate what would happen if p suddenly but tem-
porarily increased, say for a year or two. Explain what it shows. Hint: you might need to
refer back to your notes from February 10th.

s
Ec 384N, Spring 1992 The University of Texa
Exam 2
r
n
Please write your STUDENT NUMBER on your blue book and do NOT write you
ame. There are 5 questions on this exam. Please do question 1 and any TWO of
i
questions 2 through 5. There are sixty points possible so budget your time accord-
ngly.
Question 1 (30 points)
Suppose you’ve been given the task of managing a beautiful natural area which happens to
g
a
contain a valuable mineral. Your job is to decide how much of the area to allocate to minin
nd how much to leave in its natural state. However, these uses are mutually exclusive: min-
L
ing permanently destroys the scenic value of the land.
et the stock land which hasn’t been mined at time t be a (t ); let the quantity mined at t be
-
p
x (t ) and let the amenity services provided by unmined land at t be y (t ). Furthermore, sup
ose your instructions are to maximize the present value of benefits from the land, which an
econometrican tells you are the following:
( αx −
2
h
β
h
x + εy −
2
δ
h
h
y )e dt (1)
∞
0
∫
2 2 −rt
y
s
where α, β, ε and δ are parameters. In addition, the econometrican tells you that the amenit
ervices provided by a units of land are proportional to a . For simplicity, assume the constant
t
of proportionality is 1 so that y (t ) = a (t ). Thus, your problem is to maximize (1) subject to
he following constraints: a
.
= −x and y = a . After setting up the problem in the usual way,
you find that the optimum is governed by the following equations of motion:
x
.
=
β
1
h
h
I
L
ε − δa − αr + βrx
M
O
(2)
)
Y
a
.
= −x (3
ou may take (2) and (3) as given; you do NOT need to derive them. On the basis of (2) and
(
(3), please answer the following questions.
a) Solve for the steady state levels of x and a and construct an appropriate phase diagram
h
a
to represent the equations of motion. Show the steady state, the isoclines, the stable pat
nd the directions of motion.

- 2 -
n
(b) Using your diagram for part (a), discuss how you would manage the resource give
different starting quantities of land. Be sure to take into account that the destruction
(
caused by mining is permanent. Draw appropriate integral curves.
c) Draw another phase diagram and use it to discuss how the optimal use of the resource
m
would change if the a new innovation suddenly reduced the value of the mineral being
ined. In particular, show how do things change if α suddenly decreases. Discuss how
(
the new optimum plan depends on the value of a at the time of the change.
d) What conclusions can you draw from this analysis regarding the irreversible use of
E
resources in a world where unpredictable technical innovations occur from time to time?
xplain.
Please answer any TWO of the following questions. Explain your position as com-
a
pletely as you can. Use simple models or references to the literature where
ppropriate.
)
W
Question 2 (15 points
hat goes wrong in open access fisheries? Do these problems arise in all fisheries? What are
a
some policies that might be used to correct the problems? What policy would you recommend
nd why?
Question 3 (15 points)
How does the optimal management of an aquifer compare with the result produced by perfect
e
competition? Are there grounds for government intervention or does the existing legal system
ncourage optimal use?
)
I
Question 4 (15 points
n determining the optimal use of an exhaustible resource, what difference does it make if
Q
extraction costs rise with cumulative extraction?
uestion 5 (15 points)
Suppose a monopolist owns q
d units of resource which can be costlessly extracted and sold for
revenue of q dollars where q is the quantity sold and e is the elasticity of demand. The
(e −1)/e
monopolist is plans to extract the entire resource over two periods and wants to maximize
a
expected total revenue. Using a model, show how risk of expropriation in the second period
ffects the monopolist’s decision in the first period. You may assume the monopolist has a
discount rate of zero.

s
Ec 384N, Fall 1993 The University of Texa
Exam 1
S
Problem 1
uppose that in the absence of human intervention the stock of tuna (b ) would change accord-
ing to the following equation: db /dt = 10b − 0.01b . Furthermore, suppose e units of fishing
2
e
u
effort will produce a harvest of tuna (x ) given by x = 0.01be . Finally, suppose the cost of
nits of effort is $2e , where $2 is the wage rate, and that the price of tuna is constant at $1
(
dollar per unit of x .
a) 10 points. Find the profit-maximizing sustainable equilibrium for this model when the
e
f
tuna are not common property (that is, when the level of effort can be controlled). Solv
or the levels of b , e , x and profit.
Now suppose the government allows free entry into the tuna fishing industry but imposes two
s
ad valorem taxes: τ on wages, and τ on profit. The cost of e units of effort now become
w p
w p
.
F
$2(1+τ )e and after-tax profit becomes (1−τ )π, where π is profit before the profit tax
inally, assume people in the industry behave in such a way that the level of fishing effort can
be described by the following differential equation: de /dt = 1.04(1−τ )π.
p
s
(b) 10 points. Calculate the long run equilibrium levels of b , e , x and π when both tax rate
are set to zero. Evaluate this outcome relative to the one from part (a). Is it better or
(
worse? Why?
c) 10 points. Suppose the government wants to use τ and τ to improve the outcome in
w p
e
t
the tuna industry. Find a tax policy (that is, settings for the two taxes) that would mov
he long run market outcome to the efficient steady state from part (a). Discuss your
(
results.
d) 20 points. Now find a tax policy that would accomplish the goal in part (c) and would
y
s
insure that for points near the steady state the industry would move toward the stead
tate without oscillations. Explain your results.
n
(e) 10 points. Draw a phase diagram for the model from part (d). Put the level of effort o
the horizontal axis. Show the steady state, the isoclines and the directions of motion.
.
S
Also, show paths from starting points in two different quadrants to the steady state
ketch the integral curves for these paths.
... exam continues on the next page ...

F
Problem 2
- 2 -
or some resources, the cost of mining increases with the total amount of the resource
e
a
extracted. This question explores a simple example. Suppose that the profit on a silver min
t time t is given by π = pq − cq − ks , where p , c and k are exogenous constants, q is
t t t
2
t t
t l
p
the amount of silver extracted in period t , and s is the total amount of silver extracted in al
eriods before t . The evolution of s is given by s − s = q . The value of s is initially
t +1 t t
e
fi
zero. Also, the resource is infinite supply in the sense that there is no upper limit on s . Th
rm will own and operate the mine in periods 0 (now) through T −1; at T the firm expects to
e
t
be nationalized and to have no subsequent revenues or costs. The firm’s goal is to maximiz
he sum of the profits it earns in periods 0 to T −1. You may assume the interest rate is zero.
y
(a) 10 points. Set up the firm’s optimization problem and take first order conditions. Briefl
explain what each equation shows.
(b) 10 points. Solve for an expression giving the firm’s output (q ) at each point in time
t
.
from 0 to T −1. Discuss your result and give an intuitive explanation

s
Ec 384N, Fall 1993 The University of Texa
Exam 2
-
i
Global warming has been receiving a lot of attention in the press and in government pol
cy circles. One of the main contributors to global warming is the buildup of carbon dioxide in
e
e
the atmosphere. This turns out to be an interesting economic problem for several reasons: th
xternality is world wide; it comes from the stock of carbon dioxide rather than the emissions
e
themselves; and the carbon dioxide stays in the atmosphere for a long time after it has been
mitted. This exam asks you to think about the problem and to evaluate a few different poli-
cies.
Suppose there are N identical firms in the world. Let each firm’s production and cost
i
functions be summarized by a profit function f (x ) that gives the firm’s profit as a function of
ts emissions of carbon dioxide, x . You may assume that f has the following properties: it is
s
p
quadratic; f ′(0) > 0; and f ′′ < 0. However, f does NOT take into account any externalitie
roduced by the firm’s carbon dioxide emissions.
In addition, suppose that the stock of atmospheric carbon dioxide imposes world-wide
-
s
costs equal to c .g (s ), where c is a constant, s is the stock of carbon dioxide in the atmo
phere, and g (s ) is a function with the properties that g ′ > 0 and g ′′ > 0. Furthermore, sup-
i
pose that the rate of change of s is given by the following expression: s
.
= Nx − δs , where δ
s the rate at which carbon dioxide is naturally removed from the atmosphere (and is very
n
e
small). Thus, taking the externality into account the net present value of world wide carbo
missions is given by:
( N .f (x ) − c .g (s ) )e dt
(
∫
−rt
a) 12 points. Show that the optimum pattern of carbon emissions must satisfy the following
equations of motion:
x
.
=
f ′′
(r +δ)f ′ − c .g ′
h
hhhhhhhhhhhhh
(
s
.
= Nx − δs
b) 12 points. Draw a phase diagram for the equations of motion in part (a). Put s on the
e
s
horizontal axis, x on the vertical axis and label everything carefully. Be sure to show th
teady state, the isoclines, the stable path and the directions of motion.

(
- 2 -
c) 12 points. Suppose that we suddenly find out that global warming causes much more
a
s
serious problems than anyone had thought. In this model that could be represented by
udden increase in c . Use a phase diagram to determine what should happen in this
e
situation. Draw appropriate integral curves and explain your results in words. Be sure to
xplain how the current value of s affects your answer.
y
(d) 24 points. Copy your diagram from part (b) and then sketch the trajectory the econom
would follow under an unregulated market environment. In other words, show what hap-
-
t
pens when firms ignore the externality. Explain as carefully as you can why your trajec
ory looks the way it does. Also, make sure that this trajectory and your stable path from
t
s
part (b) are consistent. Finally, if the optimal and unregulated economies have differen
teady states, comment on whether these are likely to be similar to each other or not.
-
(e) 12 points. One policy often proposed for slowing global warming is to tax carbon diox
ide emissions. This would change each firm’s profit function to π = f (x )−τ(t ).x , where
t
t
τ(t ) is a tax rate that might vary over time. Proponents of this policy often suggest tha
he tax should be chosen to keep emissions constant at the long run equilibrium level. Is
this policy likely to be efficient? If so, why? If not, what would be better?

Economics 384N.1, Spring 1995 The University of Texas at Austin
Exam 1
(Please write your STUDENT NUMBER on your bluebook, NOT your name.)
Consider an ecosystem consisting of trees and spotted owls. In the absence of human
intervention, the populations of trees (T) and owls (O) evolve according to the equations:
dT
dt
= 10T − T2
100
dO
dt
= OT
1000
− O2
500
− 45
(a) Solve for the steady-state populations of owls and trees and draw an appropriate phase
diagram for the region near the steady state. Put the stock of trees on the horizontal axis.
Show the steady state, the isoclines and the directions of motion.
(b) Solve for the model's eigenvalues near the steady state. Discuss the ecosystem's stability.
(c) Now imagine that the owner of the forest begins harvesting trees at a rate x which is
constant over time. Suppose the owner sets x to the maximum sustainable yield of trees.
Find the new steady state and compare it to the old one. Discuss.
(d) Suppose the owl population is subject to normally distributed random shocks with mean
zero and standard deviation equal to 25, and that this forest is the owl's only habitat. If
you were in charge of protecting the owl from extinction with reasonable probability,
would you restrict timber harvests, and if so, to what level? In answering this, you may
interpret "reasonable probability" to mean that there is less than a 5% chance or so that the
owls will fall below the minimum viable population in any single year.

Exam 2
You may take as long as you want to complete this exam but you must do it by
yourself and you may not use any books or notes. It is due no later than 5 pm
Saturday May 13. Also, please write your STUDENT NUMBER on your
bluebook and NOT your name.
This exam explores the economics of nuclear power. Suppose the demand for electricity can be
described by the following inverse demand function, where q is the quantity of electricity (all
Greek letters throughout this problem are parameters):
p = α − βq
For the purposes of this exam, you may ignore the fixed cost of the plant. Instead, imagine that
the total cost of nuclear power is quadratic in output:
c(q) = 1
2
γq2
In addition, there are two externalities associated with nuclear power. First, any time a nuclear
plant is operation there is a small probability, , of an accident. If an accident occurs, a very large
δ
cost, , is imposed on the surrounding community. Assume that people are risk neutral with
N
respect to nuclear accidents (ha ha ha!) so the value of this externality at each point in time when
power is being produced is equal to its expected value . (Note that in this case the damage
δN
does not depend on either q or x.)
The second externality is associated with waste from the plant. If fuel is used in proportion to
output, the stock of accumulated waste, x, will be governed by the following equation:
dx
dt
= ηq
Let the externality associated with this stock be quadratic in x: .
1
2
εx2
(a) Find expressions for the price and quantity of electricity supplied, and the quantity of
accumulated waste, at each point in time when the industry takes prices as given and
ignores the externalities.
(b) Now solve for the efficient paths of q and x when the stock externality is taken into
account but is assumed to be zero. (That is, when there is a stock externality but no
δ
accident externality.) Find the steady state and draw an appropriate (and fully labeled!)

phase diagram. Compare this with the outcome in part (a) as thoroughly as you can.
Discuss differences in p, q, x, and overall social welfare. Draw both the efficient trajectory
and the path from part (a) on the phase diagram.
(c) Suppose the industry is operating efficiently (the externality problem has been cured
somehow and the industry is following the trajectory from part b) and there is an
unexpected dramatic breakthrough in an alternative technology such as solar power (or
natural gas, for that matter). This causes the cost of solar power to drop to s dollars per
unit, where , and that solar power can be produced at constant returns to scale.
s < α
Discuss thoroughly and in detail how this affects the efficient path of the nuclear power
industry. Among other things, be sure to comment on whether (or in what circumstances)
the quantity of power produced by nuclear plants would be zero.
(c) Now find the efficient outcome when both the waste problem and the accident risk are
taken into account. Compare your results with those from part (a) and discuss any
important or interesting differences.
Since this is a take home exam and there's little or no time pressure, try not to rush through it
making silly mistakes. For example, be careful about the relationship between price, marginal
benefit and total benefit. Also, here are a few other reminders:
u
max
∫f(x, u,t)dt + F(x) s.t. dx
dt
= g(x,u,t)
H = f(x, u,t) + Λg(x,u, t)
∂H
∂u
= 0,
∂H
∂x
= −dΛ
dt
,
∂H
∂Λ
= dx
dt
possibly dF
dx
= Λ(T) or x(T) = xT or other ...
possibly H(T) + dF
dT
= 0
For what it's worth, you may be interested to know that in the early days of the
U.S. nuclear power industry Congress passed a federal law severely limiting the
liability of power companies for damages caused by accidents at nuclear power
plants. It's also clear that neither the Atomic Energy Commission (predecessor to
today's Nuclear Regulatory Commission) nor the industry worried much about
what would happen to waste from the plants. To this day there is no repository
for high-level nuclear waste. Spent fuel from the nation's power plants is
currently stored in water-filled pools at the plants. The debate over nuclear
power is often carried on as though it were a moral issue but the real problem is
that the economics was botched at the beginning.

Economics 384N, Spring 1997 The University of Texas at Austin
Exam 1
yourself and you may not use any books or notes. The exam is due in class on
Tuesday, March 18th. Please write your STUDENT NUMBER on your bluebook
and NOT your name.
The salmon fishery is an important industry in the Pacific Northwest. As you no doubt
know, salmon are remarkable fish that spawn in fresh water, migrate to the ocean where they live
most of their lives, and then return to the streams in which they hatched to spawn. This life cycle
makes salmon trivial to catch: by building a trap at the mouth of a stream one could catch as many
of the adult salmon swimming upstream as one wanted at essentially zero marginal cost. Salmon
can also be caught on the open sea but it is a lot more difficult and expensive. The native people
in the area actually used such fish traps in the past but they are now illegal. This problem asks
you to think about managing a salmon fishery associated with a particular stream. To make things
concrete, suppose you are given the following information:
` Let the stock of salmon at sea be b.
` Suppose the salmon spend 5 years at sea before returning to the stream to spawn. Let s be
the number of salmon leaving the open sea each year to spawn. To keep things simple,
ignore age differences among the salmon and assume that s = b/5. Also, assume the salmon
die after spawning (not true of all species).
` Some of the salmon at sea die before spawning. Let the open ocean mortality rate be m and
suppose it is given by .
m = bb2
` Let the number of salmon caught on the open sea be h and the number caught using fish
traps be x.
` Let n be the number of newly spawned salmon added to the ocean stock each year. The
number of new salmon depends on the number of spawning adults that make it past the fish
traps and a parameter : .
a n = a(s − x)
` The number of salmon caught on the open sea depends, in the usual way, on the population
and the amount of effort, e, devoted to fishing: . Let the cost of one unit of effort be
h = cbe
w.
` The cost of catching salmon with a fish trap is zero.
` The price of a salmon is p.
... exam continues on the following page ...

Please answer the following questions:
(1) Show that the equation below describes the evolution of b. (This is really just
accounting.) Suppose that e and x are both zero (no fishing). Find the carrying capacity,
the biomass that supports the maximum sustainable yield and the maximum sustainable
yield itself.
b
.
= − ax + b a−1
5 − ce − bb2
In the remainder of the problem you may assume that the parameters and exogenous variables in
the model have the following values: , w=2000.
a = 2, b = 0.0001, c = 4/3, p = 1
(2) Suppose that traps are prohibited (x=0) and fishing at sea is an open access industry.
Suppose e is determined by the profitability of the open sea fishery, , and that
oo = ph − we
it evolves according to the equation: . Solve for the steady state values of b and e
e
.
= voo
and construct an appropriate phase diagram. Put the biomass on the horizontal axis and
be sure to label the diagram clearly.
(3) Evaluate the stability of the system from (2). Now show how the industry would evolve
starting from a point with high biomass and low fishing effort. Show the trajectory in the
phase diagram and draw the integral curves.
(4) Suppose that a perfectly competitive, profit maximizing manager could control both e and
x. What is the profit maximizing sustainable choice of e and x? Discuss in detail and
compare the outcome with that from part (2). Be sure to explain anything surprising.
(Warning: this may not be not as easy as it appears!)

Economics 384N, Spring 1997 The University of Texas at Austin
Exam 2
yourself and you may not use any books or notes. The exam is due by noon on
Friday, May 9th. You can drop it off at the department office or slip it under my
door. Please write your STUDENT NUMBER on your bluebook and NOT your
name.
Question 1
Describe Hotelling’s contribution to natural resource economics. Would you expect it to
be exactly right in practice? Why or why not? Discuss how it can be extended to make it more
realistic.
Question 2
The most difficult problem posed by global warming is that the damages are both very
uncertain and occur very far in the future. The costs, on the other hand, are fairly evident and
must be paid now. This problem asks you to consider how decisions climate change should be
made. (Caution: the problem is not exactly like anything we talked about in class so be sure to
think about what you’re doing rather than just rushing into it.)
Suppose the benefits of emitting q tons of CO2 at time t are given by ,
Bt = aqt − bqt
2
where is emissions and are constants. In addition, suppose that accumulation of excess
qt a and b
carbon dioxide (beyond the normal level) is governed by the equation: , where s
st+1 = st + qt − dst
is the stock of excess CO2 and is the rate at which it is naturally removed from the atmosphere.
d
You may assume that s is initially zero (there is no excess carbon dioxide). Next, suppose that the
cost, , of excess CO2 is uncertain: there will either be no cost at all or else there will be a large
c̃
cost D. The probability of D in any period, , depends on the amount of excess CO2 at the
o
beginning of the period: . The expectation and variance of this particular distribution are
ot = cst
and . Finally, let the periods of
E(c̃t) = ct = cstD Var(c̃t) = rt
2 = E(c̃t − ct)2 = cst(1 − cst)D2
interest be generations (t=0 is the first generation, t=1 is the next generation, and so on), and let
the interest rate from one generation to the next be 100%.
(1) Suppose each generation ignores the future when choosing q. What will the value of q be
in this case?
(... exam continues on the next page ...)

(2) One possible policy would be to maximize the present value of expected net benefits:
max St=0
T
E
aqt−bqt
2−c̃
(1+r)t
Set up an appropriate dynamic programming problem and solve it for a problem involving
three periods (T = 2). You may ignore any costs or benefits after period 2. Solve for the
optimal q in each period and discuss how your results compare to the results from (1).
(3) Now suppose there are only two generations, and that agents in each generation are risk
averse. In particular, assume that the preferences of each generation can be represented
by the following mean-variance utility function, where is a parameter and is a net
v Tt
transfer (discussed below):
Ut = E(Bt − c̃ + Tt) − vVar(Bt − c̃)
Moreover, suppose we decide that we are unwilling to define any sort of social welfare
function to aggregate the utility of different generations. Instead, suppose that we insist
only that period 1 must compensate period 2 for any increase in s by giving period 2 a
transfer (in other words, that period 1's behavior must be pareto efficient). Solve for the
optimal value of q in the first period under this restriction. What will the transfer be?
Discuss your results and compare them to those from the previous questions. You may
assume that the interest rate between generations is still 100%.

Exam 1
(Please write your STUDENT NUMBER on your bluebook, NOT your name.)
The management of real world fisheries can be complicated by externalities the fishery generates.
A prime example of this is the tuna industry, which kills a large number of dolphins accidentally in
the process of catching tuna. In the US, at least, killing dolphins is regarded as a very bad
externality and there is even a federal law -- the Marine Mammal Protection Act -- that attempts
to control it.
This problem explores how one might manage such a resource. Suppose that in the absence of
human intervention, the population of tuna, T, evolves according to the equation:
dT
dt = aT − bT2
The harvest of tuna, h, depends on the population of tuna, the level of effort, E, devoted to
fishing, and a parameter g: h=gTE. The accidental harvest of dolphins, x, depends on the harvest
of tuna and a parameter k: x=kh. Profits on tuna are given by:
o = ph − wE
where p and w are the price of tuna and the wage rate. Because of the dolphins, however, the
social value of the tuna harvest is less than the profit:
V = o − mx
where m is the externality cost of a dolphin being killed.
(a) Solve for the efficient biomass of tuna. How does this compare to the biomass
corresponding to the maximum sustainable yield? How does it compare to the optimal
biomass when no one cares about dolphins?
(... exam continues on the next page ...)

Now (and for the rest of the problem) suppose the tuna industry is initially an open access fishery
and entry depends positively on profits and negatively on the rate at which dolphins are being
killed (people don’t want to work in an industry with a bad public image):
dE
dt = vo − vx
(b) Solve for the new steady-state and draw an appropriate phase diagram for the region near
the steady state. Put the stock of tuna on the horizontal axis. Show the steady state, the
isoclines and the directions of motion. How does this compare to what would have
happened if there were no public image effect of killing dolphins (e.g., =0)?
v
(c) Solve for the model’s eigenvalues near the steady state. Discuss the fishery’s stability.
(d) Now suppose there is a sharp, permanent increase in the unpopularity of killing dolphins
(that is, rises suddenly). Illustrate the effect of this using a phase diagram. Show the
v
change in the steady state and the isoclines and draw a trajectory from the old steady state
to the new one. Finally, please draw the integral curves.

Exam 2
Please write your STUDENT NUMBER on your bluebook, NOT your name.
A very important practical problem in the oil industry is that oil does not flow quickly
and uniformly through a reservoir. It’s easy to understand why: oil is very viscous and the
cracks in the rock through which it must flow are tiny. This can lead to a phenomenon called
“coning”, which occurs when a firm tries to extract oil from a well too rapidly. Oil near the
well is drawn out quickly but oil further away in the reservoir does not flow in fast enough to
replace it. This shortens the useful life of the well and reduces the total amount of oil the firm
will eventually be able to extract. This exam explores a simple model that takes this effect into
account.
Suppose the oil industry is competitive, is in equilibrium at all times, and that everyone
in the industry knows that at a particular date in the future, say T years from now, a backstop
technology will become available at a marginal cost of B dollars. The interest rate is r, which is
constant over time.
(a) What will the path of the oil price over time look like in the period before T? After T?
Explain your reasoning. Find equations where appropriate. Please note that this is a
question about the industry equilibrium, not about the behavior of a particular firm.
Now suppose a small, perfectly competitive price-taking firm operates in this industry
and is extracting oil from a reservoir of initial size . The firm’s objective is to maximize the
R0
present value of its profits. Prices follow the path you found in part (a); for simplicity, let the
firm’s cost of extracting the oil be zero. The interesting twist is the equation describing the
evolution of the reservoir:
dR
dt = −qv
where is a parameter greater than one. In other words, when the firm extracts and sells q
v
barrels of oil it loses a larger amount of oil from the reservoir.
... exam continues on the next page ...

(b) To solve the firm’s problem, start by imagining that the firm has arrived in period T with
barrels of oil left in the reservoir. Set up and solve its optimization problem from
RT
that point on into the future. You may assume that this can be formulated as an infinite
horizon problem. Show that the value of the barrels of oil under the optimal
RT
extraction plan can be written as:
V = B(RT )
1
v rv
v−1
(1−v)
v
(c) Now step backwards and setup and solve the firm’s optimization problem from time
zero. You should treat this as a fixed-time, free-state problem where the salvage value of
the reservoir is given by the present value of the expression from part (b). Please derive
an expression for q(t) in terms of parameters and , and an expression that implicitly
RT
gives in terms of parameters and the initial stock of oil (please note that the correct
RT
equation is a polynomial in and you do not need to solve it explicitly).
RT
(d) Discuss how this result differs from what the firm would do if it didn’t have to worry
about coning -- that is, if instead. In particular, discuss what happens as
dR/dt = −q v
moves from one to larger values.
(e) What implications does this have for the volatility of the firm’s oil production? In
particular, how would the firm react to a sudden permanent change in B? (Be sure to
consider what happens to prices in the market.)

Economics 384N University of Texas at Austin
Exam 1
Spring 1999
Please write your STUDENT NUMBER on your bluebook (or whatever paper you use), NOT
your name. The exam will be due Friday March 12th at 5:00. You can turn it in by slipping it
under my office door if you can’t catch me in person. Treat it like a comp exam and don’t use
any notes or books. This is, in fact, an old comp question. If you happened to have worked it out
while studying for this exam, that’s OK – this is your lucky day.
An interesting and important new application of natural resource economics is the study
of antibiotic resistance. To see how this works, consider a population P that is affected by a
bacterial disease. To make things simple, assume that each person in the population is either (1)
well, (2) sick with the normal form of the bacteria, which is susceptible to antibiotics, or (3) sick
with an antibiotic-resistant bacteria. Let the number of people who are well at time t be W, the
number of people who are sick with the normal bacteria be N, and the number sick with the
resistant bacteria be R. You may assume the disease is not fatal (P is constant) and that W is
equal to P – N – R at all times.
Suppose that at each point in time, the number of new cases of the normal disease will
depend on W and N as follows:
( ) 2
/
1
WN
N
φ
where φN is a parameter. The number of new cases of the resistant form is similar:
( ) 2
/
1
WR
R
φ
The two forms of the disease are identical except that the resistant form of the disease is
less contagious: φR < φN. When left untreated, people recover from both forms of the disease at
rate δ. In other words, at each point in time natural recovery reduces the stock of people with the
normal disease by δN and the stock with the resistant form by δR. You may assume that δ is
much larger than φN or φR.
1. Write down the equations of motion for this system in terms of N and R and the exogenous
variables and parameters. Solve for the steady state fractions of the population who are in
each state (e.g., W/P, etc.) in terms of the parameters.
2. Construct an appropriate phase diagram for the system. Put N on the horizontal axis and
show the steady state, the isoclines and the directions of motion. Please show your work and
label everything carefully and clearly. In addition, please draw a line showing points in the

plane with the same total number of cases of the disease as the steady state. Be sure to get its
slope right relative to the isoclines.
3. Discuss the stability of the model near the steady state.
Now let’s add antibiotics to the model. Suppose that antibiotics cause a person to get
well immediately with probability ε if she has the normal form of the disease and has no effect if
she has the resistant form. However, the decision to administer the antibiotic must be made
before it is known which form of the disease she has.
4. Write down the new equations of motion when all sick people are given antibiotics. Solve
for the new steady state and discuss how it compares with the old one. Was it a good idea to
introduce antibiotics? Why?
5. Draw a new phase diagram showing the old and new steady states and a trajectory the model
might follow from the old steady state when antibiotics are introduced.
6. Discuss how the path from part (5) would be viewed by the public. How would antibiotics
look in the short run? How would they look in the long run? What would you expect to hear
from doctors and hospitals during the transition between the short and long run?
That’s the end of the question. For whatever it’s worth, an important aspect of the real world that
has been left out of this model for tractability is that some people who are given antibiotics stop
taking them before they are completely well. That tends to accelerate the development of
resistant bacteria. One way to think about this would be that giving someone antibiotics to
someone with the normal disease causes them to get well with probability ε and causes them to
develop the resistant strain with probability γ. This would be an interesting research topic but
you don’t need to say anything about it on this exam.

Economics 384N University of Texas at Austin
Exam 2
Spring 1999
Please write your STUDENT NUMBER on your bluebook (or whatever paper you use), NOT
your name. The exam will be due Friday May 14th at 12:00. You can turn it in by slipping it
under my office door if you can’t catch me in person. Treat it like half a comp exam (so you’d
have about 90 minutes to do it) and don’t use any notes or books.
Question 1
Consider the problem faced by an oil company operating in a developing country where there is
a chance that its oil reserves will be nationalized. Suppose that its profits in period t will given
by the following function as long as the reserve has not been nationalized:
2
t
t
t cq
pq −
=
π
Where p and c are exogenous (the firm is a price taker) and constant. At the end of each period,
however, there is a probability µ that the reserve will be nationalized by the government.
Nationalization is irreversible and the firm receives nothing for any reserves remaining in the
ground. The firm’s initial reserve is S0, the interest rate is zero, and you may assume that the
firm is risk neutral.
(a) Set up an appropriate dynamic programming problem, take the first order conditions and
show that in any given period the firm would extract more if it knew with certainty (µ = 1)
that it was going to be nationalized at the end of the period than if it knew there was no risk
of nationalization at all (µ = 0).
(b) Suppose that if the firm is never actually nationalized, it would run out of oil in some period
T. Solve the firm’s problem from part (a) for periods T and T-1 and discuss how the firm’s
extraction in period T-1 is affected by the risk of nationalization.
(c) Given your answer to part (b), how would expect the exhaustion date T to compare the
exhaustion date that would have arisen if there had been no risk of nationalization? Explain
in detail how you would determine T but don’t actually carry out the calculations.

of 3
Question 2
Suppose that a price-taking firm is extracting an exhaustible resource subject to increasing costs
as the stock of the resource is depleted. In particular, suppose that at time t the firm’s profit is
given by:
)
(
))
(
)
(
(
)
(
)
( t
q
t
q
t
x
c
t
pq
t +
−
=
π
where p and c are exogenous constants, q(t) is the firm’s extraction at time t and x(t) is the
amount of the resource extracted to date. The firm wants to maximize the present value of its
profits (the interest rate is r) and the mine is worth nothing after the firm ceases operations.
(a) Set up the firm’s optimization problem and take first order conditions. Be sure not to forget
about the terminal conditions.
(b) Derive the equations of motion for the problem in (q,x) space and draw an appropriate phase
diagram. This is a little tricky – you’ll have to think carefully about the terminal conditions
in order to figure out where the isoclines go. Label everything and show some possible
trajectories to the terminal point. Discuss. In doing this problem you do not need to solve
for T or the terminal value of x explicitly but your diagram must be consistent with (a).

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