Econ 427
Professor Ickes
Energy Economics and Energy Security
Spring 2013
Homework Assignment #2: Answer Sheet
1. Consider an exhaustible resource problem with demand given by  = 100 −  . Let the cost
of extraction be given by  = 10, and suppose that the rate of interest,  = 10. Suppose
that the total reserve of the resource is  = 153. Assume that Hotelling’s Rule is satisfied.
(a) Let  be the final year of production. If exactly one ton of the resource is extracted in
period  , during how many years will extraction take place (using a spreadsheet is fine
here)?
brief answer We know that  = 100−1 = 99. We also know that  −1 −10 = 99−10
=
1+1
P1
89
= 80 909, so  −1 = 91, and  −1 = 9, and =0  − = 1 + 9 = 10 Continue
11
P
this process until   − = 153, and we have the answer. So:
Figure 1:
(b) What is the price of the resource in period 1? What is output in period 1? Draw the
path price and the extraction path.
brief answer We see from the table that the initial price is 60 and initial output is 398.
So that path price and the extraction path look like:
Figure 2: Price and Output Path
(c) Suppose that  = 354. If everything else is unchanged, how do your answers change?
brief answer Now we have greater total reserves so we must produce longer. We just
P
need   − = 354 Nothing else is changed, so we now have:We now produce for
11 periods, initial price is lower and initial output is higher.
1
Figure 3:
2. Suppose that demand for an exhaustible resource is given by  = − , where    0 are
constants. Draw this demand curve. Suppose that extraction costs are constant at rate, .
The economy is competitive and the interest rate is given at .
(a) If the initial price level is 0 , and if Hotelling’s rule holds, then how does  relate to 0 ?
As  → ∞ what happens to  .
brief answer We know that  = 0 (1+) . We also know that as  → ∞ then  → ∞
as there is no choke price. The demand curve looks like
p4
2
0
0
2
4
q
Figure 4: Demand Curve for  = − , with  = 1  = 1
(b) Using the expression you have from part (a) and the demand curve, derive an expression
for  in terms of the parameters. From this expression can you determine what happens
to  as  → ∞?
brief answer We have  = − and  = 0 (1 + ) , so − = 0 (1 + ) , so solving
for  we have:
"
#1

 =
(1)
(1 + ) 0
As  → ∞ the denominator gets bigger and bigger so  → 0, which means that
 → ∞
(c) If initial reserves are given at 0 , and if reserves are exhausted as  → ∞, what is the
relationship between total extraction and reservesl?
brief answer We know that total production equals reserves, so we sum up (1) and thus
P∞ h
=0

(1+) 0
to do that.
i1
= 0 . You could solve this expression for 0 but I did not ask you
2
pt(1+r)^t parameters
0.714
0.7497
0.787185
0.826544
0.867871
0.911265
0.956828
r
1.00467
p0
1.054903
a
1.107648
b
1.163031
1.221182
1.282241
1.346353
1.413671
1.484355
1.558572
1.636501
1.718326
1.804242
1.894455
1.989177
time
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0.05
0.714
1
0.1
quantity
29.04142231
17.82891406
10.94540664
6.7195302
4.125208647
2.532520262
1.554747754
0.954480252
0.585968077
0.359733569
0.220845205
0.135579799
0.083234235
0.0510986
0.031370108
0.019258525
0.011823064
0.007258336
0.004455988
0.00273559
0.001679415
0.001031015
Tq
29.04142
46.87034
57.81574
64.53527
68.66048
71.193
72.74775
73.70223
74.2882
74.64793
74.86878
75.00436
75.08759
75.13869
75.17006
75.18932
75.20114
75.2084
75.21286
75.21559
75.21727
75.2183
Figure 5:
(d) Using a spreadsheet, assume that  = 1  = 01  = 05  = 0 and 0 = 75 Can you
figure out what 0 has to be (within a couple of decimal points)? What would happen
to the price path if  = 01? Explain.
brief answer Just use expression (1) with the parameters given, start with period 0, let
 keep increasing and total production. Try different values for 0 and notice that
 = 0 (105) . See what value of 0 works so that total production approaches 75 I
got this when I tried 714.If the interest rate was lower then the price will grow at a
slower rate. The present value of production in the future is now greater than when
 = 05. The initial price would have to be higher to lower initial quantity. I get an
initial price of 822 and 0 = 71.
(e) Suppose 0 = 200, and  = 05. What would be your new estimate of 0 ? Can you
explain why the initial price changes that way when reserves rise?
brief answer If we have  = 05 but higher reserves then we must have a lower price so
that current production is greater. That way we can approach the higher quantity of
reserves. I get an initial price of about 648 and quantity 76. The price rises at the
same rate in both cases, so if the price did not fall output would not be high enough
to approach the higher level of reserves.
3. Return to problem (1, yes problem 1) but assume now that the producer is a monopolist. If
everything else is unchanged, re-do problem 1.
brief answer The key here is to note that total revenue,   = 100 − 2 , so  =
100 − 2. Then we reason exactly as in problem 1 but we note that for the monopolist
+1 −
we have  −  = 1+
. With  = 1 in the last period we have  = 98, so
98−10
 −1 = 10 + 1+1 = 900, and we just keep proceeding accordingly. Since we have the
same reserves, we get
(a) Is the monopolist’s initial price higher or lower than in the competitive economy? What
about the time to completion?
brief answer Comparing this table with that from problem 1, we see that the initial
quantity is lower (26 versus 40) and the initial price is higher. Production takes
longer till exhaustion.
3
MR
98
90
82.7272727
76.1157025
70.1051841
64.6410764
59.6737058
55.1579144
51.0526495
47.3205904
Q
MR‐ C
1
5
8.63636364
11.9421488
14.947408
17.6794618
20.1631471
22.4210428
24.4736753
26.3397048
88
80
72.7272727
66.1157025
60.1051841
54.6410764
49.6737058
45.1579144
41.0526495
37.3205904
(MR‐c)/(1+r)
80
72.7272727
66.1157025
60.1051841
54.6410764
49.6737058
45.1579144
41.0526495
37.3205904
33.9278095
TQ
periods
1
6
14.6363636
26.5785124
41.5259204
59.2053821
79.3685292
101.789572
126.263247
152.602952
10
9
8
7
6
5
4
3
2
1
Figure 6:
45
40
35
30
tyi 25
t
an
u20
Q
15
10
5
0
1
2
3
4
5
6
Monopolist
7
8
9
10
11
12
Competitive
Figure 7:
(b) Does the monopolist or the competitive economy ”conserve” more? Explain.
brief answer The monopolist produces for a longer period of time. But this does not
raise welfare since the competitive solution re-produces the path the social planner
would produce. Marginal benefit is too high in the initial periods with the lower
production the monopolist chooses.
4. Consider the basic Hotelling model of exhaustible resources. Assume a competitive economy
with many producers, a fixed cost of extraction, , and a choke price, . The rate of interest
is given at rate . What happens to the extraction path of the resource (the plot of output,
 , against time) if:
(a) the rate of interest falls.
brief answer If  falls, it is all of a sudden better to keep a dollar’s worth of oil in
the ground than a dollar in the bank. So oil production falls. This causes 0 to rise
and 0 to fall. The Hotelling Rule requires that net rent grows at the rate of interest
which is now lower. So clearly the time to exhaustion must rise. If price starts
lower than before, and if it grows slower than before, it must take longer to reach
 Economically, the present value of future production has increased, so we should
shift extraction towards the future (see figure 8).
(b) the demand for the resource increases suddenly.
brief answer If the choke price remains unchanged this means that the demand curve
becomes flatter — greater demand at any price below  In the case of the inverse
4
qt
q0
initial optimal extraction path
new optimal extraction path
q0
T
T'
t
Figure 8: Shift in the Extraction Path
demand curve used in class,  =  − , this means that  falls. If the price path
did not change we would extract more very period and total production would exceed
. So 0 must rise to dampen down the quantity demanded. Since prices still grow
at the rate  it follows that 0 must also rise. If not, then we would reach  before
exhaustion. You can also see this from the expression we derived in class for output:
 =  [1 − (1 + )− ], so if  falls  is higher. But this means that we must reach
exhaustion at a lower  , so the new extraction path must have higher 0 and lower
.
(c) the choke price falls
brief answer This means that  is lower. If the price path were unchanged we would end
up with extra oil which cannot be optimal. So 0 must fall and 0 must increase. The
time to exhaustion must also fall, since  reaches the new lower  in less periods.
So 0 rises and  falls.
(d) a tax on the sales (gross revenue) of the resource,  per barrel is imposed on producers.
brief answer I said a tax on revenues not rents. If I sell a barrel I in period , I now
keep (1 − ) . The producer is now equating
(1 − )+1 − 
(2)
1+
the tax is clearly not neutral (as it would be if the tax were on rents, then we could
cancel out the (1 − ) terms). The impact of the tax is to lower the present value
of current profits relative to future profits, as the left-hand side of (2) fall by more
than the RHS. So the producer wants to produce less in the current period. So 0
falls. Since production is moved to the future the time to exhaustion must rise, since
prices still rise at the rate of interest.
(1 − ) −  =
(e) a new discovery of oil takes place that doubles reserves.
brief answer If reserves now equal 2, then production must rise. But if the choke
price is unchanged and prices grow at the rate of interest, then 0 must rise and 
 −1
must increase as wellh so that the total
i amount of production Σ=0  must rise. But
we know that  =  1 − (1 + )−  then the sum of production is given by:
−1
−1 
Σ=0
[1 − (1 + )− ] = 
 = Σ=0

5
so if  doubles, and  is given, the only thing that can rise is  .
6