Exhaustible Resources
Contents:
TwoPeriod Nonrenewable Resource Model with Extraction Costs
TwoPeriod Nonrenewable Resource Model with Open Access
Policies to Correct Open Access Market Failures
TwoPeriod Nonrenewable Resource Model with Monopoly
TwoPeriod Nonrenewable Resource Model with Changing Demand
TwoPeriod Nonrenewable Resource Model with a Backstop Technology
Sketch of An "nPeriod" Model of Nonrenewable Resources
Summary of Nonrenewable Resource Model Results
TwoPeriod Nonrenewable Resource Model with Extraction Costs
Utility Maximization
Assume that we are concerned with a nonrenewable resource under unsatiated
demand in two periods, t=0 and t=1.
B
(
X
t
)
is the
gross
benefit
associated with using
X
t
amount of the resource in period t.
Now, let c denote marginal extraction cost of the
resource.
Hence, the
Net Benefit
for period t becomes:
B(X
t
)
−
cX
t
We can maximize social welfare by solving:
max
X
0
,X
1
NPV SW(X
0
,X
1
)
[
]
=
B(X
0
)
−
c
⋅
X
0
+
1
1
+
r
B(X
1
)
−
c
⋅
X
1
[
]
subject to the constraint of the total available resource, S:
S
0
=
X
0
+
X
1
.
The Lagrangian equation for this problem is:
L
=
B(X
0
)
−
c
⋅
X
0
+
1
1
+
r
B(X
1
)
−
c
⋅
X
1
[
]
+
λ
(S
0
−
X
0
−
X
1
)
The F.O.C.'s are:
(1)
L
X
0
=
B
X
(X
0
)
−
c
−
λ
=
0
.
(2)
L
X
1
=
B
X
(X
1
)
−
c
1
+
r
−
λ
=
0.
(3)
L
λ
=
S
0
−
X
0
−
X
1
=
0.
where
B
X
(X
t
)
=
MB
of using
X
t
amount of the resource in period t, and c = MC.
Understanding the Relationships
Equation (1) states that the price of the mineral resource,
P
0
[which equals
B
X
(X
0
)
]
equals marginal mining cost, c, plus the shadow cost of the resource constraint,
λ
or:
P
0
=
c
+
λ
. The shadow cost of the resource constraint is also called the
user cost
in
dynamic problems. It is the opportunity cost of not being able to use the marginal unit of
the resource in the future if you use it today.
Stated in another way, the shadow price is the
benefit one could gain from increasing the constraint of initial resource by one unit.
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2
Notice from Equation (2) that higher interest rates reduce the user cost, as before.
A
reduction in user cost implies that one should use more of the resource today and less in the
future.
If one uses more of the resource today and less in the future, then the resource will
be plentiful today and scarce in the future, so the price of the resource will be lower today
and higher in the future.
Hence, an increase in the interest rate shifts a larger share of
consumption from the future to the present, lowering the price today, but raising the price in
the future.
Impacts of Extraction Costs
It is clear from both Equation (1) and Equation (2) that higher extraction cost
reduces the net marginal benefit associated with using the resource in either time period,
which in turn:
•
reduces resource use and raises prices in both periods.
•
reduces the user cost, increasing the incentive to shift some consumption from the
future to today.
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 Fall '08
 Wood
 Economics, Monopoly, Open access, nonrenewable resource, Nonrenewable Resource Model, TwoPeriod Nonrenewable Resource

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