Extraction of exhaustible resources: Economic Theory
The extraction of an exhaustible resource can be seen in its implest form as a simple
profit maximizaiton problem, for a society the same as for an individual. The scociety
or individual face the same probelm, namely that (due to mass balance):
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(1) |
where X is the stock of the resource and H is the "harvest" per unit time. The
profit is expressed very simply (we will look at costs later).
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(2) |
But the society maximizes its profit over all times, not just the present
time. For this, the value of future profits must be discounted at a
rate s that is more or less equivalent to the interest rate.
The maximization problem is thus posed:
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(3) |
This is the fundamental problem in all of exhaustible resource economics, and it
was first posed and solved by Harold Hotelling in 1931. The almost trivial
solution is that the maximum total profit is given by:
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(4). |
This formulation is known as the Hotelling Rule, and predicts that
prices inrease at an exponential rate that is equal to the exhange rate.
That means, in an economic sense, a mineral deposit in the ground has the
same significance as a bond, and is in some sense interchangeable with
such a financial instrument.
An overview of the hotelling price path is given in the figure below.
As the resource is depleted, the price rises and as the price rises
the demand, and hence the consumed quantity, falls. All the while
the quantity remaining in the reservoir sinks.
Figure 6: The Hotelling price/demand/stock curves.
All of this continues until a price is reached where an alternative
technology or a substitute for the resource becomes economically viable.
This is known as a backstop price. The backstop resource
is on the flat part of its exploitation curve, so that the price
of the resource is blocked and production ceases. At this point,
the resource is said to be economically exhausted, even though
there is some amount of the resource remaining in the ground. Since there
is always a backstop technology at a sufficiently high price, it
follows that no exhaustible resource can ever be completely
extracted. That means that in some sense, exhaustible resources
aren't exhaustible.
from this point we can begin to examine in a qualitative way some of the
consequences of the basic Hotelling theory. Figure B shows the consequence
of a change in the discount rate, the igher the discount (exchange) rate, the
quicker the resource will be depleted.
Figure 7: Effect of discount rate on extraction.
Figure 8 shows the behavior of the monopolist. His principal method of
maximizing profit is to restrict supply. This depletes the resource
more slowly than in the free market case. The monopolists is thus
the conservationist's best friend. One god example of this is the near-monopolistic
behavior of OPEC during the 1970's, which ushered in an entire era of
environmental awareness.
Figure 8: Monopolistic price path.
Finally, the discovery of new deposits, new technologies, new
strategies of conservation will introduce downward pressure on
prices and extrnd the lifetime of the resource. If this
happens at discrete times, the price path can take on a sawtooth
pattern, but always with the exponential upward tendency.
This is illustrated in Figure 9.
Figure 9: Price path with discoveries.
Nearly all of modern econmic exhaustible resource theory is an attempt to
hang more realistic sets of assumptions onto this fundamental structure.
For example considering costs:
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(5) |
Where the extraction cost is a function of X
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(5) |
This produces slightly different results, but remains fundamentally the same.
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