Lecture Notes for Economics 435: Economics of Resources
Prepared by Gunnar Knapp, Professor of Economics
January 30, 2001
NONRENEWABLE RESOURCES
Two important questions in resource economics are (1) what determines how non-renewable
resources are used over time (a positive question) and (2) how fast non-renewable resources
should be used over time (a normative question).
These are questions about resource use over time. In this lecture, we will address the question of
how a perfectly competitive market would allocate a non-renewable resource over time. In many-or perhaps most--cases, the competitive market model may not be directly applicable to real
world resources because many assumptions of the model do not hold--such as perfect
information, perfect markets, and lack of transaction costs. Nevertheless, the model offers a
great deal of insight into how different factors may affect resource use over time.
This model was originally developed by an economist named Hotelling and is sometimes
referred to as the Hotelling model.
Allocation Over Time and Allocation Over Space
In a perfectly competitive world, there would be perfect competition not only for the present
period, but also for all future periods. Everyone would know the market price not only in the
present but also in the future. In making decisions about how much to buy, sell or consume,
everyone would know future price trends. Changes in how much people planned to buy, sell or
consume in the future would change the path of future prices.
Because the real world isn't exactly like this, this model can be difficult for students to follow.
To try to make the model a little easier to understand, in this lecture I present two models
simultaneously, for which the underlying reasoning is the same.
The first model examines how a competitive market would allocate resources over space
(between different areas or different users) at a given point in time. The second examines how a
competitive market would allocate resources over time. The only difference between the two
models is that in the model of allocation over time, both buyers and sellers discount money
received or spent in the future.
Three Islands and Three Years
We will consider two similar sets of problems. In the first problem (allocation over space), there
are three islands in the middle of the ocean. None of them have any fresh water. Some supply
boats arrive with a limited quantity of water which they can supply to any of the three islands.
The questions we will examine are how much water do the boats sell on each island and at what
price? We will assume that there are many sellers, so that no individual sellers (or water
consumers) can affect the price.
Economics of Resources Lecture Notes: Nonrenewable Resources, page 1
In our second problem (allocation over time), there is only one island, and the water sellers have
to decide how much water to sell in each of three years. If they don't sell water the first year,
they can store it and sell it the next year or the following year. However, as in the first problem,
the total amount which they have to sell is limited.
In these examples, we assume that there are no physical costs of obtaining the water--there are no
extraction costs. Thus the price is the same as net price or economic rent. This assumption helps
to simplify our analysis and understand the main points of the model, but it clearly isn't realistic
for most resources. After presenting the model without extraction costs, we will relax this
assumption and see how the model results change if there are extraction costs.
Figures 1-4 (on the four pages at the end of these lecture notes) illustrate the analysis for the
problems. The models at the top of each page refer to the problem with three islands. The
models at the bottom of each page refer to the problem with one island and three years.
In model 1A, assume that the government on each island fixes a price at which water must be
sold. Water sellers may sell as much as they wish at this price. In this case, all the water would
be sold on island 1. This is because the price is highest on island 1, and none of the sellers would
have any incentive to sell on island 2 or island 3, where the prices are lower.
In model 1B, the government says that the price of water must remain the same every year, at the
price shown by the dashed line. Water sellers can sell as much as they want at this price. What
is relevant to the water sellers, however, is not the actual price that they receive, but the present
value of that price. The present value of the price which they receive in years 2 and 3 is shown
by the dashed lines. Since the present value of these prices is lower than the present value of the
price in year 1, they will choose to sell all their water in year 1.
If the price of a resource is held constant over time, sellers will do best to sell
everything immediately since this will bring them the highest present value.
In model 2A, the government has again fixed prices, but in this case the prices are the same in all
three islands or countries. In this case, sellers are indifferent as to where they sell their water.
In model 2B, the government fixes the price such that it rises at the rate of interest over time. In
this case, the present value of the price is the same for all three years. Water sellers are
indifferent as to when they sell their water because the present value which they receive is the
same.
If the net price or economic rent of a non-renewable resource rises at the rate of
interest, sellers will be indifferent between supplying in different years.
In model 3A, the government no longer fixes prices. Instead, there is a competitive market in
each country, with price being determined by supply and demand. The result of this competitive
market will be the same price in each country since if this were not the case, water sellers would
switch to selling water in the country where the price was highest, driving prices down in that
country and up in other countries. The total amount sold between the three countries must equal
Economics of Resources Lecture Notes: Nonrenewable Resources, page 2
the total supply. If the total supply had been smaller, the prices would have been higher in each
country, but still equal.
In model 3B, there is a competitive market in each year. (We can think of this as a "futures
market" for years 2 and 3, with sellers determining at the outset how much they will supply in
each year.) The market will allocate resources so that the present value of the price is equalized
for all years. If this were not the case, sellers would switch to trying to sell in the year when the
present value was highest, driving down the price and, hence, the present value of the price.
Quantities Q1, Q2 and Q3 would be sold in years 1, 2 and 3--the quantity determined by the
actual prices prevailing in each year. Note that for this solution to work there must be perfect
foresight on the part of sellers (just as the model with the three countries assumes perfect
knowledge of market conditions on all three islands on the part of sellers). Sellers are able to
look ahead and know what the price will be in future years.
If there are competitive futures markets for non-renewable resources, the present
value of future net prices or economic rent will be equalized for all years. This
means that the net price or economic rent will rise over time at the rate of interest.
In model 4A, the demand curve reflects the existence of a "backstop technology" for water
production (say a desalinization technology). Nobody will pay more than P* for water in any of
the islands because that is the price at which water can be gotten as cheaply through
desalinization. As a result, no water gets sold in island 3, because demand in islands 1 and 2 is
sufficiently high to drive the price of water above the price of the island 3 backstop technology
(which is available for a lower price than in the other islands). All the water gets sold in islands
1 and 2.
In model 4B, we assume the same P* (price of the backstop technology) for each of the three
years. Demand is sufficiently high in years 1 and 2 so that all the water gets used up in the first
two years, with the present value of the price being equalized in each of these two years (the
actual price rises at the rate of interest between years 1 and 2). No water is saved for year 3
because the present value of what people would have been willing to pay for water in year 3 is
not as high as the present value of what people are willing to pay in other years.
In a competitive market in which everyone has perfect information and perfect
foresight, the net price or economic rent of a nonrenewable fixed resource will rise
over time at the rate of interest, and the resource will be exhausted (run out) at
precisely the time at which the price (in this case actual price, not net price) of the
resource equals the price of an alternative technology (or the price at which no one
is willing to buy any of the resource).
Dynamically Efficient Allocation
The condition for dynamically efficient allocation of a scarce non-renewable resource (efficient
allocation over time) is that the present value of the marginal net benefit or economic rent should
be equal in all periods. This means that the marginal net benefit or economic rent should rise
over time at the rate of interest or discount rate. If a resource is abundant, resource consumption
should occur in each period up to the point where marginal net benefit equals zero. The higher
Economics of Resources Lecture Notes: Nonrenewable Resources, page 3
the discount rate, the earlier the resource should be used, if allocation is to be dynamically
efficient. Because the marginal net benefit should rise at the discount rate, relatively more of the
resource should be used in earlier periods.
Scarcity of a Non-Renewable Resource
Earlier in this course, we defined scarcity as a situation in which the marginal net benefit from
resource use is greater than zero. In this situation, the resource will have positive economic rent..
Put more simply, scarcity occurs when there's not enough of the resource for everyone to use as
much as they want (given the extraction costs of using the resource).
Earlier, our definition of scarcity had been limited to the situation either in the present or for
sustained use of a renewable resource over time. We can also provide a definition of scarcity for
a non-renewable resource, which takes account of the use of the resource over a period of time.
Scarcity for a non-renewable resource occurs when the present value of the marginal net
benefits provided by the resource in all time periods is greater than zero. Put more simply,
scarcity occurs when there's not enough of the resource for everyone to use as much as they want
in not only the present but also in all future time periods.
Scarcity imposes a marginal user cost. For non-renewable resources, greater current use
diminishes future opportunities. Even if we have more than enough water today--more than we
could possibly drink--we have an economic incentive to save it if we are concerned that we are
facing drought in the future. Similarly with many other non-renewable resources, the present
value of future economic rent they may provide due to future scarcity provides an incentive to
conserve the resource in the present.
The Model with Extraction Costs
A key assumption of the simple model which we presented earlier is that there are no extraction
costs for the resource. The economic rent in any period is therefore the same as the price.
Clearly, this assumption is not realistic for most resources. Extraction costs can be very high for
many non-renewable resources. In fact, for many non-renewable resources, economic rent is a
relatively small part of the price of the resource.
The same model results apply, however, if we remind ourselves to focus on economic rent, not
price. Recall that the condition for dynamically efficient allocation of a scarce non-renewable
resource is that from today's perspective, the present value of the marginal net benefit or
economic rent should be equal for all periods. This does not mean that the present value of the
price should be equal for all periods.
This means that over time it is the economic rent provided by the resource, rather than the price,
which should rise at the rate of interest.
Consider what this condition means for the price of the resource over time. Remember that the
price is equal to the extraction costs plus the economic rent. Suppose the economic rent is
Economics of Resources Lecture Notes: Nonrenewable Resources, page 4
indeed rising the rate of interest. If the extraction costs are stable, then the price will rise over
time, but at a slower rate than the rate of interest.
Now suppose that extraction costs are falling over time, perhaps because resource extraction
technologies are improving. If resource extraction costs are falling fast enough, resource prices
could be falling even while the economic rent is rising at the rate of interest.
Which Non-Renewable Resources Will be Used First?
Non-renewable resources vary widely in their extraction costs. In Saudi Arabia, oil extraction
costs are less than a dollar per barrel, while on the North Slope of Alaska, oil extraction costs are
several dollars per barrel. Yet on world markets the final selling price is essentially the same.
This means that the economic rent per barrel is higher for the Saudi Arabian oil.
Which resources does the Hotelling model predict will be extracted first? Suppose that you
owned both oil fields. Which would you sell first? For any given world price path for oil over
time, if you are trying to maximize the present value of all economic rent that you receive from
each oil field, you should produce the oil with the highest economic rent--and the lowest
extraction costs--first. Allowing for extraction costs, we may restate the fundamental result of
the Hotelling model as follows:
Assume a competitive market for nonrenewable fixed resources, in which everyone
has perfect information and perfect foresight, and for which there are multiple
classes of resource deposits with different extraction costs. Those resources which
have the lowest extraction costs--which are cheapest to be extract--will be exploited
first. Each class of resource deposits will be fully exploited before exploitation
begins for the next class of resources. The economic rent provided by each class of
deposit will rise over time at the rate of interest. The resources in each class will be
exhausted (run out) at precisely the time at which the price of the resource is
sufficiently high to make extraction of the next class of the resource economically
viable. This will continue until the price of the resource equals the price of an
alternative technology, or, alternatively, the price is so high that no one is willing to
buy any of the resource.
How Useful is the Model in Explaining the Real World?
How useful is the Hotelling model in explaining the real world? Do economic rents rise at the
rate of interest over time? Are least cost resource deposits fully extracted before extraction
begins of the next class of resource deposits? Can the Hotelling Model provide useful
predictions of long-term resource prices?
The answer, to date, appears to be that the Hotelling model is not particularly useful in
explaining the real world. The real world appears to behave differently than would be predicted
by the model. Real economic rents for non-renewable resources do not appear to be rising at the
rate of interest. Resources with widely varying extraction costs are exploited simultaneously,
instead of the cheapest being fully extracted first.
Economics of Resources Lecture Notes: Nonrenewable Resources, page 5
The reasons for this are abundant. The real world simply does not fit the assumptions of a
perfectly competitive Hotelling model. It doesn't even come close. Two assumptions
particularly don't apply in the real world:
Perfect foresight of future market conditions. This crucial assumption certainly
doesn't apply. No one is able to predict future market conditions and prices. No one
knows what future real interest rates will be, either. Wide variations in supply and
demand associated with economic cycles contribute to wide short-term fluctuations in
price that have little to do with long-run supply and demand.
Capacity constraints to adjustment. Increasing or reducing supply in any given period
is not an easy matter. For example, it is simply not possible to produce all of the cheapest
resources at once. Even if the world wished to use cheap Saudi Arabian oil, with the
purpose of extracting it all before turning to any higher-priced oil (as would be predicted
by the model) it would not be possible to produce the oil fast enough.
So Why Study the Model?
If the model doesn't seem to fit the real world, why should we study it? One reason is that
explaining why the model doesn't fit is useful in positive analysis of non-renewable resource
exploitation. A second reason is that some of the insights provided by the model do help to
explain the real world. Cheaper resource deposits do tend to be extracted earlier than more
expensive ones. Prices do tend to rise when current and future resource supplies are perceived to
be scarce. Resources do tend to cease being exploited when their prices rise to that of alternative,
"backstop" technologies.
One of the most useful insights of the model may be that most non-renewable resources are not
perceived as scarce. If suppliers really thought that non-renewable resources such as oil were
scarce, so that future demand would drive prices up as a result of limited supply, then they
probably really would hold off on present production to benefit from higher future prices. But
they don't. It may be that this is a real indication that the world is not, indeed, about to run out of
resources.
Economics of Resources Lecture Notes: Nonrenewable Resources, page 6
Figure 1A: Elastic demand at different fixed prices. All water will be sold on
Island #1 where the fixed price is highest.
P
P
P1
P
D1
P2
D2
P3
Island #1
Q
Island #2
Q
D3
Island #3
Q
Three islands, one year
Figure 1B: Elastic demand at the same fixed price in each period. Even though
the nominal price is the same in every year, the present value of the price is lower
in years 1 and 2. Therefore all water will be sold in year 1 when the present value
of the price is highest.
P
P
P1
P
D1
D2
P2
D3
D2
P2 = P1/(1+R)
P3
D3
P3 = P1/(1+R)2
Year #1
Q
Year #2
Q
Year #3
Three years, one island
Economics of Resources Lecture Notes: Nonrenewable Resources, page 7
Q
Figure 2A: Elastic demand at the same fixed price. Sellers are indifferent
between selling on different islands because the prices are the same.
P
P
P1
D1
Island #1
P
P2
Q
D2
Island #2
P3
Q
D3
Island #3
Q
Three islands, one year
Figure 2B: Elastic demand at a fixed price which rises at the rate of interest.
Sellers are indifferent between selling in different years because the present values
of the prices are the same.
P
P
P
P3 = P1(1+R)2
D3
P3
P2 = P1(1+R)
P2
P1
D2
D1
Year #1
Q
Year #2
Q
Year #3
Three years, one island
Economics of Resources Lecture Notes: Nonrenewable Resources, page 8
Q
Figure 3A: Demand is no longer elastic. Instead, there is a downward sloping
demand curve for each island. A competitive market results in prices being equalized
on all three islands. The equilibrium price is that at which the total quantity supplied
(Q1 + Q2 + Q3) is just equal to the total amount demanded across all three islands.
P
P
P
P2
P1
D1
Q
Q1
IslandQQ
#1
11
P3
Q2
D2
Island #2QQ
Three islands,11one year
D3
Q3
Q
Q
Island #3QQ
11
Figure 3B: Demand is no longer elastic. Instead, there is a downward sloping demand
curve in each year. A competitive market results in the present value of prices being
equalized in all three years. These prices are such that the total amount demanded for all
three years is just equal to the total quantity available for supply (Q1 + Q2 + Q3).
P
P
P
P3 = P1(1+R)2
P3
P2 = P1(1+R)
P2
P1
Q1
Year #1
Q
QQ
11
D1
Q2
Year #2
D2
QQ Q
11
D3
Q3
Year
QQ#3
Q
11
Three years, one island
Economics of Resources Lecture Notes: Nonrenewable Resources, page 9
Figure 4A: A backstop technology means that the maximum price at which water is demanded is P*, which
differs between islands. There is a downward sloping demand curve at prices lower than P*. A competitive
market results in prices being equalized across all three islands (P1 = P2 = P3), with total demand at that
price equaling total supply. At this price, no water is sold on Island #3, where the backstop technology price
is less than the competitive price which prevails if all water is sold on Islands #1 and #2.
P
P
P
P*1
P*2
P1
P2
D1
Q
Q1
IslandQQ
#1
11
P3
P*3
D2
Q2
Island #2
Three islands, one year
D3
Q
Q3
Q
Island #3
Figure 4B: A backstop technology means that the maximum price at which water is demanded is P*, which
is the same in each year. There is a downward sloping demand curve at prices lower than P*. A
competitive market results in the present value of prices being equalized in all three years (actual prices rise
at the rate of interest), with total demand at these prices (Q1+Q2) equaling total supply. No water is sold in
year #3, where the backstop technology price is less than the competitive price which prevails if all water is
sold in years #1 and #2.
P
P
P
P3 = P1(1+R)2
P3
P*
P2 = P1(1+R)
P2
P1
Q1
Year #1
Q
QQ
11
D1
Q2
Year #2
D3
D2
Q
Year #3
Three years, one island
Economics of Resources Lecture Notes: Nonrenewable Resources, page 10
Q