1
Updated 18.02.04
ECON/SØK500 Natural resource economics, Spring 2003
Olav Bjerkholt:
Lecture notes on the Theory of Nonrenewable Resources
3. Resource markets and imperfect competition
Due to limited supply sources market power is more common in resource markets than in
traditional markets for economic activity. Here, we will first discuss the extreme case
when there is a single resource owner supplying the market. The question to be asked is
how this supply structure affects the realization of the optimal extraction program and the
Hotelling rule derived referred to above.1
To simplify the arguments we will in the following assume that the resource may be
extracted at zero costs. This means that the market price of the resource and the resource
rent will be one and the same. The single resource owner is assumed to be faced by a
downward sloping demand function, p=p(y). From this demand structure the marginal
revenue is derived as m = d[p(y)y]/dy = p , where  = 1 - (1/ ) , and  being the elasticity
of demand (in positive value). The objective for the monopolist is the same as for the
competitive firm, i.e. maximizing discounted profits over a horizon that is also to be
determined. By using the same kind of argument as in the classical Gray model, it is
intuitive that optimal extraction, requires that marginal revenue increases at a rate equal to r.
Again, the latter is the return earned from transforming a resource unit into financial assets.
By leaving a marginal resource unit unexploited, the owner on the other hand therefore will
have to require an equal increase in marginal revenue. Formally, the equilibrium condition
for the asset market in the simple monopolist case is
(3.1)
mt
r
mt
What characterizes this condition and the extraction path in the monopolist model from
the optimal solution in the competitive case? First, it may be noted that assuming
monopolist behaviour on the supply side, what we are actually studying here is the
market solution. As opposed to the Gray model for a single competitive firm, in the
present case the equilibrium paths for the resource price and extraction are determined
simultaneously2. Second, it may be noticed that with the present assumptions, it is not
necessarily the case that the monopolist solution deviates from the competitive path, as in
static market equilibrium. Take for instance the case where the demand schedule is of the

constant elasticity type, i.e. p = Ay , where A and  are constants. In that case the rate of
1
A simple case of monopoly behaviour was analysed in Hotelling (1931).
Note that with constant (e.g. zero) extraction costs, the Gray model involves complete depletion of the whole
deposit when the Hotelling condition is fulfilled. In the monopolist case, however, even with constant unit costs
of production, the extraction will have to be stretched over a time horizon, possibly infinite, depending on the
demand structure.
2
2
change of marginal revenue equals that of the market price, so that the monopolist condition
(3.1) is actually identical to the equilibrium solution in a competitive resource market, and
the Hotelling rule prevails.
The role of the elasticity of demand
The equilibrium solution for a resource monopoly can be further illuminated by
expressing (3.1) in the following way:
pt

r t
pt
t
(3.2)
Relation (3.1) expresses the crucial role of the elasticity of demand when discussing
monopoly equilibrium. We may distinguish between three different cases:
a)  is constant.
This is the situation just studied. The monopoly and competitive equilibrium are
identical, i.e.
p
r
p
(3.3)
b)  increases with p .
As p increases along the optimal path, this implies that  > 0 , and consequently3
p
r
p
(3.4)
c)  decreases with p .
In this case, the monopoly price path satisfies
(3.5)
p
r
p
Although the result in case a) may seem surprising, it points directly to the dynamic
nature of equilibrium concepts in the theory of exhaustible resources. A resource
monopolist has the possibility to reallocate production over time, and his primary
objective is to balance the returns on different assets at various points in time.
Accordingly, it is not so much the current size of the demand elasticity that matters, it is
how consumers' reactions on supply cutbacks change over time that is important for the
resource monopolist. With an isoelastic demand structure, there is nothing to gain by
restricting extraction at a specific point in time.
In case b) the demand elasticity (in absolute value) increases as demand increases
towards "saturation". The implication of monopoly behaviour is a higher price initially,
3
This is seen by expanding
 in the following way:  
1 d  dp
.
 2 dp dt
3
and smaller production in an early period of the "extraction history" compared to a
perfectly functioning competitive market. This result is analogous to static theory, where
monopoly behaviour implies that output is restricted. This was also the case studied by
Hotelling (1931), based on a linear demand schedule. The fact that current extraction is
restricted, forms the background for the frequent statement that "the monopolist is the
conservationist's best friend".
Case c), on the other hand, implies that the curve for the monopoly resource price is
steeper than the competitive price path. There is over-utilization of the resource initially,
due to a sub-optimal price. Some authors (see e.g. Dasgupta and Heal (1979)) have
argued that market equilibrium in this case may be unstable, since the high returns on
resource stocks stimulate speculative behaviour.
Oligopolistic markets
A perhaps more interesting and realistic description of many resource markets is to
assume that the supply side is dominated - not by a monopoly - but by a limited number
of sellers. Accordingly, the literature on exhaustible resources contains a large number of
studies where the market structure is characterized by some kind of oligopoly behaviour
or game theory. By including imperfect competition in a dynamic context, the models
quickly become formally complicated. Here, we will restrict ourselves to provide some
"taste" of a couple of the oligopoly models that are discussed in the literature on resource
economics.
The most straightforward extension of monopoly behaviour is to model the resource
market as a duopoly. Assuming that the two resource owners do not coordinate their
decisions, the most common equilibrium concept is that of Nash-Cournot. This implies
symmetric non-cooperative behaviour: each producer takes the other's extraction profile
as given, and maximizes discounted profits or value of their own resource base. The
equilibrium is defined in such a way that the producers find no reason to regret their
decisions.
The equilibrium conditions in the duopoly model are analogous to the monopoly case.
The speed of extraction is regulated so that marginal revenue increases at a rate equal to
the market rate of interest. By again disregarding from extraction costs, we have
xi 1
)  i ert (i  1, 2)
y
where y = x1 + x2 . From this relation it is seen that the marginal revenue for a duopolist is
modified from that of a monopolist, as the elasticity of demand is "weighted" by the market
shares of the respective producers. Denoting the elasticity term in parenthesis in
Error! Reference source not found. by  i , and summing over the two equations for i=1,2,
yields
(3.6)
mi  p( y )(1 
p( y )( 1   2 )  (1  2 )e rt
(3.7)
This equation is formally identical to the equilibrium condition in the monopoly case, so
that it can be transformed similar to equation (3.2), but where now
4
  1   2  2 
1

Like the monopoly case, the price path in the duopoly generates a "bias" compared to
perfect competition, depending on the elasticity of demand. However, for a similar demand
structure, the bias is "dampened" in the duopoly model, expressed by the change in the
interpretation of  .
It is straightforward to extend the duopoly model to a general Nash-Cournot equilibrium
with, say, N producers. In that case, the equilibrium condition is formally unchanged, but
1
the definition of  changes to  = N - ( ) . As should be expected, as the number of

resource owners in the market (N) increases, the Nash equilibrium converges towards the
competitive solution.
So far, we have analysed the solution of the duopoly model assuming implicitly that both
producers extract simultaneously. Even in the simplified case where we disregard from
extraction costs, this will in general be true only for some part of the total horizon. More
precisely, if the sizes of the initial resource stocks differ for the two producers, it can be
shown that the smallest deposit will be emptied first, i.e. at a time where there is still
some left in the ground of the initially larger deposit4.
Rather than treating different suppliers in a resource market symmetrically, for many
resource markets it is more plausible to assume different market power among different
sellers. A common construction in the literature of exhaustible resources is to distinguish
between two groups of suppliers: a cartelized group and a competitive fringe.5 The latter
consists of a number of identical producers that all behave competitively, as they take the
market price as given. The cartel's decisions, on the other hand, take into account both the
demand reactions of the consumers and the behaviour of the competitive fringe.
The Stackelberg approach to the cartel-fringe
A frequently proposed equilibrium to this market structure is the Stackelberg solution.
This implies asymmetric responses between the two groups of suppliers. The cartel is
assumed to behave strategically. In its price setting, it recognizes that the fringe reacts to
the prices. Assuming perfect information, the cartel is accordingly able to calculate in
detail how the fringe responds to any price profile that is announced.
4
This may e.g. be seen by regarding equation Error! Reference source not found.. If we assume that
resource owner no. 1 controls the largest initial resource stock, we must have    .Furthermore, we assume
that in equilibrium
 i >0, i = 1,2 , implying
p
mi
= r > . In this case marginal revenue approaches the
p
mi
price path "from below" as extraction decreases towards zero. It can then be seen that in the period of
simultaneous extraction, the marginal revenue of producer 2 exceeds the marginal revenue of producer 1.
Consequently, marginal revenue of producer 2 will have to reach the path for the resource price at a time where
there is still positive extraction from deposit no.1.
5
See e.g. Salant (1976), Newbery (1981) and Ulph (1982).
5
The solutions for the extraction paths for the cartel and the competitive fringe and the
corresponding development of the resource price depend on the central parameters of this
kind of models: the demand structure, the initial resource stocks and the costs of
extraction. For a detailed discussion of different solutions we may refer to Dasgupta and
Heal (1979) and Newbery (1981). Here, only some basic characteristics of the
equilibrium solution will be touched upon.
A central point in this model concerns the succession of extraction for the two groups of
producers. In general, there are two features that are decisive in this respect:
i)
As usual, the cartel adjusts its extraction so that its marginal revenue increases at
the rate of discount. Under the assumption that  increases with p , we have
learned above that this implies that the resource price increases at a rate less than r.
On the other hand, fringe production requires equality between the rate of increase in
the resource price and the interest rate. From this it may be concluded that in the
special case of zero extraction costs, Stackelberg equilibrium implies that the fringe
supplies the market in an initial phase. In the same period the cartel's return from
delaying extraction exceeds the returns on other investment (r).
ii)
The matter becomes somewhat more complicated when we introduce costs of
extraction. Let the (by assumption constant) unit extraction costs for the
competitive fringe and the cartel be denoted by b f and bc respectively. If b f = bc ,
the situation is principally the same as with zero extraction costs: since the
competitive price path is steeper than the price trajectory under cartel production, the
fringe will carry out extraction in an initial phase.
It should then be obvious that b f < bc contributes to "strengthen this result":
Similarly to the case of a competitive market, asset market equilibrium implies a
tendency that low cost deposits are extracted earlier than higher cost reserves (cf..
above). Only if unit costs of the fringe are significantly higher than the extraction
costs of the cartel (sufficiently high to induce the cartel price path being steeper than
the competitive price path) will the succession of production be turned around: the
cartel takes control and supplies the market initially, while fringe production is held
back for some time. While extracting, the marginal revenue of the cartel increases at
the rate of interest. The return to the "high cost competitive fringe" from not
extracting exceeds the interest rate in an initial period.
It should be noted that all decisions are taken at time zero. The cartel has complete
information of all relevant conditions both regarding demand and the behaviour of the
competititve fringe. The fringe reacts passively to the price path announced by the cartel
at time zero. These reactions are calculated by the cartel for the whole period, and the
price path finally chosen is one that maximizes total discounted profits of the cartel.
It can be shown that the equilibrium solution involves three phases. In the first period
( [0,T 1 ] ) the fringe supplies the market. At T 1 the resource stock of the fringe is emptied
and the cartel takes over the market. For some time ( [ T 1 ,T 2 ] ), however, the cartel follows
a price and extraction policy that is a continuation of the competitive price path. Then, at
time T 2 the cartel adopts the price behaviour that corresponds to its de facto monopoly
situation in the market, with marginal revenue increasing at rate equal to r.
6
Why does not the cartel exploit monopoly power as soon as the fringe has exhausted its
reserves? The reason is that in that case, there would be a jump upwards in the market
price at T 1 . Knowing this, the firms in the fringe would hold back some of their production
potential in order to take advantage of this price rise. The price path and extraction policy is
chosen and announced by the cartel in order to speed up the exhaustion of the deposit of the
competitive fringe. However, a major question is whether the price- and extraction plans
announced by the cartel are credible. Clearly, in the lack of binding contracts there is
nothing preventing the cartel to take advantage of the situation and adopt the monopoly
behaviour immediately. This point to that the Stackelberg solution in the present case is
dynamically inconsistent. The cartel has incentives to deviate from the original plan.
Realizing this, the competitive fringe will restrain extraction more than expressed by the
Stackelberg solution, in order to gain from the potential price hike at T 1 .
The problem of dynamic consistency has been discussed extensively in the theory of
exhaustible resources, see e.g. Newbery (1981). In the Stackelberg model the problem
arises whenever equilibrium implies an initial period of fringe extraction. The core of the
problem is that the Stackelberg game in essence is static; all decisions are made at the
start of the horizon. In a dynamic context, and without any formal commitments to stick
to the announced plans, there is by necessity a major instability and inconsistency
inherent in the solution.
The asymmetric Nash-Cournot solution (Salant)
One way to avoid the inconsistency problem is to change the "rules of the game", and
assume that all agents at the supply side act strategically. This turns the market structure
into a true dynamic game. Given this market structure, the non-cooperative equilibrium is
typically assumed to be of the Nash-Cournot type (see e.g. Salant (1976)). The
equilibrium conditions include equations similar to Error! Reference source not found.,
stating that for periods with positive extraction marginal revenue must increase at the rate
of interest. (For the fringe marginal revenue equals the net price.) For the cost
configurations underlying the Stackelberg equilibrium it can be shown that the only
possible Nash-Cournot solution is one where the cartel and the fringe produces
simultaneously in an intial phase. In this time interval, both the marginal net price of the
competitive fringe and the cartel's marginal revenue increase at rate equal to r <. This is
achieved by the cartel continuously adjusting its production and thus its market share (cf.
relation Error! Reference source not found.).
The Nash-Cournot equilibrium yields a higher price path in the initial phase compared to
the Stackelberg solution (assuming binding contracts make this an alternative). The fringe
benefits from the flexibility and strategic behaviour in the Nash-Cournot solution, while
the cartel loses, since it is "forced" to extract earlier than it would have preferred given a
completely "passive" group of fringe firms. From this we may conclude that in a market
for an exhaustible resource a dominating cartel may have incentives to "move" the
organization of the market in the direction of negotiating pre-commitments with other
independent producers in the market. Some tendencies that may fit into such a pattern
have been observed, e.g. in the international oil market.
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Back-stop technologies.
Non-renewable resources can to some extent be substituted by other products. For
example, other energy carriers, such as hydro power, nuclear power and coal can replace
oil and natural gas for some purposes. Furthermore, synthetic oil produced from either
coal or tar sand represent an almost perfect substitute for crude oil. In the literature on
resource economics a perfect substitute for an exhaustible resource is denoted a back-stop
technology. This definition assumes that there is no physical constraint on the availability
of the substitute good6. Furthermore, the essential feature characterizing a perfect
substitute is that if the prices between the alternatives differ, the demand is directed either
towards the resource or against the substitute good. In other words: assuming the
substitute good is produced at constant average costs c , when the price path of the
resource (assumed to be monotonically rising) reaches c , the back-stop product will capture
the complete market.
Below, we will sketch briefly how the presence of a back-stop technology will affect the
equilibrium in a market for an exhaustible resource. The discussion will be restricted to
the cases were the properties and the market conditions for the substitute product are
known with certainty. In practice, there is considerable uncertainty of various kinds
related to substitute products. Some aspects involving uncertainty will be discussed in the
next section.
We start by considering how a back-stop technology influences the equilibrium in a
competitive resource market. All along we assume that the substitute product is available
at constant average costs, c . We again turn to the case of constant extraction costs, at
constant b , where b < c in order to have a meaningful problem. As before, equilibrium in
the asset market requires that as long as the resource supplies the market, the net marginal
price of the resource must increase at the rate of interest. Thus, the existence of the backstop affects only the end point condition. With no substitute product available, we saw
above that the length of the extraction period was determined by relation (2.5), assuming
that the demand was choked off at the price pmax . With the unit cost of the back-stop
technology representing the new "price ceiling" (assuming c < pmax ) it should be clear that
the end point condition in the present case becomes
(3.8)
( p0  b )e rT  c  b
i.e. similar to the (2.6), but with c replacing pmax. As long as the initial resource price
exceeds unit extraction costs, the competitive equilibrium (equivalent to the optimal
extraction program) implies complete depletion of the resource before production of the
substitute starts up. In this period, the net price of the resource increases at rate equal to r. At
T  , the resource stock(s) are exhausted, and the substitute product takes over supply of the
market. As is rather intuitive, the existence of a back-stop technology tends to push down
the price path of the resource in the period [0,T ] . As is quite natural, the resource is
extracted over a shorter time horizon than in a situation without access to a substitute
6
Strictly, this is seldom true. The essential point is, however, that the substitute is available in such large
quantities that for practical purposes one may disregard from the limitation.
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product, while on the other hand, the presence of a back-stop implies that the supply (and
demand) can be sustained infinitely.
Let us then turn to a discussion of consequences for the monopoly equilibrium of the
existence of a back-stop technology7. As before, the objective of the monopolist is to
maximize discounted profits of resource extraction, given that the demand side and cost
structure are known with certainty. In the present case, the monopolist has to take into
account a new restriction, namely that the market price of the resource cannot exceed the
unit cost of the back-stop technology, c . By inverting the demand equation, this is
equivalent to stating that the resource extraction cannot fall under a certain level of demand.
Formally, the optimization problem of the monopolist is restricted by
Rt  p 1 (c )
(3.9)
It can be shown (see e.g. Hoel (1978)) that a necessary condition for monopoly
equilibrium in this case is
mt  (c  b )e  r (T2 t )
(3.10)
where T 2 is the time where the resource is fully exhausted.
The equilibrium consists of two phases:
i) In the first phase [0,T 1 ] , the market equilibrium is characterized by the traditional
Hotelling rule: the marginal revenue for the monopolist increases at a rate equal to r, i.e.
equality in (3.10) prevails. The resource price increases monotonically, until it reaches the
price (unit cost) of the substitute product at T 1 .
ii) After T 1 the resource monopolist continues to supply the market for some time at the
given price c , until the resource is fully exhausted at T 2 . At this point the back-stop good
takes over. In the second period of resource production, [ T 1 ,T 2 ] , strict inequality (3.10)
prevails, while marginal revenue actually being detemined by m = m( p-1 ( c )) .
As indicated, the introduction of a back-stop technology in a monopolized resource
market induces the resource monopolist to choose a higher price path in the first phase,
[0,T 1 ] . Thus, while the effect of a back-stop alternative in a competitive market is to lower
the price path, initial prices are pushed upwards when a monopoly has control of the
resource. This somewhat surprising result may be explained in the following way: Due to
the back-stop technology, there is a limit on the prices that can be obtained by the
monopolist. In order to compensate for this restriction on future prices, prices are increased
early in the extraction period.
The first phase just described may be non-existing. This is the case if the demand
elasticity (  ) is less than one in absolute value.8 In this case it will be optimal for the
monopolist immediately to raise the price to the level of the unit cost of the back-stop
7
A common reference in the literature discussing monopoly equilibrium and the implications of a back-stop
technology is Hoel (1978).
8
Note that in this case the traditional monopoly solution breaks down, since the marginal revenue is negative.
9
product, c . A lower price is ruled out since this will actually lower the gross revenue. The
price will be kept constant at this level until the resource base is empty9. Accordingly, in the
literature the market equilibrium just described is denoted limit pricing.
The result is that part of a resource stock is held back and sold to a non-increasing price
equal to the cost of a back-stop technology, occurs in several models of resource markets
based on imperfect competition. For example, if a back-stop technology is introduced in
the Stackelberg model described above, the equilibrium price path is again pushed
upwards, implying that some part of the total resource stock is extracted after the price
path has reached the unit cost of the back-stop, c . For a more detailed discussion of this
kind of models, we refer to Dasgupta and Heal (1979) and Ulph (1982).
9
In practice, the monopolist may have to set its price slightly lower than
back-stop good out of the market.
c , in order to keep production of the