Download ECON4925 Resource economics, Autumn 2008

Updated 2 September 2008
ECON4925 Resource economics, Autumn 2008
OLAV BJERKHOLT:
Lecture note 3: Extraction under imperfect competition:
monopoly, oligopoly and the cartel-fringe model
Perman et al. (2003), Ch. 15.6; Salant (1976)
RESOURCE MARKETS AND IMPERFECT COMPETITION
Due to limited supply sources and also their location 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 optimal extraction programs
and the Hotelling rule we have discussed above. A simple case of monopoly behaviour was
analysed in Hotelling (1931).
If there is a single resource owner is assumed to be faced by a downward sloping demand
function, p  p ( y ) . From this demand schedule the marginal revenue is derived as
m  d [ p( y ) y ] / dy   p , where  = 1 - (1/ ) , and  the elasticity of demand (in positive value).
The objective for the monopolist is the same as for the competitive firms, i.e. maximizing
discounted profits over a horizon that is also to be determined.
T
(9.1)
Max  [ p( Rt ) Rt  bRt ]e  rt dt
0
Assuming here constant unit cost, constant rate of interest and a given stock of resource as in
the standard model, we get the following Hamiltonian
(9.2)
H C (t , St , Rt , t )  pt Rt  bRt  t Rt
and the first-order condition we get the net marginal revenue equal to (or less than) the
resource rent:
(9.3)
p( Rt )  p '( Rt ) Rt  b  t
( when Rt  0)
The resource rent  t will as in the standard model increase by rate r.
By using the same kind of argument as in the Hotelling Rule we derive easily guess
intuitively that optimal extraction, requires that marginal revenue increases at a rate equal to r
when the production is non-zero. The resource rent is again 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
net marginal revenue.
To simplify the analysis we will in the following assume that the resource may be extracted at
zero costs, i.e. b  0 . This means that for a competitive market that the market price of the
resource and the resource rent will be one and the same. Formally, the equilibrium condition
for the simple monopolist case is
2
(9.4)
m&t
r
mt
What distinguishes this condition and the extraction path in the monopolist model from the
optimal solution in the competitive case? First, it may be noted that by 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 simultaneously. 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 by a higher price, such as in static market
equilibrium.
Take for instance the case where the demand schedule is of the constant elasticity type, i.e.


1

1
R  Ap or p  A  R  where A and  are constants. In this case the rate of change of
marginal revenue equals that of the market price, so that the monopolist condition (9.4) 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
(9.4) in the following way, still assuming zero unit cost:
m( R )  p ( R )   
(9.5)
m& p& &
  r
m p 
Relation (9.5) 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 mentioned. The monopoly and competitive equilibrium are
identical, i.e.
p&
r
p
(9.6)
b)  increases with p .
As p increases along the optimal path, this implies that & 0 . This is seen by
1 d  dp
expanding &in the following way: & 2
. Consequently
 dp dt
p&
r
p
(9.7)
c)  decreases with p .
In this case, the monopoly price path satisfies
(9.8)
p&
r
p
3
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
schedule, 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, and smaller
production in an early part of the extraction period 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 statement that “the monopolist is the conservationist's best friend”.
Case c), on the other hand, implies that the price curve under monopoly 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 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 look at just 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 noncooperative 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
(9.9)
Rti 1
mi  p( Rt )  (1 
)  ti (i  1, 2)
Rt 
where Rt  Rt1  Rt2 . 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 (9.9) by  i , and
summing over the two equations for i=1,2, yields
(9.10)
p( Rt )( 1   2 )   1   2
4
This equation is formally identical to the equilibrium condition in the monopoly case, so that
it can be transformed similar to equation (9.5), but where now
  1  2  2
1

  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 the
1
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 deposit.
This may e.g. be seen by regarding equation (9.9). If we assume that resource owner no. 1
controls the largest initial resource stock, we must have 1   2 .Furthermore, we assume that in
equilibrium & 0, i  1, 2 , implying m&/ m  r  p&/ p . In this case marginal revenue
approaches the 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.
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 (see e.g. Salant (1976),
Newbery (1981) and Ulph (1982). 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. (We use below superscript ‘m’ for the
monopolistic supplier and ‘c’ for the competitive or fringe supplier.)
The maximization problem for the cartel may be conceived as follows:
5
T
c*
m
m
m  rt
Max
 [ p( Rt  Rt )  b ]Rt e dt
m
(10.1)
Rt
0
T
subject to
R
m
t
dt  S0m
0
c*
t
while R is the solution (worked out by the cartel) to
T
c
c  rt
Max
 [ pt  b ]Rt e dt
c
(10.2)
Rt
0
T
subject to
 R dt  S
c
t
c
0
and with pt  p( Rtc*  Rtm ) .
0
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 more detailed discussion of different solutions see Ulph (1982) and also 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 rent 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
6
competitive 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 is shown in Ulph (1982) that the equilibrium solution involves three phases, see figure 3.1.
In the first period ( [0, T1 ] ) the fringe supplies the market. At T1 the resource stock of the fringe
is emptied and the cartel takes over the market. For some time ( [T1 , T2 ] ), however, the cartel
follows a price and extraction policy that is a continuation of the competitive price path. Then, at
time T2 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.
Figure 3.1. The Stackelberg cartel-fringe solution
p
p*
$ / unit
Price
B
Marginal
revenue
A
Phase 1:
Fringe alone
Phase 2:
Cartel alone,
competitive price
T1
t
Phase 3:
Cartel alone,
monopoly price
T2
Tm
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
T1 . 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 T1 .
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.
7
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 (9.9), 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, thus the Hotelling rule.)
Salant (1976) was a landmark contribution in this literature. Salant’s picture of the market is a
cartel and a competitive fringe of identical competitive firms (or countries). All firms are in
the same market with a choke price p* . We assume zero extraction costs and thus
homogenous reserves. Salant constructs a dynamic Cournot duopoly, but with asymmetric
roles for the two sides. The competitive fringe takes the price as given and adjusts production
accordingly. The cartel sets the price and supplies whatever is needed for meeting demand,
given the quantity supplied by the fringe. Both actors act simultaneously, while taking the
decisions of their counterpart as given, thus the equilibrium is a Nash equilibrium. They do so
while maximizing the present value of their profits given the reserve constraint.
In equilibrium we can imagine three different production patterns: both types of firms produce
simultaneously, the cartel produces alone, and the fringe produces alone. For the cost
configurations underlying the Stackelberg equilibrium discussed above, i.e. with zero unit
costs, it can be shown that the only possible Nash-Cournot solution is one where the cartel
and the fringe produces simultaneously in an initial 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 (9.9)), see the figure 3.2.
Figure 3.2. The Salant cartel-fringe solution.
p
p*
Price
Marginal
revenue
Phase 1:
Cartel and
fringe produce
Phase 2:
Cartel alone
tc
tm
t
Thus for the first phase the price has to increase by rate r while at the same time the cartel’s
marginal revenue also has to increase by rate r. How is that possible?
8
Let us use superscript c for the cartel and superscript f for the fringe. There is only one price in
the market and the price must increase by r for the fringe to be in equilibrium:
p&t
r
pt
(11.1)
The cartel’s marginal revenue is according to what we have studied above
Rc 1
Rc 1
(11.2)
MR c  pt  (1  t )  p( Rt )  (1  t )
Rt 
Rt 
The marginal revenue is written as the product of two factors of which the first is the price
increasing by rate r. If the marginal revenue also should increase by rate r the second factor
must increase by rate zero!
The rate of increase of marginal revenue can be written as
Rtc 1
d (1 
) / dt
p&t
Rt 

Rc 1
pt
1 t
Rt 
(11.3)
where the second term should be zero and can be made zero by the cartel through adjusting its
own production. The fringe can be enticed to produce at any time along the r-path.
The elasticity of the demand is written as  , but the elasticity is not constant, we assume here
a choke price and, say, a linear demand function. The elasticity
(11.4)

dR p
   when
dp R
p  p* because R  0 .
We can write out the second term in (11.3) as
R&c R c 1 R&R c 1 &1 R c 1 R&tc R& &
 tc


(
  )
Rt R  R R   
R  Rtc R 

(11.5)
Rtc 1
Rc 1
1
1 t
Rt 
Rt 
We see that the cartel has to ensure that this term is zero until the fringe has emptied all its
resources. The price path will then have a kink while the marginal revenue grows smoothly
from beginning to end. The marginal revenue curve ends in the same point as the price curve
at the choke price because (well, try to figure that out!)
The Salant 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.
We can compare this asymmetric Nash-Cournot with the competitive and monopoly solutions
with regard to the price path, see figure 3.3 below. We see that the duopoly as, as should be
9
expected, in an intermediate position compared to monopoly and the competitive solution.
The starting price is intermediate, the horizon is intermediate, although there is a period in
which the duopoly price is higher than for monopoly.
Figure 3.3. Comparison of the Salant cartel-fringe with competitive and
monopoly solutions.
p
p*
Duopoly price
Monopoly
price
Competitive
price
t
tc
Competition
Duopoly
Monopoly
RESOURCE MARKETS AND 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
represents 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 good.
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 the limitation. 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 backstop 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
10
must increase at the rate of interest. Thus, the existence of the back-stop affects only the end
point condition. With no substitute product available, we saw in an earlier lecture how the length
of the extraction period was determined, assuming that the demand was choked off at the price
p max . With the unit cost of the back-stop technology representing the new “price ceiling”
(assuming c  p max ) it should be clear that the end point condition in the present case becomes
(12.1)
( p0  b )e rT  (c  b )
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 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 technology. 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
(12.2)
Rt  p 1 (c )
It can be shown (see e.g. Hoel (1978)) that a necessary condition for monopoly equilibrium in
this case is
(12.3) mt  ( c - b ) e-r( T 2 -t)
where T2 is the time where the resource is fully exhausted.
The equilibrium consists of two phases:
i) In the first phase [0, T1 ] , 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 (12.3)
prevails. The resource price increases monotonically, until it reaches the price (unit cost) of the
substitute product at T1 .
ii) After T1 the resource monopolist continues to supply the market for some time at the given
price c , until the resource is fully exhausted at T2 . At this point the back-stop good takes over.
In the second period of resource production, [T1 , T2 ] , strict inequality (12.3) prevails, while
marginal revenue actually being determined 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,T1]. Thus,
11
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.
In the first phase the marginal revenue is equal to the resource rent (shadow price) and thus
increasing by rate r. The price is higher than marginal revenue and thus will hit the backstop
price at some stage while the resource rent is lower. This is when phase two starts. The
transversality condition states that at the end of the free horizon the resource rent will be equal
to the net price at the backstop price. Hence the resource rent continues to increase at rate r
until it hits the backstop price. That is the end of phase 2.
The first phase just described may be non-existing. This is the case if the demand elasticity (  )
is less than one in absolute value. (Note that in this case the traditional monopoly solution breaks
down, since the marginal revenue is negative.) 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 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 empty. (In practice, the monopolist may have to set its price
slightly lower than c , in order to keep production of the back-stop good out of the market.)
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 may
refer to Dasgupta and Heal (1979) and Ulph (1982).