ECON4925 Resource Economics, Autumn 2008
1
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, . From this demand schedule the marginal revenue is derived as , where , 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.
(9.1)
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)
and the first-order condition we get the net marginal revenue equal to (or less than) the resource rent:
(9.3)
The resource rent 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 guessintuitivelythat 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. . 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
(9.4)
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. or where 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:
(9.5)
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.
(9.6)
b) increases with .
As increases along the optimal path, this implies that . This is seen by expanding in the following way: .Consequently
(9.7)
c) decreases with .
In this case, the monopoly price path satisfies
(9.8)
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 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
(9.9)
where . 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 , and summing over the two equations for i=1,2, yields
(9.10)
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
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 definition of changes to . 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 .Furthermore, we assume that in equilibrium , implying . 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:
(10.1)
subject to
while is the solution (worked out by the cartel) to
(10.2)
subject to and with.
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 , 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 and respectively. If , 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 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 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 () the fringe supplies the market. At the resource stock of the fringe is emptied and the cartel takes over the market. For some time (), however, the cartel follows a price and extraction policy that is a continuation of the competitive price path. Then, at time 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
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 . 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 .
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 (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 . 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 . This is achieved by the cartel continuously adjusting its production and thus its market share (cf. relation (9.9)), see the figure 3.2.