## Introduction

The treatment of duopolistic markets for raw materials from an exhaustible natural resource dates back to Hotelling’s (Hotelling, 1931) seminal paper. Because of the evident relevance to real world phenomena such as the market for crude oil, this theory was revived recently and was extended to oligopolistic markets with more than two suppliers. Lewis and Schmalensee (1980a) considered the case of similarly placed oligopolists. It was recognized by Salant (1976) that some of the markets under consideration can be characterized on the supply side by a coherent cartel and a large number of small producers called the fringe. The cartel-vs.-fringe model was further explored by Salant et al. (1979) and Lewis and Schmalensee (1980b). Pindyck (1978) did interesting empirical work on this subject. One of the important questions is whether the fringe benefits from cartelization or not. This problem is most clearly dealt with by Ulph and Folie (1980), who show that the answer can be both yes and no and who derive conditions under which each case occurs. All these authors use the Nash–Cournot equilibrium concept. Gilbert (1978) was the first to put forward that this equilibrium concept might not do enough justice to the market power of the cartel. Instead he studied the von Stackelberg equilibrium. In both models the fringe takes prices as given and chooses an extraction path so as to maximize discounted profits. In the Nash–Cournot model the cartel takes the extraction path of the fringe as given, knows the demand function and chooses either an extraction path or a price path so as to maximize discounted profits. In the von Stackelberg equilibrium the cartel announces optimal price and extraction paths taking explicitly into account the behaviour of the fringe as a price taker. It is assumed here that the fringe exactly meets the lacking supply for market equilibrium. Ulph and Folie (1981) stress that in such a von Stackelberg equilibrium dynamic inconsistency may arise. This means that the announced price and extraction paths become suboptimal when the equilibrium is reconsidered after some time has elapsed. Ulph (1982) further elaborates on the issue of dynamic inconsistency as well as Newbery (1981), who considers the cases when discount rates differ and when the demand schedule is non-linear. The basic point is that an equilibrium concept which displays dynamic inconsistency should be rejected in a framework of rational agents unless market transactions take place according to binding contracts. This last assumption is not very realistic. Newbery (1981) introduces the concept of a rational expectations von Stackelberg equilibrium. The underlying idea is that the equilibrium should have the property that none of the actors has an incentive to deviate from the equilibrium strategies at any point in time. Ulph (1982) points out that, in game theory terms, the actual problem is to find the feedback von Stackelberg equilibrium for the cartel-vs.-fringe model as an alternative for the binding-contracts open-loop von Stackelberg equilibrium. However, this problem proves to be very difficult.

In this paper the binding-contracts open-loop von Stackelberg equilibrium is reconsidered. It is shown that the results of Ulph and Folie (1981), Newbery (1981) and Ulph (1982) are not altogether correct. The equilibrium proves to be different from these results in two respects. Firstly, the specification of the marginal costs for the cartel should be corrected. As a result of this the timing of the different production stages changes somewhat. Secondly, for empirically plausible parameter values the resulting equilibrium price path may be discontinuous. This implies that not only the timing but also the order of the different production stages may differ from what was suggested in the previous literature. The possibility of the occurrence of a discontinuous price trajectory is already discussed in Groot et al. (1992). They show that a discontinuous price trajectory can yield higher cartel profits than along the solution proposed in the previous literature, but they do not provide a full proof of the equilibrium. In spite of the fact that we shall still find dynamic inconsistency in the correct von Stackelberg equilibrium, we think it is worthwhile to present a formal derivation. Not only is this not present in the literature but our method may also prove useful in other fields as well. The mathematical difficulty is that in the control problem that has to be solved the constraint qualification does not hold. We show how to deal with this in a way that is applicable to similar control problems. In the corrected equilibrium dynamic inconsistency still occurs for more or less the same parameter values as were found in the previous literature.

The paper is organized as follows. In Section 2 the cartel-vs.-fringe model is described. The model is basically the same as the one Ulph and Folie (1981) and Ulph (1982) used. The only difference is that in this paper the fringe is explicitly modelled as a large number of small resource owners having identical extraction costs and an identical initial stock. The fringe members are oligopolists, which implies that they all know the demand function and that each of them takes the extraction rate of all other resource owners as given. The cartel has only one instrument, its own extraction rate. The advantage of this approach is that it is not necessary to assume that the cartel controls the price path. Furthermore, the extraction rate of each fringe member is uniquely determined, so that no additional assumptions are needed here. In the limiting case of infinitely many fringe members the behaviour of the fringe as a group is price-taking, so that the rational reaction of the fringe is the same as in the previous literature. A qualitative characterization of the equilibrium order of exploitation and of the resulting dynamic equilibrium price path is presented in Section 3. Section 4 deals with the calculation of an equilibrium. Section 5 concludes.

## Section snippets

## The cartel-vs.-fringe model: problem statement

The cartel-vs.-fringe model is a model of the supply side of a market for an exhaustible natural resource. The demand schedule is assumed to be given. On the supply side there is a large coherent cartel and a group of many small resource owners, called the fringe. All producers are endowed with a given initial stock of the resource and choose their extraction path so as to maximize their discounted profits.

In the binding-contracts open-loop von Stackelberg equilibrium all fringe members are

## Characterization of the open-loop solution

The optimal control problem of Section 2 does not allow for a standard application of Pontryagin’s maximum principle. To see this consider , , . Assume that the optimal level of λ^{f} has already been determined. Then we have a so-called Hestenes problem (see e.g. Takayama, 1974, p. 657) or a Bolza problem (see e.g. Cesari, 1983, p. 196). In the particular case at hand, with equality and inequality constraints, a rather general constraint qualification would read that the rank of the matrix$\text{1}\text{0}\text{E}{}^{\mathrm{c}}\text{0}\text{0}\text{0}\text{0}$

## Existence and calculation of the solution

Theorem 1 gives a full qualitative characterization of the open-loop von Stackelberg equilibrium. In this section we consider the question of how to actually calculate the equilibrium trajectory.

Let us start from Fig. 2 and assume that *t*_{1}<*t*_{2} (and hence *t*_{3}<*t*_{4}). The equilibrium is fully determined by six variables, namely *t*_{1}, *t*_{2}, *t*_{3}, *t*_{4}, λ^{c} and λ^{f}, meaning that the optimal *E*^{c} and *E*^{f} are known at each instant of time if these variables are known. The six variables have to satisfy a number of

## Conclusions

In this paper we have derived the open-loop von Stackelberg equilibrium in the cartel-vs.-fringe model. Mathematically speaking, the main difficulty has been that the model specifying the problem for the cartel is an optimal control model where the constraint qualification fails to hold. As far as the qualitative characterization of the equilibrium was concerned this difficulty was circumvented by deriving ad hoc necessary conditions rather than by invoking sophisticated control theory

## Acknowledgements

Financial support to Fons Groot from the Cooperation Centre Tilburg University and Eindhoven University is gratefully acknowledged.

## Cited by (7)

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For instance, the “cartel versus fringe models”, following the seminal papers of Salant (1976) and Ulph and Folie (1980), consider a coherent cartel, as one player, facing a large number of firms acting in pure competition, see also Kagan et al. (2015) for a more recent contribution. Another branch of literature, following Gilbert (1978), Newbery (1981) and more recently Groot et al. (2000, 2003), considers the case of a first-mover cartel in a Stackelberg equilibrium. However, in these papers as well, the leader or the cartel is assumed to be a coherent structure and the agreement between cartel members is not examined.

We consider a nonrenewable resource duopoly with economic exhaustion. We characterize the set of Pareto efficient equilibria. We show that when firms are sufficiently patient, there exists no Pareto efficient agreement that yields short-run gains with respect to the noncooperative equilibrium. Given a pair of stocks, there exists a unique interior Pareto efficient agreement. We characterize the set of stocks where a Pareto efficient agreement results in larger discounted sum of profits for both players. We show that social welfare under the interior Pareto efficient agreement is smaller than under non-cooperation, despite the gains from a more cost effective extraction of the resources under an agreement.

### Imperfect cartelization in OPEC

2016, Energy Economics

Citation Excerpt :

The second effect works in the opposite direction. In this case, the cartel would like to announce ambitious output targets so as to pre-empt supply by the non-OPEC fringe, even when such a strategy would not be credible ex-post (Groot et al., 2000, 2003). We adopt two approaches for dealing with the time inconsistency issue.

A model of global oil production is applied to study cartelization by OPEC countries. We define a measure for the degree of cooperation, analogous to the market conduct parameter of Cyert et al. (1973), Geroski et al. (1987), Lofaro (1999), and Symeonidis (2000). This parameter is used to assess the incentives of different OPEC members to collude. We find that heterogeneity in OPEC and the supplies of the non-OPEC fringe create strong incentives against collusion. More specifically, OPEC's supply strategy, although observed to be substantially more restrictive than that of a Cournot–Nash oligopoly, is found to still be more accommodative than that of a perfect cartel. The strategy involves allocating larger than proportionate quotas to smaller and relatively costlier producers, as if to bribe their participation in the cartel. This is in contrast to predictions of the standard cartel model that such producers should be allocated relatively more stringent quotas. Furthermore, we demonstrate that cartel collusion is more likely to be sustained for elastic than for inelastic demand. Since global oil demand is well known to be inelastic, this observation provides another structural explanation for why OPEC behavior is inconsistent with that of a perfect cartel. Our study points to multiple headwinds that limit OPEC's ability to mark up the oil price.

### Non-renewable resource Stackelberg games

2014, Resource and Energy Economics

Citation Excerpt :

The simultaneous DF framework has been analyzed by Pindyck (1978), Salant (1982), Ulph and Folie (1980b), Groot et al. (1992), and Benchekroun et al. (2009), inter alia. The sequential DF framework has been explored by Gilbert (1978), Ulph and Folie (1980a), Newbery (1981, 1992), Groot et al. (2000, 2003), and Benchekroun and Withagen (2012). These papers distinguish between open-loop Nash equilibria, where each supplier chooses the extraction path as a function of the initial resource stock and time, and closed-loop (or feedback) subgame perfect Nash equilibria, where extraction strategies depend on current stocks.

The market structure for many mineral industries can be described as oligopoly with potential for Stackelberg leadership. This paper derives and analyzes dynamically consistent extraction equilibria in a two-period discrete-time “Truly” Stackelberg (TS) model of non-renewable resource extraction, where firms move sequentially within each period and where both the leader and follower have market power. We show how the leader may be able to manipulate extraction patterns by exploiting resource constraints. Whether the leader wants to speed up its own production relative to the Cournot–Nash (CN) equilibrium depends on the shape of its iso-profit curve, which is affected by the two firms’ relative stock endowments and relative production costs. If the leader extracts faster, then the follower extracts slower, but in aggregate the industry extracts faster. Unlike static Stackelberg games, the follower does not necessarily have a second mover disadvantage.

### Research on dynamics in a resource extraction game with bounded rationality

2014, Applied Mathematics and Computation

Citation Excerpt :

There are often several oligopolies in resource market, and each of them makes decisions in continuous time field to optimize the extraction of the limited resource. Much work have been done on oligopolistic resource competitions by dynamic differential game method in [1–5], where the paths of open-loop and closed-loop equilibriums and their stability are mainly focused on. In the work mentioned here it is supposed that all decision-makers have complete information and their behaviors are so completely rational such that they can solve the optimal strategy paths within the whole time level.

Based on bounded rationality, a linear dynamic system is proposed in this work for the duopoly game of renewable resource extraction. This dynamics reflects the decision-makers’ dynamical output strategy consisting of all the Nash equilibriums in every two adjacent periods. The resource growth rate, the discount rate and the expected extraction rate are the major parameters in this system. The system stability and the equilibrium characteristics of the stable state are analyzed. Numerical simulations are made to show the influence of all the parameters on the speed for the system to approaches its stable equilibrium. By the mathematical analysis we find that, if the discount is smaller or the expected extraction rate is larger, the system will converge in shorter time and approach a stable state with lager resource reserve and with larger resource output. So we draw the conclusion that decision-makers’ focusing on the long-term profits is helpful to sustainable development on natural resource market in the long run.

### Equilibrium path in oligopolistic market of nonrenewable resource

2008, Nonlinear Analysis: Real World Applications

In this paper we study the oligopoly model of nonrenewable resource in which the unit production cost is variable and depends on the resource reserve level. We consider both the open-loop strategy and the closed-loop strategy of this dynamical differential game. For the case of linear cost function we have observed that the open-loop equilibrium and the self-feedback equilibrium satisfy the same equilibrium conditions, which can be described as a dynamical system. The analysis shows that the equilibrium path of the model is the stable orbit of this system, and this result leads to further studies of the properties of the total extraction and reserve and the individual ones of each producer. For the total extraction rate and reserve, some of the properties are similar to those of most oligopoly models with fixed unit production cost. For the individual behaviors, we have found out the solution expressions of the individual extraction rate and resource reserve and got the main result that the producer with larger initial stock has a larger but declining market share and the share of each producer converges toward the average one when time approaches to infinite.

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