Oligopoly equilibria in nonrenewable resource markets

Abstract

Most Nash–Cournot oligopoly models of nonrenewable resources apply open-loop equilibrium concepts and are based on physical resource depletion. This paper studies feedback equilibria and economic depletion. Assuming affine-quadratic functional forms, the existence, uniqueness, and explicit solutions for the equilibria are derived for duopoly and n-player oligopoly with multiple resource stocks. For the cases of nonquadratic criteria, we develop a numeric solution scheme for the Nash feedback equilibrium. This scheme is an application of a discrete time, discrete state controlled Markov chain approximation method originally developed for solving deterministic and stochastic dynamic optimization problems. In our Nash–Cournot equilibrium, the degree of concentration in supply declines over time whereas the previous models with physical depletion and open-loop equilibrium concepts predict that a Nash–Cournot resource market will develop in the direction of monopoly supply.

Introduction

Many natural resource industries (e.g. oil, diamonds, bauxite, uranium, mercury, copper) cannot be described using pure monopoly or competitive models. Determining oligopoly equilibrium for resource markets leads to differential game models, which at the most general level belong to capital accumulation games (Reynolds, 1987, Reynolds, 1991; Dockner et al., 1996). Several recent studies have developed a particular subclass of these models where two or more players extract a common property resource stock (van der Ploeg, 1987; Benhabib and Radner, 1992; Clemhout and Wan, 1994; Haurie et al., 1994; Dutta, 1995; Dockner et al., 1996). This subclass has turned out to be tractable since the common property assumption normally leads to one state variable specifications. Compared to this development, the empirically well-justified case of well-defined property rights and two or some finite number of producers extracting their own stocks (e.g. oil) but supplying to the same market is still open in several respects. There is a growing interest in understanding the oligopolistic description of resource markets due to attempts to analyze the effects of international carbon dioxide control (or the Kioto agreement) on noncompetitive oil markets (e.g. Wirl and Dockner, 1995; Tahvonen, 1995). However, these studies have had to assume that oil is supplied by a monopoly.

Existing oligopoly models for nonrenewable resource markets and the `oil'igopoly theory (e.g. Loury, 1986; Polansky, 1992) rest on the Nash open-loop equilibrium concept and physical depletion of natural resources. This paper develops a model for Nash–Cournot feedback equilibria and for economic depletion, both in the case of duopoly and n-player oligopoly. We solve analytically the affine-quadratic specifications and extensions with more complex functional forms by the Markov chain approximation method designed for solving deterministic or stochastic dynamic optimization problems.

Oligopoly models of nonrenewable resources have an interesting history which goes back to the informal discussion included in Hotelling (1931). Later, Salant (1976) viewed oil markets in a cartel-competitive fringe setup where the cartel may take the competitive sales path as given and face a sequence of excess demand curves and the resource constraint (cf. Dasgupta and Heal, 1979). A more consistent extension of the static Nash–Cournot model is found in Lewis and Schmalensee (1980). They replace the static choice of output level by the choice of an output time path under the constraint of initial reserves. With equal, constant unit costs, the producers with larger reserves produce for a longer time than producers with smaller reserves, implying that the number of suppliers decreases over time.

The analysis by Loury (1986) continues the use of a similar model and the open-loop equilibrium concept. According to the `oil'igopoly theory, the resource rent and the agent's initial reserves are inversely related. The agent's market share is increasing if it exceeds the average market share. Typically the degree of concentration in supply increases over time and before physical depletion the resource is finally supplied by a monopoly. The theory is further developed and empirically tested in an inspiring study by Polansky (1992). `Oil'igopoly theory predicts that producers with larger stocks are more conservative, i.e. they produce larger amounts but use smaller fractions of their reserves than do producers with small reserves. These types of predictions are interesting because they can be tested with existing data. Polansky finds that the predictions are consistent with oil market data. For a large-scale computer version of this model, see Salant (1982).

The `oil'igopoly theory is problematic in two respects. First, the theory rests on the Nash open-loop equilibrium concept. Thus producers observe only initial reserves and do not use information on the levels of reserves after the initial moment of the game (Basar and Olsder, 1995, p. 231). Second, the main results are based on the `cake eating’ description of nonrenewable resources, which neglects that reserves are heterogeneous and that their complete physical depletion may be implausible.¹ We focus on economic depletion and thus study an unsolved problem that may give a more valid empirical description of resource markets. In addition, economic depletion leads to a formulation in which the feedback equilibria can either be solved analytically or in the case of complex functional forms by using numerical approximation methods.

In the Nash feedback equilibrium the players continuously observe the reserves and condition extraction on this information (Basar and Olsder, 1994, p. 317). We restrict our study to these equilibria which are prefect in Markov strategies. Since our solutions constitute threats in trigger strategy equilibria (see e.g. Mehlmann, 1994) we offer a useful basis for extending the analysis toward this direction in the future studies.

In the case of affine-quadratic functional forms, we prove the existence and uniqueness of the Nash feedback equilibrium and obtain the explicit form for both duopoly and n-player oligopoly. Such solutions and their properties have been absent in nonrenewable resource literature. Within capital accumulation games, our analytical solution may be compared to Reynolds, 1987, Reynolds, 1991 with the difference that in deriving our solution we do not need parameter restrictions like zero rate of discounting. Compared to the nonuniqueness problem of the Nash feedback equilibrium in Reynolds, 1987, Reynolds, 1991 an interesting feature of the nonrenewable resource game is that similar reasons for multiple equilibria seem to be ruled out.

Several authors have found the analysis based on affine-quadratic specifications restrictive (e.g. Reynolds, 1987; Clemhout and Wan, 1994). However, the nonzero-sum differential games literature offers quite a few examples of the analysis of more general problems, even using numerical approximation. For this purpose, we apply a numerical method for the Nash feedback equilibrium and extend the Markov chain approximation approach originally designed for solving deterministic or stochastic optimal control or dynamic programming problems (Kushner and Dupuis, 1992). In this method, the continuous time optimal control problem of each producer, given the feedback supply strategy of the other producer, is approximated by a discrete time, discrete-state-controlled Markov chain. In dynamic programming, a classical iterative method, called approximation in the policy space, is globally convergent under mild conditions, and as the approximation parameter approaches zero the value function solution converges to the value function of the continuous time problem. However, in extending the method to a Nash feedback equilibrium problem, one can no longer guarantee its global convergence. In all four of our numerical examples, the convergence of the method was very rapid, and in the affine-quadratic case it leads to the same solution (within high accuracy) as the analytic solution.² In the case of fish resources and monitoring cooperative equilibria similar approach has been developed by Haurie et al. (1994).

With affine-quadratic functional forms, our model yields a couple of reverse hypotheses compared to the `oil'igopoly theory. This is mainly explained by our replacement of the `cake eating’ property by economic depletion. The crucial differences between these two approaches are not fully noted in studies on competitive or monopoly equilibria (see Sweeney, 1993) nor by Hansen et al. (1985). In the `oil'igopoly model, small producers deplete their stocks before large producers do, implying that the degree of concentration in supply increases over time (Lewis and Schmalensee, 1980, Proposition 4). In our model, small producers enter the market when the price rise makes it profitable to extract the higher cost deposits, and thus markets become more competitive. In addition, the Nash–Cournot feedback equilibrium always approaches a stable turnpike, where the market shares of symmetric producers are equal. Such a stable turnpike is, in general, absent in the `oil'igopoly model (Loury, 1986, Theorem 4). With nonlinear demand and nonquadratic costs, our equilibria may temporarily resemble the features of the `oil'igopoly theory, given that the extraction costs are initially independent of the size of reserves. However, when economic depletion is approached, the features revealed by the affine-quadratic specification always dominate. Thus the basic predictions of the `oil'igopoly theory are conditional on the assumptions regarding physical depletion, extraction costs and functional forms. Especially the prediction that resource markets always finally end with a period of monopoly supply is conditional on the assumption of physical resource depletion. Whether these hypotheses are conditional on Nash–Cournot equilibrium depends e.g. on the properties of the Stackelberg feedback equilibrium, which is more or less unsolved.

The paper is organized as follows. Section 2 derives the duopoly and Section 3 the n-player oligopoly. Section 4 presents the numerical approximation method and results. Section 5 concludes the paper.