Losses from competition in a dynamic game model of a renewable resource oligopoly
Introduction
Is increasing competition, i.e., an exogenous increase in the number of firms, beneficial to social welfare in an oligopoly? This is one of the primary interests in economics and there is a considerable literature based on static Cournot–Nash models. When all the oligopolistic firms have an identically constant marginal cost and no fixed cost, increasing the number of firms benefits welfare. However, it is stringent whether welfare improves as a result of increasing competition under asymmetric costs among firms. In a seminal work, Lahiri and Ono (1988, Proposition 2, p. 1201) find that ‘national welfare increases if a firm with a sufficiently low share is removed from the market.’ This result has long had a great influence on the policymaking of competition.
Are these results still valid even in a resource oligopoly as well? To give an answer, this article constructs a differential game model of a renewable and open access resource oligopoly. A typical example is a transboundary fishery. Suppose that efficient Northern firms and inefficient Southern firms compete in not only the world output market but also global fishery. In such a world, there is no world government and thus extraction is completely decentralized by private firms, the number of which is fixed even though the resource has open access. Within this framework, we prove that an increase in the number of efficient firms harms welfare as is opposed to the static result. Therefore, it straightforwardly from this result that ‘helping any firms reduces welfare.’ What is worth noting is that our results need no assumption on the initial market share of efficient firms.1 It is the closed-loop property of feedback strategies that plays a central role in our arguments. While closed-loop effects are a priori absent in any static analysis, they are quite relevant in dynamic environments, particularly in dynamic games. Our result is an example where the closed-loop effects can dominate the static effects, which yields a sharp contrast between the static and dynamic outcomes.
We are not the first to identify the role of closed-loop properties of feedback strategies in dynamic games. Constructing a differential game model of a renewable resource duopoly, Benchekroun (2003) demonstrates that a unilateral production restriction on a firm can decrease the resource stock. Allowing for an arbitrary number of (symmetric) firms in the same model, Benchekroun (2008) proves that increasing the number of firms reduces the resource stock and the industry output in the long-run. Lohoues (2006) introduces heterogeneity in marginal cost and the number of firms in the Benchekroun (2008) model and characterizes the feedback Nash equilibrium. One of Lohoues’ (2006) notable results is that the low-cost firm’s feedback strategy exhibits jumps in the presence of asymmetric costs. However, he is mainly interested in characterizing the equilibrium, leaving comparative statics/dynamics out.
The article is organized as follows. Section 2 presents the model and Section 3 characterizes the feedback Nash equilibria. Section 4 states and discusses the main results. Section 5 concludes the article. The appendices prove the results in the main text.
Section snippets
A model
Consider an oligopoly consisting of efficient firms with zero marginal cost and inefficient firms with a positive marginal cost . Fixed costs are assumed away. During production, firms extract a renewable resource with the following dynamics:where is a stock of the resource, represents an efficient firm’s output and represents an inefficient firm’s output.2
Feedback Nash equilibria
We seek stationary feedback strategies of this dynamic game. Stationary feedback strategies are a decision rule such that each firm’s extraction is a function of the current resource stock only: and with and for any , and . An -tuple of strategies constitutes a feedback Nash equilibrium if it solves the problem defined above for and .3
The main results
This section establishes the main results. Let us first consider the effect of an increase in the number of efficient firms on steady state welfare. For this purpose, define steady state welfare as follows.where denotes consumer surplus, is each efficient firm’s profit and is each inefficient firm’s profit. Rearrangements after (17)
Concluding remarks
Extending Benchekroun’s (2008) model of a productive asset oligopoly to an asymmetric oligopoly, we have proved that ‘helping major firms reduces welfare.’ This yields a natural corollary that ‘helping any firm reduces welfare.’ These results have theoretically and practically important implications. From a theoretical point of view, there does exist a case in which predictions of the static theory are completely reversed. In our context, the adverse effect through the closed-loop property of
Acknowledgements
The author sincerely thanks two anonymous referees and the editor for many helpful comments and suggestions. Comments from Ngo Van Long and seminar participants at University of Tokyo are also gratefully acknowledged. Any remaining error is author’s own responsibility.
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