This post summarizes one aspect of a theorem presented and proved by Roy Radner (1980). I have previously expressed skepticism about the claim in the post title. I have also heard that, in game theory, anything can happen, but nothing need happen. So, I suppose, one should not be surprised in stumbling over a proof of the existence of almost any market behavior in the literature on game theory. But I was surprised.2.0 Selected Assumptions
2.1 Non-Cooperative Firms
In the model considered here, no mechanism exists to enforce agreements among firms. In the jargon, only (extensions of) Cournot-Nash equilibria are considered here.2.2 Firm Managers Making Approximately Optimal Output Decisions
Although not commonly stated, the textbook presentation of perfect competition assumes the managers of the firms are systematically mistaken about their optimum decisions. A homogeneous product is assumed to be produced by a finite number of firms in the industry, and the total industry output is finite. Managers are assumed to disregard any strategic reaction by other firms to variation in their own firm's output and to take the price of their product as given. But, for a given consumer demand function, the firm's (notional) variation in output results in a variation in prices. So the decisions of the managers can only be approximately optimal, in textbook theory.
Radner proposes the notion of an epsilon-equilibrium to formalize this idea that firm strategies are only approximately optimal. In such an equilibrium each firm's strategy is such that, for example, average profit is within epsilon of the maximum average profit achieved by an optimal strategy, given the strategies of all other firms. As is typical in mathematical analysis, one should think of ε as a given (small) parameter.2.3 Sequential Market Interactions
Firms are not considered as deciding on a single quantity to produce in this model. Rather, each firm decides on a sequence of T quantity outputs, one for each of T successive periods. The parameter T is known as the lifetime of the industry. Each firm decides on the output in a given period as a function of the outputs of all firms in all previous periods. A strategy is a sequence of such functions, one for each firm. The firm chooses a strategy to maximize its average or total discounted profit over the lifetime of the industry.2.4 Replication
The theorem outlined here is used to compare epsilon-equilibria for different (finite) numbers of firms in the industry. Radner defines the replication case to apply when the demand price is an unchanged function of the average output per firm. In some sense, the number of consumers increases, in the model, with the number of firms.3.0 An Informally Stated Theorem
Theorem: Consider the model with the above assumptions. Let the number of firms increase, along with the lifetime of the industry, such that the number of firms remains small enough, when compared to the lifetime of the industry. For any finite number of firms, equilibria exist in which the firms act as a cartel, and the cartel lasts for any given duration, provided the lifetime of the industry is taken large enough.4.0 Conclusion
I think of the point of this post to explore the result of tweaking textbook assumptions in the theory of perfect competition. Apparently, the results are sensitive to the exact statement and combination of assumptions. I gather that further research in microeconomic theory has confirmed that whether or not equilibria converge, as the number of firms increase, to the perfect competition model is a fine point. That is, equilibria may or may not converge to a model with a continuum of firms. Radner seems to feel exploring certain sets of assumptions is of more interest than other sets. I have chosen to emphasize a set of assumptions in which any finite number of firms may act like a monopoly, in a precise sense.
Another approach might be better in empirically describing firms in actually-existing capitalism.Reference
- Roy Radner (1980). Collusive Behavior in Noncooperative Epsilon-Equilibria of Oligopolies with Long but Finite Lives, Journal of Economic Theory, V. 22: pp. 136-154