Cournot II: Returns to Scale and Stability

Abstract

Cournot did not just invent quantity competition duopoly; he considered the entire scale of steps from monopoly to perfect competition and described the proper rules for monopoly pricing and perfect competition pricing, assuming oligopoly price to be between these extremes. He considered the route in terms of adding new competitors to an already existent market, which, by the way, explained why he focused on quantity competition. Given the simple setups for Cournot duopoly—linear demand and constant marginal cost—it came as a shock when it was discovered that the models became unstable if the competitors exceeded some rather small number. What use is it to know that an increasing number of competitors has the perfect competitive state as target if both the asymptotic state and the route are destabilized? It occurred to the present author that the problem is on the side of production. Constant marginal costs, or constant returns to scale, means that we deal with firms of potentially infinite size. Any small profit margin between price and cost could be multiplied up any number of times through expanding production. Once this is realized, destabilization is no longer surprising. Further, it was probably not such a comparison Cournot was after; he might have wanted to compare a small number of big firms to a large number of small firms. But small and big cannot be defined without decreasing returns and capacity limits. Such can be introduced through the use of a nonstandard variant of the Constant Elasticity of Substitution production function. One just has to fix capital inputs through an act of investment, and production with the remaining variable inputs is automatically provided with a capacity limit for the lifetime of invested capital. This actually removes the destabilization problem. Once one is that far, it is natural to assume that capital wears out, and one may try to formulate an endogenous process for capital wear and regeneration, which is, in fact, done. However, many alternatives to this part of modelling are conceivable. The stability issue is bound to become complicated with capital renewal included, and there are interesting scenarios with, for instance, never ending oscillations and spontaneous formation of synchronized competitor groups.