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Cournot I: Constant Returns

  • Tönu Puu
Chapter

Abstract

Cournot duopoly has now been an inspiration for mathematical modelling in economics for almost two Centuries. As the principles are very simple to grasp one needs not have a lifetime of indoctrination with economic theory to be convinced. Once nonlinear science took the stage it was mathematicians such as Rand, and Poston and Stewart, who started using applications to duopoly. Not altogether surprising, after all Cournot was a mathematician. The aim then was to create iterative processes in the style of the logistic map. Cases where both reaction functions were in this style were suggested, which, of course, could produce up to four different fixed points. Unfortunately, these were not based on economic principles, i.e., were not derived from demand functions that emerged from utility maximization. In 1991 the present author proposed the case of Cobb-Douglas utility where the resulting demand functions are always reciprocal to commodity price. Budget shares for the consumers, and hence also aggregate revenues for the producers then became constant. Dana and Montrucchio had proposed a similar case as an example 5 years before, though without fully appreciating the potential of the idea. This “isoelastic” demand function easily lends itself to explorations of dynamics. Yet, it also has its snags, above all the constancy of revenues, which leads to the absurd situation that a single supplier can reduce output to zero and sell it at an infinite price, which does not affect revenues at all. On the other hand costs can be reduced to zero by cancelling production, so the best choice is to produce nothing. This, of course, is absurd and makes the model unsuitable to deal with monopoly or collusion. The problem persists in duopoly, despite the fact that the intersection of the reaction curves in the origin is totally unstable; the curves at this intersection even have infinite slopes. Nevertheless we need some mechanism that pushes the system away from the origin if it lands there. This may sound simple, but it is not! Any such repulsion mechanism tends to take over the show, and blur the essential dynamics we want to analyze. As the reaction curves intersect the axes whereas outputs cannot be negative, it is inevitable that an occasional dropping out from production makes the system land in the origin. Once one firm chooses zero output, the other will do it as well. We need something to prevent the system from ever visiting the origin, which will be done through assuming adaptation. This means that the firms never move to the calculated best reply, just part of the way, which may be 99.99%. Given this, the assumption seems to be quite innocuous.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Tönu Puu
    • 1
  1. 1.Centre for Regional Science (CERUM)Umeå UniversityUmeåSweden

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