The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints

Abstract

Oligopoly theory, i.e., the economic theory for competition among the few, goes back to 1838 and Augustin Cournot [7]. See also [11]. Quite early it was suspected to lead to complex dynamic behaviour and chaos. See Rand 1978 [13]. The probably simplest case under which this happens with reasonable economics assumptions was suggested by one of the present authors in 1991, see [9]. It assumes an isoelastic demand function, which always arises when the consumers maximize utility functions of the Cobb-Douglas type, combined with constant marginal costs. The particular layout was a duopoly, the case of only two competitors. The model was shown to produce a period doubling sequence of ip bifurcations ending in chaos for the outputs of each of the two competitors. Later the triopoly case under these assumptions was studied. See [2], [3], and [4] for examples. An interesting fact is that with three competitors the main frame becomes the Neimark-Hopf bifurcation, which provides new and di erent scenarios. The main reason for economists to study increasing numbers of competitors is to nd out whether it is the number of competitors that uniquely decides a road from monopoly over duopoly, oligopoly, and polypoly, to perfect competition, a state where each rm is so small that its actions cannot in uence the market at all. To nd out about this it is of primary interest to know whether the number of competitors stabilizes or destabilizes the equilibrium state. Some authors have questioned the assumption, to which most economists adhered, that increasing numbers of competitors bring stabilization. However, we must be clear about what we compare. If we study increasing numbers of competitors with constant unit production costs, we are not reducing the size of the rms when their number increases. Constant marginal cost means that potentially each rm has in nite capacity, and adding such rms is not what we want for comparison. It is therefore interesting to combine an increased number of rms with decreasing size of each rm, but in order to do so we have to introduce capacity limits. Already Edgeworth [8] insisted on the importance of capacity limits. It is not so easy to nd non-constant marginal cost functions which allow us to solve for the reaction functions for the rms in explicit form, but one of the present authors, see [12], found one type of function, which models the capacity limit by letting marginal cost go to in nity at a nite output. That paper discussed the competition between two duopolists. The objective of the present paper is to nd out the facts when there are three competitors, and we still keep the assumption of capacity limits