%0 Journal Article
%A Puu, Tönu
%A Ruiz Marín, Manuel
%T The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints
%D 2006
%@ 0960-0779
%U http://hdl.handle.net/10317/998
%X 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
%K Comportamiento dinámico
%K Caos
%K Competidores
%K Coste marginal
%K Punto Cournot
%K Sistema en tres dimensiones
%K Plano de bifurcación
%K Dinamic behaviour
%K Chaos
%K Consumer
%K Competitors
%K Marginal costs
%K Cournot point
%K Three dimensional system
%K Bifurcation plane
%K Consumidor
%K Economía Aplicada
%~ GOEDOC, SUB GOETTINGEN