The Cournot–Theocharis problem reconsidered

Abstract

In 1959 Theocharis [10] showed that with linear demand and constant marginal costs Cournot equilibrium is destabilized when the competitors become more than three. With three competitors the Cournot equilibrium point becomes neutrally stable, so, even then, any perturbation throws the system into an endless oscillation. Theocharis's argument was in fact proposed already in 1939 by Palander [4]. None of these authors considered the global dynamics of the system, which necessarily becomes nonlinear when consideration is taken of the facts that prices, supply quantities, and profits of active firms cannot be negative. In the present paper we address the global dynamics.

Reducing competitors in a Cournot–Theocharis oligopoly model

“…; 0; ða 2 c n Þ=2bÞ: Remark 1.1. This statement of Theorem 1 is slightly different from that of reference [1]. This is because there is a gap in the proof of the result stated in [1].…”

“…Theocharis proved that with three competitors the Cournot equilibrium becomes neutrally stable and with four it becomes unstable. In [1], we consider the global dynamics of Theocharis model by assuming that quantities cannot be negative, something that makes the model nonlinear. Our main aim in this paper is to characterise how firms disappear from the market.…”

“…The description of the model can be seen in [1]. If we denote by q i the supplies of n competitors, the linear model states that…”

Asymptotic dynamics of a piecewise smooth map modelling a competitive market

“…To derive it we calculate the optimal capital stock at each output, and then substitute it back in the expression (6). To this end we rst dierentiate (6) with respect to k i and put the derivative equal to zero…”

“…Moreover, if one wants to consider the global dynamics (which neither Palander, nor Theocharis did), it is necessary to take into account that such a linear model being unstable is bound to result in negative outputs, so, to make sense, it must be constrained. If so, the model becomes piecewise linear which was considered, e. g., in [6,19].…”