A note on Cournot equilibrium with positive price

Consider an oligopoly in which firms compete in quantity, the market inverse demand is strictly decreasing (on the set of quantities for which the price is positive), twice differentiable and log-concave, and each of the firms has nondecreasing, twice differentiable cost of production (not necessarily convex). We extend previous literature on the existence of Cournot equilibrium by showing that, under additional mild assumptions, Cournot equilibrium with positive price is unique. This also holds if the costs are piecewise differentiable, nondecreasing, and convex with a finite number of kinks. Furthermore, if at least one firm incurs positive cost whenever the industry aggregate output implies zero market price, then the equilibrium is unique and the corresponding price is positive.