Ranking Bertrand, Cournot and supply function equilibria in oligopoly

We show that the standard argument according to which supply function equilibria rank intermediate between Bertrand and Cournot equilibria may be modified. We prove this result within a static oligopolistic game in which both supply function competition and Cournot competition yield a unique Nash equilibrium, whereas price setting yields a continuum of Nash equilibria. There are parameter regions in which Bertrand profits are higher than Cournot ones, with the latter being higher than in the supply function equilibrium. Such permutation of the typical ranking occurs when price-setting mimics collusion. We then show that the modification in the ranking of profits is responsible for a permutation in the welfare performance of the industry, for any structure of the latter. Regulation may then be needed even under price competition and free entry.