For Bertrand duopoly with linear costs, we establish via a single counterexample that: (i) A new monotone transformation of the firms’ profit functions may lead to the supermodularity of transformed profits when the standard log and identity transformations both fail, and (ii) Topkis’s notion of critical sufficient condition for monotonicity of a Bertrand firm’s best-reply correspondence cannot...

In this paper, we develop a differentiated duopoly model with endogenous cost-reducing R&D and review the argument on welfare effect of price and quantity competition in the presence of technology licensing. We show that the standard conclusion on duopoly (Singh and Vives, 1984) can be completely reversed. Cournot competition induces lower R&D investment than Bertrand competition does. Moreover...

We study how continuous time Bertrand and Cournot competitions, in which firms producing similar goods compete with one another by setting prices or quantities respectively, can be analyzed as continuum dynamic mean field games. Interactions are of mean field type in the sense that the demand faced by a producer is affected by the others through their average price or quantity. Motivated by ene...

The well-known problem of classical mechanics considered by Bertrand (1857) and Darboux (1901) is reviewed in the context of Cartan’s geometry. UDK: 514.85; MSC: 37J35, 53C05

We present a dynamic extension of the classic static model of Bertrand price competition that allows competing duopolists to undertake cost-reducing investments in an attempt to “leapfrog” their rival to attain low-cost leadership — at least temporarily. We show that leapfrogging occurs in equilibrium, resolving the Bertrand investment paradox., i.e. leapfrogging explains why firms have an ex a...