The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the BG-normalization on computer algebra, it is shown that if the perturbed harmonic oscillator with a homogeneous cubic-polynomial potential and the perturbed harmonic oscillator with a hom...

This paper focuses on the role that production subsidies play in a Bertrand mixed duopoly. The paper examines four regimes: mixed and private duopoly, each with and without subsidies. The results of this study are compared with the findings of the existing Cournot mixed market literature. As a result, the paper shows that that the introduction of production subsidies into the analyses of Bertra...

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...