Abstract
We consider a Bertrand duopoly with homogeneous goods and we allow for asymmetric marginal costs. We derive the Myopic Stable Set in pure strategies as introduced by Demuynck et al. (Econometrica 87:111–138, 2019). In contrast to the set of Nash equilibria, the unique Myopic Stable Set can be easily characterized in closed form and it provides an intuitive set-valued prediction.
Highlights
The analysis of price competition is a fundamental part of oligopoly theory since Bertrand’s contribution (1883)
The Bertrand duopoly with symmetric constant marginal costs, homogeneous goods, and continuous prices has a unique pure strategy Nash Equilibrium characterized by a strategy profile in which prices equal marginal costs
We showed that the Myopic Stable Set coincides with the set of pure strategy Nash Equilibria for supermodular games, aggregative games, and potential games. In light of these results, the Bertrand model with asymmetric costs is interesting for several reasons: it does not satisfy the compactness and continuity assumptions of Demuynck et al (2019), it does not belong to any of the aforementioned classes of games, and the set of pure strategy Nash equilibria of this game is empty
Summary
The analysis of price competition is a fundamental part of oligopoly theory since Bertrand’s contribution (1883). If marginal costs are not symmetric across firms and the market is shared if firms set equal prices, no pure strategy Nash equilibrium exists. Blume (2003) shows that there exists a Nash equilibrium in mixed strategies where the more efficient firm sets price equal to the opponent’s marginal cost and serves the entire market with probability 1. We showed that the Myopic Stable Set coincides with the set of pure strategy Nash Equilibria for supermodular games, aggregative games, and potential games In light of these results, the Bertrand model with asymmetric costs is interesting for several reasons: it does not satisfy the compactness and continuity assumptions of Demuynck et al (2019), it does not belong to any of the aforementioned classes of games, and the set of pure strategy Nash equilibria of this game is empty.
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