# Walrasian versus Cournot behavior in an oligopoly of boundedly rational firms

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## Abstract

An evolutionary oligopoly game, where firms can select between the *best-reply* rule and the *Walrasian* rule, is considered. The industry is characterized by a finite number of *ex-ante* homogeneous firms that, characterized by naïve expectations, decide next-period output by employing one of the two behavioral rules. The inverse demand function is linear and all firms have the same quadratic and convex cost function (decreasing return to scale). Based upon realized profits, the distribution of behavioral rules is updated according to a replicator dynamics. The model is characterized by two equilibria: the Cournot-Nash equilibrium, where all firms adopt the best-reply rule and produce the Cournot-Nash quantity, and the Walrasian equilibrium, where all firms adopt the Walrasian rule and produce the Walrasian quantity. The analysis reveals that the Walrasian equilibrium is globally stable as long as the rate of change of marginal cost exceeds the sum of residual market price sensitivities to output. If not, the Walrasian equilibrium loses stability and an attractor, representing complicated dynamics with *evolutionary stable heterogeneity*, arises through a bifurcation. As the propensity of firms to select the more profitable behavioral rule increases, the attractor disappears through a global bifurcation and the Cournot-Nash equilibrium can become a global Milnor attractor. To sum up, the best-reply rule can be evolutionary dominant over the Walrasian rule and this can lead an oligopoly to select the Cournot-Nash equilibrium.

## Keywords

Behavioral evolutionary oligopoly Walrasian versus best reply Bounded rationality Nonlinear dynamics## JEL Classifications

C62 C73 D21 D43 L13## Notes

### Acknowledgements

The author thanks Gian Italo Bischi, Laura Gardini, Fabio Lamantia, Ivana Tuzharova, two anonymous reviewers and all participants in the Nonlinear Economic Dynamics 2015 conference in Tokyo (Tokyo NED conference) for their insightful comments and suggestions that have lead to several improvements in the paper. The author gratefully acknowledges the Department of Economics of Chuo University in Tokyo and the University of Urbino Carlo Bo for funding his participation in the Tokyo NED conference. Special thanks to Akio Matsumoto and the Local Organizing Committee of the Tokyo NED conference for having organized a wonderful meeting.

### Compliance with Ethical Standards

### **Conflict of interests**

The author declares that he has no conflict of interest.

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