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Cournot competition

S. Leonardos & C. Melolidakis, Endogenizing the cost parameter in Cournot oligopoly, under revision, 2016. https://arxiv.org/abs/1601.07365.

Under the prevailing approach, the costs of the firms in Cournot competition are assumed to be exogenously given and there is little discussion about their origin. However, in the contemporary economic practice, the costs of the firms most of the time represent purchases from a third party (a supplier) and are anything but stable or given, rendering this assumption at least questionable. Marx & Schaffer (2015) point out the scarceness of research along this line, and even go as far to question the robustness of the results obtained in the classic Cournot oligopoly theory due to this negligence.

To address this issue, we viewed the interaction of the competing Cournot oligopolists with their supplier as a two stage game which endogenized the cost parameter of the classic Cournot model. The study of the cost-formation in this context raised several questions: Do firms have production capacities of their own and if yes, are they bounded? Does the supplier know the actual demand that the firms face, i.e. do we need to consider a complete or incomplete information market setup?

As a dynamic two-stage model, our work, combines elements from various active research areas. At the outset, we applied the standard equilibrium concepts of subgame perfect Nash equilibrium for the complete and Bayes Nash equilibrium for the incomplete information case. To maximize the flexibility of the model, we parametrized all the above factors: production capacities, demand level and demand knowledge from the supplier’s point of view and obtained solutions that were applicable to the whole range of the parameter values.

However, the initial results revealed a connection between the equilibrium strategies and the concept of log-concave probability. In view of this observation, our research shifted to the literature at the interface between probability, log-concavity and mathematical economics, see Bagnoli & Bergstrom (2005). We were able to express the supplier’s payoff function in terms of the Mean Residual Lifetime (MRL) function of the distribution of the demand parameter, which resulted in a closed expression of the derived equilibrium as a fixed point of the MRL function. The MRL function is a well known object among actuaries, reliability engineers, and applied probabilists but, to our knowledge, had never emerged before in a Cournot context.

Through this closed expression, we better understood the properties of the equilibrium solution and performed sensitivity analysis on the values of the many parameters of the model. Additionally, we drew comparisons between complete and incomplete information variations of the model and measured (or derived bounds for) the inefficiency that emerges due to lack of information on the side of the supplier. This work was presented in several talks, including the international conferences GTM16, SING12 and GAMES 2016.

To further develop our model, our current research focuses firstly in extending the main technical assumptions e.g. discrete instead of continuous beliefs, concave instead of affine inverse demand function etc. and secondly on establishing connections with other models in oligopoly theory, e.g. price instead of quantity competition in the second stage, more than one supplier in the first stage etc. In another line of research, we are interested to measure the value of information in the two-stage market, a question that intersects with a large area of contemporary game theory research and promises interesting insights. The main challenge in our future research, is to overcome the technical and conceptual difficulties that will emerge at the application of our model on the wide range of variations of oligopoly theory. Again, the great amount of variations that may studied and the wide applicability of oligopoly theory, constitute a fruitful ground to actively engage graduate students that are interested to conduct research in this area.

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