PQ oligopoly, proportional rationing, and randomly ordered consumers

Abstract

In a PQ oligopoly, firms pick prices and quantities simultaneously, and unlike with the traditional Cournot and Bertrand models, market clearing is not imposed. It is thus necessary to specify the rationing rule. Proportional rationing is one of the popular choices, often justified through a notion of randomly ordered consumers with varied reservation prices. However, such a setting would render firm-specific demand a random variable, a fact overlooked in existing models. In the paper, we (1) formalise the notion of randomly ordered consumers into a new stochastic version of the proportional rationing scheme, (2) derive the probabilistic properties of firm-specific demand under this new scheme, and (3) show that the results of the stochastic and deterministic versions are not entirely consistent.