Competition in an evolving stochastic market
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"In an efficient market all identical goods must have only one price." So states the aptly named law of one price. In the real world, however, one may easily verify that identical products are often sold for different prices. This thesis develops an extension of the Bertrand model in economics to include spatially localised competition to explain this price variation, which is then studied through simulation methods and theoretical analysis. Our model studies the effect that local heterogeneities in the environment experienced by sellers have on successful pricing strategies. Taking inspiration from models of evolutionary dynamics, we define the fitness of a seller and evolve seller prices through selection and mutation. We find three distinct steady states in our model related to the probability that a seller experiences competition for a buyer, mediated by the number of bankrupt sites in the system. When competition-free sales are unlikely, the system collapses on to a single price. If temporary monopoly situations do exist sellers can accumulate capital and variation in prices is stable. In this scenario, sellers spontaneously separate into two classes: cheap sellers – requiring sales to every potential buyer; and expensive sellers – requiring only occasional sales. Finally, we find an intermediate regime in which there is a single highly favoured price in the system which oscillates between high and low extrema. We study the properties of these steady states in detail, building a picture of how globally uncompetitive sellers can nonetheless survive if competition is strictly local.We show how the system builds up correlations, leading to niches for expensive sellers. These niches change the nature of the competition and allow for long-term survival of uncompetitive sellers. Not all expensive prices are equally likely in the steady state and we analyse why (and where) peaks in the price distribution appear.We can do this exactly for the early time dynamics of the model and extend the argument more qualitatively to the steady state. This latter analysis allows us to predict, for an observed steady distribution, the minimum price an expensive seller should charge to guarantee profit. The oscillatory ‘steady state’ is qualitatively reminiscent of boom and bust cycles in the global market. We study methods to suppress the oscillations and suggest ways of avoiding catastrophic crashes in the global economy – without negatively affecting the ability of outliers to make large profits.