Multiple equilibria in a monopoly market with heterogeneous agents and externalities

We explore the effects of social influence in a simple market model in which a large number of agents face a binary choice: to buy/not to buy a single unit of a product at a price posted by a single seller (monopoly market). We consider the case of positive externalities: an agent is more willing to buy if other agents make the same decision. We consider two special cases of heterogeneity in the individuals' decision rules, corresponding in the literature to the Random Utility Models of Thurstone, and of McFadden and Manski. In the first one the heterogeneity fluctuates with time, leading to a standard model in Physics: the Ising model at finite temperature (known as annealed disorder) in a uniform external field. In the second approach the heterogeneity among agents is fixed; in Physics this is a particular case of the quenched disorder model known as a random field Ising model, at zero temperature. We study analytically the equilibrium properties of the market in the limiting case where each agent is influenced by all the others (the mean field limit), and we illustrate some dynamic properties of these models making use of numerical simulations in an Agent based Computational Economics approach. Considering the optimization of the profit by the seller within the case of fixed heterogeneity with global externality, we exhibit a new regime where, if the mean willingness to pay increases and/or the production costs decrease, the seller's optimal strategy jumps from a solution with a high price and a small number of buyers, to another one with a low price and a large number of buyers. This regime, usually modelled with ad hoc bimodal distributions of the idiosyncratic heterogeneity, arises here for general monomodal distributions if the social influence is strong enough.