Monopoly and demand uncertainty

We develop a formal framework to analyse a monopoly’s problem when demand is determined by a Poisson-distribution and the valuations of the buyers are draws from a common distribution. The buyers have unit demand, and the good in question is discrete. The monopoly has to make its quantity and pricing decisions before demand is realised. We determine sufficient conditions for the monopoly’s pricing decision to be unique, and we demonstrate the difference to the planner’s problem by numerical examples. We also study an economy with a fixed number of buyers assuming that the valuations are draws from the uniform distribution. When the economy grows in the limit one recovers the standard case of a monopoly with linear demand and constant marginal costs.