Optimal continuous pricing with strategic consumers

Abstract

An important economic problem is that of finding optimal pricing mechanisms to sell a single item when there are a random number of buyers who arrive over time. In this paper, we combine ideas from auction theory and recent work on pricing with strategic consumers to derive the optimal continuous time pricing scheme in this situation. Under the assumption that buyers are split among those who have a high valuation and those who have a low valuation for the item, we obtain the price path that maximizes the seller's revenue. We conclude that, depending on the specific instance, it is optimal to either use a fixed price strategy or to use steep markdowns by the end of the selling season. As a complement to this optimality result, we prove that under a large family of price functions there is an equilibrium for the buyers. Finally, we derive an approach to tackle the case in which buyers' valuations follow a general distribution. The approach is based on optimal control theory and is well suited for numerical computations.