Surge Pricing and Its Spatial Supply Response

Abstract

We consider the pricing problem faced by a revenue maximizing platform matching price-sensitive customers to flexible supply units within a geographic area. This can be interpreted as the problem faced in the short-term by a ride-hailing platform. We propose a two-dimensional framework in which a platform selects prices for different locations, and drivers respond by choosing where to relocate in equilibrium based on prices, travel costs and driver congestion levels. The platform's problem is an infinite-dimensional optimization problem with equilibrium constraints. We elucidate structural properties of supply equilibria and the corresponding utilities that emerge and establish a form of spatial decomposition, which allows us to localize the analysis to regions of movement. In turn, uncovering an appropriate knapsack structure to the platform's problem, we establish a crisp local characterization of the optimal prices and the corresponding supply response. In the optimal solution the platform applies different treatments to different locations. In some locations, prices are set so that supply and demand are perfectly matched; over-congestion is induced in other locations, and some less profitable locations are indirectly priced out. To obtain insights on the global structure of an optimal solution, we derive in quasi-closed form the optimal solution for a family of models characterized by a demand shock. The optimal solution, while better balancing supply and demand around the shock, quite interestingly, also ends up inducing movement away from it.