Spatial Service Design for Public Services

Abstract

This study aims to design a service network in an urban area with continuous demand to maximize social welfare. We assume that customers are sensitive to travel and wait times, seeking service at a location that maximizes their utility. First, we demonstrate that the problem of spatial service design and pricing for an urban area with strategic customers is equivalent to a bilevel design problem, where customers can be explicitly assigned to service locations. This allows for the explicit assignment of customers to service locations and guarantees the existence of a service fee mechanism that satisfies the optimal assignment. We show that the urban area can be divided into a set of connected and disjoint regions such that, within each region, all customers seek service from the same location. We then derive the relationship between the optimal demand rate served at each service location and the optimal capacity (service rate) of the location. Our findings indicate that the optimal service capacity at each location depends on the capacity cost. For instance, the square root rule is optimal for linear capacity costs, but this does not hold for nonlinear costs. Furthermore, we characterize the relationship between the optimal service fee and observe that when the service capacity is fixed and cannot be adjusted, the optimal service fee is higher at locations with higher optimal demand rates. However, when capacity can be optimally allocated, the optimal service fee depends on the capacity cost. Specifically, if the cost is linear, optimal capacity allocation leads to optimal social welfare, resulting in all locations charging the same service fee. Conversely, if the capacity cost is nonlinear, the optimal fee decreases with the optimal demand rate when the service cost is strictly convex and increases when the cost is strictly concave.