A Continuous Equilibrium Model for Estimating Market Areas of Competitive Facilities with Elastic Demand and Market Externality

We consider a heterogeneous two-dimensional space where a given set of competitive facilities is located. Customers are assumed to be scattered continuously over the space, and each customer is assumed to choose a facility to minimize individual total cost of receiving service. The total cost consists of both the congested travel time to the facility and a cost associated with the congestion externality at the facility. Furthermore, customer demand at any location is assumed to be a function of the total cost of receiving service. Given these assumptions, it is of interest to estimate the market areas and market shares captured by each competitive facility. This problem is formulated here as a calculus of variations problem, and its optimality conditions are shown to be equivalent to the spatial customer choice equilibrium conditions with elastic demand and market externality. The model is solved by an efficient finite element method and illustrated with a numerical example.