Spatial firm competition in two dimensions with linear transportation costs: simulations and analytical results

TL;DR

This paper investigates spatial firm competition with linear transportation costs in two dimensions, demonstrating equilibrium existence under periodic boundary conditions and analyzing how profits scale with the number of firms through simulations and analytical methods.

Contribution

It provides the first analytical and simulation-based analysis of equilibrium existence and profit scaling in two-dimensional spatial competition with linear transportation costs.

Findings

Equilibrium exists for two firms at any distance with periodic boundary conditions.

Total equilibrium profit decreases proportionally to the inverse square root of the number of firms.

System behavior varies with different transportation cost exponents.

Abstract

Models of spatial firm competition assume that customers are distributed in space and transportation costs are associated with their purchases of products from a small number of firms that are also placed at definite locations. It has been long known that the competition equilibrium is not guaranteed to exist if the most straightforward linear transportation costs are assumed. We show by simulations and also analytically that if periodic boundary conditions in two dimensions are assumed, the equilibrium exists for a pair of firms at any distance. When a larger number of firms is considered, we find that their total equilibrium profit is inversely proportional to the square root of the number of firms. We end with a numerical investigation of the system's behavior for a general transportation cost exponent.