Spatial price equilibrium networks with flow-dependent arc multipliers

The spatial price equilibrium modeling framework, which emphasizes the importance of transportation costs between markets, has been utilized in agricultural, energy, mineral as well as financial applications. In this paper, we construct static and dynamic spatial price equilibrium networks with flow-dependent arc multipliers, which expand the reach of applications. The static model is formulated and analyzed as a variational inequality problem, whereas the dynamic one is formulated as a projected dynamical system, whose set of stationary points coincides with the set of solutions of the variational inequality. Qualitative results are presented, along with an algorithm, the Euler method, which yields a time-discretization of the continuous-time adjustment processes associated with the product shipments from supply markets to demand markets. The algorithm is implemented and applied to compute solutions to numerical examples with flow-dependent arc multipliers addressing losses and/or gains, inspired by perishable agricultural products, and by financial investments. The results in this paper add to the literature on generalized networks as well as that on commodity trade.