Sensitivity Analysis of Spatial Market Equilibria on Networks

Abstract

An important tool used in the equilibrium facility location models developed in this book is sensitivity analysis for variational inequalities. Sensitivity analysis is used to obtain the derivatives of the market prices with respect to the production and shipping quantity decisions made by the firm of interest. These derivatives can then be used in one of two equivalent ways. One is to use the derivatives via the chain rule to calculate the gradient of the firm’s profit function with respect to its production and shipping quantity decisions taking into account the effect of these decisions on market prices. The other is to use the derivatives to approximate linearly the reaction function and then include this approximate reaction function as a constraint in the model to constrain market prices to the reaction. In this case, the gradient of the firm’s profit function does not include price effects, but feasible directions are constrained by the reaction function constraint.

Some of the material in this chapter is adapted and expanded from Friesz, Miller and Tobin (1988A) and from Miller, Tobin and Friesz (1991).