Chapter 11. Computing All Solutions to Polynomial Equations in Economics

Multiplicity of equilibria is a common problem in many economic models. In general, it is impossible to devise methods that always find all equilibria for any type of model. A notable exception are models in which all equilibria are solutions to a system of polynomial equations since there are powerful solution methods for finding all solutions to such polynomial systems. In many economic applications, equilibria can indeed be characterized as solutions to a system of polynomial equations. This handbook article provides a hands-on introduction to two solution methods for finding all solutions to polynomial systems; the first approach relies on Gröbner bases, the second approach employs all-solution homotopy methods. Several economic examples show how to compute all equilibria using modern software implementations of these two methods.