Tackling multiplicity of equilibria with Gröbner bases

Multiplicity of equilibria is a prevalent problem in many economic models. Often equilibria are characterized as solutions to a system of polynomial equations. This paper gives an introduction to the application of Gröbner bases for finding all solutions of a polynomial system. The Shape Lemma, a key result from algebraic geometry, states under mild assumptions that a given equilibrium system has the same solution set as a much simpler triangular system. Essentially, the computation of all solutions then reduces to finding all roots of a single polynomial in a single unknown. The software package Singular computes the equivalent simple system. If all coefficients in the original equilibrium equations are rational numbers or parameters, then the Gröbner basis computations of Singular are exact. Thus, Gröbner basis methods cannot only be used for a numerical approximation of equilibria, but in fact may allow the proof of theoretical results for the underlying economic model. Three economic applications illustrate that without much prior knowledge of algebraic geometry, Gröbner basis methods can be easily applied to gain interesting insights into many modern economic models.