Bifurcations in Macroeconomic Models

Grandmont (1985) found that the parameter space of even the simplest, most classical models is stratified into bifurcation regions. But in such classical models all policies are Ricardian equivalent and all solutions are Pareto optimal. As a result he was not able to reach conclusions about policy relevance of his dramatic discovery. Barnett and He (1999,2002) subsequently found transcritical, codimension two, and Hopf bifurcation boundaries within the parameter space of the policy relevant Bergstrom and Wymer continuous time dynamic macroeconometric model of the UK economy. Because of the Lucas critique, there is increasing interest in Euler equation models with GMM estimated deep parameters. He and Barnett’s (2002) analysis of the Leeper and Sims (1994) Euler equations macroeconometric model revealed the existence of singularity-induced bifurcation within the model's parameter space. Although known in engineering, singularity-induced bifurcations have not previously been encountered in economics. The purpose of the paper is to introduce the bifurcation phenomena that we have encountered in the analysis of macroeconometric models. We include and emphasize the concept of singularity-induced bifurcation and its relationship with other forms of bifurcation. We do so for the benefit of economists who might encounter singularity bifurcation in the future, as we believe is likely with other Euler equation models similarly parameterized with deep parameters.