INTRODUCTION

Modern macroeconomics has witnessed the increasing use of dynamic models in the study of economic behavior. Among the widely recognized models are the Bergstrom and Wymer continuous time dynamic macroeconometric model of theUKeconomy (Bergstrom and Wymer, 1976), the Leeper and Sims (1994) model, and the dynamic Leontief systems (Luenberger and Arbel, 1977).

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 generalized method of moments estimated deep parameters. He and Barnett's (2003) 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.

Euler equation models represent an important class of economic systems. In addition to the Leeper and Sims model, there is also, for example, the well-known Luenberger (Luenberger and Arbel, 1977) fundamental dynamic Leontief model. Knowledge of the nature of singularity-induced bifurcations is likely to become increasingly important in understanding the dynamics of modern macroeconomic models. Bifurcation analysis of parameter space stratification is a fundamental and frequently overlooked part of understanding model properties and can provide surprising results, as we have repeatedly found.