Balanced‐budget rules: Local indeterminacy and bifurcations

Schmitt‐Grohé and Uribe (1997) illustrate that a balanced‐budget rule can lead to local indeterminacy and sunspot equilibria. I extend their local analysis by using local bifurcation techniques to investigate the possibility of cyclic and sunspot equilibria under a balanced‐budget rule in a neighborhood of a locally determinate steady state. For a class of models, I show the generic existence of two bifurcation steady‐state tax rates: flip and fold. The flip bifurcation implies the existence of 2‐cycles. Furthermore, I show analytically for a large region of the parameter space that this flip bifurcation is supercritical so the 2‐cycles are stable. This finding establishes the existence multiple equilibria and sunspot equilibria in a neighborhood of a locally determinate steady state. I also find there are model parameterizations where the fold bifurcation tax rate is lower than the flip bifurcation tax rate. In these instances, the steady state is a source for tax rates between these bifurcation tax rates. This indicates a qualitatively different type of aggregate instability due to a balanced‐budget rule.