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.

中文翻译：

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平衡预算规则：局部不确定性和分岔

Schmitt-Grohé 和 Uribe (1997) 说明平衡预算规则会导致局部不确定性和太阳黑子平衡。我通过使用局部分岔技术扩展了他们的局部分析，以研究在局部确定的稳定状态附近的平衡预算规则下循环和太阳黑子平衡的可能性。对于一类模型，我展示了两种分叉稳态税率的普遍存在：翻转和折叠。翻转分岔意味着存在 2 个循环。此外，我对参数空间的大区域进行了分析，表明这种翻转分岔是超临界的，因此 2 循环是稳定的。这一发现确立了在一个局部确定的邻域中存在多重平衡和太阳黑子平衡稳定状态。我还发现在模型参数化中，折叠分叉税率低于翻转分叉税率。在这些情况下，稳态是这些分叉税率之间的税率来源。这表明由于平衡预算规则而导致的质量不同类型的总体不稳定性。