Dancing on the Saddles: A Geometric Framework for Stochastic Equilibrium Dynamics

This paper extends deterministic saddle-path analysis to stochastic environments by introducing conditional saddle paths: the equilibrium path under frozen exogenous states. This concept yields a global geometric representation of stochastic equilibrium dynamics, in which equilibrium fluctuations decompose into movements along (endogenous propagation) and across (exogenous state transitions) conditional saddle paths. The framework delivers two theoretical results. First, state-dependent impulse responses arise from differences in the slopes of conditional saddle paths. Second, if an aggregate equilibrium variable varies strictly monotonically along conditional saddle paths, it uniquely indexes equilibrium states and thus provides an exact one-dimensional sufficient statistic. Applying this result, I prove that aggregate capital is a sufficient statistic in a canonical heterogeneous-household model (Krusell and Smith, 1998).