Endogenous business cycles via state-dependent saving and noise-induced metastability
We develop a parsimonious stochastic growth model in which state-dependent saving behavior generates endogenous business-cycle-like dynamics. The model consists of three coupled equations: a Solow-type capital accumulation equation, a linear filtering equation for the saving rate, and a bounded stochastic adjustment process. Saving is modeled as a logistic function of deviations from a balanced growth path, introducing nonlinear feedback controlled by a gain parameter. In the deterministic limit, increasing feedback strength produces a supercritical pitchfork bifurcation, splitting the balanced-growth equilibrium into two locally attracting regimes corresponding to expansion and contraction. When stochastic perturbations are introduced, these equilibria become metastable states, and the economy undergoes rare noise-induced transitions between them. The resulting dynamics exhibit persistent regimes, bimodal stationary densities, and right-skewed dwell-time distributions with approximately exponential survival tails. A discrete-time approximation is estimated using U.S. real GDP data, and Monte Carlo simulations are used to compute stationary distributions and regime persistence statistics. The results demonstrate that nonlinear state dependence, bounded multiplicative noise, and time-scale separation are sufficient to generate realistic business-cycle behavior within a low-dimensional framework.