Stochastic Stability of Endogenous Growth: The AK Case

This note studies the stochastic stability of the standard AK growth model under uncertain output technology. Capital accumulation follows a stochastic linear homogenous differential equation. It's shown that exponential balanced paths, which characterize optimal trajectories in the absence of uncertainty, are not robust to uncertainty. Precisely, it's demonstrated that the economy almost surely collapses at exponential speed even though productivity is initially arbitrarily high.