On a safe capital stock for consumption maintenance in a convex model of stochastic growth

In a stochastic one-sector optimal growth model with convex technology we show that there may be a critical safe capital stock for maintaining positive consumption over time. Consumption and capital paths are bounded away from zero with probability one if the initial stock is above this safe level, but are arbitrarily close to zero with positive probability if it is below. We derive verifiable sufficient conditions on technology and preferences under which the optimal policy generates such a critical stock. High risk aversion near zero, decreasing risk aversion and large spread of the technology shock are important factors behind this phenomenon.