Stochastic growth, conservation of capital and convergence to a positive steady state

In a general one-sector model of optimal stochastic growth where the productivity of capital is bounded but may vary widely due to technology shocks, we derive a tight estimate of the slope of the optimal policy function near zero. We use this to derive a readily verifiable condition that ensures almost sure global conservation of capital (i.e., avoidance of extinction) under the optimal policy, as well as global convergence to a positive stochastic steady state for bounded growth technology; this condition is significantly weaker than existing conditions and explicitly depends on risk aversion. For a specific class of utility and production functions, a strict violation of this condition implies that almost sure long run extinction of capital is globally optimal. Conservation is non-monotonic in risk aversion; conservation is likely to be optimal when the degree of risk aversion (near zero) is either high or low, while extinction may be optimal at intermediate levels of risk aversion.