A foundation for the solution of consumption-saving behavior with a borrowing constraint and unbounded marginal utility

Models of precautionary saving or storage include cases where the marginal value of accumulated balances is unbounded, with an invariant distribution with infinite mean. Based on a uniform continuity argument, we show that a model of saving with bounded marginal value can be used to approximate the unbounded marginal value function, and the quantiles of its invariant distribution, arbitrarily accurately. These results offer a foundation for a strategy for numerical solution of marginal values in cases where they are unbounded, and for derivation of the quantiles of their invariant distributions.