Dynamic Choices of Hyperbolic Consumers

Laboratory and field studies of time preference find that discount rates are much greater in the short-run than in the long-run. Hyperbolic discount functions capture this property. This paper solves the decision problem of a hyperbolic consumer who faces stochastic income and a borrowing constraint. The paper uses the bounded variation calculus to derive the Hyperbolic Euler Relation, a natural generalization of the standard Exponential Euler Relation. The Hyperbolic Euler Relation implies that consumers act as if they have endogenous rates of time preference that rise and fall with the future marginal propensity to consumer (e.g., discount rates that endogenously range from 5% to 41% for the example discussed in the paper). The Hyperbolic Euler Relation implies that hyperbolic consumers will engage in high interest borrowing even when the consumption path is downward sloping, will experience predictable sharp drops in consumption, and will fail to exhibit precautionary savings effects.