On hyperbolic discounting and uncertain hazard rates

The value of a future reward should be discounted where there is a risk that the reward will not be realized. If the risk manifests itself at a known, constant hazard rate, a risk-neutral recipient should discount the reward according to an exponential time-preference function. Experimental subjects, however, exhibit short-term time preferences that differ from the exponential in a manner consistent with a hazard rate that falls with increasing delay. It is shown here that this phenomenon can be explained by uncertainty in the underlying hazard. The time-preference function predicted by this analysis can be calculated by means of either (i) a direct superposition method, or (ii) Bayesian updating of the expected hazard rate. The observed hyperbolic time-preference function is consistent with an exponential prior distribution for the underlying hazard rate. Sensitivity of the predicted time-preference function to variation in the probability distribution of the underlying hazard rate is explored.