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.