Mixing discount functions: Implications for collective time preferences

It is well-known that for a group of decision makers with stationary time preferences their collective time preferences may become non-stationary. In particular, Jackson and Yariv (2014) show that aggregating heterogeneous exponential discount functions yields a function that exhibits what they call “present bias”. In a continuous-time setting “present bias” is equivalent to strictly decreasing impatience (DI). Applying the notion of comparative DI introduced by Prelec (2004), we generalise Jackson and Yariv’s result: mixing any finite number of heterogeneous discount functions from the same DI equivalence class (such as exponential discount functions) yields a mixture that exhibits uniformly more DI than each component. We also obtain necessary and sufficient conditions for mixtures of DI-ordered – but not necessarily DI-equivalent – functions to be uniformly more DI than the least DI component. For suitably smooth discount functions, Prelec (2004) defines a local index of DI. We use this to obtain local versions of our results, which apply to mixtures of any finite set of (smooth) discount functions. Our main theorems generalise a number of specialised results in the decision theory and survival analysis literatures.