  It is well-known that for a group of time-consistent decision makers their
collective time preferences may become time-inconsistent. Jackson and Yariv
(2014) demonstrated that the result of aggregation of exponential discount
functions always exhibits present bias. We show that when preferences satisfy
the axioms of Fishburn and Rubinstein (1982), present bias is equivalent to
decreasing impatience (DI). Applying the notion of comparative DI introduced by
Prelec (2004), we generalize the result of Jackson and Yariv (2014). We prove
that the aggregation of distinct discount functions from comparable DI classes
results in the collective discount function which is strictly more DI than the
least DI of the functions being aggregated. We also prove an analogue of
Weitzman's (1998) result, for hyperbolic rather than exponential discount
functions. We show that if a decision maker is uncertain about her hyperbolic
discount rate, then long-term costs and benefits will be discounted at a rate
which is the probability-weighted harmonic mean of the possible hyperbolic
discount rates.
