Aggregating time preferences with decreasing impatience

TL;DR

This paper investigates how aggregating different decreasing impatience discount functions affects collective time preferences, showing that the aggregate becomes more decreasingly impatient and deriving implications for hyperbolic discounting under uncertainty.

Contribution

It generalizes previous results by linking aggregation of DI preferences to increased DI and extends the analysis to hyperbolic discount functions under uncertainty.

Findings

Aggregating DI functions results in a more DI collective discount function.

The collective hyperbolic discount rate is the probability-weighted harmonic mean of individual rates.

Aggregation amplifies decreasing impatience in group decision-making.

Abstract

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.