# Hyperbolic Discounting of the Far-Distant Future

## TL;DR

This paper extends Weitzman's 1998 result to hyperbolic discounting, showing that when uncertain about hyperbolic rates, the far future should be discounted using a probability-weighted harmonic mean of possible rates.

## Contribution

It provides a novel theoretical result linking hyperbolic discounting uncertainty with a specific method for discounting the distant future.

## Key findings

- Far-distant future discounted by harmonic mean of rates
- Uncertainty about hyperbolic rates affects long-term valuation
- Extends exponential discounting results to hyperbolic case

## Abstract

We prove an analogue of Weitzman's (1998) famous result that an exponential discounter who is uncertain of the appropriate exponential discount rate should discount the far-distant future using the lowest (i.e., most patient) of the possible discount rates. Our analogous result applies to a hyperbolic discounter who is uncertain about the appropriate hyperbolic discount rate. In this case, the far-distant future should be discounted using the probability-weighted harmonic mean of the possible hyperbolic discount rates.