# One-Switch Discount Functions

## TL;DR

This paper extends the class of discount functions satisfying the one-switch property to include linear times exponential forms and explores their implications for preferences with increasing impatience.

## Contribution

It proves that linear times exponential discount functions satisfy the one-switch property and clarifies the relationship between weak and strong versions of this property.

## Key findings

- Linear times exponential discount functions satisfy the one-switch property.
- Preferences with these discount functions exhibit increasing impatience.
- The paper clarifies the equivalence of weak and strong one-switch properties in continuous time.

## Abstract

Bell (1988) introduced the one-switch property for preferences over sequences of dated outcomes. This property concerns the effect of adding a common delay to two such sequences: it says that the preference ranking of the delayed sequences is either independent of the delay, or else there is a unique delay such that one strict ranking prevails for shorter delays and the opposite strict ranking for longer delays. For preferences that have a discounted utility (DU) representation, Bell (1988) argues that the only discount functions consistent with the one-switch property are sums of exponentials. This paper proves that discount functions of the linear times exponential form also satisfy the one-switch property. We further demonstrate that preferences which have a DU representation with a linear times exponential discount function exhibit increasing impatience (Takeuchi (2011)). We also clarify an ambiguity in the original Bell (1988) definition of the one-switch property by distinguishing a weak one-switch property from the (strong) one-switch property. We show that the one-switch property and the weak one-switch property definitions are equivalent in a continuous-time version of the Anscombe and Aumann (1963) setting.