  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.
