Recursiveness of indifference prices and translation-invariant preferences

We consider an economic agent with dynamic preferences over a set of uncertain monetary payoffs. We assume that preferences are updated in a time-consistent way as more information is becoming available. Our main result is that the agent's indifference prices are recursive if and only if the preferences are translation-invariant. The proof is based on a characterization of time-consistency of dynamic preferences in terms of indifference sets. As a special case, we obtain that expected utility leads to recursive indifference prices if and only if absolute risk aversion is constant, that is, the Bernoulli utility function is linear or exponential.