Intertemporal Substitution and Recursive Smooth Ambiguity ...

is ambiguity averse if he prefers a sure lottery obtained as the mean value of a given act tothe act itself.Definition 3 The decision maker with {≽ s t} exhibits ambiguity aversion if for all s t , forall c ∈ C and g +1 ∈ G +1 ,(c, m(g +1 , µ s t)) ≽ s t (c, g +1 ).Similarly to this definition, we can define ambiguity loving and ambiguity neutrality inthe usual way. An immediate consequence of this definition is the following:Proposition 1 Suppose {≽ s t} satisfies Axioms A1-A7. Then {≽ s t} exhibits ambiguity aversionif φ ≡ v ◦ u −1 is concave. 16The proof of this proposition is straightforward and is omitted. Clearly, when v ◦ u −1is linear, {≽ s t} displays ambiguity neutrality. Thus, the ambiguity neutrality benchmark isthe recursive expected utility model. We need additional conditions to establish the conversestatement that ambiguity aversion implies concavity of v ◦ u −1 . The reason is that, to provethis statement, one needs to know preferences over binary bets on some P s t, but our axiomsand representation hold only for fixed P s t. To deal with this issue in the KMM model, KMM(2005) consider a family of preference relations indexed by rich supports of second-orderbeliefs, and impose an assumption that ambiguity attitude and risk attitude are invariantacross these supports (see their Assumption 4). We can adapt their assumption to establishthe converse statement. Since the proof is similar to their proof of Proposition 1 in theirpaper, we omit it here.We now turn to comparative ambiguity aversion.Definition 4 Let the representations of the preferences of persons i and j share the samesecond-order belief µ s t on the same support P s t for all s t . Say that {≽ i s} is more ambiguitytaverse than {≽ j s} if for all s t , for all c ∈ C, m ∈ M and g t +1 ∈ G +1 ,(c, m) ≽ j s(c, g t +1 ) =⇒ (c, m) ≽ i s (c, g +1),tand if this property also holds for strict preference relations ≻ i sand ≻ j t s. t16 It is easy to check if v ◦ u −1 is concave, {≽ s t} satisfies the uncertainty aversion axiom of Gilboa andSchmeidler (1989).17