Intertemporal Substitution and Recursive Smooth Ambiguity ...

3. On the subdomain C × M, we obtain a pure risk setting where only the second-stagerandomization is present. In this case, the model reduces to (12).4.4. Risk Aversion and Ambiguity AversionAs discussed before, the function W describes intertemporal substitution. Now, we discusshow ambiguity aversion is separated from risk aversion in the two-stage randomization approach.We begin by characterizing risk aversion. In doing so, we restrict attention to thesubdomain C × ∆ (L) without subjective uncertainty. In this case the utility representationtakes the form in (18). Because there is two-stage randomization, we have two risk attitudestoward the risk in the two stages (or in the first-order and the second-order).For the risk in the second stage, we remove the first-stage risk by assuming that the firststage lottery is degenerate. We then obtain the representation of recursive risk preferencegiven in (12). We can define risk aversion in the second stage in a standard way and showthat it is completely characterized by the concavity of u.Turn to risk aversion in the first stage. We define absolute risk aversion in the first stageas follows:Definition 7 The decision maker with preference {≽ s t} exhibits risk aversion in the firststage if for all s t , c ∈ C and l, l ′ ∈ L, λ ∈ [0, 1],(c, δ[λl ⊕ (1 − λ)l ′ ]) ≽ s t (c, λδ[l] + (1 − λ)δ[l ′ ]). (19)We can similarly define risk loving and risk neutrality in the first stage. In Definition7, λδ[l] + (1 − λ)δ[l ′ ] ∈ ∆ (L) represents a lottery in the first stage and δ[λl ⊕ (1 − λ)l ′ ]represents a degenerate lottery over the mixture λl ⊕(1−λ)l ′ in the second stage. Accordingto this definition, the decision maker may not be indifferent between these two lotteries, eventhough they give the same final outcome distribution. In particular, if the decision makerbelieves that the degenerate lottery is like a sure outcome and must be preferred, then hedisplays risk aversion in the first stage.Note that if we replace ≽ s t with ∼ s t in (19), we obtain a dynamic counterpart of Seo’s(2009) Reduction of Compound Lotteries axiom. Thus, according to our Definition 7, violationof the Reduction of Compound Lotteries reflects the decision maker’s attitude towardthe risk in the first stage. The following proposition characterizes this risk attitude.Proposition 3 Suppose {≽ s t} satisfies Axioms B1-B7. Then {≽ s t} exhibits risk aversionin the first-stage if and only if v ◦ u −1 is concave.23