Intertemporal Substitution and Recursive Smooth Ambiguity ...

Axiom B6 (Dynamic Consistency) For all t and s t , for all c ∈ C and h, h ′ ∈ H, if h(s)(strictly) stochastically dominates h ′ (s) with regard to ≽ s t ,s for each s ∈ S, then(c, δ[h]) ≽ s t (≻ s t) (c, δ[h ′ ]).Finally, we embed Seo’s (2009) dominance axiom to the set of one-step-ahead acts. Aone-step-ahead act is an act for which subjective uncertainty resolves just in one period. Wedefine the set of one-step-ahead acts as:H +1 = {h +1 ∈ H : h +1 (s) ∈ L, ∀s ∈ S}.Definition 6 Given h +1 ∈ H +1 and π ∈ ∆ (S), define l(h +1 , π) ∈ L by:l(h +1 , π) = ∑ s∈Sh +1 (s)π(s).Given p +1 ∈ ∆(H +1 ) and π ∈ ∆(S), define a(p +1 , π) ∈ ∆(L) by:for every Borel subset L ⊂ L.a(p +1 , π)(L) = p +1 ({h +1 ∈ H +1 : l(h +1 , π) ∈ L}) ,We take a set of one-step-ahead probability measures, P s t, as given for each history s tand impose the following Dominance axiom on this set. We allow this set to be differentfrom ∆ (S) in order to permit more flexibility in applications as discussed in Section 1.Axiom B7 (Dominance) For all t and s t , for all c ∈ C and p +1 , p ′ +1 ∈ ∆(H +1 ),(c, a(p +1 , π)) ≽ s t (c, a(p ′ +1, π)), ∀π ∈ P s t =⇒ (c, p +1 ) ≽ s t (c, p ′ +1),where P s t ⊂ ∆ (S) .To interpret this axiom, imagine that P s t is a set of probability distributions, whichcontains the ‘true’ distribution unknown to the decision maker. Given the same currentconsumption c, if the decision maker prefers the continuation two-stage lottery a(p +1 , π)induced by p +1 over another one a(p ′ +1, π) induced by p ′ +1 for each probability distributionπ ∈ P s t, then he must also prefer (c, p +1 ) over ( c, p +1) ′ .Compared to the axioms in Section 3, First-Stage Independence and Dominance arethe counterparts of the SEU on Second-Order Acts and Consistency with Preference overSecond-Order Acts. Thus, we can dispense with second-order acts.21