Intertemporal Substitution and Recursive Smooth Ambiguity ...

and so on. By induction, ∆(C ×∆(H t−1 )) and H t are compact metric spaces, for every t ≥ 1.Let H ∗ = ∏ ∞t=0 H t. It is a compact metric space with respect to the product metric.We consider sequences of acts (h 0 , h 1 , h 2 , ...) in H ∗ that are coherent. That is, h t andh t+1 must be consistent for all t ≥ 0 in the sense that will be made precise in AppendixB. The domain of coherent acts, a subset of H ∗ , is denoted by H. We put the details ofthe definition of coherent acts and formal construction of the domain in Appendix B. Thedomain H satisfies a homeomorphic property analogous to those shown in Epstein and Zin(1989), Chew and Epstein (1991), Wang (2003), Gul and Pesendorfer (2004), and Hayashi(2005).Theorem 3 The set H is homeomorphic to (∆(C × ∆(H))) S , denoted as:H ≃ (∆(C × ∆(H))) S .When restricting attention to constant acts, we obtain the subdomain consisting of twostagecompound lotteries, which satisfies the homeomorphism:L ≃ ∆(C × ∆(L)).Relations among the domains we have defined so far are summarized as follows:H ⊃ G ⊃ G ∗ ⊃ F∪ ∪ ∪ ∪L ⊃ M ⊃ ∆(C ∞ ) ⊃ (∆(C)) ∞ ⊃ C ∞ .In particular, the set of compound lottery acts G and the set of compound lotteries Mstudied in Section 3 are subsets of H and L respectively.We now introduce some useful notations. For any two-stage compound lottery acts,h, h ′ ∈ H, and any λ ∈ (0, 1) , we use λh + (1 − λ)h ′ ∈ ∆ (H) to denote a lottery thatgives h with probability λ and h ′ with probability 1 − λ. We use λh ⊕ (1 − λ)h ′ ∈ H todenote a state-wise mixture. That is, for each s ∈ S and each Borel set B ∈ B (C × ∆ (H))λh⊕(1−λ)h ′ (s)(B) = λh (s) (B)+(1−λ)h ′ (s)(B). For any p, q ∈ ∆ (H), λp+(1−λ)q ∈ ∆ (H)denotes the usual mixture.4.2. AxiomsWe impose the following axioms on the preference process {≽ s t}. The first three axioms areanalogous to Axioms A1-A3.19