Intertemporal Substitution and Recursive Smooth Ambiguity ...

Finite-step-ahead acts and densenessFinally, we define finite-step-ahead acts and show that the union of all the sets of finite-stepaheadacts is dense. LetH +1 = {h +1 ∈ (∆(C × ∆(H))) S : ∀s ∈ S, h +1 (s) ∈ ∆(C × ∆(L))}.Since H ≃ (∆(C ×∆(H))) S , we can embed H +1 into H, where the range of H +1 is embeddedinto L since L ≃ ∆(C × ∆(L)). Inductively, defineH +τ = {h +τ ∈ (∆(C × ∆(H))) S : ∀s ∈ S, h +τ (s) ∈ ∆(C × ∆(H +(τ−1) ))}.Similarly, we can embed H +τ into H. We call ⋃ τ≥1 H +τ the domain of finite-step-ahead acts.Lemma 9 The domain of finite-step-ahead acts ⋃ τ≥1 H +τ⋃τ≥1 ∆(C × ∆(H +τ)) is a dense subset of ∆(C × ∆(H)).is a dense subset of H. Also,This result is analogous to Proposition 1 in Hayashi (2005) and its proof is omitted. Itis useful to establish the existence of a risk equivalent as in Lemma 9 of Hayashi (2005). Wehave implicitly applied a similar result in Appendix A.43