Intertemporal Substitution and Recursive Smooth Ambiguity ...

compute:∫˜ū (m) =∫=∫=∫= AC×MC×MC×MC×M)ũ(Ṽ (c ′ , m ′ ) dm(c ′ , m ′ )ũ ◦ Φ (V (c ′ , m ′ )) dm(c ′ , m ′ )Au (V (c ′ , m ′ )) dm(c ′ , m ′ ) + Bu (V (c ′ , m ′ )) dm(c ′ , m ′ ) + B = Aū (m) + B.Let ū (m) = w. Then we haveṽ ◦ ũ −1 (Aw + B) = v ◦ u −1 (w).Since ũ ◦ Φ(w) = Au(w) + B, it follows that:ṽ ◦ ũ −1 (Aw + B) = ṽ ◦ ũ −1 (Au ◦ u −1 (w) + B) = ṽ ◦ Φ(u −1 (w)).Thus, we obtain:ṽ ◦ Φ(u −1 (w)) = v ◦ u −1 (w).By replacing u −1 (w) by x, we obtain ṽ ◦ Φ(x) = v(x). Finally, uniqueness of µ s tAxiom A5.follows fromB Appendix: Proof of Theorem 3Given a compact metric space Y , let B(Y ) be the family of Borel subsets of Y , and ∆(Y )be the set of Borel probability measures defined over B(Y ), which is again a compact metricspace with respect to the weak convergency topology. Inductively define the family ofdomains {H 0 , H 1 , · · · } by:H 0 = (∆(C)) S ,H 1 = (∆(C × ∆(H 0 ))) S ,.H t = (∆(C × ∆(H t−1 ))) S ,and so on. By induction, ∆(C × ∆(H t−1 )) is a compact metric space and so is H t , for everyt ≥ 0. Let d t be the metric over H t . Let H ∗ = ∏ ∞t=0 H t. This is a compact metric spacewith respect to the product metric d(h, h ′ ) = ∑ ∞t=0371 d t(h t,h ′2 t t )1+d t (h t ,h ′ ). t