Intertemporal Substitution and Recursive Smooth Ambiguity ...

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where the second equality follows from equation (42).Finally, we extend the above representation to the whole domain C × ∆(H).A riskequivalent always exists as discussed before. By a similar argument as in Appendix A2, forevery s t and every h ∈ H, there exists a one-step-ahead act h +1 ∈ H +1 such that h (s) andh +1 (s) are stochastically equivalent. We call h +1 the equivalent one-step-ahead act of h. ByAxiom B6 (Dynamic Consistency), we deduce(c, δ[h]) ∼ s t (c, δ[h +1 ]). (48)Suppose that p ∈ ∆(H) has a finite support {h 1 , h 2 , · · · , h m }, with p = ∑ i α iδ[h i ],α i ∈ (0, 1) , and ∑ i α i = 1. For each h i , i = 1, · · · , m, let h i +1 ∈ H +1 be its equivalentone-step-ahead act. Let p +1 ∈ ∆(H +1 ) be a probability measure with a finite supportsuch that the support is {h 1 +1, h 2 +1, · · · , h m +1}, and for each i = 1, · · · , m, p +1 ({h i +1}) =p({h i }).obtain (c, p) ∼ s tBy repeated applications of Axiom B4 (First-stage Independence) and (48), we(c, p +1 ). This relation is also true for arbitrary c ∈ C because of Axiom B2(Current Consumption Separability). By continuity of ≽ s t, the claim extends to arbitraryp. Hence, we haveV s t(c, p)= V s t(c, p +1 )( ∫ ∫(c, u −1 ◦ φ −1 H +1= W= W(c, u −1 ◦ φ −1 ( ∫H∫P s t( ∑∫)))φ π(s) u(V (c ′ , a ′ ))dh +1 (s)(c ′ , a ′ ) dµ s t(π)dp +1 (h +1 )s∈SC×∆(L)( ∑∫)))φ π(s) u(V s t ,s(c ′ , a ′ ))dh(s)(c ′ , a ′ ) dµ s t(π)dp(h) ,s∈SC×∆(H)P s twhere we have used the fact that h (s) and h +1 (s) are stochastically equivalent to derive thelast equality. Defining v = φ ◦ u, we obtain the representation as in the theorem.C3. Proof of UniquenessSuppose ({Ṽs t}, ˜W , ũ, ṽ, {˜µ s t}) and ({V s t}, W, u, v, {µ s t}) represent the same preference. Onthe domain of deterministic consumption streams C ∞ , all Ṽ s tfunction Ṽ and all V s tcoincide with the commoncoincide with the common function V . Since Ṽ and V are ordinallyequivalent over C ∞ , there is a monotone transformation Φ such thatṼ (y) = Φ ◦ V (y).for all y ∈ C ∞ . By (43), we have Ṽs t = Φ ◦ V s t.47
Since˜W (c, Ṽ (y)) = Ṽ (c, y) = Φ(V (c, y)) = Φ(W (c, V (y))) = Φ(W (c, Φ−1 (Ṽ (y)))),we have ˜W (c, z) = Φ(W (c, Φ −1 (z))).On L, ∫ C×∆(L) u(V (c′ , a ′ ))dl(c ′ , a ′ ) and ∫ C×∆(L) ũ(Ṽ (c′ , a ′ ))dl(c ′ , a ′ ) are equivalent mixturelinearrepresentations of the second-stage risk preference conditional on the fixed currentconsumption ĉ defined in Appendix C1. Therefore, there exist constants A, B with A > 0such that:)ũ(Ṽ (c ′ , a ′ ) = Au(V (c ′ , a ′ )) + B,for all (c ′ , a ′ ) ∈ C × ∆(L). Since Ṽ = Φ ◦ V , we obtain ũ ◦ Φ = Au + B.On ∆(L),∫Lṽ ◦ ũ −1 (∫∫Lv ◦ u −1 (∫C×∆(L)C×∆(L)) )ũ(Ṽ (c ′ , a ′ ) dl(c ′ , a ′ ) da(l) and)u (V (c ′ , a ′ )) dl(c ′ , a ′ ) da(l)are equivalent mixture-linear representations of the first-stage risk preference conditional onthe fixed current consumption ĉ. Hence, there exist constants D, E with D > 0 such that(∫) )ṽ ◦ ũ −1 ũ(Ṽ (c ′ , a ′ ) dl(c ′ , a ′ )= Dv ◦ u −1 (∫C×∆(L)C×∆(L)From the previous result, we have∫)∫ũ(Ṽ (c ′ , a ′ ) dl(c ′ , a ′ ) =C×∆(L)∫= A)u (V (c ′ , a ′ )) dl(c ′ , a ′ ) + E.C×∆(L)C×∆(L)Let ∫ C×∆(L) u (V (c′ , a ′ )) dl(c ′ , a ′ ) = x. Then we haveFrom ũ ◦ Φ(x) = Au(x) + B, it follows thatṽ ◦ ũ −1 (Ax + B) = Dv ◦ u −1 (x) + E.ũ ◦ Φ (V (c ′ , a ′ )) dl(c ′ , a ′ )u (V (c ′ , a ′ )) dl(c ′ , a ′ ) + Bṽ ◦ ũ −1 (Ax + B) = ṽ ◦ ũ −1 (Au ◦ u −1 (x) + B) = ṽ ◦ Φ(u −1 (x)).Thus, combining the above two equations, we obtainSo ṽ ◦ Φ = Dv + E.ṽ ◦ Φ(u −1 (x)) = Dv ◦ u −1 (x) + E.48
Page 1: Intertemporal Substitution andRecur
Page 6 and 7: As in KMM (2005, 2009a), we impose
Page 8 and 9: 2. Review of the Atemporal ModelsIn
Page 10 and 11: defined over it. Notice that by res
Page 12: For any c ∈ C, we use δ[c] to de
Page 15: The axiom below states that the pre
Page 18: is ambiguity averse if he prefers a
Page 22 and 23: Axiom B6 (Dynamic Consistency) For
Page 24 and 25: 3. On the subdomain C × M, we obta
Page 26 and 27: We can similarly define ambiguity l
Page 28 and 29: epresentations under these two appr
Page 30 and 31: where R t+1 is the market return fr
Page 32 and 33: Segal (1987, 1990) and Seo (2009).
Page 34 and 35: A Appendix: Proof of Theorems 1 and
Page 36 and 37: Define v = ψ ◦ ū −1 ◦ u, wh
Page 38 and 39: compute:∫˜ū (m) =∫=∫=∫= A
Page 40 and 41: DefineH = {h = (h 0 , h 1 , h 2 ,
Page 42 and 43: Lemma 5 We have the homeomorphic re
Page 44 and 45: Finite-step-ahead acts and densenes
Page 46 and 47: which is strictly increasing in the
Page 50 and 51: D Appendix: Proofs for Section 4.4P
Page 52 and 53: This relation holds true because i
Page 54 and 55: In complete markets, the following
Page 56 and 57: Ergin, H. I. and F. Gul (2009): “
Page 58: Weil, P. (1989): “The Equity Prem

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