Intertemporal Substitution and Recursive Smooth Ambiguity ...

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This relation holds true because i is more risk averse than j in the first stage.Turn to the proof of the converse statement. Fix a set E ⊂ S such that λ = ∫ π(E)dµ P s t(π) ∈s t(0, 1). Suppose (c, δ[l]) ≽ j s(c, λδ[l ′ ] + (1 − λ)δ[l ′′ ]) for l, l ′ , l ′′ ∈ L. Let h t +1 be the one-stepaheadact that gives l ′ if event E happens and gives l ′′ , otherwise. Then by definitionwe can show that b(h +1 , µ s t) = λδ[l ′ ] + (1 − λ)δ[l ′′ ]. Using the representation in Theorem3, we can verify that (c, b(h +1 , µ s t)) ∼ j s(c, δ[h t +1 ]) or (c, λδ[l ′ ] + (1 − λ)δ[l ′′ ]) ∼ j s t(c, δ[h +1 ]), which implies (c, δ[l]) ≽ j s(c, δ[h t +1 ]). By comparative ambiguity aversion, wehave (c, δ[l]) ≽ i s(c, δ[h t +1 ]). Since (c, λδ[l ′ ] + (1 − λ)δ[l ′′ ]) ∼ i s(c, δ[h t +1 ]) holds as well, weobtain (c, δ[l]) ≽ i s(c, λδ[l ′ ] + (1 − λ)δ[l ′′ ]). Hence we havet(c, δ[l]) ≽ j s(c, λδ[l ′ ] + (1 − λ)δ[l ′′ ]) =⇒ (c, δ[l]) ≽ i t s (c, t λδ[l′ ] + (1 − λ)δ[l ′′ ]).We can extend this result to all λ ∈ (0, 1) by continuity (Axiom B1) and Axiom B4 (FirstStage Independence). We can also extend this result to all finite lotteries over L by repeatedlyapplying the above argument. We finally extend it to all lotteries over L by continuity ofpreferences (Axiom B1).E Appendix: Proofs for Section 5Proof of Proposition 7:Defineφ t (α) = V t (c + αδ) ,for an adapted process (δ t ) . Using equations (24)-(26), we have:φ t (α) = W (c t + αδ t , R t (V t+1 (c + αδ))) .Taking derivatives in the preceding equation yieldsφ ′ t (0) = W 1 (c t , R t (V t+1 )) δ t{+ W 2 (c t , R t (V t+1 )) v ′ ◦ u ( −1 E πz,t [u (V t+1 )] )Ev ′ µt(R t (V t+1 )) u ( ′ u ( −1 E πz,t [u ′ (V t+1 )] ))E [π z,t u ′ (V t+1 ) φ ′ t+1 (0) ]}Define λ t as in (22) and E z tas in (23). We obtain:[ ]Eφ ′ zt (0) = λ t δ t + E t+1t φ ′Etz t+1 (0) ,where E t is the conditional expectation operator with respect to the predictive distribution∑z µ t (z) π z (·|s t ) . From this equation and the definition in (21), we can derive that ξ z t =E z t λ t .51
Proof of Proposition 8:When the utility function takes the homothetic form, we useProposition 7 and the definition of the pricing kernel to derive (27). Alternatively, we maywrite the pricing kernel in terms of the market return as in Epstein and Zin (1989). In acomplete market, wealth X t satisfiesX t = E t[ ∞∑s=tThat is, time t wealth is equal to the present value of the consumption stream. By Lemma6.25 in Skiadas (2009), we haveBy equation (22), we have the relation:c tV t=Thus, the consumption-wealth ratio satisfies:c tX t= c tλ tV t=ξszξtzc s].V t = λ t X t . (50)( ) −1/ρ λt.1 − β(λt1 − βEliminating λ t from equations (50) and (51) yields:V t = λ t X t = (1 − β) 1= (1 − β) 1−ρ1−ρ c1−ρt X 11−ρ−ρ1−ρ c1−ρt R 11−ρ) −1/ρλ t . (51)t (52)t (X t−1 − c t−1 ) 1where we have used equation (29) to derive the last equality. Note that the second equalityimplies thatX t= 1 ( ) 1−ρ Vt. (53)c t 1 − β c tAs a result, for unitary EIS (ρ = 1), the consumption-wealth ratio is equal to 1 − β.1−ρ ,Now, substituting equation (52) into (27) and manipulating, we derive:⎛⎞ ⎛ [ρ−γ −ρ(1−γ) 1−γ−ρ( ) −ρ 1−ρMt+1 z ct+1= β ⎜ct+1 R 11−ρ 1−ρ1−ρ⎜(E πz,t ct+1 Rt+1t+1c t⎝) ⎟ ⎜⎜⎝−ρ ⎠)−ρ1−ρ 1−ρ1−ρ 1−ρR t(cR t(cM z t+1 =β(ct+1c tt+1 R 1t+1t+1 R 1t+1Writing in terms of consumption growth and manipulating, we obtain:( ) ) 1−γ⎛ (−ρ 1−ρ⎡ ⎛(× ⎣R t⎝ β1−γ1−ρR−1t+1⎡⎝E πz,t⎣( ) ) 1−ρ 1−ρct+1R t+1c t⎞⎤⎠⎦βη−ρ(ct+1.c t]) 11−γ) −ρR t+1) 1−γ1−ρ⎞⎟⎠⎤⎞⎦⎠−(η−γ)−(η−γ)1−γ.52
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 48 and 49: where the second equality follows f
Page 50 and 51: D Appendix: Proofs for Section 4.4P
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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