Intertemporal Substitution and Recursive Smooth Ambiguity ...

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epresentations under these two approaches give identical functional in the domain of adaptedconsumption processes. In addition, these two approaches give identical characterizations ofambiguity attitude in terms of the function v for fixed u or v ◦ u −1 .Unlike the second-order act approach in Section 3 or KMM (2005), the two-stage randomizationapproach does not need to have a rich support of µ s tto establish that absoluteor comparative ambiguity aversion implies concavity or comparative concavity of v ◦ u −1 .The reason is that the presence of two-stage randomization provides rich choices of lotteries,which allow us to use the standard analysis for objective risk.5. ApplicationWe use the representation in (3) to illustrate the application of our general model in finance.In that model, the decision maker does not observe a finite parameter z ∈ Z and hasambiguous beliefs about the possible consumption distributions π z indexed by z (P s t in (2)is a set indexed by z). We first derive the utility gradient (Duffie and Skiadas (1994)) forthe utility function defined in (3). The utility gradient is useful for solving an individual’soptimal consumption and investment problem. It is also useful for equilibrium asset pricing.We define the gradient of a utility function V 0 at c given z as the adapted process (ξ z t ) suchthat:V 0 (c + αδ) − V 0 (c)limα↓0 αLet V t denote V s t (c) in (3) and define[ ∞]∑= E ξt z δ t . (21)t=0R t (V t+1 ) = v −1 ( E µt v ◦ u −1 ( E πz,t u (V t+1 ) )) ,where we use µ t and π z,t to denote the posterior distribution µ s tand the conditional distributionπ z (·|s t ) , respectively.Proposition 7 Suppose W, u and v are differentiable. Then the utility gradient (ξ z t ) at cfor the generalized smooth ambiguity model is given by ξ z t = λ t E z tfor all t, whereE z tλ t = W 1 (c t , R t (V t+1 )) , (22)= Π t−1 W 2 (c s , R s (V s+1 )) v ′ ◦ u ( −1 E πz,s [u (V s+1 )] )s=0v ′ (R s (V s+1 )) u ( ′ u ( −1 E πz,s [u (V s+1 )] ))u′ (V s+1 ) , E0 z = 1. (23)This proposition demonstrates that under some regularity conditions, our generalizedrecursive smooth ambiguity model delivers a unique utility gradient, which is tractable for27
applications. By contrast, the widely adopted recursive multiple-priors model implies a set ofutility supergradients due to its kinked indifference curves (see Epstein and Wang (1994)).After we obtain the utility gradient, we can easily derive the pricing kernel. The pricingkernel M z t+1 between date t and t + 1 is defined as: M z t+1 = ξ z t+1/ξ z t . The pricing kernel isoften referred to as the intertemporal marginal rate of substitution or the stochastic discountfactor in the literature.In applications, it proves important to work with tractable parametric utility functionals.Our model permits flexible parametric specifications. Inspired by Epstein and Zin (1989),we consider the following homothetic functional forms in equation (3):W (c, y) = [ (1 − β) c 1−ρ + βy 1−ρ] 11−ρ, ρ > 0, (24)u (c) = c1−γ, γ > 0, ≠ 1, (25)1 − γv (x) = x1−η, η > 0, ≠ 1, (26)1 − ηwhere β ∈ (0, 1) is the subjective discount factor, 1/ρ represents the elasticity of intertemporalsubstitution (EIS), γ is the risk aversion parameter, and η is the ambiguity aversionparameter. If η = γ, the decision maker is ambiguity neutral and our model reduces tothe recursive utility model of Epstein and Zin (1989) and Weil (1989). The decision makerdisplays ambiguity aversion if and only if η > γ. By the property of certainty equivalent, amore ambiguity averse agent with a higher value of η has a lower utility level. The precedinginterpretations are justified by our axiomatic foundations in previous sections. We refer thereader to Ju and Miao (2010) for more discussions on the specification in (24)-(26).The key to understanding asset pricing puzzles in a representative-agent consumptionbasedframework is to understand the pricing kernel. We now derive the pricing kernel forthe homothetic generalized recursive ambiguity model. As is well known in the literature ofrecursive utility, we can write the pricing kernel in two ways.Proposition 8 The pricing kernel in terms of continuation values satisfies:⎛( ) −ρ ( ) ρ−γ( [ ])⎞1Mt+1 z ct+1 Vt+1 Eπz,t V1−γ 1−γ= β⎝t+1⎠c t R t (V t+1 )R t (V t+1 )−(η−γ). (27)Alternatively, the pricing kernel in terms of the market return under complete markets satisfies:M z t+1 =(β(ct+1c t) ) 1−γ⎛ ⎡−ρ 1−ρ ( ) 1−1−γ(11−ρ⎝E πz,t⎣ βR t+1(ct+1c t) ) 1−γ⎤⎞−ρ 1−ρR t+1⎦⎠− η−γ1−γ, (28)28
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 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 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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