Intertemporal Substitution and Recursive Smooth Ambiguity ...

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Because we allow lotteries as outcomes of acts whereas preference at each period isdefined over pairs of current consumption and continuation acts, our preceding formulationof dynamic consistency is more general than that in the literature (e.g., Epstein and Zin(1989), Epstein and Schneider (2003), Hayashi (2005), and KMM (2009a)). When we restrictattention to smaller domains used in the literature, we obtain the standard definition. Forexample, suppose the choice domain is the adapted consumption processes C × F and theutility representation is given by (2). Our Axiom A5 implies the following: For all c ∈ C andd, d ′ ∈ F ≃ (C × F) S , if d(s) ≽ s t (≻ s t) d ′ (s) for each s ∈ S, then (c, d) ≽ s t (≻ s t) (c, d ′ ).Now, we introduce two axioms on {≽ 2 s} in order to embed the atemporal KMM modeltin the dynamic setting. First, we follow KMM (2009a) and assume that the preference oversecond-order acts falls in the subjective expected utility (SEU) theory of Savage (1954), inwhich P s t is the state space and M is the set of pure outcomes.Axiom A6 (SEU Representation of Preference over Second-order Acts) For each s t , thereexists a unique countably additive probability measure µ s t : P s t → [0, 1] and a continuousand strictly increasing function ψ : M → R such that for all f, g ∈ I (P s t),∫∫f ≽ 2 s g ⇐⇒ ψ(f(π))dµ t s t(π) ≥ ψ(g(π))dµ s t(π).P s tP s tMoreover, ψ is unique up to a positive affine transformation if µ s t (J) ∈ (0, 1) for someJ ⊂ P s t. 14Second, we introduce an axiom that connects preference relations {≽ s t} and {≽ 2 s} usingtone-step-ahead acts and their corresponding second-order acts. A one-step-ahead act g +1 ∈ Gis a compound lottery act in which subjective uncertainty resolves just in one period. Definethe set of one-step-ahead acts as:G +1 = {g +1 ∈ G : g +1 (s) ∈ M, ∀s ∈ S}.Definition 2 Given a one-step-ahead act g +1 ∈ G +1 , its corresponding second-order act onP s tis given by g+1 2 : P s t → M, whereg+1(π) 2 = ∑ g +1 (s)π(s),s∈Sfor each π ∈ P s t.14 Because ψ is independent of history s t in this axiom, we implicitly assume that ≽ 2 srestricted to constanttacts in I (P s t) is independent of s t .13
The axiom below states that the preference {≽ s t} over the subdomain of one-step-aheadacts and the preference {≽ 2 s t } over the subdomain of the corresponding second-order actsare consistent with each other.Axiom A7 (Consistency with the Preference over Second-order Acts): For each s t , forevery c ∈ C and g +1 , h +1 ∈ G +1 ,3.4. RepresentationNow we state our first representation theorem.(c, g +1 ) ≽ s t (c, h +1 ) ⇐⇒ g 2 +1 ≽ 2 s t h2 +1.Theorem 1 (Representation) The preference process {≽ s t, ≽ 2 s t } satisfies Axioms A1-A7 ifand only if there exist representation ({V s t}, W, u, v, {µ s t}) such that:(i) On C × G, each ≽ s t is represented by: V s t(c, g) =( ( ∫ ( ∑∫) ))W c, v −1 v ◦ u −1 π(s) u (V s t ,s(c ′ , g ′ )) dg(s)(c ′ , g ′ ) dµ s t(π) , (11)P s ts∈SC×Gfor each (c, g) ∈ C × G, where W : C × R → R is continuous and strictly increasing in thesecond argument, u, v : R → R are continuous and strictly increasing functions. 15(ii) On C × M, each V s t coincides with:(∫))V (c, m) = W(c, u −1 u (V (c ′ , m ′ )) dm(c ′ , m ′ ) , ∀(c, m) ∈ C × M. (12)C×M(iii) On I (P s t) , each ≽ 2 s t is represented by the function:∫Vs 2 v ◦ u −1 ◦ u (g (π)) dµ s t(π), ∀g ∈ I (P s t) ,P s twhere v ◦ u −1 ◦ u = ψ and u : M → R is a mixture linear function:∫u (m) = u (V (c ′ , m ′ )) dm(c ′ , m ′ ), ∀m ∈ M. (13)C×MIn addition, we have the following uniqueness result, up to some monotonic transformations:15 Note that the domains of W , u and v may be smaller than those specified in the theorem. We do notmake this explicit in order to avoid introducing additional notations.14
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 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 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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