Intertemporal Substitution and Recursive Smooth Ambiguity ...

Intertemporal Substitution andRecursive Smooth Ambiguity Preferences ∗Takashi Hayashi † and Jianjun Miao ‡November 7, 2010AbstractIn this paper, we establish an axiomatically founded generalized recursive smoothambiguity model that allows for a separation among intertemporal substitution, riskaversion, and ambiguity aversion. We axiomatize this model using two approaches:the second-order act approach à la Klibanoff, Marinacci, and Mukerji (2005) and thetwo-stage randomization approach à la Seo (2009). We characterize risk attitude andambiguity attitude within these two approaches. We then discuss our model’s applicationin asset pricing. Our recursive preference model nests some popular models inthe literature as special cases.Keywords: ambiguity, ambiguity aversion, risk aversion, intertemporal substitution,model uncertainty, recursive utility, dynamic consistencyJEL Classification: D80, D81, D90Forthcoming in Theoretical Economics∗ We are grateful to three anonymous referees and the coeditor (Gadi Barlevy) for helpful comments andsuggestions. We also thank Bart Lipman, Massimo Marinacci, and Kyoungwon Seo for helpful comments.First version: February 2010.† Department of Economics, University of Texas at Austin, BRB 1.116, Austin, TX 78712, USA. Email:th925@eco.utexas.edu. Phone: (512) 475 8543.‡ Department of Economics, Boston University, CEMA, Central University of Finance and Economics,and Xinhua School of Finance and Insurance, Zhongnan University of Economics and Law. 270 Bay StateRoad, Boston, MA 02215. Email: miaoj@bu.edu. Tel.: 617-353-6675.1

where µ s t (z) is the posterior distribution of z given s t . More generally, the learning modelin (3) allows z to be a hidden state that follows a Markov process because P s t can be historydependent.Our generalized recursive smooth ambiguity model nests some popular models in theliterature as special cases:• The subjective version of the recursive expected utility model of Kreps and Porteus(1978) and Epstein and Zin (1989) is obtained by setting v = u in (2). In this case, thetwo distributions µ s t and π can be reduced to a one-step-ahead predictive distribution:p ( s t+1 |s t) ∫= π (s t+1 ) dµ s t (π) . (4)P s tThis is the standard Bayesian approach which rules out ambiguity-sensitive behavior.If we further set v (x) = u (x) = − exp (−x/θ) , we obtain the multiplier preferencemodel or the risk-sensitivity model discussed in Hansen and Sargent (2001). 6is a robustness parameter, which enhances risk aversion.Here θ• The generalized recursive multiple-priors model of Hayashi (2005) is obtained as thelimit of (2) under some technical regularity conditions when ambiguity aversion goesto infinity:V s t(c) = W( ∫(c t , u −1 min u ( V (sπ∈P t ,s t+1 )(c) ) ))dπ (s t+1 ) . (5)s t SThis model nests the recursive multiple-priors model of Epstein and Wang (1994) andEpstein and Schneider (2003) as a special case, as discussed in Hayashi (2005).• The recursive smooth ambiguity model of KMM (2009a) has a discounted aggregatorand takes the form:V s t(c) = u (c t ) + βφ −1 (∫Z(∫(φ V (s t ,s t+1 )(c)dπ z st+1 |s t)) )dµ s t (z) . (6)SThe concavity of φ characterizes ambiguity aversion. The curvature of u describes bothintertemporal substitution and risk aversion. Thus, they are intertwined.6 This multiplier model is dynamically consistent according to the standard definition and the definitionin this paper. Hansen and Sargent (2001, 2008) also propose several other models of robustness. Some ofthem, e.g., “constraint preferences,” are dynamically inconsistent according to the standard definition aspointed out by Epstein and Schneider (2003). However, the constraint preferences satisfy a different notionof dynamic consistency defined in Section 19.4 of Hansen and Sargent (2008, p. 407-412).3

As in KMM (2005, 2009a), we impose the last axiom that connects preferences oversecond-order acts and the original preferences over pairs of current consumption and continuationcompound lottery acts. In doing so, we introduce the notion of one-step-ahead act–acompound lottery act in which subjective uncertainty resolves just in one period. We thenconstruct a second-order act associated with a one-step-ahead act, that maps a probabilitymeasure on S to a compound lottery on the consumption space. This compound lottery isobtained by averaging out states in the one-step-ahead act using the probability measure onS. The last axiom says that the decision maker orders pairs of current consumption and theone-step-ahead act identically to the second-order acts associated with the one-step-aheadacts. The intuition is that the decision maker’s ranking of the former choices reflects hisuncertainty about the underlying distribution of the choices, which is the domain of thesecond-order acts.One critique of the KMM (2005) model raised by Seo (2009) is that second-order actsand preferences over second-order acts are typically unobservable in the financial markets.For example, investors typically bet on the realization of stock prices, but not on the truedistribution underlying stock prices. A similar critique applies to the Anscombe-Aumannacts as well because these acts are also unobservable in financial markets: the realizations ofstock prices are monetary values, not lotteries. However, both Anscombe-Aumann acts andsecond-order acts are useful modelling devices and available from laboratory and thoughtexperiments. 8 More concretely, when measures in P s t correspond to conditional distributionsindexed by an unknown parameter as in (3), the second-order acts are bets on the value ofthe parameter. In an asset pricing application studied by Ju and Miao (2010), P s t consists oftwo distributions for consumption growth in a boom and in a recession so that the secondorderacts are bets on the economic regime. In a portfolio choice application studied byChen, Ju and Miao (2009), P s tconsists of two distributions for the possibly misspecifiedstock return models so that the second-order acts are simply bets on the statistical modelof stock returns.It is possible to dispense with the auxiliary domain of second-order acts following Seo’s(2009) axioms. Building on his insight, we provide an alternative axiomatization for (2)modelling devices to deliver Ellsbergian choices on the state space S of primary interest. To accommodateEllsbergian choices for second-order acts, one can simply expand the state space to incorporate measures onS.8 To further illustrate this point, we quote Kreps (1988, p.101): “This procedure of enriching the setof items to which preference must apply is quite standard. It makes perfectly good sense in normativeapplications, as long as the Totrep involved is able to envision the extra objects and agree with the axiomapplied to them. This need be no more than a thought experiment for Totrep, as long as he is willing to saythat it is a valid (i.e., conceivable) thought experiment.”5

without second-order acts. Adapting Seo’s (2009) static setup, we introduce an extra stageof randomization. As a by-product contribution, we construct a set of two-stage compoundlottery acts, which allows for randomization both before and after the realization of the stateof the world. We then define the product space of current consumption and the continuationlotteries over two-stage compound lottery acts as the single domain of preferences. Weimpose five axioms analogous to the first five axioms in the second-order act approach. Wereplace the last two axioms in that approach with a first-stage independence axiom anda dominance axiom adapted from Seo (2009). Given these seven axioms, we establish adynamic version of Seo’s static model. To the best of our knowledge, our paper provides thefirst dynamic extension of Seo’s static model.We should mention that each of our adopted two different axiomatic approaches is debatable.For example, some researchers (e.g., Seo (2009) and Epstein (2010)) argue thatsecond-order acts or preferences on these acts are either unobservable or may not be totallyplausible. In the two-stage randomization approach, a failure of the reduction of compoundlotteries may not be normatively appealing.After providing axiomatic foundations, we characterize risk attitude and ambiguity attitude.Our characterization in the second-order act approach is similar to that of KMM(2005), suitably adapted to our dynamic setting with Anscombe-Aumann-type acts. In thisapproach, ambiguity aversion is associated with aversion to the variation of ex ante evaluationsof one-step-ahead acts due to model uncertainty. In the two-stage randomizationapproach, we distinguish between attitudes toward risks in the two stages. We define absoluteambiguity aversion as an aversion to a first-stage mixture of acts before the realizationof the state of the world, compared to the second-stage mixture of these acts after the realizationof the state. We show that this notion of ambiguity aversion is equivalent to riskaversion in the first stage. In particular, ambiguity aversion is associated with the violationof reduction of compound lotteries. We also show that in both approaches, risk attitude andambiguity attitude are characterized by the shapes of the functions u and v, respectively.The remainder of the paper proceeds as follows. Section 2 reviews the atemporal modelsof KMM (2005) and Seo (2009). Section 3 embeds the KMM (2005) model in a dynamicsetting and axiomatizes it using the second-order act approach. Section 4 embeds the Seo(2009) model in a dynamic setting and axiomatizes it using the two-stage randomizationapproach. Section 5 applies our model to asset pricing. Section 6 discusses related literature.Appendices A-E contain proofs.6

2. Review of the Atemporal ModelsIn this section, we provide a brief review of the atemporal models of ambiguity proposed byKMM (2005) and Seo (2009). We will embed these models in a dynamic setting in Sections3-4. Both atemporal models when restricted to the space of random consumption deliver anidentical representation in (1). The two models differ in domain and axiomatic foundation.For both models, we take a complete, transitive, and continuous preference relation ≽ asgiven.Consider the KMM model first. KMM originally study Savage acts over S × [0, 1], wherethe auxiliary state space [0, 1] is used for describing objective lotteries. Here we translatetheir model in the Anscombe-Aumann domain. Let S be the set of states, which is assumedto be finite for simplicity. Let C be a compact metric space and ∆(C) be the set of lotteriesover C. 9An Anscombe-Aumann act is defined as a mapping g : S → ∆(C). Let G denotethe set of all such acts.In order to pin down second-order beliefs, KMM introduce anauxiliary preference ordering ≽ 2 over second-order acts. A second-order act is a mappingg : P → ∆(C), where P ⊂ ∆(S). Let G 2 denote the set of all second-order acts.The preference ordering ≽ over G and the preference ordering ≽ 2 over G 2 are representedby the following form:and∫U (g) =Pφ∫U 2 (g) =( ∑s∈SPπ (s) ū (g (s)))φ (ū (g (π))) dµ (π) , ∀g ∈G 2 ,dµ (π) , ∀g ∈ G, (7)where φ : u (∆(C)) → R is a continuous and strictly increasing function and ū : ∆(C) → Ris a mixture-linear function. 10The previous representation is characterized by the following axioms: 11 (i) The preference≽ satisfies the mixture-independence axiom over the set of constant acts ∆(C). (ii) Thepreference over second-order acts ≽ 2 is represented by subjective expected utility of Savage(1954). (iii) The two preference relations ≽ and ≽ 2 are consistent with each other in the sensethat g ≽ h if and only if g 2 ≽ 2 h 2 , where g 2 is the second-order acts associated with g defined9 We use the following notations and assumptions throughout the paper. Given a compact metric spaceY , let B(Y ) be the family of Borel subsets of Y , and ∆(Y ) be the set of Borel probability measures definedover B(Y ). Endow ∆(Y ) with the weak convergence topology. Then ∆(Y ) is a compact metric space.10 A function f is mixture linear on some set X if f (λx + (1 − λ) y) = λf (x)+(1 − λ) f (y) for any x, y ∈ Xand any λ ∈ [0, 1] .11 The proof can be obtained from our proof of Theorem 1 in Appendix A.7

y g 2 (π) = ∑ s∈S g(s)π(s) for each π ∈ P, and h2 is defined similarly. The interpretationfor the last axiom is the following. If the decision maker prefers f to g, then the average off across sates over all possible beliefs (distributions) should also be preferred to that of g.The reverse is also true.The last two axioms are controversial as argued by Epstein (2010). To illustrate theplausibility of these axioms, consider the following example. Suppose there is an Ellsbergurn containing 90 balls. A decision maker is told that there are 30 black balls and 60 whiteor red balls in the urn. But he does not know the composition of white or red balls. Thereare four bets as in Table 1. The Ellsbergian choice isg 1 ≻ g 2 but g 4 ≻ g 3 .One justification is that the decision maker is unsure about the probabilities of white andred balls and averse to this ambiguity.Table 1.b w rg 1 10 0 0g 2 0 10 0g 3 10 0 10g 4 0 10 10Table 2.π 1 π 2g1 2 10/3 10/3g2 2 20/3 0g3 2 10/3 10g4 2 20/3 20/3Suppose there are two possible distributions over the set of ball color S = {b,w,r}:π 1 = (1/3, 2/3, 0) , π 2 = (1/3, 0, 2/3) . Consider the second-order acts associated with g i ,i = 1, ..., 4, gi 2 (π j ) = ∑ s∈{b,w,r} g i (s) π j (s) , where j = 1, 2. We write their payoffs in Table2. The previous consistency axiom implies that:g1 2 ≻ 2 g2 2 but g4 2 ≻ 2 g3.2This behavior can be consistent with expected utility over second-order acts as long as thedecision maker is risk averse because g1 2 and g4 2 give sure outcomes, but g2 2 and g3 2 are riskybets. The intuition is that second-order acts average out uncertain states (ball color) bydefinition and such hedging may eliminate ambiguity (see Gilboa and Schmeidler (1989)).So it is possible that the decision maker is ambiguity neutral for second-order acts, butambiguity averse for bets on the Ellsberg urn. Of course, one can design thought experimentsto display Ellsbergian choices for second-order acts, which are ruled out by the KMM model.Seo (2009) provides a different axiomatic foundation for (7) by dispensing with the auxiliaryset of second-order acts and the associated preferences over this set. He considers thedomain of lotteries over Anscombe-Aumann acts, ∆(G), and a single preference relation ≽8

defined over it. Notice that by restricting attention to lotteries over constant acts we havethe domain of two-stage lotteries ∆(∆(C)) as a subset of ∆(G), and by further making thefirst-stage randomization degenerate we have ∆(C) as a subset of ∆(∆(C)), hence of ∆(G)too.Seo (2009) shows that the representation of preference ≽ takes the form: 12∫U (p) =G∫Pφ( ∑s∈Sπ (s) ū (g (s))When p is degenerate at some g ∈ G, (8) reduces to (7).)dµ (π) dp(g), ∀p ∈ ∆ (G) (8)In Seo’s approach, the representation is characterized by the following axioms: (i) Thepreference ≽ satisfies the mixture-independence axiom on the set of one-stage lotteries ∆(C).(ii) The preference ≽ satisfies the mixture-independence axiom on the set of lotteries overacts, ∆(G). (iii) A dominance condition holds. To state the dominance axiom formally,we define a two-stage lottery a(p, π) ∈ ∆ (∆ (C)) induced by p ∈ ∆ (G) and π ∈ P asa(p, π) (B) = p ({g ∈ G : g 2 (π) ∈ B}) for any Borel set B on ∆ (C) . Dominance says thatfor any p, q ∈ ∆ (G) , p ≽ q if a(p, π) ≽ a(p, π) for all π ∈ P. Seo’s approach does not deliveruniqueness of the second-order belief µ in general. For example, if φ is linear, then any µwith an identical mean ∫ πdµ(π) yields the same ranking. It is unique in some special cases,Pfor example, if φ is some exponential function. We refer to Seo (2009) for a characterizationof the uniqueness of µ.3. Axiomatization with Second-Order ActsWe embed the atemporal KMM model reviewed in Section 2 in a discrete-time infinitehorizonenvironment. Time is denoted by t = 0, 1, 2, .... Let S be a finite set of states ateach period. The full state space is S ∞ . Let C be a complete, separable, and compact metricspace, which is the set of consumption choices in each period.3.1. Primary DomainTo introduce the domain of choices of primary interest, we consider the set of compoundlottery acts introduced by Hayashi (2005). A compound lottery act is identified as a randomvariable that maps a state of the world into a joint lottery over consumption and a compound12 In Seo’s (2009) original representation, he takes P = ∆(S). When adapting his dominance axiom for P,we can allow P to be an arbitrary subset of ∆(S). For example, the proof in his appendix gives an exampleof a finite set P.9

lottery act for the next period. Hayashi (2005) shows that the set of all such acts G satisfiesthe following homeomorphic relation: 13G ≃ (∆(C × G)) S .It is a compact metric space with respect to the product metric. By abuse of notation, wemay refer S to the set of states as well as its cardinality.Up to homeomorphic transformations, the domain G of compound lottery acts includessubdomains G ∗ , F, M, C ∞ , that are defined as follows:1. adapted processes of consumption lotteries:G ∗ ≃ (∆(C) × G ∗ ) S ,2. adapted processes of consumption levels:F ≃ (C × F) S , (9)3. compound lotteries:M ≃ ∆(C × M),4. deterministic consumption streams:C ∞ ≃ C × C ∞ .The subdomain G ∗ is obtained by randomizing current consumption only, but not overacts. This domain corresponds to the one adopted by Epstein and Schneider (2003). Thesubdomain F is obtained when there is no randomization. It is adopted by Wang (2003).The subdomain M is obtained by taking constant acts.Epstein and Zin (1989) definerecursive utility under objective risk over the domain C × M, while Chew and Epstein(1991) axiomatize this utility over the domain M ≃ ∆(C × M). The space of deterministicconsumption plans, C ∞ is obtained by taking constant acts with no randomization.Relations among these subdomains are expressed as follows:G ⊃ G ∗ ⊃ F∪ ∪ ∪M ⊃ ∆(C ∞ ) ⊃ (∆(C)) ∞ ⊃ C ∞13 Two topological spaces X and Y are called homeomorphic (denoted X ≃ Y ) if there is a one-to-onecontinuous map f from X onto Y such that f −1 is continuous too. The map f is called a homeomorphism.10

For any c ∈ C, we use δ[c] to denote the degenerate lottery over c. When no confusionarises, we tend to omit the symbol of degenerate lottery and write down the deterministiccomponent as it is. For example, a deterministic sequence y = (c 0 , c 1 , c 2 , · · · ) is used as it is,instead of being denoted like (c 0 , δ[(c 1 , δ[(c 2 , δ[· · · ])])]).3.2. PreferencesWe consider two preference relations over two domains. Of primary interest are the decisionmaker’s preferences ≽ s t at each history s t over pairs of current consumption and continuationcompound lottery acts, C ×G. Each pair is called a consumption plan. In order to recover thedecision maker’s second-order beliefs, we introduce another preference ordering over secondorderacts. Take a set of one-step-ahead probability measures P s t ⊂ ∆(S) as a primitive foreach history s t . A second-order act on P s t is a mapping f : P s t → M. Let I (P s t) denotethe set of all the second-order acts on P s t. Let ≽ 2 sdenote the conditional second-ordertpreference defined over I (P s t) at each history s t .3.3. AxiomsWe start by introducing five standard axioms for the preference process {≽ s t}. First, weassume weak order (complete and transitive), continuity, and sensitivity. This ensures theexistence of a continuous functional representation of preference (see Debreu (1954)).Axiom A1 (Order) For all t and s t , ≽ s texist y, y ′ ∈ C ∞ such that y ≻ s t y ′ .is a continuous weak order over C × G, and thereSecond, we assume that preference over acts for the future is independent of currentconsumption. This axiom is adapted from Koopmans (1960) and is essential for the representationto have a form in which current consumption and continuation value are separable.Axiom A2 (Current Consumption Separability) For all t and s t , for all c, c ′g, g ′ ∈ G,(c, g) ≽ s t (c, g ′ ) ⇐⇒ (c ′ , g) ≽ s t (c ′ , g ′ ).∈ C andThird, we assume that preference over risky consumption is independent of history. Thisaxiom ensures that utility is stationary (or time invariant) in the pure risk domain C × M.It also implies that preference over deterministic consumption streams is independent ofhistory.11

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

where π z (s t+n |s t ) is the conditional probability of s t+n given s t , ¯c is a constant consumptionplan, and c (s t+n ) and c (s t ) are consumption levels at histories s t+n and s t , respectively. In) )addition, β (c) = W 2(¯c, ¯V is the discount factor, where ¯V = W(¯c, ¯V . As in KMM (2009a),we can show that if the full rank condition holds, then the marginal rate of substitutionassumption is equivalent to Bayesian updating of µ s t.Theorem 1 does not say anything about the existence and uniqueness of a solution for{V s t} to the recursive equation (11). Following a similar argument to that in the proof ofTheorem 2 in KMM (2009a), we can show that {V s t} exists. We need additional conditions forthe uniqueness. Epstein and Zin (1989) provide sufficient conditions for recursive expectedutility. KMM (2009a) give sufficient conditions for their recursive smooth ambiguity model.Marinacci and Montrucchio (2009) derive sufficient conditions for general recursive equationsthat may be applied to our model.3.5. Ambiguity AttitudeBecause our model nests the deterministic case (14) and the pure risk case (12), we immediatelydeduce that the function W characterizes intertemporal substitution and the functionu characterizes risk aversion in the usual way. We turn to the characterization of ambiguityaversion. We adopt the behavioral foundation of ambiguity attitude developed by Ghirardatoand Marinacci (2002) and KMM (2005). Epstein (1999) provides a different foundation. Themain difference is that the benchmark ambiguity neutral preference is the expected utilitypreference according to Ghirardato and Marinacci (2002), while Epstein’s (1999) benchmarkis the probabilistic sophisticated preference.We first consider absolute ambiguity aversion. According to our first axiomatization,ambiguity comes from the multiplicity of distributions in the set P s t. The decision maker’sambiguity attitude is toward uncertainty about the possible distributions in P s t. To characterizethis attitude, we define the lottery m(g +1 , µ s t) ∈ M associated with the one-step-aheadact g +1 and the second-order belief µ s t on P s t as:∫∑m(g +1 , µ s t) = g +1 (s) π (s) dµ s t(π).P s t s∈SSince ∑ s∈S g +1 (s) π (s) is the outcome of the second-order act g+1 2 (π) associated with g +1 ,m (g +1 , µ s t) is simply the mean value of g+1 2 with respect to the second-order belief µ s t.Alternatively, from the definition of predictive distribution in (4), we observe that the lotterym(g +1 , µ s t) is also the mean value of the act g +1 with respect to the predictive distributioninduced by µ s t. The following definition of ambiguity aversion states that the decision maker16

is ambiguity averse if he prefers a sure lottery obtained as the mean value of a given act tothe act itself.Definition 3 The decision maker with {≽ s t} exhibits ambiguity aversion if for all s t , forall c ∈ C and g +1 ∈ G +1 ,(c, m(g +1 , µ s t)) ≽ s t (c, g +1 ).Similarly to this definition, we can define ambiguity loving and ambiguity neutrality inthe usual way. An immediate consequence of this definition is the following:Proposition 1 Suppose {≽ s t} satisfies Axioms A1-A7. Then {≽ s t} exhibits ambiguity aversionif φ ≡ v ◦ u −1 is concave. 16The proof of this proposition is straightforward and is omitted. Clearly, when v ◦ u −1is linear, {≽ s t} displays ambiguity neutrality. Thus, the ambiguity neutrality benchmark isthe recursive expected utility model. We need additional conditions to establish the conversestatement that ambiguity aversion implies concavity of v ◦ u −1 . The reason is that, to provethis statement, one needs to know preferences over binary bets on some P s t, but our axiomsand representation hold only for fixed P s t. To deal with this issue in the KMM model, KMM(2005) consider a family of preference relations indexed by rich supports of second-orderbeliefs, and impose an assumption that ambiguity attitude and risk attitude are invariantacross these supports (see their Assumption 4). We can adapt their assumption to establishthe converse statement. Since the proof is similar to their proof of Proposition 1 in theirpaper, we omit it here.We now turn to comparative ambiguity aversion.Definition 4 Let the representations of the preferences of persons i and j share the samesecond-order belief µ s t on the same support P s t for all s t . Say that {≽ i s} is more ambiguitytaverse than {≽ j s} if for all s t , for all c ∈ C, m ∈ M and g t +1 ∈ G +1 ,(c, m) ≽ j s(c, g t +1 ) =⇒ (c, m) ≽ i s (c, g +1),tand if this property also holds for strict preference relations ≻ i sand ≻ j t s. t16 It is easy to check if v ◦ u −1 is concave, {≽ s t} satisfies the uncertainty aversion axiom of Gilboa andSchmeidler (1989).17

and so on. By induction, ∆(C ×∆(H t−1 )) and H t are compact metric spaces, for every t ≥ 1.Let H ∗ = ∏ ∞t=0 H t. It is a compact metric space with respect to the product metric.We consider sequences of acts (h 0 , h 1 , h 2 , ...) in H ∗ that are coherent. That is, h t andh t+1 must be consistent for all t ≥ 0 in the sense that will be made precise in AppendixB. The domain of coherent acts, a subset of H ∗ , is denoted by H. We put the details ofthe definition of coherent acts and formal construction of the domain in Appendix B. Thedomain H satisfies a homeomorphic property analogous to those shown in Epstein and Zin(1989), Chew and Epstein (1991), Wang (2003), Gul and Pesendorfer (2004), and Hayashi(2005).Theorem 3 The set H is homeomorphic to (∆(C × ∆(H))) S , denoted as:H ≃ (∆(C × ∆(H))) S .When restricting attention to constant acts, we obtain the subdomain consisting of twostagecompound lotteries, which satisfies the homeomorphism:L ≃ ∆(C × ∆(L)).Relations among the domains we have defined so far are summarized as follows:H ⊃ G ⊃ G ∗ ⊃ F∪ ∪ ∪ ∪L ⊃ M ⊃ ∆(C ∞ ) ⊃ (∆(C)) ∞ ⊃ C ∞ .In particular, the set of compound lottery acts G and the set of compound lotteries Mstudied in Section 3 are subsets of H and L respectively.We now introduce some useful notations. For any two-stage compound lottery acts,h, h ′ ∈ H, and any λ ∈ (0, 1) , we use λh + (1 − λ)h ′ ∈ ∆ (H) to denote a lottery thatgives h with probability λ and h ′ with probability 1 − λ. We use λh ⊕ (1 − λ)h ′ ∈ H todenote a state-wise mixture. That is, for each s ∈ S and each Borel set B ∈ B (C × ∆ (H))λh⊕(1−λ)h ′ (s)(B) = λh (s) (B)+(1−λ)h ′ (s)(B). For any p, q ∈ ∆ (H), λp+(1−λ)q ∈ ∆ (H)denotes the usual mixture.4.2. AxiomsWe impose the following axioms on the preference process {≽ s t}. The first three axioms areanalogous to Axioms A1-A3.19

Axiom B6 (Dynamic Consistency) For all t and s t , for all c ∈ C and h, h ′ ∈ H, if h(s)(strictly) stochastically dominates h ′ (s) with regard to ≽ s t ,s for each s ∈ S, then(c, δ[h]) ≽ s t (≻ s t) (c, δ[h ′ ]).Finally, we embed Seo’s (2009) dominance axiom to the set of one-step-ahead acts. Aone-step-ahead act is an act for which subjective uncertainty resolves just in one period. Wedefine the set of one-step-ahead acts as:H +1 = {h +1 ∈ H : h +1 (s) ∈ L, ∀s ∈ S}.Definition 6 Given h +1 ∈ H +1 and π ∈ ∆ (S), define l(h +1 , π) ∈ L by:l(h +1 , π) = ∑ s∈Sh +1 (s)π(s).Given p +1 ∈ ∆(H +1 ) and π ∈ ∆(S), define a(p +1 , π) ∈ ∆(L) by:for every Borel subset L ⊂ L.a(p +1 , π)(L) = p +1 ({h +1 ∈ H +1 : l(h +1 , π) ∈ L}) ,We take a set of one-step-ahead probability measures, P s t, as given for each history s tand impose the following Dominance axiom on this set. We allow this set to be differentfrom ∆ (S) in order to permit more flexibility in applications as discussed in Section 1.Axiom B7 (Dominance) For all t and s t , for all c ∈ C and p +1 , p ′ +1 ∈ ∆(H +1 ),(c, a(p +1 , π)) ≽ s t (c, a(p ′ +1, π)), ∀π ∈ P s t =⇒ (c, p +1 ) ≽ s t (c, p ′ +1),where P s t ⊂ ∆ (S) .To interpret this axiom, imagine that P s t is a set of probability distributions, whichcontains the ‘true’ distribution unknown to the decision maker. Given the same currentconsumption c, if the decision maker prefers the continuation two-stage lottery a(p +1 , π)induced by p +1 over another one a(p ′ +1, π) induced by p ′ +1 for each probability distributionπ ∈ P s t, then he must also prefer (c, p +1 ) over ( c, p +1) ′ .Compared to the axioms in Section 3, First-Stage Independence and Dominance arethe counterparts of the SEU on Second-Order Acts and Consistency with Preference overSecond-Order Acts. Thus, we can dispense with second-order acts.21

4.3. RepresentationThe following theorem gives our second representation result.Theorem 4 (Representation) The preference process {≽ s t} satisfies Axioms B1-B7 if andonly if there exists a family of functions ({V s t}, W, u, v) and a process of probability measures{µ s t} over P s t, such that for each s t , the function V s t : C × ∆(H) → R represents ≽ s tandhas the form: V s t(c, p) =( ( ∫ ∫ ( ∑W c, v −1 v ◦ u −1H P s ts∈S∫)))π(s) u(V s t ,s(c ′ , a ′ ))dh(s)(c ′ , a ′ ) dµ s t(π)dp(h) ,C×∆(H)(17)for (c, p) ∈ C ×∆(H), where W is continuous and strictly increasing in the second argument,u and v are continuous and strictly increasing.We also have the following uniqueness result, up to some monotonic affine transformations:Theorem 5 (Uniqueness) Let {≽ s t} satisfy Axioms B1-B7. If both ({Ṽs t}, ˜W , ũ, ṽ, {˜µ s t})and ({V s t}, W, u, v, {µ s t}) represent {≽ s t}, then there exist a strictly increasing function Φand constants A, B, D, E with A, D > 0, such that:Ṽ s t = Φ ◦ V s t, ˜W (·, ·) = Φ(W (·, Φ −1 (·))),ũ ◦ Φ = Au + B, ṽ ◦ Φ = Dv + E.As in the static model of Seo (2009), the process of second-order beliefs {µ s t} is not uniquein general. For example, when φ = v ◦ u −1 is linear, {µ s t} is indeterminate. It is unique if φis some exponential function. The existence of a solution for {V s t} to the recursive equation(17) follows a similar argument in the proof of Theorem 2 in KMM (2009a). We may applysufficient conditions in Marinacci and Montrucchio (2009) to establish uniqueness. The listbelow shows how the above model nests the existing models:1. On the subdomain C × G, the representation reduces to (11), which further reduces to(2) on C × F.2. On the subdomain C × ∆(L), we obtain a pure risk setting where the two-stage randomizationis present. In this case, each V s t coincides with the common representation:(∫ (∫) ))V (c, a) = W(c, v −1 v ◦ u −1 u(V (c ′ , a ′ ))dl(c ′ , a ′ ) da(l) , (18)LC×∆(L)where (c, a) ∈ C × ∆(L).22

3. On the subdomain C × M, we obtain a pure risk setting where only the second-stagerandomization is present. In this case, the model reduces to (12).4.4. Risk Aversion and Ambiguity AversionAs discussed before, the function W describes intertemporal substitution. Now, we discusshow ambiguity aversion is separated from risk aversion in the two-stage randomization approach.We begin by characterizing risk aversion. In doing so, we restrict attention to thesubdomain C × ∆ (L) without subjective uncertainty. In this case the utility representationtakes the form in (18). Because there is two-stage randomization, we have two risk attitudestoward the risk in the two stages (or in the first-order and the second-order).For the risk in the second stage, we remove the first-stage risk by assuming that the firststage lottery is degenerate. We then obtain the representation of recursive risk preferencegiven in (12). We can define risk aversion in the second stage in a standard way and showthat it is completely characterized by the concavity of u.Turn to risk aversion in the first stage. We define absolute risk aversion in the first stageas follows:Definition 7 The decision maker with preference {≽ s t} exhibits risk aversion in the firststage if for all s t , c ∈ C and l, l ′ ∈ L, λ ∈ [0, 1],(c, δ[λl ⊕ (1 − λ)l ′ ]) ≽ s t (c, λδ[l] + (1 − λ)δ[l ′ ]). (19)We can similarly define risk loving and risk neutrality in the first stage. In Definition7, λδ[l] + (1 − λ)δ[l ′ ] ∈ ∆ (L) represents a lottery in the first stage and δ[λl ⊕ (1 − λ)l ′ ]represents a degenerate lottery over the mixture λl ⊕(1−λ)l ′ in the second stage. Accordingto this definition, the decision maker may not be indifferent between these two lotteries, eventhough they give the same final outcome distribution. In particular, if the decision makerbelieves that the degenerate lottery is like a sure outcome and must be preferred, then hedisplays risk aversion in the first stage.Note that if we replace ≽ s t with ∼ s t in (19), we obtain a dynamic counterpart of Seo’s(2009) Reduction of Compound Lotteries axiom. Thus, according to our Definition 7, violationof the Reduction of Compound Lotteries reflects the decision maker’s attitude towardthe risk in the first stage. The following proposition characterizes this risk attitude.Proposition 3 Suppose {≽ s t} satisfies Axioms B1-B7. Then {≽ s t} exhibits risk aversionin the first-stage if and only if v ◦ u −1 is concave.23

An immediate corollary of this proposition is that, given Axioms B1-B7, the Reductionof Compound Lotteries axiom is satisfied if and only if v ◦ u −1 is a strictly increasing affinefunction. In this case, the two lotteries l and a in (18) can be reduced to a compound lotteryand hence (18) reduces to a model belonging to the class of recursive expected utility underobjective risk.Next, we consider comparative risk aversion.Definition 8 Say that {≽ i s t } is more risk averse than {≽ j s t } in the first stage if for all s t ,c ∈ C, l ∈ L and a ∈ ∆(L),(c, δ[l]) ≽ j s(c, a) =⇒ (c, δ[l]) ≽ i t st (c, a),and if this property also holds true for strict preference relations ≻ j s t and ≻ i s t .Take current consumption c as given. Suppose person j prefers a ‘sure’ outcome (withthe outcome being a lottery) to an arbitrary lottery for tomorrow. This must be due to j’saversion to risk. Facing the same choices, if person i is more risk averse than person j in thefirst stage, then person i should dislike what person j dislikes.Proposition 4 Suppose {≽ i s t } and {≽ j s t } satisfy Axioms B1-B7. Then {≽ i s t } is more riskaverse than {≽ j s t } in the first stage if and only if there exist corresponding utility representationssuch that V i | C×∆(L) = V j | C×∆(L) , W i = W j , u i = u j and v i = Ψ ◦ v j where Ψ is astrictly increasing and concave transformation.By Definition 8, persons i and j rank deterministic consumption plans in the same wayand rank lotteries in the second stage in the same way. Thus, (W i , u i ) and (W j , u j ) areordinally equivalent. Proposition 4 shows that person i is more risk averse than person j inthe first stage if and only if v i is a monotone concave transformation of v j .Now, we consider ambiguity attitude. Because ambiguity attitude deals with subjectiveuncertainty, we focus on the subdomain C × ∆ (H +1 ) in which uncertainty resolves in justone period. We define absolute ambiguity aversion as follows.Definition 9 The decision maker with {≽ s t} exhibits ambiguity aversion if for all s t , c ∈ C,h +1 , h ′ +1 ∈ H +1 , and λ ∈ [0, 1],(c, δ[λh +1 ⊕ (1 − λ)h ′ +1]) ≽ s t (c, λδ[h +1 ] + (1 − λ)δ[h ′ +1]). (20)24

We can similarly define ambiguity loving and ambiguity neutrality. Definition 9 says thatif a first-stage mixture of acts is preferred to their second-stage mixture, then the decisionmaker is ambiguity averse. The intuition for this definition is that hedging across ambiguousstates is valuable compared to randomization of acts before the realization of the states. Itis related to Gilboa and Schmeidler’s (1989) definition of ambiguity aversion, which statesthat hedging across states for two indifferent acts is valuable to an ambiguity averse decisionmaker. 17 When ≽ s t is replaced with ∼ s t in (20), then it becomes the dynamic counterpartof Seo’s Reversal of Order axiom. Thus, ambiguity attitude is associated with the violationof the Reversal of Order axiom.An example taken from Seo (2009) illustrates Definition 9. Restrict attention to a staticsetting. Consider an Ellsberg urn which contains 100 black or white balls, but the exactcomposition is unknown. The state of the world is the color of the ball. Let f be the actthat gives $100 if the chosen ball is black and nothing otherwise. Let g be the act thatgives $100 if the chosen ball is white and nothing otherwise. Let p be a lottery with 50%of winning $100. Experimental evidence reveals that most people are indifferent between fand g, but prefer p to f and p to g. The first-stage mixture 1f + 1 g is still an ambiguous2 2act. But the second-stage mixture 1f ⊕ 1 g gives an identical lottery p no matter whether the2 2chosen ball is black or white. Thus, it is intuitive that an ambiguity averse decision makerprefers 1 2 f ⊕ 1 2 g to 1 2 f + 1 2 g.As Seo (2009) and Segal (1987, 1990) point out, ambiguity attitude is associated withthe violation of the Reduction of Compound Lotteries. 18We now characterize this relationship.In his atemporal model, Seo (2009) shows that Reduction of Compound Lotteriesand Reversal of Order are equivalent under Dominance. Adapting his argument to our dynamictwo-stage compound lottery acts framework, we show below that ambiguity aversionis identical to risk aversion in the first stage.Proposition 5 Suppose {≽ s t} satisfies Axioms B1-B7. Then {≽ s t} exhibits ambiguity aversionif and only if {≽ s t} exhibits risk aversion in the first stage.An immediate implication of this proposition is that ambiguity aversion is equivalent toconcavity of v ◦u −1 . In addition, the decision maker is ambiguity neutral if and only if v ◦u −117 Given Axiom B4 (First-Stage Independence), our definition implies the following Gilboa-Schmeidler’sdefinition of ambiguity aversion: (c, δ[h +1 ]) ∼ s t (c, δ[h ′ +1]) =⇒ (c, δ[λh +1 ⊕ (1 − λ)h ′ +1]) ≽ s t (c, δ[h +1 ]), forall s t , c ∈ C, h +1 , h ′ +1 ∈ H +1 , and λ ∈ [0, 1],18 Halevy (2007) finds experimental evidence to support this view. This view is controversial becausenonreduction of compound lotteries is arguably a ‘mistake.’25

is a strictly increasing affine function. As a result, the four distributions h, π, µ s t and p canbe reduced to a compound distribution and the model reduces to recursive expected utilityunder uncertainty.Finally, we study comparative ambiguity aversion.Definition 10 Let the utility representations of {≽ i s} and {≽ j t s} share the same secondorderbelief µ s t on the same support P s t. Say that {≽ i st } is more ambiguity averse than {≽ j t s} tif for all s t , all c ∈ C, l ∈ L and h +1 ∈ H +1 ,(c, δ[l]) ≽ j s(c, δ[h t +1 ]) =⇒ (c, δ[l]) ≽ i s (c, δ[h +1]),tand if this property also holds true for strict preference relations ≻ j sand ≻ i t s. tTo interpret this definition, fix current consumption at c and consider two sure outcomesfor tomorrow with one outcome being a lottery and the other outcome being a one-stepaheadact. Suppose person j prefers the sure lottery outcome to the sure one-step-aheadact. This must be due to person j’s aversion to subjective uncertainty or ambiguity. Facingthe same choices, if person i dislikes what person j dislikes, then person i must be moreambiguity averse than person j because differences in beliefs are ruled out.The following proposition states that in the framework of two-stage randomization comparativeambiguity aversion is identical to comparative risk aversion in the first stage.Proposition 6 Suppose that {≽ i s} and {≽ j t s} satisfy Axioms B1-B7 and that their representationsshare the same second-order belief µ s t on the same support P s t for all s t . Thent{≽ i s} is more ambiguity averse than {≽ j t s} if and only if {≽ i t s} is more risk averse thant{≽ j s} in the first stage.tAn immediate corollary of this proposition is that a decision maker whose preferencessatisfy Axiom B1-B7 and have a representation with a concave function v ◦ u −1 if and onlyif he is more ambiguity averse than a decision maker whose preferences also satisfy AxiomB1-B7 and are represented by recursive expected utility. This result connects our definitionof ambiguity aversion in Definition 5 to our definition of comparative ambiguity aversion inDefinition 6. It shows that recursive expected utility is the dividing line between ambiguityloving and ambiguity aversion.What is the relationship between the notion of ambiguity aversion defined in this sectionand that in Section 3? Because the preference domain of choices is different under the twoapproaches in these two sections, ambiguity aversion reflects different natures. But the utility26

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

where R t+1 is the market return from periods t to t + 1 that satisfies:where (X t ) is the wealth process.X t+1 = R t+1 (X t − c t ) , (29)When η = γ, the homothetic recursive ambiguity model reduces to the Epstein-Zin-Weil model. In this case, the pricing kernel in (27) or (28) reduces to that in Epstein andZin (1989) and Hansen et al. (2008). Why is our generalized recursive smooth ambiguitymodel useful in explaining asset pricing puzzles? Equation (27) reveals that there are twoadjustments to the standard pricing kernel β (c t+1 /c t ) −ρ . The first adjustment is presentfor recursive expected utility of Epstein-Zin (1989). This adjustment is the second term onthe right-hand side of (27). The second adjustment is due to ambiguity aversion, which isgiven by the last term on the right-hand side of (27). This adjustment has the feature thatan ambiguity averse agent with η > γ puts a higher weight on the pricing kernel when hiscontinuation value is low in recessions. This pessimistic behavior helps explain the equitypremium puzzle and the riskfree rate puzzle and also generates time-varying equity premium.Ju and Miao (2010) study the quantitative implications of the above homothetic specificationusing the pricing kernel in (27), when z is governed by a regime switching process.They show that our model proves successful in explaining many asset pricing puzzles quantitatively.6. Related LiteratureOur paper is related to a small literature on axiomatically founded dynamic models of ambiguity.Our second-order act approach is closely related to KMM (2009a). 19Unlike thatpaper, we adopt a hierarchical Anscombe-Aumann-type domain. This domain allows us toimpose simple and intuitive axioms. More importantly, it permits a separation of intertemporalsubstitution from attitudes toward risk or uncertainty. Our utility representation allowsfor flexible parametric specifications and nests KMM (2009a) model and some other popularmodels in the literature as special cases such as the recursive expected utility model (Krepsand Porteus (1978) and Epstein and Zin (1989)) and the multiplier preference model withhidden states (Hansen (2007) and Hansen and Sargent (2007)). In addition, this representa-19 Hanany and Klibanoff (2009) also provide a dynamic extension of the KMM (2005) model. Theirapproach is non-recursive in that they first define preference over consumption plans and then determineconditional preferences by updating beliefs.29

tion permits an information structure with hidden states that could be unknown parametersas in KMM (2009a) or Markov processes.Our preference domain in the second-order act approach is built on Hayashi (2005) whofirst constructs the domain of compound lottery acts G. He chooses ∆ (C × G) as the preferencedomain, while we adopt C × G as the primary domain of preference {≽ s t} . It provesthat our domain choice is more convenient in our setting. Hayashi (2005) embeds Gilboa andSchmeidler’s (1989) static multiple-priors model in a dynamic environment and establishes ageneralized recursive multiple-priors model. His model permits a separation of intertemporalsubstitution from attitudes toward risk or uncertainty. It generalizes the recursive multiplepriorsmodel of Epstein and Wang (1994) and Epstein and Schneider (2003). Epstein andSchneider (2003) axiomatize the recursive multiple-priors model and prove that dynamicconsistency leads to rectangular sets of priors and to prior-by-prior Bayesian updating asthe updating rule for such sets of priors. Wang (2003) also axiomatizes this model and someupdating rules for preferences that are not necessarily in the expected utility class. Hananyand Klibanoff (2007) follow a non-recursive approach and extend the Epstein and Schneidermodel by allowing updating of the set of priors to violate conseqentialism. As KMM (2005)point out, one limitation of the multiple-priors model is that there is no separation betweenambiguity and ambiguity attitude. The set of priors may reflect the decision maker’s perceivedambiguity or his attitude toward ambiguity. This confounding makes comparativestatic analysis hard to interpret.Our second axiomatization using the two-stage randomization approach extends Seo’s(2009) static model to a dynamic setting. To the best of our knowledge, our paper providesthe first dynamic extension of Seo’s static model. As a by-product contribution, we constructa domain of two-stage compound lottery acts H, which contains G and allows for randomizationboth before and after the realization of the state of the world. We then dispensewith second-order acts and the associated preferences over these acts. We define a singlepreference relation {≽ s t} over C × ∆ (H) .Our characterization of ambiguity attitude in the two axiomatic approaches is based onthe foundation of Ghirardato and Marinacci (2002). Ambiguity aversion reflects somewhatdifferent natures in the two approaches because of different choice domains, though thecharacterization in terms of concavity of v◦u −1 is identical. In the second-order act approach,ambiguity aversion is an aversion to the subjective uncertainty about ex ante evaluationsof one-step-ahead acts. In the two-stage randomization approach, ambiguity aversion isassociated with the violation of reduction of compound lotteries, as also pointed out by30

Segal (1987, 1990) and Seo (2009). Segal uses the anticipated utility model and considersobjective lotteries, while the Ellsberg paradox is often viewed as a phenomenon associatedwith subjective uncertainty. Seo does not provide a formal definition of ambiguity aversionand characterizations of ambiguity attitude. We provide such an analysis and characterizethe link between ambiguity aversion and reduction of compound lotteries. Our result thatambiguity aversion is identical to risk aversion in the first stage is similar to Theorem 5in Ergin and Gul (2009) who refer to risk aversion in the first stage as second-order riskaversion.Maccheroni, Marinacci, and Rustichini (2006b) provide a dynamic extension of the staticvariational model of ambiguity developed by Maccheroni, Marinacci, and Rustichini (2006a).The static variational model includes multiple-priors model and the multiplier preferencemodel of Hansen and Sargent (2001) as special cases. The dynamic extension does notseparate intertemporal substitution from attitude toward risk or ambiguity. Variationalpreferences are also subject to the limitation concerning the separation of ambiguity fromambiguity attitude.Our model is also related to the literature on recursive utility under risk or uncertainty(Chew and Epstein (1991), Epstein and Zin (1989), Klibanoff and Ozdenoren (2007), Krepsand Porteus (1978), and Skiadas (1998)). This literature does not deal with ambiguity. Inthe framework of Klibanoff and Ozdenoren (2007) or Skiadas (1998), preferences depend onthe filtration. Unlike their framework, we take the filtration as given and, thus, cannot makecomparisons of representations across filtrations.Recursive utility models allow for preferences for the timing of the temporal resolutionof uncertainty. As is well known in the literature on recursive expected utility preferences, anonlinear time aggregator is needed to permit nonindifference to the timing of the temporalresolution of uncertainty. Strzalecki (2009) shows that even without a nonlinear aggregator,or with standard discounting, most dynamic models of ambiguity aversion (including themodels discussed above) result in timing nonindifference. In particular, decision makerswith such preferences prefer earlier resolution of uncertainty. The only model of ambiguityaversion that exhibits indifference to timing is the multiple-priors utility model. Our paperdoes not study this issue. Presumably, Strzalecki’s analysis can be applied to our setting.Finally, like most papers in the literature on dynamic models, we follow a recursiveapproach and maintain dynamic consistency. This approach is normative appealing andcomputationally simple in applications because the usual dynamic programming method canbe applied. This approach typically shares the drawback of lacking a ‘reduction’ or ‘closure’31

property discussed in KMM (2009a). 20 This means that our recursive model (2) over adaptedconsumption processes does not have the reduced static KMM smooth ambiguity functionalform. Siniscalchi (2010) follows a different approach to formulating dynamic models ofambiguity. He takes an individual’s preferences over decision trees, rather than acts, asprimitive. His approach allows for dynamic inconsistency. He formalizes sophistication asan assumption about the way individuals resolve conflicts between preferences at differentdecision points. It remains to see whether dynamic smooth ambiguity preferences can beformulated in his framework.20 Epstein and Schneider (2003) and Maccheroni et al. (2006b) are exceptions.32

A Appendix: Proof of Theorems 1 and 2We prove the sufficiency of the axioms. The proof of necessity is routine.A1. Representation of Risk PreferenceWhen {≽ s t} is restricted to the domain C × M, Axiom A3 (History Independence of RiskPreference) implies that {≽ s t} induces a single preference relation ≽ defined on C × M.By Axiom A1 (Order) and Debreu’s (1954) theorem, there is a continuous representationV : C × M → R of ≽ . We fix such a representation.Fix some arbitrary ĉ ∈ C throughout the proof. By Axiom A2 (Current ConsumptionSeparability), V (c, ·) and V (ĉ, ·) represent the same ranking over M, hence V has the form:for some functionV (c, m) = Ŵ (c, V (ĉ, m)), (30)Ŵ which is strictly increasing in the second argument. By Axiom A4(Independence for Timeless Lotteries), V (ĉ, m) is ordinally equivalent to an expected utilityrepresentation on C × M. Thus, we have the form:(∫)V (ĉ, m) = ζ û(c ′ , m ′ )dm(c ′ , m ′ ) , (31)C×Mwhere û is a v-NM index and ζ is a monotone transformation.Lemma 1 Given Axioms A1 and A3, Axiom A5 implies that:for any c ∈ C and m, m ′ ∈ M.(c, δ[(c ′ , m ′ )]) ≽ (c, δ[(c ′′ , m ′′ )]) ⇐⇒ (c ′ , m ′ ) ≽ (c ′′ , m ′′ ) ,Proof. We restrict attention to the subdomain C × M. By Axiom A3 (History Independenceof Risk Preference), we can replace {≽ s t} with ≽ in Axiom A5 (Dynamic Consistency).Suppose (c ′ , m ′ ) ≽ (c ′′ , m ′′ ) . Then (c ′′ , m ′′ ) ≽ (c 0 , m 0 ) =⇒ (c ′ , m ′ ) ≽ (c 0 , m 0 )for any (c 0 , m 0 ) ∈ C × M. Thus, δ[(c ′ , m ′ )] stochastically dominates δ[(c ′′ , m ′′ )]. By AxiomA5, (c, δ[(c ′ , m ′ )]) ≽ (c, δ[(c ′′ , m ′′ )]). Suppose (c, δ[(c ′ , m ′ )]) ≽ (c, δ[(c ′′ , m ′′ )]), but (c ′ , m ′ ) ≺(c ′′ , m ′′ ) . By continuity of ≽ from Axiom A1, there exists some (c 0 , m 0 ) ∈ C × M such that(c ′ , m ′ ) ≺ (c 0 , m 0 ) ≼ (c ′′ , m ′′ ) . Thus, δ[(c ′′ , m ′′ )] strictly stochastically dominates δ[(c ′ , m ′ )].By Axiom A5, (c, δ[(c ′ , m ′ )]) ≺ (c, δ[(c ′′ , m ′′ )]), which is a contradiction.Now, we deduce:û(c ′ , m ′ ) ≥ û(c ′′ , m ′′ )33

⇐⇒ (ĉ, δ[(c ′ , m ′ )]) ≽ (ĉ, δ[(c ′′ , m ′′ )]) by (31), (32)⇐⇒ (c ′ , m ′ ) ≽ (c ′′ , m ′′ ) by Lemma 1,⇐⇒ V (c ′ , m ′ ) ≥ V (c ′′ , m ′′ ).Hence, û and V are ordinally equivalent representations of ≽ on C × M, implying that thereis a monotone transformation u such that û = u ◦ V . Plugging this equation into (31) yields:(∫)V (ĉ, m) = ζ u(V (c ′ , m ′ ))dm(c ′ , m ′ ) . (33)C×MDefine W by W (c, x) = Ŵ (c, ζ(u(x))), which is strictly increasing in the second argument.Then,( (∫))V (c, m) = Ŵ c, ζ u(V (c ′ , m ′ ))dm(c ′ , m ′ )C×M( ( ∫))= Ŵ c, ζ u ◦ u −1 ◦ u(V (c ′ , m ′ ))dm(c ′ , m ′ )C×M(∫))= W(c, u −1 u(V (c ′ , m ′ ))dm(c ′ , m ′ ) .C×MA2. Extension to the Whole DomainBy an argument similar to the proof of Lemmas 8-9 in Hayashi (2005), we can use continuityof ≽ s t from Axiom A1, Dynamic Consistency Axiom A5, and compactness of C to show thatfor each (c, g) ∈ C×G, there exists a risk equivalent (c, m) ∈ C×M such that (c, g) ∼ s t (c, m)for each s t . Thus, for each s t , define V s t : C × G → R by:V s t(c, g) = V (c, m), (34)where m is such that (c, g) ∼ s t (c, m). Using this definition and (30), we obtain:V s t(c, g) = V (c, m) = Ŵ (c, V (ĉ, m)) = Ŵ (c, V st(ĉ, g)). (35)From Axioms A6 (SEU Representation of Preference over Second-order Acts) and A7(Consistency with the Preference over Second-order Acts), we obtain:( ∫ ( ) )∑V s t(ĉ, g +1 ) = ξ s t ψ g +1 (s)π(s) dµ s t(π) , (36)P s t s∈Swhere ξ s t is a monotone transformation. By restricting attention to M, we can use AxiomA3 (History Independence of Risk Preference) to set ξ s t = ξ for all s t .34

Define v = ψ ◦ ū −1 ◦ u, where ū is defined in (13). 21Using Axiom A6, we immediatelyobtain part (iii) of the theorem. Plugging this definition of v in (36) yields:( ∫ ( ) )∑V s t(ĉ, g +1 ) = ξ v ◦ u −1 ◦ ū g +1 (s)π(s) dµ s t(π)P s ts∈S( ∫ ( ∑∫) )= ξ v ◦ u −1 π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π) ,P s ts∈SC×Mwhere the second equality follows from (13).When restricting V s t to the domain M, we obtain:(∫)V s t(ĉ, m) = ξ ◦ v ◦ u −1 u(V (c ′ , m ′ ))dm(c ′ , m ′ )C×M(∫)= V (ĉ, m) = ζ u(V (c ′ , m ′ ))dm(c ′ , m ′ ) ,C×Mfor all m ∈ M, where the last equality follows from (33). Therefore, we have ξ ◦ v ◦ u −1 = ζ,implying u −1 ◦ ζ −1 ◦ ξ = v −1 .Define:y ≡ V s t(ĉ, g +1 ) = ξ( ∫P s tUsing equation (35), we obtain:v ◦ u −1 ( ∑s∈S∫) )π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π) .C×MV s t(c, g +1 ) = Ŵ (c, y) = Ŵ ( c, ζ ◦ u(u −1 ◦ ζ −1 (y)) ) = W ( c, u −1 ◦ ζ −1 (y) )(( ∫ ( ∑∫) ))= W c, u −1 ◦ ζ −1 ◦ ξ v ◦ u −1 π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π)P s ts∈SC×M( ( ∫ ( ∑∫) ))= W c, v −1 v ◦ u −1 π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π) ,P s ts∈SC×Mwhere the third equality follows from the definition of W in Appendix A1.For any g ∈ G, for each s ∈ S, and each (c ′ , g ′ ) in the support of g(s) ∈ ∆(C × G),there exists a risk equivalent (c ′ , m ′ ) ∈ C × M such that (c ′ , m ′ ) ∼ s t ,s (c ′ , g ′ ). Let g +1 bea one-step-ahead act such that g +1 (s)(L ′ ) = g(s)(L) holds for all pairs L ⊂ C × G andL ′ ⊂ C × M where L ′ consists of all risk equivalents (c ′ , m ′ ) of corresponding elements(c ′ , g ′ ) in L. By construction, g +1 (s) and g (s) are stochastically equivalent. By Axiom A521 Note that ū is increasing on M when M is ordered by first-order stochastic dominance. So its inverseexists.35

(Dynamic Consistency), (c, g) ∼ s t (c, g +1 ). Therefore,V s t(c, g) = V s t(c, g +1 )(( ∫ ( ∑= W c, v −1 v ◦ u −1P s ts∈S( ( ∫ ( ∑∫= W c, v −1 ◦ v ◦ u −1 π(s)P s ts∈S∫) ))π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π)C×M) ))C×Gu(V s t ,s(c ′ , g ′ ))dg(s)(c ′ , g ′ )dµ s t(π)where we have used the fact that g (s) and g +1 (s) are stochastically equivalent to derive thesecond equality.A3. 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 ∞ , each Ṽs tcoincides with the commonfunction Ṽ and each V st coincides with the common function V . Since Ṽ and V are ordinallyequivalent over C ∞ , there is a monotone transformation Φ such that:By (34), we have Ṽs t = Φ ◦ V s t.SinceṼ (y) = Φ ◦ V (y), for all y ∈ C ∞ .˜W (c, Ṽ (y)) = Ṽ (c, y) = Φ(V (c, y)) = Φ(W (c, V (y))) = Φ(W (c, Φ−1 (Ṽ (y)))),,we deduce that ˜W (c, ·) = Φ(W (c, Φ −1 (·))).On M, ∫ C×M u(V (c′ , m ′ ))dm(c ′ , m ′ ) and ∫ C×M ũ(Ṽ (c′ , m ′ ))dm(c ′ , m ′ ) are equivalent mixturelinearrepresentations of the risk preference conditional on the fixed current consumption ĉ.Therefore, there exist constants A, B with A > 0 such that:)ũ(Ṽ (c ′ , m ′ ) = Au(V (c ′ , m ′ )) + B, for all (c ′ , m ′ ) ∈ M.Since Ṽ = Φ ◦ V , we obtain ũ ◦ Φ = Au + B.By construction from Appendix A2, ṽ ◦ ũ −1 ◦ ˜ū = ψ = v ◦ u −1 ◦ ū. By equation (13), we36

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

The domain to be constructed is a subset of H ∗ , which consists of coherent acts. Definea mapping π 0 : C × ∆(H 0 ) → C by:π 0 (c, p 0 ) = c,for each (c, p 0 ) ∈ C × ∆(H 0 ). Define a mapping ρ 0 : H 1 → H 0 by:ρ 0 (h 1 )(s)[B 0 ] = h 1 (s)[π −10 (B 0 )],for each h 1 ∈ H 1 , s ∈ S and B 0 ∈ B(C). Define a mapping ˜ρ 0 : ∆(H 1 ) → ∆(H 0 ) by:for each p 1 ∈ ∆(H 1 ) and H 0 ∈ B(H 0 ).˜ρ 0 (p 1 )[H 0 ] = p 1 [ρ −10 (H 0 )],Similarly, define π 1 : C × ∆(H 1 ) → C × ∆(H 0 ) by:π 1 (c, p 1 ) = (c, ˜ρ 0 (p 1 )),for each (c, p 1 ) ∈ C × ∆(H 1 ), ρ 1 : H 2 → H 1 by:ρ 1 (h 2 )(s)[B 1 ] = h 2 (s)[π −11 (B 1 )],for each h 2 ∈ H 2 , s ∈ Ω and B 1 ∈ B(C × ∆(H 0 )), and ˜ρ 1 : ∆(H 2 ) → ∆(H 1 ) by:for each p 2 ∈ ∆(H 2 ) and H 1 ∈ B(H 1 ).˜ρ 1 (p 2 )[H 1 ] = p 2 [ρ −11 (H 1 )],Inductively, given π t−1 : C × ∆(H t−1 ) → C × ∆(H t−2 ), ρ t−1 : H t → H t−1 and ˜ρ t−1 :∆(H t ) → ∆(H t−1 ), define π t : C × ∆(H t ) → C × ∆(H t−1 ) by:π t (c, p t ) = (c, ˜ρ t−1 (p t )),for each (c, p t ) ∈ C × ∆(H t ), ρ t : H t+1 → H t by:ρ t (h t+1 )(s)[B t ] = h t+1 (s)[π −1t (B t )],for each h t+1 ∈ H t+1 , s ∈ S and B t ∈ B(C × ∆(H t−1 )), and ˜ρ t : ∆(H t+1 ) → ∆(H t ) by:for each p t+1 ∈ ∆(H t+1 ) and H t ∈ B(H t ).˜ρ t (p t+1 )[H t ] = p t+1 [ρ −1t (H t )],38

DefineH = {h = (h 0 , h 1 , h 2 , · · · ) ∈ H ∗ : h t = ρ t (h t+1 ), t ≥ 0} .For each s ∈ S, the sequence (h 0 (s), h 1 (s), h 2 (s), · · · ) ∈ ∏ ∞t=0 ∆(C × ∆(H t−1)) is viewed as asequence of constant acts sinceand so on.h 0 (s) ∈ ∆(C) ⊂ H 0 ,h 1 (s) ∈ ∆(C × ∆(H 0 )) ⊂ H 1 ,.h t (s) ∈ ∆(C × ∆(H t−1 )) ⊂ H t ,The lemmas below verify that such constant acts are also coherent. They are immediatefrom the definition of H.Lemma 2 For every h ∈ H and s ∈ S, the sequence (h 0 (s), h 1 (s), h 2 (s), · · · ) ∈ ∏ ∞t=0 ∆(C ×∆(H t−1 )) satisfies h t (s) = ρ t (h t+1 (s)).Lemma 3 For every t ≥ 0, h t ∈ H t and h t+1 ∈ H t+1 , if h t (s) = ρ t (h t+1 (s)) for every s ∈ S,then h t = ρ t (h t+1 ).Let{}∞∏Q = (q t ) ∈ ∆(H t ) : q t = ˜ρ t (q t+1 ), ∀t ≥ 0 ,{A = (a t ) ∈t=0}∞∏∆(C × ∆(H t−1 )) : a t = ρ t (a t+1 ), ∀t ≥ 0 .t=0Lemma 4 We have the homeomorphic relation:A ≃ ∆(C × Q).Proof. Given (a t ) ∈ A ⊂ ∏ ∞t=0 ∆(C × ∆(H t−1)), by the Kolmogorov extension theoremthere exists a unique a ∈ (C × ∏ ∞t=0 ∆(H t−1)) such that:mrg C×∆(Ht−1 )a = a t ,for each t ≥ 0. Define a mapping ξ : A → ∆(C × Q) by ξ ((a t )) = aWe need to show a ∈ ∆(C × Q). For each t ≥ 0, letQ t = {(q t , q t+1 ) ∈ ∆(H t ) × ∆(H t+1 ) : q t = ˜ρ t (q t+1 )} ×39∏τ≠t,t+1∆(H τ ).

We derive that:a(C × Q t ) = mrg C×∆(Ht)×∆(H t+1 )a(C × {(q t , q t+1 ) ∈ ∆(H t ) × ∆(H t+1 ) : q t = ˜ρ t (q t+1 )})Therefore,= mrg C×∆(Ht)a(C × ˜ρ t (∆(H t+1 )))= a t+1 (C × ˜ρ t (∆(H t+1 )))= a t+1 (π t+1 (C × ∆(H t+1 )))= ρ t+1 (a t+2 )(π t+1 (C × ∆(H t+1 )))= a t+2 (π −1t+1(π t+1 (C × ∆(H t+1 ))))= a t+2 (C × ∆(H t+1 )) = 1.( ∞) (⋂T)a(C × Q) = a (C × Q t ) = lim a ⋂(C × Q t ) = 1.T →∞t=0ξ is one to one: It follows from the uniqueness of Kolmogorov extension theorem.ξ is onto: For every a ∈ ∆(C × Q), the inverse is given by (a t ) ∈ ∏ ∞t=0 ∆(C × ∆(H t−1))such that:a t = mrg C×∆(Ht−1 )a,for each t ≥ 0. To show (a t ) ∈ A, take any B t ∈ B(C × ∆(H t−1 )). We deduce the following:(a t (B t ) = a B t × ∏ )∆(H t )Sinceτ≠t−1t=0≥ a ({(c, (q τ )) ∈ C × Q : (c, q t ) ∈ B t })= a ({(c, (q τ )) ∈ C × Q : (c, ˜ρ t (q t+1 )) ∈ B t })= a ( {(c, (q τ )) ∈ C × Q : (c, q t+1 ) ∈ πt −1 (B t )} )= a t+1 (π −1t (B t ))= ρ t (a t+1 )(B t ).1 = a t (C×∆(H t−1 )) = a t (B t )+a t (B c t ) ≥ ρ t (a t+1 )(B t )+ρ t (a t+1 )(B c t ) = ρ t (a t+1 )(C×∆(H t−1 )) = 1,we obtain:a t (B t ) = ρ t (a t+1 )(B t ).ξ and ξ −1 are continuous: Immediate from the nature of the product topology.40

Lemma 5 We have the homeomorphic relation:H ≃ A S .Proof. Define ξ : H → A S byξ(h)(s) = (h 0 (s), h 1 (s), h 2 (s), · · · ).It follows from Lemma 2 that ξ(h) ∈ A.ξ is one to one: Suppose ξ(h) = ξ(h ′ ). By definition of ξ, we have (h 0 (s), h 1 (s), h 2 (s), · · · ) =(h ′ 0(s), h ′ 1(s), h ′ 2(s), · · · ) for all s ∈ S, which implies h = h ′ .ξ is onto: Take any ˜h ∈ A S . By definition,˜h(s) = (˜h0 (s), ˜h 1 (s), ˜h 2 (s), · · · ) ∈∞∏∆(C × ∆(H t−1 )),t=0for each s ∈ S. Then ξ −1 (˜h) = (h 0 , h 1 , h 2 , · · · ) ∈ H ∗ satisfies h t (s) = ˜h t (s) for each t and s.By Lemmas 2-3, the sequence (h 0 , h 1 , h 2 , · · · ) is coherent and hence ξ −1 (˜h) ∈ H.ξ and ξ −1 are continuous: Immediate from the nature of the product topology.LetP ∗ ={(p t ) ∈(∞∏ t∏)∆ H ττ=0t=0: mrg ∏ tτ=0 H τ p t+1 = p t}.Lemma 6 For any (p t ) ∈ P ∗ , there exists a unique p ∈ ∆(H ∗ ) such thatmrg ∏ tτ=0 H τ p = p t.Moreover, there exists a homeomorphism χ : P ∗ → ∆(H ∗ ).Proof. It follows from Lemma 1 in Brandenberger and Dekel (1993).LetH t =for each t ≥ 0, and{(h 0 , · · · , h t ) ∈}t∏H τ : h τ = ρ τ (h τ+1 ), τ = 0, · · · , t − 1 ,τ=0P = { (p t ) ∈ P ∗ : p t (H t ) = 1, t ≥ 0 } .Lemma 7 χ(P ) = ∆(H). As a result, P ≃ ∆(H) holds through χ.41

Proof. ⊂ part: Let p = χ((p t )) for some (p t ) ∈ P . Let∞∏Γ t = H t × H τ ,τ=t+1for each t ≥ 0. Then ,we have H ⊂ Γ t ⊂ H ∗ for each t ≥ 0, (Γ t ) is decreasing, and⋂t≥0 Γ t = H.Since p is the Kolmogorov extension of (p t ), we havep(Γ t ) = p t (H t ) = 1,for every t ≥ 0. Thus, p(H) = p( ⋂ t≥0 Γ t) = lim p(Γ t ) = 1.⊃ part: Pick any p ∈ ∆(H), which satisfies p(H) = 1. Let (p t ) be the sequence of marginalsdefined by p t = mrg ∏ tτ=0 Hτ p for each t ≥ 0. Then, p t(H t ) = p(Γ t ) ≥ 1, where the secondinequality follows from Γ t ⊃ H. Since p t is a probability measure, we have p t (H t ) = 1. Sincep is the Kolmogorov extension of (p t ), we have p = χ((p t )).Lemma 8 For every (q t ) ∈ Q, there exists a unique (p t ) ∈ P such that:Moreover, Q and P are homeomorphic.mrg Ht p t = q t .Proof. Define a sequence of mappings (ξ t ), ξ t : H t → ∏ tτ=0 H τ for each t ≥ 0, by:ξ t (h t ) = (ĥ0, · · · , ĥt),where ĥt = h t , ĥτ = ρ τ (ĥτ+1) for τ = 0, 1, · · · , t − 1.By construction, each (ξ t ) is a one-to-one mapping and ξ t (H t ) = H t . Therefore,we can definethe sequence of inverse mappings (ξt−1 ), ξt−1 : H t → H t given by:ξ −1t (h 0 , · · · , h t ) = h t ,which is a projection mapping that is continuous.For (q t ) ∈ Q, define the corresponding sequence (p t ) ∈ P by:p t (E t ) = q t (ξ −1t (E t )),for each E t ∈ B( ∏ tτ=0 H τ) and t ≥ 0. We can see that (p t ) ∈ P since p t (H t ) = q t (ξ −1t (H t )) =q t (H t ) = 1. By construction, mrg Ht p t = q t for each t ≥ 0.Now, Theorem 3 follows from the fact that H ≃ A S , A ≃ ∆(C × Q), Q ≃ P, andP ≃ ∆(H).42

Finite-step-ahead acts and densenessFinally, we define finite-step-ahead acts and show that the union of all the sets of finite-stepaheadacts is dense. LetH +1 = {h +1 ∈ (∆(C × ∆(H))) S : ∀s ∈ S, h +1 (s) ∈ ∆(C × ∆(L))}.Since H ≃ (∆(C ×∆(H))) S , we can embed H +1 into H, where the range of H +1 is embeddedinto L since L ≃ ∆(C × ∆(L)). Inductively, defineH +τ = {h +τ ∈ (∆(C × ∆(H))) S : ∀s ∈ S, h +τ (s) ∈ ∆(C × ∆(H +(τ−1) ))}.Similarly, we can embed H +τ into H. We call ⋃ τ≥1 H +τ the domain of finite-step-ahead acts.Lemma 9 The domain of finite-step-ahead acts ⋃ τ≥1 H +τ⋃τ≥1 ∆(C × ∆(H +τ)) is a dense subset of ∆(C × ∆(H)).is a dense subset of H. Also,This result is analogous to Proposition 1 in Hayashi (2005) and its proof is omitted. Itis useful to establish the existence of a risk equivalent as in Lemma 9 of Hayashi (2005). Wehave implicitly applied a similar result in Appendix A.43

C Appendix: Proof of Theorems 4 and 5We prove the sufficiency of the axioms. The proof of necessity is routine.C1. Representation of Risk PreferenceWhen {≽ s t} is restricted to the domain C × ∆(L), Axiom B3 (History Independence of RiskPreference) implies that {≽ s t} induces a single preference relation ≽ defined on C × ∆(L).By Axiom B1 (Order) and Debreu’s (1954) theorem, there is a continuous representationV : C × ∆(L) → R of ≽ . We fix such a representation.By Axiom B2 (Current Consumption Separability), V (c, ·) and V (ĉ, ·) represent the sameranking over ∆(L), hence V has the form:for some functionV (c, a) = Ŵ (c, V (ĉ, a)), ∀(c, a) ∈ C × ∆(L), (37)Ŵ which is strictly increasing in the second argument. Because of B4(First-stage Independence), V (ĉ, a) has the form(∫ )V (ĉ, a) = ζ U(l)da(l) , (38)Lwhere ζ is a strictly increasing function and U is a v-NM index.Because of Axiom B5 (Second-stage Independence), U has the form:(∫)U(l) = φ û(c ′ , a ′ )dl(c ′ , a ′ ) , (39)C×∆(L)where φ is a strictly increasing function and û is a v-NM index.By Axiom B6 (Dynamic Consistency) and a similar argument as in Appendix A1, û andV are ordinally equivalent. Hence, we deduce that:where u is a strictly increasing function.û(c ′ , a ′ ) = u(V (c ′ , a ′ )), (40)Plugging equations (38), (39), (40) into equation (37) yields:( (∫ (∫V (c, a) = Ŵ c, ζ φ u(V (c ′ , a ′ ))dl(c ′ , a ′ )L C×∆(L))))da(l) . (41)Now define W byW (c, x) = Ŵ (c, ζ ◦ φ ◦ u(x)),44

which is strictly increasing in the second argument. Then we haveand hence,(∫V (c, a) = W(c, u −1 ◦ φ −1Ŵ (c, ζ(z)) = W (c, u −1 ◦ φ −1 (z)), (42)φL(∫Let v = φ ◦ u. We obtain representation (18).C2. Extension to the Whole Domainu(V (c ′ , a ′ ))dl(c ′ , a ′ )C×∆(L))))da(l) .Define V s t : C × ∆(H) by:V s t(c, p) = V (c, a), (43)for each (c, p) ∈ C × ∆(H), where a ∈ ∆(L) is such that (c, p) ∼ s t(c, a). The existenceof such a risk equivalent a follows from Lemma 9, Dynamic Consistency, compactness of C,and continuity of ≽ s tderive:(see Lemma 9 in Hayashi (2005)). Using definition (43) and (37), weV s t(c, p) = V (c, a) = Ŵ (c, V (ĉ, m)) = Ŵ (c, V st(ĉ, p)). (44)Our axioms B1, B4, B5 and B7 when restricted to ∆(H +1 ) satisfies the conditions inTheorem 4.2 in Seo (2009).By this theorem, V s t(ĉ, ·) restricted to ∆(H +1 ) is ordinallyequivalent to a second-order subjective expected utility representation, and hence has theform:and∫U s t(h +1 ) =(∫)V s t(ĉ, p +1 ) = ζ s t U s t(h +1 )dp +1 (h +1 ) , (45)H +1P s tφ s t( ∑s∈S∫)π(s) û s t(c ′ , a ′ )dh +1 (s)(c ′ , a ′ ) dµ s t(π), (46)C×∆(L)where ζ s t and φ s t are strictly increasing functions and û s t is a v-NM index. By Axiom B6(Dynamic Consistency) and a similar argument in Appendix A1, û s tequivalent over C × ∆(L). Thus, there is a monotone transformation u s tfor every (c ′ , a ′ ) ∈ C × ∆(L).and V are ordinallysuch that:û s t(c ′ , a ′ ) = u s t(V (c ′ , a ′ )), (47)Equations (38) and (43) imply that on ∆(L),(∫ )(∫)ζ U(l)da(l) = V (ĉ, a) = V s t(ĉ, a) = ζ s t U s t(l)da(l) ,LL45

which in turn implies that on L,Hence, we deduce that:which implies that:ζ (U(l)) = V (ĉ, δ[l]) = V s t(ĉ, δ[l]) = ζ s t (U s t(l)) .U s t = ζ −1s t ◦ ζ ◦ U,(∫ ) (∫)ζ U(l)da(l) = ζ s t ζ −1s◦ ζ ◦ U(l)da(l) .tLLBy the additivity of integral formula, we haveζ −1s t◦ ζ (αx + (1 − α) y) = αζ −1s t◦ ζ(x) + (1 − α) ζ −1s t◦ ζ(y)for all x, y in the range of U and all α ∈ [0, 1]. Therefore ζ s t and ζ are identical up to positiveaffine transformations. Thus without loss of generality we can take ζ s t = ζ and U s t = U forall s t .Equations (39), (40), (46), and (47) imply that on L,(∫)φ u ◦ V (c ′ , a ′ )dl(c ′ , a ′ ) = U(l) = U s t(l)C×∆(L)(∫)= φ s t u s t ◦ V (c ′ , a ′ )dl(c ′ , a ′ ) ,C×∆(L)which in turn implies that:φ ◦ u ◦ V (c ′ , a ′ ) = U(δ[c ′ , a ′ ]) = U s t(δ[c ′ , a ′ ]) = φ s t ◦ u s t ◦ V (c ′ , a ′ ).Hence, we have φ ◦ u = φ s t ◦ u s t, which implies that:(∫) (∫)φ u ◦ V (c ′ , a ′ )dl(c ′ , a ′ ) = φ s t φ −1s◦ φ ◦ u ◦ V (c ′ , a ′ )dl(c ′ , a ′ ) .tC×∆(L)C×∆(L)By the same reasoning as above, φ s t and φ are identical up to positive affine transformations.Therefore without loss of generality we can set φ s t = φ and u s t = u for all s t .Now, plugging equations (45), (46), and (47) into (44), we obtain that, on C × ∆(H +1 ),= Ŵ= WV s t(c, p +1 )( ( ∫ ∫H +1c, ζP s tφ( ∑s∈S(c, u −1 ◦ φ −1 ( ∫H +1∫P s t∫π(s)φ( ∑s∈SC×∆(L)∫π(s)u(V (c ′ , a ′ ))dh +1 (s)(c ′ , a ′ )46C×∆(L))u(V (c ′ , a ′ ))dh +1 (s)(c ′ , a ′ )dµ s t(π)dp +1 (h +1 ))))dµ s t(π)dp +1 (h +1 ))),

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

D Appendix: Proofs for Section 4.4Proof of Proposition 3: If v ◦ u −1 is concave, it is straightforward to check that {≽ s t}is risk averse in the first stage. We now prove the reverse direction. Pick any l 1 , l 2 ∈ L andλ ∈ [0, 1]. By Theorem 3, we havewhereV (c, λl 1 + (1 − λ)l 2 ) = W ( c, v −1 ( λv ◦ u −1 (V ∗ (l 1 )) + (1 − λ)v ◦ u −1 (V ∗ (l 2 )) )) ,∫V ∗ (l) =C×∆(L)u(V (c ′ , a ′ ))dl(c ′ , a ′ )is a mixture linear function on L. Also we have(∫))V (c, δ[λl 1 ⊕ (1 − λ)l 2 ]) = W(c, v −1 v ◦ u −1 (λV ∗ (l 1 ) + (1 − λ)V ∗ (l 2 )) .LFrom the definition of risk aversion in the first stage, we have (c, δ[λl 1 ⊕ (1 − λ)l 2 ]) ≽ s t(c, λl 1 + (1 − λ)l 2 ). Thus,λv ◦ u −1 (V ∗ (l 1 )) + (1 − λ)v ◦ u −1 (V ∗ (l 2 )) ≤ v ◦ u −1 (λV ∗ (l 1 ) + (1 − λ)V ∗ (l 2 )).We may vary l 1 and l 2 to cover the whole domain of v ◦ u −1 . The above inequality impliesthat v ◦ u −1 is concave.Proof of Proposition 4: Suppose {≽ i s} is more risk averse than {≽ j t s} in the first stage.tBy definition they rank deterministic consumption streams in the same way and rank lotteriesin the second stage in the same way. So there exist representations such that V i = V j ,W i = W j and u i = u j .Since v i (V i (·)) and v j (V j (·)) are ordinally equivalent over C ∞ , there is a monotonetransformation Ψ such that v i = Ψ ◦ v j . The rest is to show that Ψ is concave. Let y, y ′ , y ′′ ∈C ∞ be such that (c, δ[δ[y]]) ∼ j s(c, λδ[δ[y ′ ]]+(1−λ)δ[δ[y ′′ ]]). This is possible due to continuitytof preference ordering. Thus, we have v j (V j (y)) = λv j (V j (y ′ )) + (1 − λ)v j (V j (y ′′ )). Since≽ i sis more risk averse than ≽ j t sin the first stage, we have (c, δ[δ[y]]) ≽ i t s(c, λδ[δ[y ′ ]] + (1 −tλ)δ[δ[y ′′ ]]), which implies v i (V i (y)) ≥ λv i (V i (y ′ )) + (1 − λ)v i (V i (y ′′ )).Since v i (V i (·)) = v i ((V j (·))) = Ψ(v j (V j (·))), we obtainΨ(u j (V j (y))) ≥ λΨ(u j (V j (y ′ ))) + (1 − λ)Ψ(u j (V j (y ′′ ))).One can choose y, y ′ , y ′′ so as to cover the whole range of u j ◦ V j . Thus, Ψ is concave. Theproof of the other direction of the proposition is routine.49

Proof of Proposition 5: When restricting to the subdomain C × ∆(L), we immediatelydeduce that ambiguity aversion implies risk aversion in the first stage. Now, we consider thereverse statement. Given h +1 , h ′ +1 ∈ H +1 , we define a ∈ ∆ (L) as in Definition 6. We thenhave( (c, a λδ[h+1 ] ⊕ (1 − λ) δ[h ′ +1], π ))for all π ∈ P s t, where the relation ≽ s t= ( c, λa (δ[h +1 ], π) ⊕ (1 − λ) a ( δ[h ′ +1], π )) ≽ s t(c, λa (δ[h+1 ], π) + (1 − λ) a ( δ[h ′ +1], π ))= ( c, a ( λδ[h +1 ] + (1 − λ) δ[h ′ +1], π )) ,follows from the definition of risk aversion in the firststage. By Axiom A7 (Dominance), we obtain:(c, λδ[h+1 ] ⊕ (1 − λ) δ[h ′ +1] ) (≽ s t c, λδ[h+1 ] + (1 − λ) δ[h ′ +1] ) .It follows from definition that the decision maker is ambiguity averse.Proof of Proposition 6:First, we show that comparative risk aversion in the first stageimplies comparative ambiguity aversion. To show i is more ambiguous averse than j, weneed to show:(c, δ[l]) ≽ j s t (c, δ[h +1 ]) =⇒ (c, δ[l]) ≽ i s t (c, δ[h +1]). (49)Given h +1 ∈ H +1 and µ s t ∈ ∆ (P s t), define b(h +1 , µ s t) ∈ ∆(L) asb(h +1 , µ s t)(L) = µ s t({π ∈ P s t : l(h +1 , π) ∈ L}),for every Borel set L ⊂ L, where l(h +1 , π) = ∑ s h +1 (s) π (s).For any preferences {≽ s t} satisfying Axioms B1-B7, we use Theorem 3 to compute:V (c, b(h +1 , µ s t))(∫ (∫)))= W(c, v −1 v ◦ u −1 u(V (c ′ , a ′ ))dl(c ′ , a ′ ) db(h +1 , µ s t)(l)LC×∆(L)( ( ∫ ( ∑∫) ))= W c, v −1 v ◦ u −1 π(s) u(V (c ′ , a ′ ))dh +1 (s)(c ′ , a ′ ) dµ s t(π)P s ts∈SC×∆(H +1 )= V s t(c, δ[h +1 ]).where we have used the change of variables theorem (Aliprantis and Border (1999, p. 452))to derive the second equality. This implies that (c, b(h +1 , µ s t)) ∼ s thave (c, b(h ′ +1, µ s t)) ∼ s t (c, δ[h ′ +1]). Thus, equation (49) is equivalent to(c, δ[l]) ≽ j s t (c, b(h +1 , µ)) =⇒ (c, δ[l]) ≽ i s t (c, b(h +1, µ)).50(c, δ[h +1 ]). Likewise, we

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

In complete markets, the following Euler equation holds:E t[Mzt+1 R t+1]= 1.Substituting the preceding pricing kernel into this Euler equation, we obtain:1 = E t⎧⎪ ⎨⎪ ⎩(β(ct+1c t⎡ ⎛(× ⎣R t⎝ βNoting that E t = E µt E πz,t) ) 1−γ⎛ ⎡(−ρ 1−ρR t+1⎝E πz,t⎣ β( ) ) 1−ρ 1−ρct+1R t+1c t⎞⎤⎠⎦η−ρ.(ct+1c t) ) 1−γ⎤−ρ 1−ρR t+1⎦and using the definition of R t , we obtain:⎡ ⎛(( ) ) ⎞⎤1−ρ 1−ρ⎣R t⎝ ct+1β R t+1⎠⎦c t1−ρ= 1.⎞⎠⎫−(η−γ)1−γ ⎪⎬⎪⎭Thus,⎛(( ) ) ⎞1−ρ 1−ρR t⎝ ct+1β R t+1⎠ = 1,c tso that we can write the pricing kernel as equation (28).53

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