Intertemporal substitution and recursive smooth ambiguity preferences
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Miao, Jianjun; Hayashi, Takashi Article Intertemporal substitution and recursive smooth ambiguity preferences Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Miao, Jianjun; Hayashi, Takashi (2011) : Intertemporal substitution and recursive smooth ambiguity preferences, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 6, Iss. 3, pp. 423-472, https://doi.org/10.3982/TE843 This Version is available at: https://hdl.handle.net/10419/150160 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/
Theoretical Economics 6 (2011), 423–472 1555-7561/20110423 Intertemporal substitution and recursive smooth ambiguity preferences Takashi Hayashi Department of Economics, University of Texas at Austin Jianjun Miao Department of Economics, Boston University, CEMA, Central University of Finance and Economics, and AFR, Zhejiang University In this paper, we establish an axiomatically founded generalized recursive smooth ambiguity model that allows for a separation among intertemporal substitution, risk aversion, and ambiguity aversion. We axiomatize this model using two approaches: the second-order act approach à la Klibanoff et al. (2005) and the twostage randomization approach à la Seo (2009). We characterize risk attitude and ambiguity attitude within these two approaches. We then discuss our model’s application in asset pricing. Our recursive preference model nests some popular models in the literature as special cases. Keywords. Ambiguity, ambiguity aversion, risk aversion, intertemporal substitution, model uncertainty, recursive utility, dynamic consistency. JEL classification. D80, D81, D90. 1. Introduction The rational expectations hypothesis is a workhorse assumption in macroeconomics and finance. However, it rules out ambiguity-sensitive behavior. In addition, it faces serious difficulties when confronted with experimental evidence (Ellsberg 1961) or asset markets data (Hansen and Singleton 1983 and Mehra and Prescott 1985). Since Gilboa and Schmeidler’s (1989)andSchmeidler’s (1989) seminal contributions, there is a growing body of literature that develops theoretical models of decision making under ambiguity.1In addition, there is also a growing body of literature that applies these utility models to finance and macroeconomics.2This literatures demonstrates that these models are useful for explaining many economic phenomena. In this paper, we establish an axiomatically founded generalized recursive smooth ambiguity model that allows for a separation among intertemporal substitution, risk Takashi Hayashi: [email protected] Jianjun Miao: [email protected] We are grateful to three anonymous referees and a co-editor (Gadi Barlevy) for helpful comments and suggestions. We also thank Bart Lipman, Massimo Marinacci, and Kyoungwon Seo for helpful comments. 1See Cerreia-Vioglio et al. (2008) for a comprehensive study and the references cited therein. 2See Backus et al. (2005) and Hansen and Sargent (2007b)forsurveys. Copyright ©2011 Takashi Hayashi and Jianjun Miao. Licensed under the Creative Commons AttributionNonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE843
424 Hayashi and Miao Theoretical Economics 6 (2011) aversion, and ambiguity aversion.3An axiomatic foundation is important because the choice-based assumptions on preferences make the model testable in principle. We axiomatize our model using two approaches: the second-order act approach à la Klibanoff et al. (henceforth KMM) (2005) and the two-stage randomization approach à la Seo (2009). We characterize risk attitude and ambiguity attitude within these two approaches. We then apply our model to asset pricing and derive its pricing kernel using a homothetic specification. We show that an ambiguity averse agent attaches more weight on the pricing kernel when his continuation value is low in a recession. This feature generates countercyclical market price of uncertainty and is useful in explaining asset pricing puzzles (Hansen 2007,Hansen and Sargent 2010,andJu and Miao forthcoming). Our dynamic model is built on the static smooth ambiguity model developed by KMM (2005). This static model delivers a utility function over the space of random consumption as4 V(c)=v−1P v◦u−1S u(c) dπ(s)dμ(π)c:S→R+(1) where Sis the state space, Pis a set of probability measures on S,μis a probability measure over P,udescribes risk attitude, and v◦u−1describes ambiguity attitude. The set Preflects model uncertainty or the decision maker’s ambiguity about the “true” distribution of consumption. This model permits a separation between ambiguity and ambiguity attitude, and allows smooth, rather than kinked, indifference curves. Both features are conceptually important and empirically useful. In addition, KMM (2005)show that this model includes the multiple-priors model of Gilboa and Schmeidler (1989)as a special case when ambiguity aversion goes to infinity under some technical regularity conditions. Embedding their static model in a dynamic environment, KMM (2009a) develop a recursive smooth ambiguity preference model. This dynamic model suffers from a limitation that intertemporal substitution and attitudes toward risk or uncertainty are intertwined. This inflexibility limits its empirical applications and makes comparative statics of risk aversion hard to interpret.5For example, calibrating this model in a representative–agent consumption-based asset-pricing setting, Ju and Miao (2007) show that somewhat implausible parameter values are needed to explain the equity premium puzzle. By contrast, after separating out intertemporal substitution as in our generalized recursive smooth ambiguity model, Ju and Miao (forthcoming) show that the empirical performance improves significantly. We summarize our preference model when restricted to the space of adapted consumption processes as follows. Consider an infinite-horizon setting and denote time by 3Roughly speaking, risk refers to situations where known probabilities are available to guide choices, while ambiguity refers to situations where probabilities are vague so that multiple probabilities may be available. Ambiguity aversion means that individuals dislike ambiguity. 4For alternative axiomatizations of an essentially identical functional form, see Chew and Sagi (2008), Ergin and Gul (2009), Nau (2006), and Seo (2009). 5See Epstein and Zin (1989) for an early discussion of the importance of the separation between intertemporal substitution and risk aversion in a pure risk setting.
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 425 t=012. The state space in each period is S.Attimet, the decision maker’s information consists of histories st={s1s2st}with s0∈Sgiven and st∈S. The decision maker ranks adapted consumption plans c=(ct)t≥0.Thatis,ctis a measurable function of st. His preferences are represented by the recursive form Vst(c) =Wctv−1Pst v◦u−1S u(V(stst+1)(c)) dπ(st+1)dμst(π)(2) where Vst(c) is conditional utility or continuation value at history st,W:R2→Ris a time aggregator, Pstis a set of one-step-ahead probability measures on Sat history st,and μstis a probability measure over Pst. The measure μstrepresents second-order beliefs about distributions governing one-step-ahead resolution of uncertainty. Given some assumptions similar to those in KMM (2009a), we show that μstis obtained by Bayesian updating from an initial prior. When the set Pstconsists of a set of conditional likelihood distributions πz(·|st)indexed by an unknown parameter z∈Z,weuse(2) to derive a model with learning, Vst(c) =Wctv−1Z v◦u−1S u(V(stst+1)(c)) dπz(st+1|st)dμst(z)(3) where μst(z) is the posterior distribution of zgiven st. More generally, the learning model in (3) allows zto be a hidden state that follows a Markov process because Pst can be history dependent. Our generalized recursive smooth ambiguity model nests some popular models in theliteratureasspecialcases. •The subjective version of the recursive expected utility model of Kreps and Porteus (1978)andEpstein and Zin (1989) is obtained by setting v=uin (2). In this case, the two distributions μstand πcan be reduced to a one-step-ahead predictive distribution: p(st+1|st)=Pst π(st+1)dμst(π) (4) This is the standard Bayesian approach that rules out ambiguity-sensitive behavior. If we further set v(x) =u(x) =−exp(−x/θ), we obtain the multiplier preference model or the risk-sensitivity model discussed in Hansen and Sargent (2001).6 Here θis a robustness parameter, which enhances risk aversion. •The generalized recursive multiple-priors model of Hayashi (2005) is obtained as the limit of (2) under some technical regularity conditions when ambiguity aver6This multiplier model is dynamically consistent according to the standard definition and the definition in this paper. Hansen and Sargent (2001, 2007b) also propose several other models of robustness. Some of them, e.g., “constraint preferences,” are dynamically inconsistent according to the standard definition as pointed out by Epstein and Schneider (2003). However, the constraint preferences satisfy a different notion of dynamic consistency defined in Section 19.4 of Hansen and Sargent (2007b, pp. 407–412).
426 Hayashi and Miao Theoretical Economics 6 (2011) sion goes to infinity: Vst(c) =Wctu−1min π∈PstS u(V(stst+1)(c)) dπ(st+1) This model nests the recursive multiple-priors model of Epstein and Wang (1994) and Epstein and Schneider (2003) as a special case, as discussed in Hayashi (2005). •The recursive smooth ambiguity model of KMM (2009a) has a discounted aggregator and takes the form Vst(c) =u(ct)+βφ−1Z φS V(stst+1)(c) dπz(st+1|st)dμst(z) The concavity of φcharacterizes ambiguity aversion. The curvature of udescribes both intertemporal substitution and risk aversion. Thus, they are intertwined. •The multiplier preference model with hidden states of Hansen (2007)and Hansen and Sargent (2007a) is obtained by setting W(cy)=h(c) +βy,u(x) = −exp(−x/θ1),andv(x) =−exp(−x/θ2),θ1θ2>0,in(3). In this model, there are two risk-sensitivity adjustments. The first risk-sensitivity adjustment for the distribution πz(·|st)reflects the decision maker’s concerns about the misspecification in the conditional distribution given the parameter value z.Thesecond risk-sensitivity adjustment for the distribution μstreflects the decision maker’s concerns about the misspecification of the posterior distribution. To provide an axiomatic foundation for the model in (2), we need to choose a suitable domain for preferences. As is well known from Kreps and Porteus (1978)andEpstein and Zin (1989), one needs to define a hierarchical domain of choices so as to separate intertemporal substitution from risk aversion. In our second-order act approach, we take the product space of current consumption and the continuation compound lottery acts as the primary preference domain. Hayashi (2005) first introduces the domain of compound lottery acts to provide an axiomatic foundation for a generalized recursive multiple-priors model. A compound lottery act is a random variable that maps today’s state of the world into a joint lottery over current consumption and a compound lottery act for tomorrow. It is the dynamic counterpart of the horse-race roulette-wheel act introduced by Anscombe and Aumann (1963). Our first axiomatic characterization consists of five standard axioms to deliver recursive expected utility under uncertainty and two additional axioms related to ambiguity. The five standard axioms deliver Wand uin (2). The two additional axioms deliver v and μst. To pin down a unique vand a unique μst, we need more choices available to the decision maker. Because μstis a second-order probability measure over the first-order probability measures on S, to elicit this belief, it seems natural and intuitive to assume that choices contingent on the first-order probability measures are observable. These choices are modelled as second-order acts in KMM (2005) in a static setting. Extending their insight to our dynamic Anscombe–Aumann setting, we define a second-order act
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 427 as a mapping that maps a probability measure on Sto a compound consumption lottery. We then define auxiliary preferences over second-order acts and impose an axiom that these preferences are represented by subjective expected utility. This representation can be delivered by imposing additional primitive axioms from the standard subjective expected utility theory.7 As in KMM (2005, 2009b), we impose the last axiom that connects preferences over second-order acts and the original preferences over pairs of current consumption and continuation compound lottery acts. In doing so, we introduce the notion of a one-stepahead act—a compound lottery act in which subjective uncertainty resolves in just one period. We then construct a second-order act associated with a one-step-ahead act that maps a probability measure on Sto a compound lottery on the consumption space. This compound lottery is obtained by averaging out states in the one-step-ahead act using the probability measure on S. The last axiom says that the decision maker orders pairs of current consumption and the one-step-ahead act identically to the second-order acts associated with the one-step-ahead acts. The intuition is that the decision maker’s ranking of the former choices reflects his uncertainty about the underlying distribution of the choices, which is the domain of the second-order acts. One critique of the KMM (2005) model raised by Seo (2009) is that second-order acts and 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 true distribution underlying stock prices. A similar critique applies to the Anscombe–Aumann acts as well because these acts are also unobservable in financial markets: the realizations of stock prices are monetary values, not lotteries. However, both Anscombe–Aumann acts and second-order acts are useful modelling devices and are available from laboratory and thought experiments.8More concretely, when measures in Pstcorrespond to conditional distributions indexed by an unknown parameter as in (3), the second-order acts are bets on the value of the parameter. In an asset pricing application studied by Ju and Miao (forthcoming), Pstconsists of two distributions for consumption growth in a boom and in a recession so that the second-order acts are bets on the economic regime. In a portfolio choice application studied by Chen et al. (2009), Pstconsists of two distributions for the possibly misspecified stock return models so that the second-order acts are simply bets on the statistical model of 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) without second-order acts. Adapting Seo’s (2009) static setup, we introduce an 7In a recent critique of the static KMM model, Epstein (2010) argues that Ellsbergian choices on Sshould lead to Ellsbergian choices for second-order acts. In response, KMM (2009a) argue that second-order acts are modelling devices to deliver Ellsbergian choices on the state space Sof primary interest. To accommodate Ellsbergian choices for second-order acts, one can simply expand the state space to incorporate measures on S. 8To further illustrate this point, we quote Kreps (1988, p. 101): “This procedure of enriching the set of items to which preference must apply is quite standard. It makes perfectly good sense in normative applications, as long as the Totrep involved is able to envision the extra objects and agree with the axiom applied to them. This need be no more than a thought experiment for Totrep, as long as he is willing to say that it is a valid (i.e., conceivable) thought experiment.”
428 Hayashi and Miao Theoretical Economics 6 (2011) extra stage of randomization. As a by-product contribution, we construct a set of twostage compound lottery acts, which allows for randomization both before and after the realization of the state of the world. We then define the product space of current consumption and the continuation lotteries over two-stage compound lottery acts as the single domain of preferences. We impose five axioms analogous to the first five axioms in the second-order act approach. We replace the last two axioms in that approach with a first-stage independence axiom and a dominance axiom adapted from Seo (2009). Given these seven axioms, we establish a dynamic version of Seo’s static model. To the best of our knowledge, our paper provides the first dynamic extension of Seo’s static model. We should mention that each of our two different adopted axiomatic approaches is debatable. For example, some researchers (e.g., Seo 2009 and Epstein 2010) argue that second-order acts or preferences on these acts are either unobservable or may not be totally plausible. In the two-stage randomization approach, a failure of the reduction of compound lotteries 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 this approach, ambiguity aversion is associated with aversion to the variation of ex ante evaluations of one-step-ahead acts due to model uncertainty. In the twostage randomization approach, we distinguish between attitudes toward risks in the two stages. We define absolute ambiguity aversion as an aversion to a first-stage mixture of acts before the realization of the state of the world compared to the second-stage mixture of these acts after the realization of the state. We show that this notion of ambiguity aversion is equivalent to risk aversion in the first stage. In particular, ambiguity aversion is associated with the violation of reduction of compound lotteries. We also show that in both approaches, risk attitude and ambiguity attitude are characterized by the shapes of the functions uand v, respectively. The remainder of the paper proceeds as follows. Section 2 reviews the atemporal models of KMM (2005)andSeo (2009). Section 3 embeds the KMM (2005)modelin a dynamic setting 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 twostage randomization approach. Section 5 applies our model to asset pricing. Section 6 discusses related literature. Appendices A–Econtain proofs. 2. Review of the atemporal models In this section, we provide a brief review of the atemporal models of ambiguity proposed by KMM (2005)andSeo (2009). We embed these models in a dynamic setting in Sections 3and 4. Both atemporal models when restricted to the space of random consumption deliver an identical representation in (1). The two models differ in domain and axiomatic foundation. For both models, we take a complete, transitive, and continuous preference relation as given. Consider the KMM model first. KMM originally study Savage acts over S×[01], where the auxiliary state space [01]is used to describe objective lotteries. Here we
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 429 translate their model into the Anscombe–Aumann domain. Let Sbe the set of states, which is assumed to be finite for simplicity. Let Cbe a compact metric space and let (C)be the set of lotteries over C.9An Anscombe–Aumann act is defined as a mapping g:S→(C).LetGdenote the set of all such acts. To pin down second-order beliefs, KMM introduce an auxiliary preference ordering 2over second-order acts. A secondorder act is a mapping g:P→(C),whereP⊂(S).LetG2denote the set of all secondorder acts. The preference ordering over Gand the preference ordering 2over G2are represented by U(g)=P φ s∈S π(s)¯ u(g(s))dμ(π) ∀g∈G(5) and U2(g)=P φ(¯ u(g(π))) dμ(π) ∀g∈G2 where φ:¯ u((C)) →Ris a continuous and strictly increasing function and ¯ u:(C)→R is a mixture-linear function.10 The 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) The preference over second-order acts 2is represented by the subjective expected utility of Savage (1954). (iii) The two preference relations and 2are consistent with each other in the sense that ghif and only if g22h2,whereg2is the second-order acts associated with gdefined by g2(π) =s∈Sg(s)π(s) for each π∈Pand h2is defined similarly. The interpretation for the last axiom is the following. If the decision maker prefers fto g, then the average of facross states 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 the plausibility of these axioms, consider the following example. Suppose there is an Ellsberg urn containing 90 balls. A decision maker is told that there are 30 black balls and 60 white or red balls in the urn. But he does not know the composition of white or red balls. There are four bets as in Table 1. The Ellsbergian choice is g1g2but g4g3 One justification is that the decision maker is unsure about the probabilities of white and red balls and is averse to this ambiguity. Suppose there are two possible distributions over the set of ball color S={bwr}: π1=(1/32/30)and π2=(1/302/3). Consider the second-order acts associated with 9We use the following notations and assumptions throughout the paper. Given a compact metric space Y,letB(Y) be the family of Borel subsets of Yand let (Y ) be the set of Borel probability measures defined over B(Y).Endow(Y ) with the weak convergence topology. Then (Y ) is a compact metric space. 10Afunctionfis mixture linear on some set Xif f(λx+(1−λ)y) =λf (x) +(1−λ)f (y) for any x y ∈X and any λ∈[01]. 11The proof can be obtained from our proof of Theorem 1 in Appendix A.
430 Hayashi and Miao Theoretical Economics 6 (2011) bwr g110 0 0 g2010 0 g310 0 10 g401010 Table 1. This table shows four acts, g1,g2,g3,andg4, with payoffs contingent on three events {b},{w},and{r}. π1π2 g2 110/310/3 g2 220/30 g2 310/310 g2 420/320/3 Table 2. This table shows four second-order acts, g2 1,g2 2,g2 3,andg2 4, associated with four acts g1,g2,g3,andg4, respectively. Their payoffs are contingent on two distributions π1and π2. gi,i=14,g2 i(πj)=s∈{bwr}gi(s)πj(s),wherej=12. We write their payoffs in Table 2. The previous consistency axiom implies that g2 12g2 2but g2 42g2 3 This behavior can be consistent with expected utility over second-order acts as long as the decision maker is risk averse, because g2 1and g2 4give sure outcomes, but g2 2and g2 3 are risky bets. The intuition is that second-order acts average out uncertain states (ball color) by definition and such hedging may eliminate ambiguity (see Gilboa and Schmeidler 1989). So it is possible that the decision maker is ambiguity neutral for secondorder acts, but ambiguity averse for bets on the Ellsberg urn. Of course, one can design thought experiments to display Ellsbergian choices for second-order acts, which are ruled out by the KMM model. Seo (2009) provides a different axiomatic foundation for (5) by dispensing with the auxiliary set of second-order acts and the associated preferences over this set. He considers the domain of lotteries over Anscombe–Aumann acts, (G), and a single preference relation defined over it. Notice that by restricting attention to lotteries over constant acts, we have the domain of two-stage lotteries ((C)) as a subset of (G),and by further making the first-stage randomization degenerate, we have (C)as a subset of ((C)),henceof(G)too. Seo (2009) shows that the representation of preference takes the form12 U(p)=GP φ s∈S π(s)¯ u(g(s))dμ(π)dp(g) ∀p∈(G) (6) 12In Seo’s (2009) original representation, he takes P=(S). When we adapt his dominance axiom for P, we can allow Pto be an arbitrary subset of (S). For example, the proof in Seo’s appendix gives an example of a finite set P.
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 437 of Theorem 2 in KMM (2009a), we can show that {Vst}exists. We need additional conditions for the uniqueness. Epstein and Zin (1989) provide sufficient conditions for recursive expected utility. KMM (2009a) give sufficient conditions for their recursive smooth ambiguity model. Marinacci and Montrucchio (2010) derive sufficient conditions for general recursive equations that may be applied to our model. 3.5 Ambiguity attitude Because our model nests the deterministic case (11) and the pure risk case (9), we immediately deduce that the function Wcharacterizes intertemporal substitution and the function ucharacterizes risk aversion in the usual way. We turn to the characterization of ambiguity aversion. We adopt the behavioral foundation of ambiguity attitude developed by Ghirardato and Marinacci (2002)andKMM(2005). Epstein (1999)providesa different foundation. The main difference is that the benchmark ambiguity neutral preference is the expected utility preference according to Ghirardato and Marinacci (2002), while Epstein’s (1999) benchmark is 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 Pst. The decision maker’s ambiguity attitude is toward uncertainty about the possible distributions in Pst. To characterize this attitude, we define the lottery m(g+1μst)∈Massociated with the one-step-ahead act g+1and the second-order belief μston Pstas m(g+1μst)=Pst s∈S g+1(s)π(s) dμst(π) Since s∈Sg+1(s)π(s) is the outcome of the second-order act g2 +1(π) associated with g+1,m(g+1μst)is simply the mean value of g2 +1with respect to the second-order belief μst. Alternatively, from the definition of predictive distribution in (4), we observe that the lottery m(g+1μst)is also the mean value of the act g+1with respect to the predictive distribution induced by μst. The following definition of ambiguity aversion states that the decision maker is ambiguity averse if he prefers a sure lottery obtained as the mean value of a given act to the act itself. Definition 3. The decision maker with {st}exhibits ambiguity aversion if for all st, for all c∈Cand g+1∈G+1, (c m(g+1μst)) st(c g+1) Similarly to this definition, we can define ambiguity loving and ambiguity neutrality in the usual way. An immediate consequence of this definition is the following proposition. Proposition 1. Suppose {st}satisfies Axioms A1–A7.Then{st}exhibits ambiguity aversion if φ≡v◦u−1is concave.16 16If v◦u−1is concave, it is easy to check that {st}satisfies the uncertainty aversion axiom of Gilboa and Schmeidler (1989).
438 Hayashi and Miao Theoretical Economics 6 (2011) The proof of this proposition is straightforward and is omitted. Clearly, when v◦u−1 is linear, {st}displays ambiguity neutrality. Thus, the ambiguity neutrality benchmark is the recursive expected utility model. We need additional conditions to establish the converse statement that ambiguity aversion implies concavity of v◦u−1.Thereasonis that, to prove this statement, one needs to know preferences over binary bets on some Pst, but our axioms and representation hold only for fixed Pst. To deal with this issue in the KMM model, KMM (2005) consider a family of preference relations indexed by rich supports of second-order beliefs, and impose an assumption that ambiguity attitude and risk attitude are invariant across these supports (see their Assumption 4). We can adapt their assumption to establish the converse statement. Since the proof is similar to the proof of Proposition 1 in their paper, we omit it here. We now turn to comparative ambiguity aversion. Definition 4. Let the representations of the preferences of persons iand jshare the same second-order belief μston the same support Pstfor all st. Say that {i st}is more ambiguity averse than {j st}if for all st, for all c∈C,m∈M,andg+1∈G+1, (c m) j st(c g+1)⇒ (c m) i st(c g+1) and if this property also holds for strict preference relations i stand j st. The interpretation of this definition is similar to that of Definition 5 in KMM (2005). The idea is that if person iprefers a lottery over an uncertain act whenever person jdoes so, then this must be due to person i’s comparatively higher aversion to uncertainty. This cannot be due to aversion to risk, because the act g+1itself may be a lottery and the conditions in the definition imply that persons iand jrank lotteries in the same way. Because the difference in beliefs is ruled out in the definition, the behavior in the definition must be due to differences in ambiguity attitude. The following proposition is a partial characterization. We omit its straightforward proof. Proposition 2. Suppose {i st}and {j st}satisfy Axioms A1–A7 and their representations share the same second-order belief μstonthesamesupportPstfor all st.Then{i st}is more ambiguity averse than {j st}if there exist corresponding utility representations such that Vi|C×M=Vj|C×M,Wi=Wj,ui=uj,andvi=◦vj,whereis a strictly increasing and concave function. As in the case of absolute ambiguity aversion, one needs more information to establish the converse statement that comparative ambiguity aversion implies concavity of . As discussed earlier, we may make an assumption similar to Assumption 4 in KMM (2005) to establish this statement. 4. Axiomatization with two-stage compound lottery acts To embed Seo’s (2009) atemporal model in a dynamic setting, we adapt his atemporal domain—the set of lotteries over Anscombe–Aumann acts—to a dynamic setting. This leads us to consider the set of two-stage compound lottery acts.
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 439 4.1 Domain We consider preference relations stat each history stdefined on the domain C×(H), where His a set of two-stage compound lottery acts constructed as follows. Inductively define the family of sets {H0H1}by H0=((C))S H1=(C×(H0))S Ht=(C×(Ht−1))S and so on. By induction, (C×(Ht−1)) and Htare compact metric spaces, for every t≥1.LetH∗=∞ t=0Ht.Itisacompactmetricspacewithrespecttotheproductmetric. We consider sequences of acts (h0h1h2)in H∗that are coherent.Thatis,htand ht+1must be consistent for all t≥0in the sense that is made precise in Appendix B. The domain of coherent acts, a subset of H∗, is denoted by H. The details of the definition of coherent acts and formal construction of the domain are given in Appendix B. The domain Hsatisfies 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 His homeomorphic to ((C×(H)))S, denoted as H≃(C×(H))S When attention is restricted to constant acts, we obtain the subdomain consisting of two-stage compound lotteries, which satisfies the homeomorphism L≃(C×(L)) Relations among the domains defined so far are summarized as H⊃G⊃G∗⊃F ∪∪ ∪ ∪ L⊃M⊃(C∞)⊃((C))∞⊃C∞ In particular, the set of compound lottery acts Gand the set of compound lotteries M studied in Section 3 are subsets of Hand L, respectively. We now introduce some useful notations. For any two-stage compound lottery acts, hh∈H,andanyλ∈(01),weuseλh +(1−λ)h∈(H)to denote a lottery that gives hwith probability λand hwith probability 1−λ.Weuseλh ⊕(1−λ)h∈Hto denote a statewise mixture. That is, for each s∈Sand each Borel set B∈B(C×(H)),λh ⊕ (1−λ)h(s)(B) =λh(s)(B) +(1−λ)h(s)(B).Foranypq ∈(H),λp +(1−λ)q ∈(H) denotes the usual mixture.
440 Hayashi and Miao Theoretical Economics 6 (2011) 4.2 Axioms We impose the following axioms on the preference process {st}. The first three axioms are analogous to Axioms A1–A3. Axiom B1 (Order). For all tand st,stis a continuous weak order over C×(H)and there exist yy∈C∞such that ysty. Axiom B2 (Current Consumption Separability). For all tand st, for all cc∈Cand pq ∈(H), (c p) st(cq) ⇐⇒ (cp)st(cq) Axiom B3 (History Independence of Risk Preference). For all t,˜ tand st,˜ s˜ t, for all (c a)(ca)∈C×(L), (c a) st(ca)⇐⇒ (ca) ˜ s˜ t(ca) Next, we assume independence conditions for “timeless” gambles, similar to Axiom A4. There are two kinds of such timeless gambles here: One is made before the realization of the one-step-ahead subjective uncertainty, and the other is made after that. Axiom B4 (First-stage Independence). For all tand st, for all p qr ∈(H)and λ∈(01), (c p) st(cq) ⇐⇒ (cλp +(1−λ)r) st(cλq +(1−λ)r) Axiom B5 (Second-Stage Independence). For all tand st, for all c∈C, for all lmn ∈ (C×(H)) and λ∈(01), (c δ[l])st(cδ[m])⇐⇒ cδ[λl ⊕(1−λ)n]stcδ[λm ⊕(1−λ)n] To connect preferences across histories, we impose a dynamic consistency axiom, similar to Axiom A5. Definition 5. Given a b ∈(C×(H)), say that astochastically dominates bwith regard to stif a{(cp)∈C×(H):(cp)st(cp)}≥b{(cp)∈C×(H):(cp)st(cp)} for all (cp) ∈C×(H). If in addition there is some (c p) ∈C×(H)such that ≥is replaced with >,thenwesayastrictly stochastically dominates b.Ifaand bstochastically dominate each other, we say that aand bare stochastically equivalent with regard to st. Note that in this definition, we allow aor bto be a measure on C×(L),say a∈(C×(L)).Inthiscase,weviewa∈(C×(H)) with the support C×(L).
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 441 Axiom B6 (Dynamic Consistency). For all tand st, for all c∈Cand h h∈H,if h(s) (strictly) stochastically dominates h(s) with regard to sts for each s∈S,then (c δ[h])st(st)(cδ[h]). Finally, we embed Seo’s (2009) dominance axiom to the set of one-step-ahead acts. A one-step-ahead act is an act for which subjective uncertainty resolves in just one period. We define 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+1and π∈(S),definel(h+1π)∈Lby l(h+1π)= s∈S h+1(s)π(s) Given p+1∈(H+1)and π∈(S),definea(p+1π)∈(L)by a(p+1π)(L) =p+1{h+1∈H+1:l(h+1π)∈L} for every Borel subset L⊂L. We take a set of one-step-ahead probability measures, Pst, as given for each history st and impose the following dominance axiom on this set. We allow this set to be different from (S) to permit more flexibility in applications as discussed in Section 1. Axiom B7 (Dominance). For all tand st, for all c∈Cand p+1p +1∈(H+1), (c a(p+1π))st(c a(p +1π)) ∀π∈Pst⇒ (cp+1)st(cp +1) where Pst⊂(S). To interpret this axiom, imagine that Pstis a set of probability distributions, which contains the “true” distribution unknown to the decision maker. Given the same current consumption c, if the decision maker prefers the continuation two-stage lottery a(p+1π)induced by p+1over another one a(p +1π)induced by p +1for each probability distribution π∈Pst, then he must also prefer (cp+1)over (c p +1). Compared to the axioms in Section 3, First-Stage Independence (Axiom B4)and Dominance (Axiom B7) are the counterparts of the SEU Representation of Preference Over Second-Order Acts (Axiom A6) and Consistency With Preference Over SecondOrder Acts (Axiom A7). Thus, we can dispense with second-order acts. 4.3 Representation The following theorem gives our second representation result.
442 Hayashi and Miao Theoretical Economics 6 (2011) Theorem 4 (Representation). The preference process {st}satisfies Axioms B1–B7 if and only if there exists a family of functions ({Vst}Wuv)and a process of probability measures {μst}over Pstsuch that for each st, the function Vst:C×(H)→Rrepresents stand has the form Vst(c p) =Wcv−1HPst v◦u−1 s∈S π(s) ×C×(H) u(Vsts(ca))dh(s)(ca)dμst(π)dp(h) (13) for (c p) ∈C×(H),whereWis continuous and strictly increasing in the second argument, and uand vare continuous and strictly increasing. We also have the following uniqueness result, up to some monotonic affine transformations. Theorem 5 (Uniqueness). Let {st}satisfy Axioms B1–B7.Ifboth({˜ Vst}˜ W˜ u ˜ v{˜μst}) and ({Vst}Wuv{μst})represent {st}, then there exist a strictly increasing function and constants A,B,D,Ewith AD > 0,suchthat ˜ Vst=◦Vst˜ W(··)=(W (·−1(·))) ˜ u◦=Au +B ˜ v◦=Dv +E As in the static model of Seo (2009), the process of second-order beliefs {μst}is not unique in general. For example, when φ=v◦u−1is linear, {μst}is indeterminate. It is unique if φis some exponential function. The existence of a solution for {Vst}to the recursive equation (13) follows a similar argument in the proof of Theorem 2 in KMM (2009a). We may apply sufficient conditions in Marinacci and Montrucchio (2010)to establish uniqueness. The following list shows how the above model nests the existing models. 1. On the subdomain C×G, the representation reduces to (8), which further reduces to (2) on C×F. 2. On the subdomain C×(L), we obtain a pure risk setting where the two-stage randomization is present. In this case, each Vstcoincides with the common representation V(ca)=Wcv−1L v◦u−1C×(L) u(V (ca)) dl(ca)da(l)(14) where (c a) ∈C×(L). 3. On the subdomain C×M, we obtain a pure risk setting where only the secondstage randomization is present. In this case, the model reduces to (9).
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 443 4.4 Risk aversion and ambiguity aversion As discussed before, the function Wdescribes intertemporal substitution. Now, we discuss how 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 the subdomain C×(L)without subjective uncertainty. In this case, the utility representation takes the form in (14). Because there is two-stage randomization, we have two risk attitudes toward 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 first-stage lottery is degenerate. We then obtain the representation of recursive risk preference given in (9). We can define risk aversion in the second stage in a standard way and show that 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 stage as follows. Definition 7. The decision maker with preference {st}exhibits risk aversion in the first stage if for all st,c∈Cand ll∈L,λ∈[01], cδ[λl ⊕(1−λ)l]st(cλδ[l]+(1−λ)δ[l]) (15) We can similarly define risk loving and risk neutrality in the first stage. In Definition 7,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)lin the second stage. According to this definition, the decision maker may not be indifferent between these two lotteries, even though they give the same final outcome distribution. In particular, if the decision maker believes that the degenerate lottery is like a sure outcome and must be preferred, then he displays risk aversion in the first stage. Note that if we replace stwith ∼stin (15), we obtain a dynamic counterpart of Seo’s (2009) Reduction of Compound Lotteries axiom. Thus, according to our Definition 7, violation of the Reduction of Compound Lotteries reflects the decision maker’s attitude toward the risk in the first stage. The following proposition characterizes this risk attitude. Proposition 3. Suppose {st}satisfies Axioms B1–B7.Then{st}exhibits risk aversion in the first-stage if and only if v◦u−1is concave. An immediate corollary of this proposition is that, given Axioms B1–B7, the Reduction of Compound Lotteries axiom is satisfied if and only if v◦u−1is a strictly increasing affine function. In this case, the two lotteries land ain (14) can be reduced to a compound lottery and hence (14) reduces to a model belonging to the class of recursive expected utility under objective risk. Next, we consider comparative risk aversion.
444 Hayashi and Miao Theoretical Economics 6 (2011) Definition 8. Say that {i st}is more risk averse than {j st}in the first stage if for all st, c∈C,l∈L,anda∈(L), (c δ[l])j st(c a) ⇒ (c δ[l])i st(c a) and if this property also holds true for strict preference relations j stand i st. Take current consumption cas given. Suppose person jprefers a “sure” outcome (with the outcome being a lottery) to an arbitrary lottery for tomorrow. This must be due to j’s aversion to risk. Facing the same choices, if person iis more risk averse than person jin the first stage, then person ishould dislike what person jdislikes. Proposition 4. Suppose {i st}and {j st}satisfy Axioms B1–B7. Then {i st}is more risk averse than {j st}in the first stage if and only if there exist corresponding utility representations such that Vi|C×(L)=Vj|C×(L),Wi=Wj,ui=uj,andvi=◦vj,whereis a strictly increasing and concave transformation. By Definition 8,personsiand jrank deterministic consumption plans in the same way and rank lotteries in the second stage in the same way. Thus, (W iui)and (W juj) are ordinally equivalent. Proposition 4 shows that person iis more risk averse than person jin the first stage if and only if viis a monotone concave transformation of vj. Now, we consider ambiguity attitude. Because ambiguity attitude deals with subjective uncertainty, we focus on the subdomain C×(H+1)in which uncertainty resolves in just one period. We define absolute ambiguity aversion as follows. Definition 9. The decision maker with {st}exhibits ambiguity aversion if for all st, c∈C,h+1h +1∈H+1,andλ∈[01], cδ[λh+1⊕(1−λ)h +1]st(c λδ[h+1]+(1−λ)δ[h +1]) (16) We can similarly define ambiguity loving and ambiguity neutrality. Definition 9 says that if a first-stage mixture of acts is preferred to their second-stage mixture, then the decision maker is ambiguity averse. The intuition for this definition is that hedging across ambiguous states is valuable compared to randomization of acts before the realization of the states. It is related to Gilboa and Schmeidler’s (1989) definition of ambiguity aversion, which states that hedging across states for two indifferent acts is valuable to an ambiguity averse decision maker.17 When stis replaced with ∼stin (16), then it becomes the dynamic counterpart of Seo’s Reversal of Order axiom. Thus, ambiguity attitude is associated with the violation of the Reversal of Order axiom. An example taken from Seo (2009) illustrates Definition 9. Restrict attention to a static setting. Consider an Ellsberg urn that contains 100 black or white balls, but the exact composition is unknown. The state of the world is the color of the ball. Let fbe the 17Given Axiom B4 (First-Stage Independence), our definition implies the following Gilboa and Schmeidler definition of ambiguity aversion: (c δ[h+1])∼st(cδ[h +1])⇒ (c δ[λh+1⊕(1−λ)h +1])st(c δ[h+1]) for all st,c∈C,h+1h +1∈H+1, and λ∈[01].
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 445 act that gives $100 if the chosen ball is black and nothing otherwise. Let gbe the act that gives $100 if the chosen ball is white and nothing otherwise. Let pbe a lottery with 50% chance of winning $100. Experimental evidence reveals that most people are indifferent between fand g, but prefer pto fand pto g. The first-stage mixture 1 2f+1 2gis still an ambiguous act. But the second-stage mixture 1 2f⊕1 2ggives an identical lottery pno matter whether the chosen ball is black or white. Thus, it is intuitive that an ambiguity averse decision maker prefers 1 2f⊕1 2gto 1 2f+1 2g. As Seo (2009)andSegal (1987, 1990) point out, ambiguity attitude is associated with violation of the Reduction of Compound Lotteries.18 We now characterize this relationship. In his atemporal model, Seo (2009) shows that Reduction of Compound Lotteries and Reversal of Order are equivalent under Dominance. Adapting his argument to our dynamic two-stage compound lottery acts framework, we show below that ambiguity aversion is identical to risk aversion in the first stage. Proposition 5. Suppose {st}satisfies Axioms B1–B7.Then{st}exhibits ambiguity aversion if and only if {st}exhibits risk aversion in the first stage. An immediate implication of this proposition is that ambiguity aversion is equivalent to concavity of v◦u−1. In addition, the decision maker is ambiguity neutral if and only if v◦u−1is a strictly increasing affine function. As a result, the four distributions h,π,μst,andpcan be reduced to a compound distribution and the model reduces to recursive expected utility under uncertainty. Finally, we study comparative ambiguity aversion. Definition 10. Let the utility representations of {i st}and {j st}share the same secondorder belief μston the same support Pst. Say that {i st}is more ambiguity averse than {j st}if for all st,allc∈C,l∈L,andh+1∈H+1, (c δ[l])j st(c δ[h+1])⇒ (cδ[l])i st(c δ[h+1]) and if this property also holds true for strict preference relations j stand i st. To interpret this definition, fix current consumption at cand consider two sure outcomes for tomorrow, with one outcome being a lottery and the other outcome being a one-step-ahead act. Suppose person jprefers the sure lottery outcome to the sure one-step-ahead act. This must be due to person j’s aversion to subjective uncertainty or ambiguity. Facing the same choices, if person idislikes what person jdislikes, then person imust be more ambiguity averse than person jbecause differences in beliefs are ruled out. The following proposition states that in the framework of two-stage randomization, comparative ambiguity aversion is identical to comparative risk aversion in the first stage. 18Halevy (2007) finds experimental evidence to support this view. This view is controversial because nonreduction of compound lotteries is arguably a “mistake.”
446 Hayashi and Miao Theoretical Economics 6 (2011) Proposition 6. Suppose that {i st}and {j st}satisfy Axioms B1–B7 and that their representations share the same second-order belief μstonthesamesupportPstfor all st. Then {i st}is more ambiguity averse than {j st}if and only if {i st}is more risk averse than {j st} in the first stage. Given Axioms B1–B7, an immediate corollary of this proposition is that a decision maker’s preferences have a representation with a concave function v◦u−1if and only if he is more ambiguity averse than a decision whose preferences are represented by recursive expected utility. This result connects our definition of ambiguity aversion in Definition 5 to our definition of comparative ambiguity aversion in Definition 6.Itshowsthat recursive expected utility is the dividing line between ambiguity loving and ambiguity aversion. What is the relationship between the notion of ambiguity aversion defined in this section and that in Section 3? Because the preference domain of choices is different under the two approaches in these two sections, ambiguity aversion reflects different natures. But the utility representations under these two approaches give identical functionals in the domain of adapted consumption processes. In addition, these two approaches give identical characterizations of ambiguity attitude in terms of the function v for fixed uor v◦u−1. Unlike the second-order act approach in Section 3 or KMM (2005), the two-stage randomization approach does not need to have a rich support of μstto establish that absolute or 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. Application We 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 has ambiguous beliefs about the possible consumption distributions πzindexed by z(Pstin (2)isasetindexedbyz). We first derive the utility gradient (Duffie and Skiadas 1994) for the utility function defined in (3). The utility gradient is useful for solving an individual’s optimal consumption and investment problem. It is also useful for equilibrium asset pricing. We define the gradient of a utility function V0at cgiven z as the adapted process (ξz t)such that lim α↓0 V0(c +αδ) −V0(c) α=E∞ t=0 ξz tδt(17) Let Vtdenote Vst(c) in (3)anddefine Rt(Vt+1)=v−1Eμtv◦u−1(Eπzt u(Vt+1)) whereweuseμtand πzt to denote the posterior distribution μstand the conditional distribution πz(·|st), respectively.
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 453 Define y≡Vst( cg+1)=ξPst v◦u−1 s∈S π(s)C×M u(V (cm)) dg+1(s)(cm)dμst(π) Using (30), we obtain Vst(c g+1)= W(cy)= Wcζ ◦u(u−1◦ζ−1(y))=W(cu −1◦ζ−1(y)) =Wcu−1◦ζ−1◦ξPst v◦u−1 s∈S π(s) ×C×M u(V (cm)) dg+1(s)(cm)dμst(π) =Wcv−1Pst v◦u−1 s∈S π(s) ×C×M u(V (cm)) dg+1(s)(cm)dμst(π) where the third equality follows from the definition of Win Appendix A1. For any g∈G,foreachs∈Sand each (cg)in the support of g(s) ∈(C×G), there exists a risk equivalent (cm)∈C×Msuch that (cm)∼sts (cg).Letg+1be a one-step-ahead act such that g+1(s)(L)=g(s)(L) holds for all pairs L⊂C×Gand L⊂C×M,whereLconsists of all risk equivalents (cm)of corresponding elements (cg)in L. By construction, g+1(s) and g(s) are stochastically equivalent. By Axiom A5 (Dynamic Consistency), (cg) ∼st(c g+1). Therefore, Vst(c g) =Vst(c g+1) =Wcv−1Pst v◦u−1 s∈S π(s) ×C×M u(V (cm)) dg+1(s)(cm)dμst(π) =Wcv−1◦Pst v◦u−1 s∈S π(s) ×C×G u(Vsts(cg))dg(s)(cg)dμst(π) wherewehaveusedthefactthatg(s) and g+1(s) are stochastically equivalent to derive the second equality. A3 Proof of uniqueness Suppose ({˜ Vst}˜ W˜ u ˜ v{˜μst})and ({Vst}Wuv{μst})represent the same preference. On the domain of deterministic consumption streams C∞, each ˜ Vstcoincides with the
454 Hayashi and Miao Theoretical Economics 6 (2011) common function ˜ Vand each Vstcoincides with the common function V. Since ˜ Vand Vare ordinally equivalent over C∞, there is a monotone transformation such that ˜ V(y)=◦V(y) for all y∈C∞ By (29), we have ˜ Vst=◦Vst. Since ˜ W(c ˜ V(y))=˜ V(cy)=(V (cy)) =(W (cV (y))) =(W (c−1(˜ V (y)))) we deduce that ˜ W(c·)=(W (c−1(·))). On M,C×Mu(V (cm)) dm(cm)and C×M˜ u( ˜ V(c m)) dm(cm)are equivalent mixture-linear representations of the risk preference conditional on the fixed current consumption c. Therefore, there exist constants AB with A>0such that ˜ u( ˜ V(c m)) =Au(V (cm)) +Bfor all (cm)∈M Since ˜ V=◦V,weobtain ˜ u◦=Au +B. By construction from Appendix A2,˜ v◦˜ u−1◦˜ ¯ u=ψ=v◦u−1◦¯ u.By(10), we compute ˜ ¯ u(m) =C×M ˜ u( ˜ V(c m)) dm(cm) =C×M ˜ u◦(V (cm)) dm(cm) =C×M Au(V (cm)) dm(cm)+B =AC×M u(V (cm)) dm(cm)+B=A¯ u(m) +B Let ¯ u(m) =w.Thenwehave ˜ v◦˜ u−1(Aw +B) =v◦u−1(w) Since ˜ u◦(w) =Au(w) +B, it follows that ˜ v◦˜ u−1(Aw +B) =˜ v◦˜ u−1(Au ◦u−1(w) +B) =˜ v◦(u−1(w)) Thus, we obtain ˜ v◦(u−1(w)) =v◦u−1(w) By replacing u−1(w) with x,weobtain˜ v◦(x) =v(x) Finally, uniqueness of μstfollows from Axiom A5. Appendix B: Proof of Theorem 3 Given a compact metric space Y,letB(Y) be the family of Borel subsets of Yand let (Y) be the set of Borel probability measures defined over B(Y), which is again a compact metric space with respect to the weak convergence topology. Inductively define the
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 455 family of domains {H0H1}by H0=((C))S H1=(C×(H0))S Ht=(C×(Ht−1))S and so on. By induction, (C×(Ht−1)) is a compact metric space and so is Ht,for every t≥0.Letdtbe the metric over Ht.LetH∗=∞ t=0Ht. This is a compact metric space with respect to the product metric d(hh)=∞ t=0(1/2t)dt(hth t)/(1+dt(hth t)). Thedomaintobeconstructedisasubset of H∗, which consists of coherent acts. Defineamappingπ0:C×(H0)→Cby π0(c p0)=c for each (cp0)∈C×(H0). Define a mapping ρ0:H1→H0by ρ0(h1)(s)[B0]=h1(s)[π−1 0(B0)] for each h1∈H1,s∈S,andB0∈B(C). Define a mapping ˜ρ0:(H1)→(H0)by ˜ρ0(p1)[H0]=p1[ρ−1 0(H0)] for each p1∈(H1)and H0∈B(H0). Similarly, define π1:C×(H1)→C×(H0)by π1(c p1)=(c ˜ρ0(p1)) for each (cp1)∈C×(H1),defineρ1:H2→H1by ρ1(h2)(s)[B1]=h2(s)[π−1 1(B1)] for each h2∈H2,s∈,andB1∈B(C×(H0)),anddefine ˜ρ1:(H2)→(H1)by ˜ρ1(p2)[H1]=p2[ρ−1 1(H1)] for each p2∈(H2)and H1∈B(H1). Inductively, given πt−1:C×(Ht−1)→C×(Ht−2),ρt−1:Ht→Ht−1,and ˜ρt−1:(Ht)→(Ht−1),defineπt:C×(Ht)→C×(Ht−1)by πt(c pt)=(c ˜ρt−1(pt)) for each (cpt)∈C×(Ht),defineρt:Ht+1→Htby ρt(ht+1)(s)[Bt]=ht+1(s)[π−1 t(Bt)]
456 Hayashi and Miao Theoretical Economics 6 (2011) for each ht+1∈Ht+1,s∈S,andBt∈B(C×(Ht−1)),anddefine ˜ρt:(Ht+1)→(Ht)by ˜ρt(pt+1)[Ht]=pt+1[ρ−1 t(Ht)] for each pt+1∈(Ht+1)and Ht∈B(Ht). Define H={h=(h0h1h2)∈H∗:ht=ρt(ht+1) t ≥0} For each s∈S, the sequence (h0(s)h1(s) h2(s) ) ∈∞ t=0(C×(Ht−1)) is viewed as a sequence of constant acts since h0(s) ∈(C)⊂H0 h1(s) ∈(C×(H0)) ⊂H1 ht(s) ∈(C×(Ht−1)) ⊂Ht and so on. The lemmas below verify that such constant acts are also coherent. They are immediate from the definition of H. Lemma 2. For every h∈Hand s∈S, the sequence (h0(s) h1(s) h2(s) ) ∈ ∞ t=0(C×(Ht−1)) satisfies ht(s) =ρt(ht+1(s)). Lemma 3. For every t≥0,ht∈Ht,andht+1∈Ht+1,ifht(s) =ρt(ht+1(s)) for every s∈S, then ht=ρt(ht+1). Let Q=(qt)∈ ∞ t=0 (Ht):qt=˜ρt(qt+1)∀t≥0 A=(at)∈ ∞ t=0 (C×(Ht−1)):at=ρt(at+1)∀t≥0 Lemma 4. We have the homeomorphic relation A≃(C×Q) Proof.Given(at)∈A⊂∞ t=0(C×(Ht−1)), by the Kolmogorov extension theorem there exists a unique a∈(C×∞ t=0(Ht−1)) such that mrgC×(Ht−1)a=at for each t≥0, where mrg denotes marginal. Define a mapping ξ:A→(C×Q) by ξ((at)) =a.
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 457 We need to show a∈(C×Q).Foreacht≥0,let Qt={(qtqt+1)∈(Ht)×(Ht+1):qt=˜ρt(qt+1)}× τ=tt+1 (Hτ) We derive that a(C×Qt)=mrgC×(Ht)×(Ht+1)aC×{(qtqt+1)∈(Ht)×(Ht+1):qt=˜ρt(qt+1)} =mrgC×(Ht)aCטρt((Ht+1)) =at+1Cטρt((Ht+1)) =at+1πt+1(C×(Ht+1)) =ρt+1(at+2)πt+1(C×(Ht+1)) =at+2π−1 t+1πt+1(C×(Ht+1)) =at+2(C×(Ht+1)) =1 Therefore, a(C×Q) =a∞ t=0 (C×Qt)=lim T→∞ aT t=0 (C×Qt)=1 •Mapping ξis one-to-one: This follows from the uniqueness of Kolmogorov extension theorem. •Mapping ξis onto: For every a∈(C×Q), the inverse is given by (at)∈ ∞ t=0(C×(Ht−1)) such that at=mrgC×(Ht−1)a for each t≥0.Toshow(at)∈A,takeanyBt∈B(C×(Ht−1)).Wededucethat at(Bt)=aBt× τ=t−1 (Ht) ≥a(c (qτ)) ∈C×Q:(cqt)∈Bt =a(c (qτ)) ∈C×Q:(c ˜ρt(qt+1)) ∈Bt =a(c (qτ)) ∈C×Q:(cqt+1)∈π−1 t(Bt) =at+1(π−1 t(Bt)) =ρt(at+1)(Bt) Since 1=at(C×(Ht−1)) =at(Bt)+at(Bc t) ≥ρt(at+1)(Bt)+ρt(at+1)(Bc t)=ρt(at+1)(C×(Ht−1)) =1
458 Hayashi and Miao Theoretical Economics 6 (2011) we obtain at(Bt)=ρt(at+1)(Bt) •Mappings ξand ξ−1are continuous: This is immediate from the nature of the product topology. Lemma 5. We have the homeomorphic relation H≃AS Proof.Defineξ:H→ASby ξ(h)(s) =(h0(s)h1(s) h2(s) ) It follows from Lemma 2that ξ(h) ∈A. •Mapping ξis one-to-one: Suppose ξ(h) =ξ(h). By definition of ξ,wehave (h0(s) h1(s) h2(s) ) =(h 0(s)h 1(s)h 2(s) ) for all s∈S, which implies h=h. •Mapping ξis onto:Takeany˜ h∈AS. By definition, ˜ h(s) =(˜ h0(s) ˜ h1(s) ˜ h2(s) ) ∈ ∞ t=0 (C×(Ht−1)) for each s∈S.Thenξ−1(˜ h) =(h0h1h2)∈H∗satisfies ht(s) =˜ ht(s) for each tand s. By Lemmas 2and 3, the sequence (h0h1h2) is coherent and hence ξ−1(˜ h) ∈H. •Mappings ξand ξ−1are continuous: This is immediate from the nature of the product topology. Let P∗=(pt)∈ ∞ t=0 t τ=0 Hτ:mrgt τ=0Hτpt+1=pt Lemma 6. For any (pt)∈P∗, there exists a unique p∈(H∗)such that mrgt τ=0Hτp=pt Moreover, there exists a homeomorphism χ:P∗→(H∗). The proof follows from Lemma 1 in Brandenberger and Dekel (1993). Let Ht=(h0ht)∈ t τ=0 Hτ:hτ=ρτ(hτ+1)τ =0t−1 for each t≥0and let P={(pt)∈P∗:pt(Ht)=1t ≥0}
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 459 Lemma 7. The equality χ(P) =(H)holds. As a result, P≃(H)holds through χ. Proof.The⊂part:Letp=χ((pt)) for some (pt)∈P.Let t=Ht× ∞ τ=t+1 Hτ for each t≥0.ThenwehaveH⊂t⊂H∗for each t≥0,(t)is decreasing, and t≥0t=H. Since pis the Kolmogorov extension of (pt),wehave p(t)=pt(Ht)=1 for every t≥0.Thus,p(H)=p(t≥0t)=lim p(t)=1. The ⊃part: Pick any p∈(H)that satisfies p(H)=1.Let(pt)be the sequence of marginals defined by pt=mrgt τ=0Hτpfor each t≥0.Thenpt(Ht)=p(t)≥1,where the second inequality follows from t⊃H. Since ptis a probability measure, we have pt(Ht)=1. Since pis the Kolmogorov extension of (pt),wehavep=χ((pt)). Lemma 8. For every (qt)∈Q, there exists a unique (pt)∈Psuch that mrgHtpt=qt Moreover, Qand Pare homeomorphic. Proof. Define a sequence of mappings (ξt),ξt:Ht→t τ=0Hτfor each t≥0,by ξt(ht)=( h0 ht) where ht=htand hτ=ρτ( hτ+1)for τ=01t−1. By construction, each (ξt)is a one-to-one mapping and ξt(Ht)=Ht. Therefore, we can define the sequence of inverse mappings (ξ−1 t),ξ−1 t:Ht→Htgiven by ξ−1 t(h0ht)=ht which is a projection mapping that is continuous. For (qt)∈Q, define the corresponding sequence (pt)∈Pby pt(Et)=qt(ξ−1 t(Et)) for each Et∈B(t τ=0Hτ)and t≥0.Wecanseethat(pt)∈Psince pt(Ht)= qt(ξ−1 t(Ht)) =qt(Ht)=1. By construction, mrgHtpt=qtfor each t≥0. Now, Theorem 3 follows from the fact that H≃AS,A≃(C×Q),Q≃P,and P≃(H).
460 Hayashi and Miao Theoretical Economics 6 (2011) Finite-step-ahead acts and denseness Finally, we define finite-step-ahead acts and show that the union of all the sets of finitestep-ahead acts is dense. Let H+1=h+1∈(C×(H))S:∀s∈Sh+1(s) ∈(C×(L)) Since H≃((C×(H)))S,wecanembedH+1into H, where the range of H+1is embedded into Lsince L≃(C×(L)). Inductively, define H+τ=h+τ∈(C×(H))S:∀s∈Sh+τ(s) ∈(C×(H+(τ−1))) Similarly, we can embed H+τinto H.Wecallτ≥1H+τthe domain of finite-step-ahead acts. Lemma 9. The domain of finite-step-ahead acts τ≥1H+τis a dense subset of H.Also, τ≥1(C×(H+τ)) is a dense subset of (C×(H)). This result is analogous to Proposition 1 in Hayashi (2005) and its proof is omitted. It is useful to establish the existence of a risk equivalent as in Lemma 9 of Hayashi (2005). We implicitly applied a similar result in Appendix A. Appendix C: Proofs of Theorems 4and 5 We prove the sufficiency of the axioms. The proof of necessity is routine. C1 Representation of risk preference When {st}is restricted to the domain C×(L),Axiom B3 (History Independence of Risk Preference) implies that {st}induces a single preference relation defined on C×(L). By Axiom B1 (Order) and Debreu’s (1954) theorem, there is a continuous representation V:C×(L)→Rof . We fix such a representation. By Axiom B2 (Current Consumption Separability), V(c·)and V( c·)represent the same ranking over (L),henceVhas the form V(ca)= W(cV( ca)) ∀(ca) ∈C×(L)(32) for some function Wthat is strictly increasing in the second argument. Because of Axiom B4 (First-Stage Independence), V( ca) has the form V( ca) =ζL U(l)da(l)(33) where ζis a strictly increasing function and Uis a vNM index. Because of Axiom B5 (Second-Stage Independence), Uhas the form U(l)=φC×(L) u(ca)dl(ca)(34)
Theoretical Economics 6 (2011) Substitution and ambiguity preferences 461 where φis a strictly increasing function and uis a vNM index. By Axiom B6 (Dynamic Consistency) and a similar argument as in Appendix A1, uand Vare ordinally equivalent. Hence, we deduce that u(ca)=u(V (ca)) (35) where uis a strictly increasing function. Plugging (33), (34), and (35) into (32) yields V(ca)= WcζL φC×(L) u(V (ca)) dl(ca)da(l) Now define Wby W(cx)= W(cζ◦φ◦u(x)) which is strictly increasing in the second argument. Then we have W(cζ(z))=W(cu −1◦φ−1(z)) (36) and hence, V(ca)=Wcu−1◦φ−1L φC×(L) u(V (ca)) dl(ca)da(l) Let v=φ◦u. We obtain representation (14). C2 Extension to the whole domain Define Vst:C×(H)by Vst(c p) =V(ca) (37) for each (cp) ∈C×(H),wherea∈(L)is such that (cp) ∼st(ca). The existence of such a risk equivalent afollows from Lemma 9, Dynamic Consistency (Axiom B6), compactness of C, and continuity of st(see Lemma 9 in Hayashi 2005). Using definition (37)and(32), we derive Vst(c p) =V(ca)= W(cV( cm)) = W(cV st( c p)) (38) When our Axioms B1,B4,B5,andB7 are restricted to (H+1), they satisfy the conditions in Theorem 4.2 in Seo (2009). By this theorem, Vst( c·)restricted to (H+1)is ordinally equivalent to a second-order subjective expected utility representation, and hence has the form Vst( cp+1)=ζstH+1 Ust(h+1)dp+1(h+1)(39) and Ust(h+1)=Pst φst s∈S π(s)C×(L) ust(ca)dh+1(s)(ca)dμst(π) (40)
462 Hayashi and Miao Theoretical Economics 6 (2011) where ζstand φstare strictly increasing functions and ustis a vNM index. By Axiom B6 (Dynamic Consistency) and a similar argument in Appendix A1, ustand Vare ordinally equivalent over C×(L). Thus, there is a monotone transformation ustsuch that ust(ca)=ust(V (ca)) (41) for every (ca)∈C×(L). Equations (33)and(37) imply that on (L), ζL U(l)da(l)=V( ca) =Vst( ca) =ζstL Ust(l)da(l) whichinturnimpliesthatonL, ζ(U(l)) =V( cδ[l])=Vst( cδ[l])=ζst(Ust(l)) Hence, we deduce that Ust=ζ−1 st◦ζ◦U which implies that ζL U(l)da(l)=ζstL ζ−1 st◦ζ◦U(l)da(l) By the additivity of integral formula, we have ζ−1 st◦ζ(αx +(1−α)y) =αζ−1 st◦ζ(x) +(1−α)ζ−1 st◦ζ(y) for all x y in the range of Uand all α∈[01]. Therefore, ζstand ζare identical up to positive affine transformations. Thus, without loss of generality, we can take ζst=ζand Ust=Ufor all st. Equations (34), (35), (40), and (41) imply that on L, φC×(L) u◦V(c a)dl(ca)=U(l)=Ust(l) =φstC×(L) ust◦V(c a)dl(ca) whichinturnimpliesthat φ◦u◦V(c a)=U(δ[ca])=Ust(δ[ca])=φst◦ust◦V(c a) Hence, we have φ◦u=φst◦ust, which implies that φC×(L) u◦V(c a)dl(ca)=φstC×(L) φ−1 st◦φ◦u◦V(c a)dl(ca) By the same reasoning as above, φstand φare identical up to positive affine transformations. Therefore, without loss of generality, we can set φst=φand ust=ufor all st.
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