Intertemporal Substitution and Recursive Smooth Ambiguity ...

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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
Page 1: Intertemporal Substitution andRecur
Page 8 and 9: 2. Review of the Atemporal ModelsIn
Page 10 and 11: defined over it. Notice that by res
Page 12: For any c ∈ C, we use δ[c] to de
Page 15: The axiom below states that the pre
Page 18: is ambiguity averse if he prefers a
Page 22 and 23: Axiom B6 (Dynamic Consistency) For
Page 24 and 25: 3. On the subdomain C × M, we obta
Page 26 and 27: We can similarly define ambiguity l
Page 28 and 29: epresentations under these two appr
Page 30 and 31: where R t+1 is the market return fr
Page 32 and 33: Segal (1987, 1990) and Seo (2009).
Page 34 and 35: A Appendix: Proof of Theorems 1 and
Page 36 and 37: Define v = ψ ◦ ū −1 ◦ u, wh
Page 38 and 39: compute:∫˜ū (m) =∫=∫=∫= A
Page 40 and 41: DefineH = {h = (h 0 , h 1 , h 2 ,
Page 42 and 43: Lemma 5 We have the homeomorphic re
Page 44 and 45: Finite-step-ahead acts and densenes
Page 46 and 47: which is strictly increasing in the
Page 48 and 49: where the second equality follows f
Page 50 and 51: D Appendix: Proofs for Section 4.4P
Page 52 and 53: This relation holds true because i
Page 54 and 55: In complete markets, the following
Page 56 and 57:
Ergin, H. I. and F. Gul (2009): “
Page 58:
Weil, P. (1989): “The Equity Prem
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