R ecursiv e Sm ooth A mbiguit y Preferences1Peter Klib an o ffME DS Dept., Kellogg School of Managem ent, North western [email protected] estern.eduMa ssim o M arinacciCollegio C a rlo A lberto,Università di Torinomassimo.ma rin [email protected] y MukerjiDept. of Economics, University of Oxfordsujo [email protected] v ersion: August 3, 2007First version: July 20061We thank Nabil Al-Najjar, Paolo Ghirardato, Christian Gollier, Eran Hanany, Ian Jewitt,Mark Machina, Luigi Montrucchio, and Kevin Sheppard for helpful discussion, Andreas Langefor point ing out an error in a previous version and seminar audiences at the Cowles Foundationworkshop “Uncertainty in Economic Theory,” the University of Iowa, Harvard/MIT, the Univ ersityof Texas, Austin, the RUD 2005 conference, the 9thWorld Congress of the Econometric Societyand the FUR XII conference for their comments.AbstractThis paper axiomatizes an in tertemporal v ersion of the Smooth Ambiguit y decision modeldevelopedinKlibanoff, Marinacci, and Muk erji (2005). A key feature of the model is that itac h ieves a separation betw een ambiguity, iden tified as a c ha racteristic of the decision maker’ssubjectiv e beliefs, and ambiguity attitude, a characteristic of the decision ma ker’s tastes. Inapplications one may thus specify/vary these two characteristics independent of each other,thereby facilitatin g ric her comp ara tive sta tics and m odeling flexibility than possible und erother models whic h accomodate am biguity sensitive preferen ces. Another k ey feature is thatthe preferences are dynamically consistent and ha ve a recursiv e representation. Thereforetec h niq ues of dyn am ic program m ing can be applied when using th is model.JEL Classification N u mbers: D800, D810.Keywords: Am biguity, Uncertainty, Knigh tian Uncertainty, Ambiguity Aversion, Uncer-tain ty Aversion, Ellsberg P arad ox, Dynam ic Decision Making, Dynamic Program m in g underAmbiguity, Sm ooth A mbiguity.Corresponding A u thor: Massim o Marina cci, C ollegio Carlo Alberto, Via Real Collegio,30, 10024 M o ncalieri (Torino), Italy; e-mail: massimo .m a rina cci@ u nito.it1 IntroductionThis paper axiomatizes and in vestigates a model of recursive preferences o ver in tertemporalplans, extendin g the smooth ambiguit y m odel developed in Klibanoff, Marinacci, and Mu k erji(2005) (henceforth KMM ) to a setting invo lving dynamic decision m a king.In KMM we propose and axiom atize a model of preferences over acts such that thedecision m a ker prefers act f to act g if an d only if Eµφ (Eπu ◦ f) ≥ Eµφ (Eπu ◦ g),whereEis the expectation operator, u is a v N -M utility functio n, φ is an increasing tran sform atio n,and µ is a subjectiv e probability ov er the set Π o f p ro ba bility m ea sures π that the decisionmaker thinks are relevan t given his subjective information. A key feature of our m odel isthat it achieves a separation between ambiguit y, identified as a c ha racteristic of the decisionmaker’s su bjective beliefs, and am bigu ity attitu de, a chara cter istic of the decision maker’stastes. We sho w that attitudes towards pure risk are c ha racterized b y the shape of u,asusual, while attitudes to wards ambiguit y are characterized by the shape of φ. Am biguityitself is defined beha v iorally and is show n to be c ha ra cterized b y properties of the subjectiv eset of measures Π. One advantage of this model is that the w ell-d eveloped machinery fordealing with risk attitudes can be applied as well to ambiguity attitudes. The m odel isalso distinct from ma ny in the literature on ambiguity in t ha t it allo w s sm ooth, rather thankinked, indifference curv es. This leads to differ ent beha vior and improved tractability, w hilestill sharing the main feature s (e.g., Ellsberg’s P a radox). The maxm in expected utilit y model(e.g., Gilboa and Sc hmeidler (1989)) with a given set of measu res may be seen as a limitingcase o f ou r model w ith infinite a mbiguity aversion.1The functional rep resentation obtained in KMM is particularly useful in economic mod-eling in answ ering comparativ e statics questions in volving ambiguity. Tak e an economicmodelwhereagents’beliefsreflect some am biguit y. Next, without perturbing the infor-mation structure, it is useful to know how the equilibrium w ould change if the exten t ofambiguity aversion were to decrease; e.g., if we were to replace ambiguit y aversion withambiguity neutrality, ho lding inform a tion and risk attitude fixe d. (See, fo r exam p le, Gollier(2005) for a portfolio choice app lication.) Another useful comparative statics exercise isto hold ambiguity attitudes fixed and ask how the equilibrium is affected if the perceiv edambiguit y is varied (see Jewitt and Muk erji (2006) for a definition and c haracterization ofthe notion of "more am bigu ous"). Working out such comp arative statics pro perly requires amodel which allo w s a conceptual/parametric separation of (possibly) ambiguous beliefs andambiguit y attitude, analogous to the distinction usually mad e bet ween risk and risk attitude.The model and functional representation in KM M allows that, whereas suc h a separationis not eviden t in the pioneerin g and mo st popular decision making models that incorporateambiguity, namely, the multiple priors/ m a x m in expected utilit y (M EU) preferences (G ilboaand Schmeidler (1989)) an d the Choquet expected utilit y (C EU) model of Sch m eidle r (1989).Wh ile the preference model in KM M achiev es the task of separating ambiguit y and am -biguity attitude, the scope of application of this model is limited b y the fact that it is atimeless framew o rk. Yet man y economic question s inv o lving uncertain en vironmen ts, es-pecially in ma croeconomics and finance, are more in tu itively modeled using intertemporal1For alternative developments of similar models see Ergin and Gul (2004), Nau (2006), Neilson (1993)and Seo (2006). All of these models draw inspiration from Segal (1987), the earliest paper relating ambiguitysensitive behavior to a two-stage functional relaxing reduction.1decision making frameworks. It is of interest to re-examine such questions b y adding anambiguit y dim ension. C om putation an d analysis of in tertemporal choices is greatly facili-tated by applying recur sive methods. For these m eth ods to be applicable, preferen ces hav eto satisfy a certain dynam ic consistency propert y. A number of recent papers,