DYNAMIC PREFERENCE FOR FLEXIBILITY. R. VIJAY KRISHNA Duke University, Durham, NC 27708, U.S.A.

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1 Econometrica, Vol. 82, No. 2 (March, 2014), DYNAMIC PREFERENCE FOR FLEXIBILITY R. VIJAY KRISHNA Duke Univerity, Durham, NC 27708, U.S.A. PHILIPP SADOWSKI Duke Univerity, Durham, NC 27708, U.S.A. The copyright to thi Article i held by the Econometric Society. It may be downloaded, printed and reproduced only for educational or reearch purpoe, including ue in coure pack. No downloading or copying may be done for any commercial purpoe without the explicit permiion of the Econometric Society. For uch commercial purpoe contact the Office of the Econometric Society (contact information may be found at the webite or in the back cover of Econometrica). Thi tatement mut be included on all copie of thi Article that are made available electronically or in any other format.

2 Econometrica, Vol. 82, No. 2 (March, 2014), DYNAMIC PREFERENCE FOR FLEXIBILITY BY R. VIJAY KRISHNA AND PHILIPP SADOWSKI 1 We conider a deciion maker who face dynamic deciion ituation that involve intertemporal trade-off, a in conumption aving problem, and who experience tate hock that are tranient contingent on the tate of the world. We axiomatize a recurive repreentation of choice over tate contingent infinite horizon conumption problem, where uncertainty about conumption utilitie depend on the obervable tate and the tate follow a ubjective Markov proce. The parameter of the repreentation are the ubjective proce that govern the evolution of belief over conumption utilitie and the dicount factor; they are uniquely identified from behavior. We characterize a natural notion of greater preference for flexibility in term of a dilation of belief. An important pecial cae of our repreentation i a recurive verion of the Ancombe Aumann model with parameter that include a ubjective Markov proce over tate and tate-dependent utilitie, all of which are uniquely identified. KEYWORDS: Dynamic choice, tate hock, continuation trategic rationality, recurive Ancombe Aumann model. 1. INTRODUCTION UNCERTAINTY ABOUT FUTURE CONSUMPTION UTILITIES influence how economic agent make deciion. A deciion maker (DM) who i uncertain about future conumption utilitie, perhap due to uncertain future rik averion or other tate hock, prefer not to commit to a coure of future action today and, therefore, ha a preference for flexibility. For example, DM might be willing to forfeit current conumption if thi allow him to delay a deciion about future conumption. While thi intuition i inherently dynamic, tandard model that accommodate preference for flexibility, mot prominently Krep (1979) and Dekel, Lipman, and Rutichini (2001) (henceforth DLR 2 ), are tatic in the ene that there i no intertemporal trade-off (although Krep (1979) ugget that an infinite horizon model of preference for flexibility i deirable). In thi paper, we conider a dynamic environment, allow DM tate hock to have an unobervable tranient a well a an obervable peritent component, and provide foundation for a recurive repreentation of DM preference that i fully identified. We conider tate contingent infinite horizon conumption problem (S-IHCP) a the domain of choice. Given the collection of relevant tate of the world, S-IHCP are defined recurively a act that pecify, for each tate of the world, a deciion problem, which i a menu (i.e., a cloed et) of lot- 1 We would like to thank, without implicating, Haluk Ergin, Sujoy Mukerji, Wolfgang Peendorfer, Todd Sarver, Norio Takeoka, ix anonymou referee, and Philippe Jehiel for helpful comment and uggetion, and Vivek Bhattarcharya, Matt Horne, and Jutin Valaek for valuable reearch aitance. Krihna gratefully acknowledge upport from the National Science Foundation (Grant SES ). 2 A relevant corrigendum i Dekel, Lipman, Rutichini, and Sarver (2007) (henceforth DLRS) The Econometric Society DOI: /ECTA10072

3 656 R. V. KRISHNA AND P. SADOWSKI terie that yield a conumption prize in the preent period and a new S-IHCP tarting in the next period. For a imple example of an S-IHCP that feature only degenerate lotterie, uppoe DM ha fixed income y in every period and current wealth w. The relevant tate of the world i the per period price of conumption, ρ>0. In each period, DM can chooe to conume an amount ] at cot mρ. Thi will leave him with wealth w = w + y mρ for the next period. Given thi technology, we can formulate the conumption problem he face recurively a a price contingent collection of feaible pair of current conumption and a new conumption problem for the next period, m [0 w+y ρ f w (ρ) := { (m f w ) : m [ 0 w + y ρ ] w = w + y mρ In the cae where the tate pace i degenerate (in the example thi correpond to ρ being contant), our domain reduce to that of Infinite Horizon Conumption Problem (IHCP) introduced by Gul and Peendorfer (2004), henceforth GP. The domain of S-IHCP i rich and allow for complicated future choice behavior. 3 For example, future choice may depend directly on time or on conumption hitory. We rule out uch dependencie o a to tay a cloe to the tandard model a poible. 4 A deciion maker who doe not anticipate any tate hock, contingent on the tate of the world, i not avere to committing to a tate contingent plan of choice. Following Krep (1979), we refer to thi property a State Contingent Strategic Rationality. In contrat, a deciion maker who anticipate tate hock hould atify Monotonicity, that i, he alway weakly prefer to retain additional option. We conider tate hock that are tranient (relevant only for one period) contingent on the tate of the world. DM hould then be trategically rational with repect to hi choice of continuation problem beginning in the next period. Thi i the content of our main behavioral axiom State Contingent Continuation Strategic Rationality (S-CSR). Conceptually, full tate contingent trategic rationality would greatly implify the analyi of DM behavior. If it i violated with repect to all S-IHCP becaue of unobervable tate hock, then it i deirable to undertand the nature of thi violation by aking if DM i trategically rational with repect to ome maller cla of S-IHCP, for example, a in S-CSR. Baed on S-CSR, we provide a repreentation of dynamic preference for flexibility (DPF) that i the olution to a Bellman equation and can, therefore, be analyzed uing tandard dynamic programming technique. The evolution } 3 We only model the initial choice of a conumption problem, but our repreentation ugget dynamically conitent future choice. 4 Dependence of choice on the conumption hitory i central to the notion of habit. We are independently working on a model of habit formation on the domain of IHCP.

4 DYNAMIC PREFERENCE FOR FLEXIBILITY 657 of uncertainty about future conumption utilitie i driven by the evolution of the (obervable) tate of the world, which in turn follow a ubjective Markov proce. All the parameter of the repreentation, which are the tranition operator on the tate pace, tate contingent belief over conumption utilitie, and the dicount factor, are uniquely identified. In contrat, the tatic model of DLR cannot identify belief over conumption utilitie. The domain of S-IHCP accommodate important pecial cae. If the obervable tate pace i degenerate (o there are no obervable hock), then S-CSR become Continuation Strategic Rationality (CSR) on the domain of IHCP. The reulting repreentation feature unobervable tranient tate hock that are independent and identically ditributed (i.i.d.). If, intead, the analyt can only oberve nontrivial preference over tate contingent infinite horizon conumption tream (which provide no flexibility), rather than preference over all S-IHCP, then the model reduce to a fully identified recurive verion of the Ancombe Aumann model with tate dependent utilitie. The DPF repreentation i a generalization of tate hock model that are commonly ued in applied work. Our axiom pell out the tetable implication of uch model. In particular, if the model feature i.i.d. hock, then choice hould atify CSR. To tet thi aumption, conider an agent who mut order inventory for the coming eaon. For example, a farm may need to order eed for the coming planting eaon and face hock to their tate for eed due to unmodelled environmental condition. Suppoe the order can either be placed long in advance or hortly before the eaon tart. If the agent i willing to pay a premium to delay placing the order, then CSR i violated and, hence, the i.i.d. model i mipecified: Tate hock mut then have a peritent component. It i poible that thi peritent component can be captured by eaily obervable environmental condition, uch a rainfall prior to the tart of the eaon. In that cae, choice hould atify S-CSR, which can be teted in a imilar fahion. Our identification reult allow u to characterize a notion of greater preference for flexibility in term of a dilation (which i a multidimenional mean preerving pread) of belief over conumption utilitie. Identification i alo relevant for model baed forecating, which i a central reaon for the ue of formal model. It involve etimating a relevant parameter in one context o a to forecat outcome in another. Model baed forecating, thu, require that (i) the relevant parameter i uniquely identified and (ii) the modeller i willing to aume that the parameter i meaningful outide the context of the oberved data. In our model, belief over conumption utilitie are uniquely identified. Thoe belief might, for example, forecat future choice frequencie of alternative from continuation problem, a dicued in Sadowki (2013). The remainder of the paper i tructured a follow. Section 2 review related literature. Section 3 lay out the baic framework, introduce the repreentation, explain the axiom, and derive our main theorem, which etablihe that the axiom are the behavioral content of the repreentation. Section 4 dicue

5 658 R. V. KRISHNA AND P. SADOWSKI important pecial cae. Section 5 concern the empirical content of our theory, where Section 5.1 dicue the lack of identification in the tatic model of DLR and provide a direct intuition for the identification of the parameter in our dynamic model, Section 5.2 ugget pecific choice experiment from which an experimenter could elicit thoe parameter, and Section 5.3 provide a characterization of greater preference for flexibility in term of the parameter. Section 6 conclude. All proof, a well a a general repreentation of preference for flexibility with an infinite prize pace, are provided in the Appendice. 2. RELATED LITERATURE Our work build on tandard axiomatic model of preference for flexibility, which follow Krep (1979) and invetigate choice over menu of conumption outcome. The econd eminal paper in thi literature, DLR, modifie the domain to conider menu of lotterie over outcome a object of choice. Thi facilitate the interpretation of the ubjective tate a tate, where a tate i imply a (twice normalized) von Neumann Morgentern (vn-m) ranking over conumption outcome. While menu choice can capture DM attitude toward the future, implied choice in thee model i actually tatic, in the ene that there i no opportunity for any intertemporal trade-off. We provide a model of preference for flexibility due to uncertain conumption utilitie in the tradition of thee earlier paper, where the implied choice i dynamic. A mentioned before, if the et of obervable tate i degenerate, then our recurive domain of S-IHCP reduce to that of IHCP, firt analyzed by GP, who how that thi recurive domain i well defined. GP provide a dynamic model of conumption with temptation preference. We allow, intead, for uncertain utilitie. The pace of poible utilitie on our dynamic domain i infinite dimenional, which complicate the analyi. Higahi, Hyogo,and Takeoka (2009) alo conider preference for flexibility on the domain of IHCP. Their model only accommodate preference for flexibility due to a random dicount factor that follow an i.i.d. proce, but not due to, ay, uncertainty about rik averion, a explained in Section 4.2. Our model net their in term of behavior. Takeoka (2006) conider choice between menu of menu of lotterie and derive a three period verion of DLR. The model cannot capture intertemporal trade-off and, conequently, the repreentation i only jointly identified, a in DLR. Rutichini (2002) conider preference over lotterie over et of infinite conumption path. While an additive repreentation can be obtained, the domain preclude a recurive value function and doe not allow unique identification of the repreentation. Static model of preference for flexibility lack identification becaue the et of poible utilitie i the ubjective tate pace, which mean that utilitie are (trivially) tate dependent. Thee model, therefore, uffer from the ame lack of identification a the Ancombe and Aumann (1963) model with tatedependent utilitie. DLR ugget the introduction of a numeraire good (with

6 DYNAMIC PREFERENCE FOR FLEXIBILITY 659 tate-independent valuation) to their model, o a to identify belief uniquely in the ame ene that they are uniquely identified in the Ancombe and Aumann model with tate independence. 5 If the objective tate pace i degenerate, then the ource of identification in our repreentation i imilar, where the numeraire arie naturally in the form of continuation problem. In contrat, continuation utilitie in the DPF repreentation are tate dependent, and identification relie directly on the recurive tructure of the repreentation (Section 5.1 provide intuition for thi reult). Sadowki (2013) identifie belief in a ituation where preference for flexibility depend on the tate of the world by requiring the obervable tate to contain information that i ufficiently relevant for future preference. Intead of choice between menu, Gul and Peendorfer (2006) tudy random choice from menu. Oberved choice frequencie naturally correpond to a unique meaure over utilitie, but the caling of utilitie remain arbitrary in their model. Ahn and Sarver (2013) imultaneouly model choice between menu and random choice from menu. They achieved full identification by requiring the belief in the repreentation of choice between menu to correpond to frequencie of choice from menu. Dillenberger, Llera, Sadowki, and Takeoka (2013) fully identify repreentation of preference for flexibility that feature uncertain future belief, rather than uncertain future tate. That a deciion maker can be uncertain about future preference in a dynamic etting i noted by Koopman (1964), who ditinguihe between onceand-for-all planning, where the agent elect an action for all poible future contingencie, and piecemeal planning, where, in each period, the agent chooe an action for the current period while imultaneouly narrowing the et of alternative for the future. Jone and Otroy (1984) conider a nonaxiomatic (but dynamic) model of choice, where an agent prefer flexibility due to uncertainty about future utilitie. They point out that preference for flexibility, for example, in the form of preference for liquidity, i a pervaive theme in economic, and they relate many intance in macroeconomic and finance where DM dynamic behavior exhibit preference for liquidity in particular and preference for flexibility in general, a in Goldman (1974). Theyalodicu a notion of greater variability of belief, which roughly correpond to our notion of a dilation of belief. A pecial cae of our repreentation (ee Corollary 6) i ued in empirical work by Hendel and Nevo (2006), who tudy the problem of a deciion maker who maintain an inventory. A growing literature argue that variation in preference over time in particular, in rik averion are central in explaining variou market behavior. Indeed, etimated rik averion ha been uggeted a a ueful index of market entiment (ee, for example, Bollerlev, Gibon, and Zhou (2011)), and there i evidence that the larget component of change of the equity rik premium i variation in rik averion, rather than the quantity of rik (ee Smith and 5 Schenone (2010) formalize thi argument.

7 660 R. V. KRISHNA AND P. SADOWSKI Whitelaw (2009)). More generally, variation of rik averion over time ha been conidered in repreentative agent etting to improve our undertanding of aet pricing phenomena. For intance, Campbell and Cochrane (1999) identify variation in rik premia that correlate with the fundamental of the economy a a crucial apect of many dynamic aet pricing phenomena. 6 Bekaert, Engtrom, and Grenadier (2010) how that tochatic rik averion that i not driven by, or perfectly correlated with, the fundamental of the economy can explain a wide range of aet pricing phenomena, a well a fit important feature of bond and tock market imultaneouly. Our DPF repreentation can accommodate both kind of tochatic rik averion. An immediate application of our model i provided in Krihna and Sadowki (2012), who invetigate a Luca tree economy with an invetment tage and a repreentative agent whoe preference atify our axiom. They how that greater variation in rik averion reult in greater price volatility. Perhap more urpriingly, they apply our characterization of greater preference for flexibility to find that thi variation alo reduce invetment in a productive aet with liquidity contraint. Thi type of prediction could help dicipline the ue of tate hock model in applied work. 3. MODEL AND REPRESENTATION In thi ection, we decribe the environment and provide a model of dynamic preference for flexibility with obervable peritent and unobervable tranient tate hock Environment For a compact metric pace Y,letP (Y ) denote the pace of probability meaure endowed with the topology of weak convergence, o that P (Y ) i compact and metrizable. Let K(Y ) denote the pace of cloed ubet of a compact metric pace Y, endowed with the Haudorff metric, which make K(Y ) a compact metric pace. Let S := {1 n} be a finite et of tate of the world. For any metric pace Y, H(Y ) := Y S i the pace of act from S to Y. Let M be a finite et of conumption prize with typical member m. Atate contingent infinite horizon conumption problem (S-IHCP) i an act that pecifie, for each tate of the world, an (infinite horizon) conumption problem. Such a conumption problem i a menu (i.e., a et) of lotterie that yield a conumption prize in the preent period and a new S-IHCP tarting in the next period. 6 In Campbell and Cochrane (1999), the correlation i generated indirectly via habit forming conumption choice. Empirically, however, it i the correlation itelf that i important and not the particular mechanim (uch a habit formation) that drive it, a Bekaert, Engtrom, and Grenadier (2010) point out.

8 DYNAMIC PREFERENCE FOR FLEXIBILITY 661 Let H be the collection of all S-IHCP. It can be hown that H i linearly homeomorphic to the pace of all act that take value in K(P (M H)). 7 We denote thi linear homeomorphim a H H(K(P (M H))). Typical S-IHCP are f g H; typical element x y z K(P (M H)) (alo written a K) are menu of lotterie over conumption and S-IHCP; p q P (M H) are typical lotterie; and p m and p h denote the marginal ditribution of p on M and H, repectively. Each f H can be identified with an act that yield a compact et of probability meaure over M H in every tate. We take the convex um of et to be the Minkowki um; thu, for x y K and λ [0 1], λx + (1 λ)y := {λp + (1 λ)q : p x q y}. Notice that if x y K, then λx + (1 λ)y i alo cloed and, hence, i in K. Then, for f g H, wedefine(λf + (1 λ)g)() := λf () + (1 λ)g() K for each. Clearly, H i convex. When there i no rik of confuion, we identify prize and continuation problem with degenerate lotterie and lotterie with ingleton menu. For example, we denote the lottery over continuation problem that yield x with certainty by x, and denote the lottery that yield current conumption m and continuation problem f with certainty by (m f ). Similarly, x H alo refer to the act that give x K in every tate. We explicitly model choice between conumption problem from an ex ante perpective, before conumption begin. That i, we analyze a binary relation H H, which we refer to a a preference. Welet and denote, repectively, the aymmetric and ymmetric part of. The interpretation i that, following the initial choice of an S-IHCP, the tate of the world, S, i realized and DM get to chooe an alternative from the correponding menu. The recurive domain of S-IHCP i amenable to analyi by tochatic dynamic programming. It contruction follow the decriptive approach of Krep and Porteu (1978) in that it more cloely decribe how economic agent act and, a mentioned above, embodie what Koopman (1964) refer to a piecemeal planning: intead of chooing a conumption tream that determine conumption for all time, at each intant the deciion maker chooe immediate conumption a well a a et of alternative for the future Repreentation of Dynamic Preference for Flexibility We now introduce a repreentation of dynamic preference for flexibility for. The relevant ubjective tate pace (for each ) itheetofallvn-m utility function over M that are identified up to an additive contant, U := {u R M : u i = 0}, endowed with the Borel σ-algebra. Subjective tate u U are naturally referred to a conumption utilitie. 8 Probability meaure on 7 We decribe the contruction of H in Appendix B. 8 Subjective tate in tatic model of tate-dependent expected utility, uch a the model in DLRS, are conumption utilitie that are normalized up to the addition of contant (i.e., requir-

9 662 R. V. KRISHNA AND P. SADOWSKI U are referred to a ubjective belief. To enure that expected conumption utility under a meaure μ on U i well defined, the expected utility from every prize m M mut be finite. We ay a meaure μ i nice if it atifie μu := u dμ(u) U;itinontrivial if μ({0}) 1. U DEFINITION 1: Let U be defined a above and let (μ ) S be a collection of nice probability meaure on U uch that for ome S, μ i nontrivial. Let Π repreent the tranition probabilitie for a fully connected Markov proce on S 9 and let π denote time-0 belief about the tate in S. Alo, let δ (0 1). A preference on H ha a repreentation of dynamic preference for flexibility (a DPF repreentation) ((μ ) S Π δ)if V 0 ( ) := V( )π() repreent, where V( )i defined recurively a V(f )= Π ( [ )[ˆU max u(pm ) + δv ( p h )] ] (3.1) dμ (u) p f( ) S and where π i the unique tationary ditribution of Π. In the repreentation above, V i linear on H, andv(p h ) denote the linear extenion (by continuity) of V from H to P (H), that i, V(p h ) = V(g ) dph (g). Notice that V 0 (f ) take the form V 0 (f ) = ˆ π() [ max u(pm ) + δv (p h ) ] dμ (u) U p f() which follow from the fact that π( ) = π()π( ) for all S, becaue π i the unique tationary ditribution of Π. The Markov proce repreented by Π capture obervable and poibly peritent tate hock, while the tate contingent meaure (μ ) S account for additional unobervable tranient hock. Initial time-0 belief are given by π and correpond to DM long-run (ergodic) belief about the ditribution of tate. Finally, DM patience i captured by the time preference parameter δ (0 1). The DPF repreentation ugget that DM i uncertain about the per period conumption utility, which capture how much enjoyment he derive from today conumption, but not about hi patience. ing i u i = 0 for every utility u) and the caling (e.g., i u2 i = 1 for every poible utility u). In contrat, a Section 5.1 dicue in detail, the dynamic context allow u to identify the caling from behavior. Anticipating thi, the pace U contain all poible conumption utilitie up to the addition of contant. In particular, u U implie λu U for all λ 0. Importantly, additive contant have no behavioral implication in the context of a linear aggregator, uch a the expected utility repreentation in Definition 1. 9 The probability of tranitioning to tate from tate i Π( ). The Markov proce with tate pace S i fully connected if Π( )>0forall S.

10 DYNAMIC PREFERENCE FOR FLEXIBILITY 663 PROPOSITION 2: Each DPF repreentation ((μ ) S Π δ) induce a unique continuou function V C(H S) that atifie equation (3.1). The proof i in Appendix A Axiom We now dicu our axiom without reference to the DPF repreentation, but Theorem 1 in Section 3.4 etablihe that the axiom are it behavioral content. To define the induced ranking of menu contingent on the tate,,fixx K and for any x K, conider the act f x ( ) { x if =, := x otherwie. For any x y K,letx y if and only if f x f y. Under our axiom below, turn out to be independent of the particular x choen in the definition. Our firt two axiom on collect tandard requirement. AXIOM 1 Statewie Nontriviality: Every tate i nonnull in the ene that there exit x y F uch that x y. AXIOM 2 Continuou Order: i (a) complete and tranitive and (b) continuou in the ene that the et {f : f g} and {f : g f } are cloed. The following axiom i the uual independence axiom a applied to our domain. AXIOM 3 Independence: f g implie λf + (1 λ)h λg + (1 λ)h for all λ (0 1) and h H. We are intereted in a deciion maker who anticipate preference hock contingent on the tate and, hence, value flexibility. AXIOM 4 Monotonicity: x y x for all x y K and S. Thi i the central axiom in Krep (1979). It ay that additional alternative are alway weakly beneficial. The next axiom ay that DM doe not care about correlation between outcome in M and H, but only care about the marginal ditribution induced by the lotterie in the menu. In particular, if two lotterie induce the ame marginal ditribution over M and H, then DM doe not value the flexibility of having both lotterie available for choice.

11 664 R. V. KRISHNA AND P. SADOWSKI AXIOM 5 Separability: For p q P (M H), if p m = q m and p h = q h, then {p q} {p} for all S. Our verion of Separability i tronger than the verion aumed by GP. Variant of the axiom alo appear in Higahi, Hyogo, and Takeoka (2009) and Schenone (2010). It i intructive to conider a very pecific example of a potential complementarity ruled out by Separability. Conider two S-IHCP that correpond to meal plan for conecutive evening, f 1 f 2 H. Both plan provide the diner with a degenerate choice the firt two night, and do not retrict the available meal in any other period. The plan f 1 yield, with equal probability, either pizza today and alad tomorrow or vice vera, while f 2 yield either pizza both night or alad both night, alo with equal probability. It doe not eem unreaonable that DM would prefer a more varied diet and o rank f 1 f 2. However, both plan clearly generate the ame marginal ditribution over meal in each period, and o Separability rule thi out. We note that virtually all dynamic model ued in application atify Separability. Verion of the next two axiom appear in GP, who provided a more detailed dicuion. We are intereted in tationary preference, where the ranking of continuation problem doe not depend on time. The recurive nature of the domain allow u to capture thi notion via the following axiom, which ay that if f g, then f i alo better than g a a certain continuation problem. AXIOM 6 Aggregate Stationarity: f g if and only if {(m f )} {(m g)} for all m M. The domain of S-IHCP i rich enough to decribe temporal lotterie, a in Krep and Porteu (1978). We abtract from any preference for the timing of reolution of uncertainty by impoing the following axiom. AXIOM 7 Singleton Indifference to Timing: {λ(m f ) + (1 λ)(m g)} {(m λf + (1 λ)g)} for all λ [0 1] and S. The axiom tate that in every tate, DM i indifferent between (i) receiving lottery λ(m f ) + (1 λ)(m g), which yield conumption m with certainty and determine whether the degenerate continuation problem will be f or g (early reolution), and (ii) receiving with certainty conumption m and the continuation menu λf + (1 λ)g, where uncertainty only reolve in the following period (late reolution) In particular, uppoe tate i realized tomorrow. The hypothetical choice of lotterie p from f( ) and q from g( ), when conidered before the reolution of the lottery over (m f ) and (m g) in (i), generate the ditribution λp + (1 λ)q over outcome (m f ) M H. Thii the ame ditribution a that of lottery λp + (1 λ)q from (λf + (1 λ)g)( ) in (ii). The only difference i that the uncertainty in (i) reolve in two tage, while the uncertainty in (ii) reolve entirely in the econd tage.

12 DYNAMIC PREFERENCE FOR FLEXIBILITY 665 If all uncertainty i captured by the objective tate, then hould be trategically rational. In the preence of unobervable hock, trategic rationality will be violated, even contingent on the tate. Our central new axiom, State Contingent Continuation Strategic Rationality (S-CSR), require all peritent hock that are relevant for future conumption tate to be commonly oberved (i.e., captured by the tate of the world), while allowing additional unoberved tranient tate hock. AXIOM 8 S-CSR: {(m f )} {(m g)} implie {(m f ) (m g)} {(m f )} for all m M and S. The axiom conider the choice of a menu contingent on. All menu compared here fix firt period conumption in tate at m. The axiom ay that, contingent on, there i no preference for flexibility with repect to continuation problem. The axiom doe not imply that DM i certain about hi conumption utility following the firt period. It only implie that he doe not expect thi uncertainty to be reolved prior to the ubequent period choice. In particular, the axiom i ilent on how DM value the tate contingent option of retaining the alternative from both x and y for choice two period ahead. We interpret the ranking {(m f x y )} {(m f x y )} {(m f )}, which i not precluded by Axiom 8, a the manifetation of DM uncertainty about hi tate contingent conumption utility two period ahead. Our lat axiom hold if the tate pace i large enough, o that the current objective tate capture all pat peritent hock. In the more familiar context of objective uncertainty, tate are routinely aumed to follow a (firt order) Markov proce, and thi i jutified by auming that the tate pace i ufficiently large. AXIOM 9 Hitory Independence: {(m f x)} {(m f y )} implie {(m f x)} y {(m f )} for all x y K, m M, and S. In analogy to the definition of x y adm rankingofmenux over y contingent on the firt period tate being, one can interpret {(m f x)} {(m f y )} a DM ranking of x over y contingent on the firt and econd period tate being and, repectively. The axiom ay that thi contingent preference i independent of the firt tate realization Repreentation Theorem DEFINITION 3: Two probability meaure μ and μ on U are identical up to caling if there i λ>0 uch that μ(d) = μ (λd) for all meaurable D U, where λd := {λu : u D}. Two collection of meaure (μ ) S and (μ ) S are identical up to a common caling if, for each, μ and μ are identical up to a caling that i independent of.

13 666 R. V. KRISHNA AND P. SADOWSKI Given a DPF repreentation ((μ ) S Π δ)of,caling(μ ) S correpond to a caling of the induced value function V. THEOREM 1: The binary relation atifie Axiom 1 9 if and only if it ha a DPF repreentation, ((μ ) S Π δ). Moreover, the meaure (μ ) S are unique up to a common caling, and Π and δ are unique. We begin the proof (Appendix A.1) by etablihing a repreentation that i additively eparable over S in a traightforward manner. The contruction of the DPF repreentation then ha three main tep. Firt, we find a DLR-type repreentation of the induced binary relation on menu of lotterie. Since our lotterie are defined on an infinite dimenional pace of outcome, thi require a generalization of the repreentation theorem in DLR for infinite dimenional prize pace. Theorem 3 in Appendix A.1 provide uch a repreentation that feature an infinite dimenional tate pace with a meaure that i normal (i.e., i outer and inner regular, in the terminology of Alipranti and Border (1999)), but only finitely additive, and that further lack the identification propertie of the repreentation in DLR. The econd tep relie heavily on S-CSR in contructing a recurive repreentation where δ may depend on the obervable tate. We etablih that our axiom in particular,s-csr (Axiom8) and Separability (Axiom 5) allowu to confine attention to a ubjective tate pace that conit of intantaneou conumption utilitie and o i finite dimenional. The carrier of the meaure can then be taken to be finite dimenional. Thi enable u to how that the meaure i tight with repect to a compact cla of et and o can be regarded a a regular, countably additive probability meaure. We can, therefore, confine attention to a finite dimenional tate pace and countably additive meaure. The final tep conit of howing that there exit exactly one uch repreentation where patience, δ, i tate independent. We how thi via an application of the Perron Theorem (Theorem 4 in Appendix D). Note, again, that Theorem 1 only identifie conumption utilitie up to the addition of contant. Importantly, becaue additive contant have no behavioral implication in the context of the DPF repreentation (which i linear), the uniquely identified meaure (μ ) are independent of the normalization of thee additive contant; in our cae, u i = 0forallu U. Intuition for the unique identification that highlight the role of recurivity i given in Section 5.1. The Markovproce Π ha a unique tationary ditribution becaue it i fully connected. To ee why thi tationary ditribution i the appropriate prior π for period 0 behavior, recall that Aggregate Stationarity (Axiom 6) require that f g if and only if {(m f )} {(m g)}. Therefore, V 0 ( ) and π()v ( ) hould repreent the ame preference. Let u compute both term in turn: V 0 (f ) := π ( ) ˆ [ max u(pm ) + δv ( p h )] dμ (u) U p f()

14 DYNAMIC PREFERENCE FOR FLEXIBILITY 667 and π()v (f ) = π() Π ( ) ˆ = π()π ( ) ˆ [ max u(pm ) + δv ( p h )] dμ (u) U p f() [ max u(pm ) + δv ( p h )] dμ (u) U p f() It i clear that if π i a tationary ditribution of Π, that i, if π( ) = π()π( ) for all S, then V 0 ( ) = π()v ( ) for every f H, and, hence, V 0 ( ) and π()v ( ) do repreent the ame preference. Uing Singleton Indifference to Timing (Axiom 7), we how in Propoition 31 in Appendix D.2.4 that thi i the only poibility, that i, π mut be the unique tationary ditribution of Π. The requirement that Π be a fully connected Markov proce may eem unnecearily trong. A natural, weaker requirement would be that the Markov proce on S i irreducible. 11 The cla of fully connected Markov procee i dene in the cla of irreducible procee. The mall gain in generality doe not eem to warrant impoing a weaker, but more involved, verion of Axiom 9. It i intructive to conider other plauible repreentation of preference over IHCP that are not a pecial cae of the DPF repreentation. Firt, uppoe that can be repreented by a value function V with the ame tructure a in Theorem 1, wheres ={1 2} and π = ( 1 1 ), but, contrary to the theorem, DM patience i tate dependent, δ 1 δ 2,andΠ =[ ] i not poitive. 0 1 The interpretation i that today DM i uncertain about tomorrow obervable tate, but once he learn the tate, he doe not expect it to ever change again. In that cae, there i no recurive repreentation with tate-independent patience, δ, becaue δ mut play the role of the (obviouly unique) dicount factor, contingent on being in tate forever. The meaure μ 1 and μ 2 are not identified in thi repreentation. It i eay to verify that the correponding preference atify all our axiom except Hitory Independence (Axiom 9). Second, the agent tate hock may be peritent or otherwie correlated, even contingent on the tate. For example, rik averion might vary over time with poitive correlation. The reulting preference would violate S-CSR (Axiom 8). We provide foundation for uch a repreentation in Krihna and Sadowki (2013). Third, the agent utility may depend on what he conumed in the pat, for example, due to habit forming preference or a preference 11 The Markov proce on S with tranition probabilitie given by Π i irreducible if for each S, there exit t uch that Π t ( )>0. An irreducible proce ha a unique tationary ditribution.

15 668 R. V. KRISHNA AND P. SADOWSKI for intertemporal diverity a dicued after Separability (Axiom 5). Such conumption-dependent utility i not eparable over time and, hence, i ruled out by Axiom SPECIAL CASES We now conider important pecial cae of the DPF repreentation The DPF Repreentation With Contant Belief In an environment without obervable tate, the domain of S-IHCP reduce to the pace of IHCP, Z K(P (M Z)), and the DPF repreentation reduce to the pair (μ δ), which induce a continuou function V : Z R that atifie ˆ [ V(x)= max u(pm ) + δv (p z ) ] dμ(u) U p x and repreent on Z. We refer to (μ δ) a a repreentation of dynamic preference for flexibility with contant belief. Notice that the identification reult in Theorem 1 doe not depend on any ort of richne aumption on (μ ) S, o that μ in the DPF repreentation with contant belief i identified up to caling Random Dicount Factor In the environment of Section 4.1, we call a DPF repreentation with contant belief (μ δ) of on Z a random dicount factor repreentation if there i a utility function u U uch that μ({λu : λ 0}) = 1. In that cae, we can fix u and label ubjective tate by λ, and following, ay, Cochrane (2005),wecan interpret (δ λ dμ)/λ a a random dicount factor. COROLLARY 4: The binary relation atifie Axiom 1 9 and K(P (M {f })) i trategically rational if and only if ha a random dicount factor repreentation. Thi repreentation i unique up to a caling of μ. The proof i in Appendix E. Strategic rationality here applie to conumption menu in K(P (M {f })), where the continuation problem i fixed at f. Intuitively, there i then no ubjective uncertainty about the per period conumption preference, but only about the intenity of thoe preference. It follow immediately from the uniquene of μ up to caling that the implied ditribution over random dicount factor a defined above i unique and independent of the choice of u. Higahi, Hyogo, and Takeoka (2009) alo conider a repreentation where the effective dicount factor can be random. While our model net the behavior generated by their model, the parametrization of their model render it outide the cla of model we conider here.

16 DYNAMIC PREFERENCE FOR FLEXIBILITY The Recurive Ancombe Aumann Repreentation One may wonder what happen if the analyt can only oberve nontrivial preference over L, the pace of tate contingent infinite horizon conumption tream (S-IHCS). Thi i a domain commonly ued in application. In our framework, a conumption tream l L i jut an S-IHCP where all menu are degenerate. 12 We thu refer to l() a a lottery, and denote by l m () and l l () it marginal on M and L, repectively. Obviouly, thi maller domain provide no opportunity to elicit unobervable tate hock, but it i poible to conider a preference on L that atifie the retriction of our axiom to L.Weay ha a recurive, tate dependent, Ancombe Aumann repreentation (recurive Ancombe Aumann repreentation for hort), ((u ) S Π δ), if V 0 ( ) := π()v ( )repreent,wherev( )i defined recurively a V (l ) = S Π ( )[ u ( lm ( )) + δv ( l l ( ) )] where u U for each with u 0 for ome S,andΠ and δ are a defined in the DPF repreentation. COROLLARY 5: Let be a binary relation on L. Then i tatewie nontrivial on L, and atifie Axiom 2, 3, and 5 9 retricted to L if and only if ha a recurive Ancombe Aumann repreentation ((u ) S Π δ). Moreover, thi repreentation i unique up to a common caling of (u ) S. The proof i in Appendix E. Given Theorem 1, the intuition for the reult i imple. If on L i tatewie nontrivial and atifie the retriction of Axiom 2, 3, and5 9 to L, then there i a unique binary relation that extend to H, atifie Axiom 1 9, and alo atifie State Contingent Strategic Rationality (i.e., each i fully trategically rational). But then on H admit a unique DPF repreentation, ((μ ) S Π δ), and it follow immediately that thi repreentation atifie μ ({u }) = 1foromeu U for each. Therefore, ((u ) S Π δ) i a recurive Ancombe Aumann repreentation of that coincide with L. Now uppoe that ((u ) S Π δ) and ((u ) S Π δ ) are two recurive Ancombe Aumann repreentation of uch that (u ) and (u ) differ by more than a common caling. Define tate-dependent probability meaure μ and μ uch that μ ({u }) = 1 = μ ({u }). It i immediate that ((μ ) S Π δ) and ((μ ) S Π δ ) are two DPF repreentation of that differ by more than a common caling of (μ ), contradicting the uniquene tatement in Theorem 1. Hence, the uniquene in the corollary follow. 12 The pace L i defined recurively a L H(P (M L)). ItieaytoeethatL i a cloed and convex ubet of H.

17 670 R. V. KRISHNA AND P. SADOWSKI The argument above illutrate the advantage, in general, of the domain H over L: Eliciting preference over L correpond to finding unconditional expectation over conumption utilitie, while eliciting preference over H correpond to finding (date t) conditional expectation. Identification of ubjective probabilitie on the tate pace in the tandard (tatic) Ancombeand Aumann (1963)model require two aumption. Firt, the ordinal ranking of lotterie mut be independent of the tate. Thi aumption i referred to a the State Independence axiom. Second, the cardinal repreentation of thee ranking (the utility) i normalized o a to be tate independent. In contrat, the recurive Ancombe Aumann repreentation of Corollary5 feature intantaneou a well a continuation utilitie that are tate dependent. Recurivity and a tate-independent dicount factor are ufficient to imply that belief (which are generated by a ubjective Markov proce) and tate-dependent utilitie are fully identified The Independent Shock Repreentation We now dicu how the model in Hendel and Nevo (2006) map into our etting. They conider an agent who maintain an inventory of a particular good. In each period, he mut chooe to purchae combination of the good in quetion (the inide good) and a bundle of other good (the outide good) from hi budget et. He mut further decide whether to replenih hi inventory, or to draw it down by conuming more or le of the inide good. The price of the good, which follow a Markov proce, naturally affect the feaible choice, which mean that the conumer face an S-IHCP, where the price i the obervable tate of the world. In each period, the agent alo face tate hock that affect hi utility from conumption of the inide good. In Hendel and Nevo pecification, thee hock do not depend on the realized price. It i eay to ee that their agent preference have a DPF repreentation, with the additional contraint that the utility hock are drawn from a tate-independent ditribution, that i, μ = μ for all S, whereμu 0. We refer to thi a an independent hock repreentation. In what follow, let L 0 denote the pace of tate-independent conumption tream and let X := K(P (M L 0 )). Notice that L 0 L H(X). COROLLARY 6: The binary relation atifie Axiom 2 9, L0 i nontrivial, and X = X for all S if and only if admit an independent hock repreentation. Moreover, thi repreentation i unique up to a caling of μ. The nontriviality of L0 implie π()μ u 0, that i, the average expected utility i nontrivial. Requiring X = X implie that (i) the repreentation of and are identical up to caling, and (ii) that the value of the continuation tream l L 0 i independent of the tate. Combining (i) and (ii), the two repreentation mut be identical with μ = μ for all S.

18 DYNAMIC PREFERENCE FOR FLEXIBILITY 671 Hendel and Nevo etting lead the agent to maintain an inventory, which i to ay, it give him an unconditional preference for flexibility. There are two channel for thi preference: The feaible et change with the tochatic price and the conumer face additional tate hock. Our identification reult mean, for example, that Hendel Nevo aumption that conumption hock are log-normally ditributed ha tetable implication. 5. EMPIRICAL CONTENT In thi ection, we dicu the empirical content of our model. Section 5.1 explain the lack of identification in the tatic model of DLR and provide a direct intuition for the ource of identification in the dynamic model. Section 5.2 ugget a detailed algorithm to elicit the identified parameter from choice data and Section 5.3 characterize behavior in term of the parameter Identification in Static veru Dynamic Model DLR conidered preference over tatic conumption problem without obervable tate, that i, over K(P (M)) with typical element a. The relevant verion of DLR reult i that if M on K(P (M)) i monotone, continuou, and atifie Independence, then there exit a probability meaure μ on U uch that the preference functional ˆ U (a) := M max u(α) α a dμ (u) U repreent. In thi cae, we ay that M μ repreent M. Conider an example where μ i upported on {u 1 u 2 } with μ (u 1 ) = μ (u 2 ) = 1. Suppoe alo that 2 M repreent different level of monetary prize and that u 1 correpond to a tate with high rik averion, while u 2 correpond to a tate with low rik averion. Clearly, it would be deirable to interpret μ, which repreent, M a aying that the agent ubjectively aee both high and low rik averion a being equally likely. Such an interpretation i not poible in the tatic etting becaue there are many meaure that repreent M. To ee thi, for i = 1 2, let λ i > 0andλ 1 + λ 2 = 1, and define ũ i := u i /2λ i (o that ũ i ha the ame rik averion a u i ). Alo, define the meaure μ on {ũ 1 ũ 2 } a μ(ũ i ) := λ i.itieay to ee that for any a K(P (M)), wehave U (a) = M max α a u(α)μ (u) = max ũ(α) μ(ũ) α a u U ũ U Since λ 1 and λ 2 are arbitrary, we ee that there i a continuum of meaure μ that alo repreent the ame preference M and for each election of (λ 1 λ 2 ), we have different ubjective probabilitie of either high or low rik averion. In contrat, in the dynamic etting without obervable tate a in Section 4.1,

19 672 R. V. KRISHNA AND P. SADOWSKI defined over Z ha the unique DPF repreentation with contant belief (μ δ), uch that the induced preference, M, over firt period conumption problem, K(P (M)), i then repreented (in the DLR ene) by a unique μ. Therefore, in our dynamic etting, we can interpret μ a a probability meaure over, ay, different level of rik averion. We now provide ome intuition for unique identification of a DPF repreentation. Let ((μ ) S Π δ)and ((μ ) S Π δ ) be two DPF repreentation of. Unique identification mean that (i) for each S, μ and μ are identical up to a caling that i independent of, (ii) the time preference parameter are identical (i.e., δ = δ ), and (iii) the objective tate evolve according to the ame ubjective proce, namely Π = Π. (a) Uniquene of μ up to caling. Firt, notice that induced preference over menu in tate are repreented by U max p f()[u(p m ) + δv (p h )] dμ (u). Thi i an additive expected utility (EU) repreentation in the fahion of DLR, with the pecial feature that continuation problem are evaluated independent of u and, hence, provide a natural numeraire, albeit a numeraire that depend on the tate S. A mentioned in the Introduction, the preence of a numeraire allow the unique identification of the meaure μ up to caling, jut a tate-independent utilitie allow the identification of belief in the Ancombe Aumann model. Therefore, μ and μ are identical up to caling, but we have yet to etablih that thi caling i independent of. (b) Uniquene of Π and δ. For implicity, let u aume that retricted to the ubdomain L of S-IHCS i tatewie nontrivial. To identify Π and δ, we can then ignore tate contingent preference for flexibility by oberving only choice over S-IHCS in L (a defined in Section 4.3). By Corollary 5, the DPF repreentation ((μ ) S Π δ) and ((μ ) S Π δ ) become recurive Ancombe Aumann repreentation ((ū ) S Π δ) and ((ū ) S Π δ ), repectively, where a before, ū = μ u and ū = μ u. Becaue V and V both repreent, it i eay to ee that ū and ū mut repreent the ame preference for intantaneou conumption in tate, and o mut be collinear. That i, for each, there exit λ R + uch that ū = λ ū. Crucially, recurivity allow u to expre V( ) L in term of (ū ) S. Recall that p i the lottery uch that m u(p m ) = 0forallu U. Fix a lottery α p m. Suppoe that, contingent on being in tate today, DM i indifferent between (i) receiving conumption lottery α today and p m in all other period, and (ii) receiving β tomorrow in tate,andp m in all other tate and period. 13 In that cae ū (α) = δπ( )ū (β) and analogouly for ((ū ) S Π δ ). It i a matter of linear algebra to verify that λ mut be independent of. For completene, thi i formally etablihed in Lemma 33 in Appendix F. It then follow immediately that Π = Π and δ = δ. (Our proof of Theorem 1 doe not rely on Lemma 33, but intead invoke the Perron Theorem.) 13 In the DPF repreentation, all tate are nonnull. There are only finitely many pair of tate and, hence, for α ufficiently cloe to p m,uchβ exit for all S.

20 DYNAMIC PREFERENCE FOR FLEXIBILITY 673 (c) Uniquene of (μ ) S up to common caling. We oberved above that μ i identified up to the ame caling a V( ), which, therefore, provide a numeraire in tate that doe not depend on u. We have jut een that ū = λū for ome λ>0, δ = δ,andπ = Π for all S. Thi, in turn, implie that on the ubdomain of conumption tream, V ( )= λv ( ), and, hence, μ and μ are identified up to the ame caling λ>0forall S. Recurivity play two important part in the argument outlined above. Firt, for each, it provide continuation problem a a natural numeraire to imply that each μ i unique up to a caling (that may depend on ). Second, the recurivity and the poitivity of the tranition matrix tie together all the different numeraire, one for each, to imply that the meaure (μ ) S are unique up to a common caling and that Π and δ are unique. We emphaize that thi argument depend crucially on the time horizon being infinite. The uniquene of the identification doe not hold in any finite horizon etting for eentially the ame reaon that identification fail in the tatic etting. Roughly put, becaue we do not have identification in the lat period, the choice of repreentation in the lat period i arbitrary, and each uch choice then reult in a different repreentation of preference at time 0. The infinite horizon obviate the poibility of uch arbitrary choice Elicitation Given the identification reult in Theorem 1, one may wonder how the parameter in the repreentation, ((μ ) Π δ), could be elicited experimentally if the experimenter know (or aume) that DM atifie all our axiom and, hence, ha a unique DPF repreentation. A all the parameter have a continuou range, eliciting any parameter exactly from binary choice data require an infinite amount of data. Thi i a common problem that experimenter addre either by directly eliciting indifference point or by approximating indifference point iteratively. 14 We now dicu how to elicit the parameter of the model via indifference point. Given the true DPF repreentation, ((μ ) Π δ), the average vn-m utility in tate i ū := μ u. Directly eliciting each utility index ū up to caling (where, a before, we ignore additive contant) i a tandard exercie on which we do not elaborate here. Theorem 1 implie that elicitation of the parameter i poible even in knife-edge cae where ome, or even all, of the ū are trivial (a long a at leat one μ i nontrivial), but we aume here for eae of expoition that every ū i nontrivial. In thi cae, there i a unique normalized utility function r {r U : i r2 i = 1} uch that ū = ū r for each, where we are not yet cognizant of the caling ū. 14 A Ma-Colell (1978) note (in the context of tandard conumer preference), it i poible to preciely pin down the conumer preference with ever-increaing data. (Of coure, the data have to increae in a regular way.) A imilar reult can be proved in our etting.