Dynamic Preference for Flexibility�

R. Vijay Krishna� Philipp Sadowski�

Wednesday 20th November, 2013

Abstract

We consider a decision maker who faces dynamic decision situations that involveintertemporal tradeoffs, as in consumption-savings problems, and who experiencestaste shocks that are transient contingent on the state of the world. We axiomatize arecursive representation of choice over state contingent infinite horizon consumptionproblems, where uncertainty about consumption utilities depends on the observablestate and the state follows a subjective Markov process. The parameters of the represen-tation are the subjective process governing the evolution of beliefs over consumptionutilities and the discount factor; they are uniquely identified from behavior. We char-acterize a natural notion of greater preference for flexibility in terms of a dilation ofbeliefs. An important special case of our representation is a recursive version of theAnscombe-Aumann model with parameters that include a subjective Markov processover states and state dependent utilities, all of which are uniquely identified.

1. Introduction

Uncertainty about future consumption utilities influences how economic agentsmake decisions. A decision maker (DM) who is uncertain about future consumptionutilities, perhaps due to uncertain future risk aversion or other taste shocks, prefersnot to commit to a course of future action today and therefore has a preference forflexibility. For example, DM might be willing to forfeit current consumption if thisallows him to delay a decision about future consumption. While this intuition isinherently dynamic, standard models that accommodate preference for flexibility,most prominently Kreps [1979] and Dekel, Lipman and Rustichini [2001] (hence-forth DLR1), are static in the sense that there is no intertemporal tradeoff (although

(�) We would like to thank, without implicating, Haluk Ergin, SujoyMukerji, Wolfgang Pesendorfer,Todd Sarver, Norio Takeoka, six anonymous referees, and a co-editor for helpful comments andsuggestions, and Vivek Bhattarcharya, Matt Horne, and Justin Valasek for valuable researchassistance. Krishna gratefully acknowledges support from the National Science Foundation(grant SES-1132187).

(�) Duke University <vijay.krishna@duke.edu>(�) Duke University <p.sadowski@duke.edu>(1) A relevant corrigendum is Dekel, Lipman, Rustichini and Sarver [2007] (henceforth DLRS).

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Kreps [1979] suggests that an infinite horizon model of preference for flexibilityis desirable). In this paper, we consider a dynamic environment, allow DM’s tasteshocks to have an unobservable transient as well as an observable persistent compo-nent, and provide foundations for a recursive representation of DM’s preferencesthat is fully identified.We consider state contingent infinite horizon consumption problems (S-IHCPs) as

the domain of choice. Given the collection of relevant states of the world, S-IHCPsare defined recursively as acts that specify, for each state of the world, a decisionproblem, which is a menu (ie, a closed set) of lotteries that yield a consumptionprize in the present period and a new S-IHCP starting in the next period. For asimple example of an S-IHCP that features only degenerate lotteries, suppose DMhas fixed income y in every period and current wealth w. The relevant state of theworld is the per period price of consumption, � > 0. In each period DM can chooseto consume an amount m 2

�0; wCy

�

�at cost m�. This will leave him with wealth

w0 D w C y � m� for next period. Given this technology, we can formulate theconsumption problem he faces recursively as a price contingent collection of feasiblepairs of current consumption and a new consumption problem for next period,

fw .�/ WD¶.m; fw 0/ W m 2 Œ0;

wCy

��; w0 D w C y �m�

·In the case where the state space is degenerate (in the example this correspondsto � being constant), our domain reduces to that of Infinite Horizon ConsumptionProblems (IHCPs) introduced by Gul and Pesendorfer [2004], henceforth GP.The domain of S-IHCPs is rich and allows for complicated future choice behavior.2

For example, future choices may depend directly on time or on consumption history.We rule out such dependencies in order to stay as close to the standard model aspossible.3 A decision maker who does not anticipate any taste shocks, contingenton the state of the world, is not averse to committing to a state contingent planof choice. Following Kreps [1979], we refer to this property as state contingentstrategic rationality. In contrast, a decision maker who anticipates taste shocksshould satisfyMonotonicity, that is, he always weakly prefers to retain additionaloptions.We consider taste shocks that are transient (relevant only for one period) contingent

on the state of the world. DM should then be strategically rational with respect to hischoice of continuation problem beginning in the next period. This is the content ofour main behavioural axiom State Contingent Continuation Strategic Rationality(S-CSR).Conceptually, full state contingent strategic rationality would greatly simplify

the analysis of DM’s behavior. If it is violated with respect to all S-IHCPs because

(2) We only model the initial choice of a consumption problem, but our representation suggestsdynamically consistent future choice.

(3) Dependence of choice on the consumption history is central to the notion of habits. We areindependently working on a model of habit formation on the domain of IHCPs.

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of unobservable taste shocks, then it is desirable to understand the nature of thisviolation by asking if DM is strategically rational with respect to some smaller classof S-IHCPs, for example as in S-CSR.Based on S-CSR, we provide a representation of dynamic preference for flexibil-

ity (DPF) that is the solution to a Bellman equation, and can therefore be analyzedusing standard dynamic programming techniques. The evolution of uncertainty aboutfuture consumption utilities is driven by the evolution of the (observable) state of theworld, which in turn follows a subjective Markov process. All the parameters of therepresentation, which are the transition operator on the state space, state contingentbeliefs over consumption utilities, and the discount factor, are uniquely identified. Incontrast, the static model of DLR can not identify beliefs over consumption utilities.The domain of S-IHCPs accommodates important special cases. If the observable

state space is degenerate (so there are no observable shocks), then S-CSR becomesContinuation Strategic Rationality (CSR) on the domain of IHCPs. The resultingrepresentation features unobservable transient taste shocks that are iid. If, instead,the analyst can only observe non-trivial preferences over state contingent infinitehorizon consumption streams (which provide no flexibility), rather than preferencesover all S-IHCPs, then the model reduces to a fully identified recursive version ofthe Anscombe-Aumann model with state dependent utilities.The DPF representation is a generalization of ‘taste shock’ models that are com-

monly used in applied work. Our axioms spell out the testable implications of suchmodels. In particular, if the model features iid shocks, then choice should satisfyCSR. To test this assumption, consider an agent who must order inventory for thecoming season. For example, a farm may need to order seeds for the coming plantingseason and face shocks to their taste for seeds due to unmodelled environmentalconditions. Suppose the order can either be placed long in advance or shortly beforethe season starts. If the agent is willing to pay a premium in order to delay placingthe order, then CSR is violated, and hence the iid model is misspecified: Taste shocksmust then have a persistent component. It is possible that this persistent componentcan be captured by easily observable environmental conditions, such as rainfallsprior to the start of the season. In that case choice should satisfy S-CSR, which canbe tested in a similar fashion.Our identification results allow us to characterize a notion of greater preference

for flexibility in terms of a dilation (which is a multi-dimensional mean preservingspread) of beliefs over consumption utilities.Identification is also relevant for model based forecasting, which is a central

reason for the use of formal models. It involves estimating a relevant parameter inone context in order to forecast outcomes in another. Model based forecasting, thus,requires that (i) the relevant parameter is uniquely identified and (ii) the modeler iswilling to assume that the parameter is meaningful outside the context of the observeddata. In our model, beliefs over consumption utilities are uniquely identified. Thosebeliefs might, for example, forecast future choice frequencies of alternatives from

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continuation problems, as discussed in Sadowski [2013].The remainder of the paper is structured as follows. Section 2 reviews related

literature. Section 3 lays out the basic framework, introduces the representation,explains the axioms, and derives our main theorem which establishes that the axiomsare the behavioral content of the representation. Section 4 discusses important specialcases. Section 5 concerns the empirical content of our theory, where section 5.1discusses the lack of identification in the static model of DLR and provides a directintuition for the identification of the parameters in our dynamic model, section 5.2suggests specific choice experiments from which an experimenter could elicit thoseparameters, and section 5.3 provides a characterization of greater preference forflexibility in terms of the parameters. Section 6 concludes. All proofs, as well as ageneral representation of preference for flexibility with an infinite prize space, arein the appendices.

2. Related Literature

Our work builds on standard axiomatic models of preference for flexibility, whichfollow Kreps [1979] and investigate choice over menus of consumption outcomes.The second seminal paper in this literature, DLR, modifies the domain to considermenus of lotteries over outcomes as objects of choice. This facilitates the interpreta-tion of the subjective states as tastes, where a taste is simply a (twice normalized)vN-M ranking over consumption outcomes. While menu choice can capture DM’sattitude towards the future, implied choice in these models is actually static, in thesense that there is no opportunity for any intertemporal tradeoff. We provide a modelof preference for flexibility due to uncertain consumption utilities in the tradition ofthese earlier papers, where the implied choice is dynamic.As mentioned before, if the set of observable states is degenerate, then our re-

cursive domain of S-IHCPs reduces to that of IHCPs, first analyzed by GP, whoshow that this recursive domain is well defined. GP provide a dynamic model ofconsumption with temptation preferences. We allow, instead, for uncertain utili-ties. The space of possible utilities on our dynamic domain is infinite dimensional,which complicates the analysis. Higashi, Hyogo and Takeoka [2009] also considerpreference for flexibility on the domain of IHCPs. Their model only accommodatespreference for flexibility due to a random discount factor that follows an iid process,but not due to, say, uncertainty about risk aversion, as explained in section 4.2. Ourmodel nests theirs in terms of behavior. Takeoka [2006] considers choice betweenmenus of menus of lotteries, and derives a three period version of DLR. The modelcannot capture intertemporal tradeoffs, and consequently the representation is onlyjointly identified, as in DLR. Rustichini [2002] considers preferences over lotteriesover sets of infinite consumption paths. While an additive representation can beobtained, the domain precludes a recursive value function and does not allow uniqueidentification of the representation.

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Static models of preference for flexibility lack identification because the set ofpossible utilities is the subjective state space, which means that utilities are (trivially)state dependent. These models, therefore, suffer from the same lack of identificationas the Anscombe and Aumann [1963] model with state dependent utilities. DLRsuggest the introduction of a numeraire good (with state independent valuation)to their model, so as to identify beliefs uniquely in the same sense that they areuniquely identified in the Anscombe and Aumann model with state independence.4If the objective state space is degenerate, then the source of identification in ourrepresentation is similar, where the numeraire arises naturally in the form of con-tinuation problems. In contrast, continuation utilities in the DPF representation arestate dependent, and identification relies directly on the recursive structure of therepresentation (section 5.1 provides intuition for this result).Sadowski [2013] identifies beliefs in a situation where preference for flexibility

depends on the state of the world by requiring the observable states to containinformation that is ‘sufficiently relevant’ for future preferences. Instead of choicebetween menus, Gul and Pesendorfer [2006] study random choice from menus.Observed choice frequencies naturally correspond to a unique measure over utilities,but the scaling of utilities remains arbitrary in their model. Ahn and Sarver [2013]simultaneously model choice between menus and random choice from menus. Theyachieve full identification by requiring the beliefs in the representation of choicebetween menus to correspond to frequencies of choice from menus. Dillenberger etal [2013] fully identify representations of preference for flexibility which featureuncertain future beliefs, rather than uncertain future tastes.That a decision maker can be uncertain about future preferences in a dynamic

setting is noted by Koopmans [1964], who distinguishes between once-and-for-allplanning, where the agent selects an action for all possible future contingencies, andpiecemeal planning where, in each period, the agent chooses an action for the currentperiod, while simultaneously narrowing the set of alternatives for the future. Jonesand Ostroy [1984] consider a non-axiomatic (but dynamic) model of choice wherean agent prefers flexibility due to uncertainty about future utilities. They point outthat preference for flexibility, for example in the form of preference for liquidity, is apervasive theme in economics, and relate many instances in macroeconomics andfinance where DM’s dynamic behavior exhibits preference for liquidity in particular,and preference for flexibility in general, as in Goldman [1974]. They also discuss anotion of greater variability of beliefs which roughly corresponds to our notion of adilation of beliefs. A special case of our representation (see corollary 6) is used inempirical work by Hendel and Nevo [2006], who study the problem of a decisionmaker who maintains an inventory.A growing literature argues that variations in preferences over time, and in par-

ticular in risk-aversion, are central in explaining various market behaviors. Indeed,estimated risk aversion has been suggested as a useful index of market sentiment

(4) Schenone [2010] formalizes this argument.

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(see, for example, Bollerslev et al [2011]), and there is evidence that the largestcomponent of changes of the equity risk premium is variation in risk aversion,rather than the quantity of risk (see Smith and Whitelaw [2009]). More generally,variation of risk aversion over time has been considered in representative agentsettings to improve our understanding of asset pricing phenomena. For instance,Campbell and Cochrane [1999] identify variations in risk premia that correlatewith the fundamentals of the economy as a crucial aspect of many dynamic assetpricing phenomena.5 Bekaert et al [2010] show that stochastic risk aversion that isnot driven by, or perfectly correlated with, the fundamentals of the economy canexplain a wide range of asset pricing phenomena, as well as fit important features ofbond and stock markets simultaneously. Our DPF representation can accommodateboth kinds of stochastic risk aversion.An immediate application of our model is provided in Krishna and Sadowski

[2012], who investigate a Lucas tree economy with an investment stage and arepresentative agent whose preferences satisfy our axioms. They show that greatervariation in risk aversion results in greater price volatility. Perhaps more surprisingly,they apply our characterization of greater preference for flexibility to find that thisvariation also reduces investment in a productive asset with liquidity constraint. Thistype of prediction could help discipline the use of taste shock models in appliedwork.

3. Model and Representation

In this section we describe the environment and provide a model of dynamic pref-erence for flexibility with observable persistent and unobservable transient tasteshocks.

3.1. Environment

For a compact metric space Y , let P .Y / denote the space of probability measuresendowed with the topology of weak convergence, so that P .Y / is compact andmetrizable. Let K.Y / denote the space of closed subsets of a compact metric spaceY , endowed with the Hausdorff metric, which makes K.Y / a compact metric space.Let S WD f1; : : : ; ng be a finite set of states of the world. For any metric space Y ,H .Y / WD Y S is the space of acts from S to Y .Let M be a finite set of consumption prizes with typical member m. A State

Contingent Infinite Horizon Consumption Problem (S-IHCP) is an act that specifies,for each state of the world, an (infinite horizon) consumption problem. Such a

(5) In Campbell and Cochrane [1999] the correlation is generated indirectly, via habit formingconsumption choices. Empirically, however, it is the correlation itself that is important and notthe particular mechanism (such as habit formation) that drives it, as Bekaert et al [2010] pointout.

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consumption problem is a menu (ie, a set) of lotteries that yield a consumption prizein the present period and a new S-IHCP starting in the next period.Let H be the collection of all S-IHCPs. It can be shown that H is linearly homeo-

morphic to the space of all acts that take values in K�P .M �H/

�.6 We shall denote

this linear homeomorphism as H ' H�K�P .M � H/

��. Typical S-IHCPs are

f; g 2 H ; typical elements x; y; ´ 2K�P .M �H/

�(also written as K) are menus

of lotteries over consumption and S-IHCPs; p; q 2 P .M �H/ are typical lotteries;and pm and ph denote the marginal distributions of p on M and H respectively.Each f 2 H can be identified with an act that yields a compact set of probabilitymeasures overM �H in every state.We take the convex sum of sets to be the Minkowski sum; thus, for x; y 2K and

� 2 Œ0; 1�, �xC .1��/y WD f�pC .1��/q W p 2 x; q 2 yg. Notice that if x; y 2K ,then �x C .1� �/y is also closed, and hence is in K . Then, for f; g 2 H , we define��f C .1 � �/g

�.s/ WD �f .s/C .1 � �/g.s/ 2K for each s. Clearly, H is convex.

When there is no risk of confusion, we identify prizes and continuation problemswith degenerate lotteries and lotteries with singleton menus. For example, we denotethe lottery over continuation problems that yields x with certainty by x, and thelottery that yields current consumptionm and continuation problem f with certaintyby .m; f /. Similarly, x 2 H also refers to the act that gives x 2K in every state.We explicitly model choice between consumption problems from an ex-ante

perspective, before consumption begins. That is, we analyze a binary relation %� H �H , which we refer to as a preference. We let � and � denote, respectively,the asymmetric and symmetric parts of %. The interpretation is that, following theinitial choice of an S-IHCP, the state of the world, s 2 S , is realized and DM gets tochoose an alternative from the corresponding menu.The recursive domain of S-IHCPs is amenable to analysis by stochastic dynamic

programming. Its construction follows the descriptive approach of Kreps and Porteus[1978] in that it more closely describes how economic agents act and, as mentionedabove, embodies what Koopmans [1964] refers to as piecemeal planning: insteadof choosing a consumption stream that determines consumption for all time, ateach instant the decision maker chooses immediate consumption as well as a set ofalternatives for the future.

3.2. Representation of Dynamic Preference for Flexibility

We now introduce a representation of dynamic preference for flexibility for %. Therelevant subjective state space (for each s) is the set of all vN-M utility functionsoverM that are identified up to an additive constant, U WD

¶u 2 RM W

Pui D 0

·,

endowed with the Borel �-algebra. Subjective states u 2 U are naturally referred to

(6) We describe the construction of H in appendix B.

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as consumption utilities.7 Probability measures on U are referred to as subjectivebeliefs. To ensure that expected consumption utility under a measure � on U iswell defined, the expected utility from every prize m 2 M must be finite. Wesay a measure � is nice if it satisfies �u WD

RUu d�.u/ 2 U; it is non-trivial if

�.f0g/ ¤ 1.

Definition 1. Let U be defined as above, and for each s 2 S , let �s be a nice proba-bility measure on U such that for some s0 2 S , �s0 is non-trivial. Let … representthe transition probabilities for a fully connected Markov process on S ,8 and let �denote time-0 beliefs about the states in S . Also, let ı 2 .0; 1/. A preference % onHhas a representation of Dynamic Preference for Flexibility (a DPF representation),�.�s/s2S ;…; ı

�, if V0.�/ WD

Ps V.�; s/�.s/ represents %, where V.�; s/ is defined

recursively as

(3.1) V.f; s/ DXs02S

….s; s0/

�ZU

maxp2f .s0/

�u.pm/C ıV .ph; s

0/�d�s0.u/

�and where � is the unique stationary distribution of….

In the representation above, V is linear on H , and V.ph; �/ denotes the linearextension (by continuity) of V fromH to P .H/, that is, V.ph; �/ D

RV.g; �/ dph.g/.

Notice that V0.f / takes the form

V0.f / DXs

�.s/

ZU

maxp2f .s/

�u.pm/C ıV .ph; s/

�d�s.u/

which follows from the fact that �.s0/ DPs �.s/….s; s

0/ for all s0 2 S , because �is the unique stationary distribution of….The Markov process represented by… captures observable and possibly persistent

taste shocks, while the state contingent measures .�s/s2S account for additionalunobservable transient shocks. Initial, time-0 beliefs are given by � , and correspondsto DM’s long-run (ergodic) beliefs about the distribution of states. Finally, DM’spatience is captured by the time preference parameter ı 2 .0; 1/.The DPF representation suggests that DM is uncertain about the per period

consumption utility, which captures how much enjoyment he derives from today’sconsumption, but not about his patience.

(7) Subjective states in staticmodels of state dependent expected utility, such as the model in DLRS,are consumption utilities that are normalized up to the addition of constants (ie, requiringPi ui D 0 for every utility u) and the scaling (eg,

Pi u2i D 1 for every possible utility u). In

contrast, as section 5.1 discusses in detail, the dynamic context allows us to identify the scalingfrom behavior. Anticipating this, the space U contains all possible consumption utilities upto the addition of constants. In particular, u 2 U implies �u 2 U for all � > 0. Importantly,additive constants have no behavioral implications in the context of a linear aggregator, such asthe expected utility representation in Definition 1 below.

(8) The probability of transitioning to state s0 from state s is ….s; s0/. The Markov process withstate space S is fully connected if….s; s0/ > 0 for all s; s0 2 S .

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Proposition 2. Each DPF representation�.�s/s2S ;…; ı

�induces a unique continu-

ous function V 2 C.H � S/ that satisfies equation (3.1) above.

The proof is in appendix D.

3.3. Axioms

We now discuss our axioms without reference to the DPF representation, but Theo-rem 1 in section 3.4 establishes that the axioms are its behavioral content.To define the induced ranking of menus contingent on the state, %s, fix x� 2K

and for any x 2K , consider the act

f xs .s0/ WD

¼x if s D s0

x� otherwise

For any x; y 2K , let x %s y if, and only if, f xs % fys . Under our axioms below, %s

turns out to be independent of the particular x� chosen in the definition.Our first two axioms on % collect standard requirements.

Axiom 1 (Statewise Non-triviality). Every state s is non-null, in the sense that thereexist x; y 2 F such that x �s y.

Axiom 2 (Continuous Order). % satisfies the following:

(a) % is complete and transitive.

(b) % is continuous in the sense that the sets ff W f % gg and ff W g % f g areclosed.

The following axiom is the usual Independence axiom as applied to our domain.

Axiom 3 (Independence). f � g implies �f C .1 � �/h � �g C .1 � �/h for all� 2 .0; 1/ and h 2 H .

We are interested in a decisionmaker who anticipates preference shocks contingenton the state, and hence values flexibility.

Axiom 4 (Monotonicity). x [ y %s x for all x; y 2K and s 2 S .

This is the central axiom in Kreps [1979]. It says that additional alternatives arealways weakly beneficial.The next axiom says that DM does not care about correlations between outcomes

inM andH , but only cares about the marginal distributions induced by the lotteriesin the menu. In particular, if two lotteries induce the same marginal distributions overM and H , then DM does not value the flexibility of having both lotteries availablefor choice.

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Axiom 5 (Separability). For p; q 2 P .M � H/, if pm D qm and ph D qh, thenfp; qg �s fpg for all s 2 S .

Our version of Separability is stronger than the version assumed by GP. Variants ofthe axiom also appear in Higashi, Hyogo and Takeoka [2009] and Schenone [2010].It is instructive to consider a very specific example of a potential complementarityruled out by Separability. Consider two S-IHCPs that correspond to meal plans forconsecutive evenings, f1; f2 2 H . Both plans provides the diner with a degeneratechoice the first two nights, and do not restrict the available meals in any other period.The plan f1 yields, with equal probability, either pizza today and salad tomorrowor vice versa, while f2 yields either pizza both nights or salad both nights, alsowith equal probability. It does not seem unreasonable that DM would prefer a morevaried diet, and so rank f1 � f2. However, both plans clearly generate the samemarginal distribution over meals in each period, and so Separability rules this out.We note that virtually all dynamic models used in applications satisfy Separability.Versions of the next two axioms appear in GP, who provide a more detailed discus-

sion. We are interested in stationary preferences, where the ranking of continuationproblems does not depend on time. The recursive nature of the domain allows us tocapture this notion via the following axiom, which says that if f % g, then f is alsobetter than g as a certain continuation problem.

Axiom 6 (Aggregate Stationarity). f % g if, and only if, f.m; f /g % f.m; g/g forall m 2M .

The domain of S-IHCPs is rich enough to describe temporal lotteries, as in Krepsand Porteus [1978]. We abstract from any preference for the timing of resolution ofuncertainty by imposing the following axiom.

Axiom 7 (Singleton Indifference to Timing). f�.m; f /C.1��/.m; g/g �s f.m; �fC.1 � �/g/g for all � 2 Œ0; 1� and s 2 S .

The axiom states that in every state s, DM is indifferent between (i) receivinglottery �.m; f /C .1 � �/.m; g/, which yields consumption m with certainty anddetermines whether the degenerate continuation problem will be f or g, (earlyresolution) and (ii) receiving with certainty consumption m and the continuationmenu �f C .1 � �/g where uncertainty only resolves in the following period (lateresolution).9If all uncertainty is captured by the objective state s, then%s should be strategically

rational. In the presence of unobservable shocks, strategic rationality will be violated,

(9) In particular, suppose state s0 is realized tomorrow. The hypothetical choice of lotteries p fromf .s0/ and q from g .s0/, when considered before the resolution of the lottery over .m; f / and.m; g/ in (i), generates the distribution �p C .1 � �/ q over outcomes .m0; f 0/ 2M �H . Thisis the same distribution as that of lottery �p C .1 � �/ q from .�f C .1 � �/g/ .s0/ in (ii). Theonly difference is that the uncertainty in (i) resolves in two stages, while the uncertainty in (ii)resolves entirely in the second stage.

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even contingent on the state. Our central new axiom requires all persistent shocks thatare relevant for future consumption tastes to be commonly observed (ie, captured bythe state of the world), while allowing additional unobserved transient taste shocks.

Axiom 8 (S-CSR). For s 2 S , f.m; f /g %s f.m; g/g implies f.m; f /; .m; g/g �sf.m; f /g.

The axiom considers the choice of a menu contingent on s. All menus comparedhere fix first period consumption in state s atm. The axiom says that, contingent on s,there is no preference for flexibility with respect to continuation problems. The axiomdoes not imply that DM is certain about his consumption utility following the firstperiod. It only implies that he does not expect this uncertainty to be resolved prior tothe subsequent period’s choice. In particular, the axiom is silent on how DM valuesthe state contingent option of retaining the alternatives from both x and y for choicetwo periods ahead. We interpret the ranking f.m; f x[ys /g � f.m; f xs /g % f.m; f

ys /g,

which is not precluded by Axiom 8, as the manifestation of DM’s uncertainty abouthis state contingent consumption utility, two periods ahead.Our last axiom holds if the state space is large enough, so that the current objective

state captures all past persistent shocks. In the more familiar context of objectiveuncertainty, states are routinely assumed to follow a (first order) Markov process,and this is justified by assuming that the state space is sufficiently large.

Axiom 9 (History Independence). f.m; f xs /g %s0 f.m; fys /g implies f.m; f xs /g %s00

f.m; fys /g for all s; s0; s00 2 S .

In analogy to the definition of x %s y as the DM’s ranking of menu x overy contingent on the first period’s state being s, one can interpret f.m; f xs /g %s0f.m; f

ys /g as the DM’s ranking of x over y contingent on the first and second

period’s states being s0 and s, respectively. The axiom says that this contingentpreference is independent of the first state realization.

3.4. Representation Theorem

Definition 3. Two probability measures � and �0 on U are identical up to scalingif there is � > 0 such that �.D/ D �0.�D/ for all measurable D � U, where�D WD f�u W u 2 Dg. Two collections of measures .�s/s2S and .�0s/s2S are identicalup to a common scaling if each �s and �0s are identical up to a scaling that isindependent of s.

Given a DPF representation�.�s/s2S ;…; ı

�of %, scaling .�s/s2S corresponds to

a scaling of the induced value function V .

Theorem 1. The binary relation % satisfies Axioms 1–9 if, and only if, it has aDPF representation,

�.�s/s2S ;…; ı

�. Moreover, the measures .�s/s2S are unique up

to a common scaling, and… and ı are unique.

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The proof is in appendix D, where we begin by establishing a representationthat is additively separable over S in a straightforward manner. The constructionof the DPF representation then has three main steps. First, we find a DLR-typerepresentation of the induced binary relation %s on menus of lotteries. Since ourlotteries are defined on an infinite dimensional space of outcomes, this requires ageneralization of the representation theorem in DLR for infinite dimensional prizespaces. Theorem 3 in appendix A.1 provides such a representation that featuresan infinite dimensional state space with a measure that is normal (ie, is outer andinner regular, in the terminology of Aliprantis and Border [1999]), but only finitelyadditive, and that further lacks the identification properties of the representation inDLR.The second step relies heavily on S-CSR in constructing a recursive representation

where ı may depend on the observable state s. We establish that our axioms – inparticular, S-CSR (Axiom 8) and Separability (Axiom 5) – allow us to confineattention to a subjective state space that consists of instantaneous consumptionutilities, and so is finite dimensional. The carrier of the measure can then be taken tobe finite dimensional. This enables us to show that the measure is tight with respectto a compact class of sets, and so can be regarded as a regular, countably additiveprobability measure. We can, therefore, confine attention to a finite dimensionalstate space and countably additive measures.The final step consists of showing that there exists exactly one such representation

where patience, ı, is state independent. We show this via an application of the PerronTheorem (Theorem 4 in the appendix). Note, again, that Theorem 1 only identifiesconsumption utilities up to the addition of constants. Importantly, because additiveconstants have no behavioral implications in the context of the DPF representation(which is linear), the uniquely identified measures .�s/ are independent of thenormalization of these additive constants, in our case

Pui D 0 for all u 2 U.

Intuition for the unique identification that highlights the role of recursivity is givenin section 5.1.The Markov process … has a unique stationary distribution, because it is fully

connected. To see why this stationary distribution is the appropriate prior � forperiod 0 behavior, recall that Aggregate Stationarity (Axiom 6) requires that f % g

if, and only if, f.m; f /g % f.m; g/g. Therefore, V0.�/ andPs �.s/V .�; s/ should

represent the same preferences. Let us compute both terms in turn:

V0.f / WDXs0

�.s0/

ZU

maxp2f .s/

�u.pm/C ıV .ph; s

0/�d�s0.u/

andXs

�.s/V .f; s/ DXs

�.s/Xs0

….s; s0/

ZU

maxp2f .s/

�u.pm/C ıV .ph; s

0/�d�s0.u/

D

Xs0

Xs

�.s/….s; s0/

ZU

maxp2f .s/

�u.pm/C ıV .ph; s

0/�d�s0.u/

12

It is clear that if � is a stationary distribution of …, ie, if �.s0/ DPs �.s/….s; s

0/

for all s0 2 S , then V0.�/ DPs �.s/V .�; s/ for every f 2 H and hence V0.�/ andP

s �.s/V .�; s/ do represent the same preferences. Using Singleton Indifference toTiming (Axiom 7), we show in Proposition 31 in Appendix D.2.4 that this is theonly possibility, that is, � must be the unique stationary distribution of….The requirement that… be a fully connected Markov process may seem unneces-

sarily strong. A natural, weaker requirement would be that the Markov process onS is irreducible.10 The class of fully connected Markov processes is dense in theclass of irreducible processes. The small gain in generality does not seem to warrantimposing a weaker, but more involved version of Axiom 9.It is instructive to consider other plausible representations of preferences over

IHCPs that are not a special case of the DPF representation. First, suppose that %can be represented by a value function V with the same structure as in Theorem 1,where S D f1; 2g, and � D .1

2; 12/, but, contrary to the theorem, DM’s patience is

state dependent, ı1 ¤ ı2, and… Dh 1 0

0 1

iis not positive. The interpretation is that

today DM is uncertain about tomorrow’s observable state, but once he learns thestate, he does not expect it to ever change again. In that case, there is no recursiverepresentation with state independent patience, ı, because ıs must play the role ofthe (obviously unique) discount factor, contingent on being in state s forever. Themeasures �1 and �2 are not identified in this representation. It is easy to verify thatthe corresponding preferences satisfy all our axioms except History Independence(Axiom 9).Second, the agent’s taste shocks may be persistent or otherwise correlated, even

contingent on the state. For example, risk aversion might vary over time with positivecorrelation. The resulting preferences would violate S-CSR (Axiom 8). We providefoundations for such a representation in a supplement to this paper, Krishna andSadowski [2013]. Third, the agent’s utility may depend on what he consumed in thepast, for example, due to habit forming preferences or a preference for intertemporaldiversity as discussed after Separability (Axiom 5). Such consumption-dependentutility is not separable over time, and hence is ruled out by Axiom 5.

4. Special Cases

We now consider important special cases of the DPF representation.

(10) The Markov process on S with transition probabilities given by … is irreducible if for eachs; s0 2 S , there exists t such that…t .s; s0/ > 0. An irreducible process has a unique stationarydistribution.

13

4.1. The DPF representation with constant beliefs

In an environment without observable states, the domain of S-IHCPs reduces to thespace of IHCPs, Z ' K

�P .M � Z/

�, and the DPF representation reduces to the

pair .�; ı/ which induces a continuous function V W Z ! R that satisfies

V.x/ D

ZU

maxp2x

�u.pm/C ıV .p´/

�d�.u/

and represents% onZ. We refer to .�; ı/ as a representation ofDynamic Preferencefor Flexibility with Constant Beliefs. Notice that the identification result in Theorem1 does not depend on any sort of ‘richness’ assumption on .�s/s2S , so that � in theDPF representation with constant beliefs is identified up to scaling.

4.2. Random discount factors

In the environment of section 4.1, we call a DPF representation with constantbeliefs .�; ı/ of % on Z a random discount factor representation, if there is autility function u 2 U such that � .f�u W � > 0g/ D 1. In that case we can fix u andlabel subjective states by � and following, say, Cochrane [2005], we can interpret.ıR� d�/=� as a random discount factor.

Corollary 4. The binary relation % satisfies Axioms 1–9 and % jK.P .M�ff �g// isstrategically rational if, and only if, % has a random discount factor representation.This representation is unique up to a scaling of �.The proof is in appendix E. Strategic rationality here applies to consumptionmenus

in K.P .M � ff �g//, where the continuation problem is fixed at f �. Intuitively,there is then no subjective uncertainty about the per period consumption preferences,but only about the intensity of those preferences. It follows immediately from theuniqueness of � up to scaling that the implied distribution over random discountfactors as defined above is unique and independent of the choice of u. Higashi, Hyogoand Takeoka [2009] also consider a representation where the effective discount factorcan be random. While our model nests the behavior generated by their model, theparametrization of their model renders it outside the class of models we considerhere.

4.3. The Recursive Anscombe-Aumann representation

One may wonder what happens if the analyst can only observe non-trivial prefer-ences over L, the space of state contingent infinite horizon consumption streams(S-IHCSs). This is a domain commonly used in applications. In our framework, aconsumption stream ` 2 L is just an S-IHCP where all menus are degenerate.11 We(11) The space L is defined recursively as L ' H

�P .M � L/

�. It is easy to see that L is a closed

and convex subset of H .

14

thus refer to `.s/ as a lottery and denote by `m.s/ and ``.s/ its marginals onM andL respectively. Obviously, this smaller domain provides no opportunity to elicitunobservable taste shocks, but it is possible to consider a preference & on L that sat-isfies the restriction of our axioms to L. We say & has a recursive, state dependent,Anscombe-Aumann representation (recursive Anscombe-Aumann representation,in short),

�.us/s2S ;…; ı

�, if V0.�/ WD

Ps �.s/V .�; s/ represents &, where V.�; s/ is

defined recursively as

V.`; s/ DXs02S

….s; s0/�us0.`m.s

0//C ıV .``.s0/; s0/

�where us 2 U for each s with us ¤ 0 for some s 2 S , and… and ı are as defined inthe DPF representation.

Corollary 5. Let & be a binary relation on L. Then, & is state-wise non-trivial onL and satisfies Axioms 2–3 and 5–9 restricted to L if, and only if, & has a recursiveAnscombe-Aumann representation

�.us/s2S ;…; ı

�. Moreover, this representation is

unique up to a common scaling of .us/s2S .

The proof is in appendix E. Given Theorem 1, the intuition for the result issimple. If & on L is state-wise non-trivial and satisfies the restriction of Axioms2–3 and 5–9 to L, then there is a unique binary relation % that extends & to H ,satisfies Axioms 1–9, and also satisfies state contingent strategic rationality (ie,each %s is fully strategically rational). But then % on H admits a unique DPFrepresentation,

�.�s/s2S ;…; ı

�, and it follows immediately that this representation

satisfies �s.fusg/ D 1 for some us 2 U for each s. Therefore,�.us/s2S ;…; ı

�is a

recursive Anscombe-Aumann representation of & which coincides with % jL.Now suppose that

�.us/s2S ;…; ı

�and

�.u�s /s2S ;…

�; ı��are two recursiveAnscombe-

Aumann representations of & such that .us/ and .u�s / differ by more than a com-mon scaling. Define state dependent probability measures �s and ��s such that�s.fusg/ D 1 D �

�s .fu

�s g/. It is immediate that

�.�s/s2S ;…; ı

�and

�.��s /s2S ;…

�; ı��

are two DPF representations of % that differ by more than a common scaling of.�s/, contradicting the uniqueness statement in Theorem 1. Hence, the uniquenessin the corollary follows.The argument above illustrates the advantage in general of the domain H over L:

Eliciting preferences over L corresponds to finding unconditional expectations overconsumption utilities, while eliciting preferences over H corresponds to finding(date t) conditional expectations.Identification of subjective probabilities on the state space in the standard (static)

Anscombe and Aumann [1963] model requires two assumptions. First, the ordinalranking of lotteries must be independent of the state. This assumption is referredto as the State Independence Axiom. Second, the cardinal representation of theserankings (the utility) is normalized so as to be state independent. In contrast, therecursive Anscombe-Aumann representation of corollary 5 features instantaneous

15

as well as continuation utilities that are state dependent. Recursivity and a state-independent discount factor are sufficient to imply that beliefs (which are generatedby a subjective Markov process) and state dependent utilities are fully identified.

4.4. The Independent Shock representation

We now discuss how the model in Hendel and Nevo [2006] maps into our setting.They consider an agent who maintains an inventory of a particular good. In eachperiod, he must choose to purchase combinations of the good in question (the insidegood) and a bundle of other goods (the outside good) from his budget set. He mustfurther decide whether to replenish his inventory, or draw it down by consumingmore or less of the inside good. The price of the good, which follows a Markovprocess, naturally affects the feasible choices, which means that the consumer facesan S-ICHP, where the price is the observable state of the world.In each period the agent also faces taste shocks that affect his utility from con-

sumption of the inside good. In Hendel and Nevo’s specification these shocks donot depend on the realized price. It is easy to see that their agent’s preferences havea DPF representation, with the additional constraint that the utility shocks are drawnfrom a state independent distribution, ie, �s D � for all s 2 S where �u ¤ 0. Wewill refer to this as an Independent Shock representation.In what follows, letL0 denote the space of state-independent consumption streams,

let X WDK�P .M � L0/

�. Notice that L0 � L � H .X/.

Corollary 6. The binary relation % satisfies Axioms 2–9, % jL0is non-trivial, and

%s jX D %s0 jX for all s; s0 2 S if, and only if, % admits an Independent Shockrepresentation. Moreover, this representation is unique up to a scaling of �.

The proof is in appendix E. The non-triviality of % jL0implies

Ps �.s/�su ¤ 0,

ie, the average expected utility is non-trivial. Requiring %s jX D %s0 jX implies that(i) the representations of %s and %s0 are identical up to scaling and (ii) that the valueof continuation stream ` 2 L0 is independent of the state. Combining (i) and (ii),the two representations must be identical with �s D �s0 for all s; s0 2 S:Hendel and Nevo’s setting leads the agent to maintain an inventory, which is to say,

it gives him an unconditional preference for flexibility. There are two channels forthis preference: The feasible set changes with the stochastic price, and the consumerfaces additional taste shocks. Our identification results mean, for example, thatHendel-Nevo’s assumption that consumption shocks are log-normally distributedhas testable implications.

5. Empirical Content

In this section we discuss the empirical content of our model. Section 5.1 explainsthe lack of identification in the static model of DLR and provides a direct intuition

16

for the source of identification in the dynamic model. Section 5.2 suggests a de-tailed algorithm to elicit the identified parameters from choice data, and section 5.3characterizes behavior in terms of the parameters.

5.1. Identification in static versus dynamic models

DLR consider preferences over static consumption problems without observablestates, ie, over K

�P .M/

�with typical element a. The relevant version of DLR’s

result is that if %�M on K�P .M/

�is monotone, continuous, and satisfies Indepen-

dence, then there exists a probability measure �� on U such that the preferencefunctional

U �M .a/ WD

ZU

max˛2a

u.˛/ d��.u/

represents %�M . In this case, we say that �� represents %�M . Consider an examplewhere �� is supported on fu1; u2g with ��.u1/ D ��.u2/ D 1

2. Suppose also thatM

represents different levels of monetary prizes, and that u1 corresponds to a state withhigh risk aversion, while u2 corresponds to a state with low risk aversion. Clearly, itwould be desirable to interpret ��, which represents %�M , as saying that the agentsubjectively assesses both high and low risk aversion as being equally likely. Suchan interpretation is not possible in the static setting because there are many measuresthat represent %�M . To see this, for i D 1; 2, let �i > 0 and �1 C �2 D 1, and defineQui WD ui=2�i (so that Qui has the same risk aversion as ui ). Also, define the measureQ� on f Qu1; Qu2g as follows: Q�. Qui/ WD �i . It is easy to see that for any a 2 K

�P .M/

�,

we haveU �M .a/ D

Xu2U

max˛2a

u.˛/��.u/ DXQu2U

max˛2aQu.˛/ Q�. Qu/

Since �1; �2 are arbitrary, we see that there is a continuum of measures Q� thatalso represent the same preference %�M and for each selection of .�1; �2/, we havedifferent subjective probabilities of either high or low risk aversion. In contrast,in the dynamic setting without observable states as in section 4.1, % defined overZ has the unique DPF representation with constant beliefs .�; ı/, such that theinduced preference, %M , over first period consumption problems, K

�P .M/

�, is

then represented (in the DLR sense) by a unique �. Therefore, in our dynamicsetting, we can interpret � as a probability measure over, say, different levels of riskaversion.We now provide some intuition for unique identification of a DPF representation.

Let�.�s/s2S ;…; ı

�and

�.�0s/s2S ;…

0; ı0�be two DPF representations of %. Unique

identification means that (i) for each s 2 S , �s and �0s are identical up to a scalingthat is independent of s, (ii) the time preference parameters are identical, ie, ı D ı0,and (iii) the objective states evolve according to the same subjective process, namely… D …0.

17

(a) Uniqueness of �s up to scaling. First, notice that induced preferences over menusin state s are represented by

RUmaxp2f .s/

�u.pm/C ıV .ph; s/

�d�s.u/. This is an

additive EU representation in the fashion of DLR, with the special feature thatcontinuation problems are evaluated independent of u and hence provide a naturalnumeraire, albeit a numeraire that depends on the state s 2 S . As mentioned in theintroduction, the presence of a numeraire allows the unique identification of themeasure �s up to scaling, just as state independent utilities allow the identificationof beliefs in the Anscombe-Aumann model. Therefore, �s and �0s are identicalup to scaling, but we have yet to establish that this scaling is independent of s.

(b) Uniqueness of … and ı. For simplicity, let us assume that % restricted to thesubdomain L of S-IHCSs is state-wise non-trivial. To identify … and ı, wecan then ignore state contingent preference for flexibility by observing onlychoice over S-IHCSs in L (as defined in section 4.3). By corollary 5, the DPFrepresentations

�.�s/s2S ;…; ı

�and

�.�0s/s2S ;…

0; ı0�become recursive Anscombe-

Aumann representations�. Nus/s2S ;…; ı

�and

�. Nu0s/s2S ;…

0; ı0�, respectively, where

as before, Nus D �su and Nu0s D �0su. Because V and V 0 both represent %, it iseasy to see that Nus and Nu0s must represent the same preference for instantaneousconsumption in state s, and so must be collinear. That is, for each s, there exists�s 2 RC such that Nu0s D �s Nus. Crucially, recursivity allows us to express V.�; s/jLin terms of . Nus/s2S .Recall that p�m is the lottery such that u.p�m/ D 0 for all u 2 U. Fix a lottery˛ ¤ p�m. Suppose that, contingent on being in state s today, DM is indifferentbetween (i) receiving consumption lottery ˛ today, and p�m in all other periods,and (ii) receiving ˇ tomorrow in state s0, and p�m in all other states and periods.12In that case Nus .˛/ D ı… .s; s0/ Nus0 .ˇ/ and analogously for

�. Nu0s/s2S ;…

0; ı0�: It

is a matter of linear algebra to verify that �s must be independent of s. Forcompleteness this is formally established in lemma 33 in appendix F. It thenfollows immediately that… D …0 and ı D ı0. (Our proof of Theorem 1 does notrely on lemma 33, but instead invokes the Perron Theorem.)

(c) Uniqueness of .�s/s2S up to common scaling. We observed above that �s isidentified up to the same scaling as V.�; s/, which, therefore, provides a numerairein state s that does not depend on u. We have just seen that Nus D � Nu0s for some� > 0, ı D ı0 and … D …0 for all s 2 S . This, in turn, implies that on thesub-domain of consumption streams, V 0.�; s/ D �V.�; s/, and hence �s and �0s areidentified up to the same scaling � > 0 for all s 2 S .

Recursivity plays two important parts in the argument outlined above. First, foreach s, it provides continuation problems as a natural numeraire to imply that each�s is unique up to a scaling (that may depend on s). Second, the recursivity and the

(12) In the DPF representation all states are non-null. There are only finitely many pairs of states,and hence, for ˛ sufficiently close to p�m, such ˇ exists for all s; s0 2 S .

18

positivity of the transition matrix tie together all the different numeraires, one foreach s, to imply that the measures .�s/s2S are unique up to a common scaling andthat… and ı are unique. We emphasize that this argument depends crucially on thetime horizon being infinite. The uniqueness of the identification does not hold inany finite horizon setting for essentially the same reasons that identification failsin the static setting. Roughly put, because we do not have identification in the lastperiod, the choice of representation in the last period is arbitrary, and each suchchoice then results in a different representation of preference at time 0. The infinitehorizon obviates the possibility of such arbitrary choices.

5.2. Elicitation

Given the identification result in Theorem 1, one may wonder how the parameters inthe representation,

�.�s/;…; ı

�, could be elicited experimentally, if the experimenter

knows (or assumes) that DM satisfies all our axioms, and hence has a unique DPFrepresentation. As all the parameters have a continuous range, eliciting any parameterexactly from binary choice data requires an infinite amount of data. This is a commonproblem that experimenters address either by directly eliciting indifference pointsor by approximating indifference points iteratively.13 We now discuss how to elicitthe parameters of the model via indifference points.Given the true DPF representation,

�.�s/;…; ı

�, the average vN-M utility in state

s is Nus WD �su. Directly eliciting each utility index Nus up to scaling (where, as before,we ignore additive constants) is a standard exercise on which we do not elaboratehere. Theorem 1 implies that elicitation of the parameters is possible even in theknife-edge cases where all but one of the Nus’s are trivial, but we assume here for easeof exposition that every Nus is non-trivial. In this case, there is a unique normalizedutility function rs 2 fr 2 U W

Pi r2i D 1g such that Nus D k Nusk rs for each s, where

we are not yet cognizant of the scaling k Nusk.

(a) To simultaneously elicit k Nusk (so that Nus D k Nusk rs is then known), the transitionprobabilities….s; s0/, and the discount factor ı, we can consider the same indif-ferences as in step b of the identification argument in section 5.1: Fix a lottery˛ ¤ p�m. Given s and s0, find a lottery ˇ such that, contingent on being in state s to-day, DM is indifferent between (i) receiving consumption lottery ˛ today, and p�min all other periods, and (ii) receiving ˇ tomorrow in state s0, and p�m in all otherstates and periods. As before this indifference implies Nus .˛/ D ı… .s; s0/ Nus0 .ˇ/,and hence

….s; s0/ D1

ı

Nus.pm/

Nus0.p0m/D1

ı

k Nusk

k Nus0k s;s0

(13) As Mas-Colell [1978] notes (in the context of standard consumer preferences), it is possible toprecisely pin down the consumer’s preferences with ever-increasing data. (Of course, the datahas to increase in a ‘regular’ way.) A similar result can be proved in our setting.

19

where s;s0 D rs.pm/=rs0.p0m/ is known. Since ….s; �/ must be a probability

measure for all s; we must havePs0 s;s0= k Nus0k D ı= k Nusk. Let ‰ be the S �

S matrix ‰ WD Œ s;s0�. We then need to solve the problem of finding the lefteigenvector � 2 �S�1 of ‰ and the corresponding eigenvalue ı so that �‰ Dı�.14 Upon solving this linear programming problem, set k Nusk WD 1=�s, and….s; s0/ WD Nus.pm/=ı Nus0.p

0m/ for each s; s0 2 S .

(b) Eliciting the measure�s on U up to scaling is a little more complicated, as we firsthave to elicit its support up to scaling. To do so, fix s 2 S and let vs W P .M/! R

be defined as vs.pm/ WDPs0….s; s

0/ Nus0.pm/. Let B" be the "-ball around p�m. Forany � > 0 that is sufficiently small, it is easy to determine a lottery ˛.�/ 2 P .M/

such that ıvs.˛.�// D �. Then x WD¸.pm; ˛.�// W .pm 2 B"/ ^

�p"2 C �2 D �

�¹with � small enough features a different maximizer, q�.u/ WD argmaxx .uC ıvs/,for every possible taste uC ıvs with u 2 U.15 Given the menu x, we ask the DMto identify the minimal menu x0 (in the sense of set inclusion) such that x0 � xand x0 �s x. There is then an obvious bijection that maps the elements of x0 tothe support of �s in U, and the mapping is unique up to scaling.In order to simplify our argument, let us assume x0, and thus the support of �s,is finite. If, contingent on state s, DM is indifferent between (i) receiving extracontinuation utility � when choosing q�.u/, and (ii) receiving extra continuationutility � 0 when choosing q� .u0/, then ��s.u/ D � 0�s.u0/,16 and thus we determinethe relative weight �s.u/=�s.u0/. Fixing u, this can be done for all u0 ¤ u, so thatthe probability measure �s is known up to scaling.

5.3. Behavioral Comparison

Intuitively, one decision maker has more preference for flexibility than another ifhe has a stronger preference for menus over singletons in every state. In a manneranalogous to the characterization of risk aversion where lotteries are comparedto certain amounts of money, characterizing preference for flexibility requires acomparison between S-IHCPs and S-IHCSs in L. This comparison is meaningfulonly if preferences restricted to L are non-trivial.

Axiom 10 (Consumption Non-triviality). There exist `; `0 2 L such that ` � `0.

(14) Theorem 1 says that there is a unique � 2 �S�1 that solves this problem, that this � � 0, andthat the eigenvalue corresponding to � is ı 2 .0; 1/.

(15) To see this, let ."; �/�.u/ WD argmax¸.";�/Wp"2C�2D�

¹ kuk "C �. There is an obvious bijection

between ."; �/� and kuk. Furthermore, for u; u0 2 U with kuk D ku0k and u ¤ u0 the is a uniquemaximizer for any vN-M ranking of consumption prizes, where the maximal consumption valuein B" is kuk " for all u 2 U.

(16) Formally, using the notation from the previous footnote, x [�q�m.u/; ˛ .�

� .u/C �/�� x [�

q�m.u0/; ˛ .�� .u0/C �0/

�, where � and �0 must be small enough not to upset the bijection between

the maximizers and the support of �s :

20

If % has DPF representation�.�s/s2S ;…; ı

�, then by corollary 5 the restriction

of % to L is the recursive Anscombe-Aumann representation�.�su/s2S ;…; ı

�. It

follows that if % has a DPF representation, then it satisfies Axiom 10 if, and onlyif, �su ¤ 0 for some s 2 S (where 0 2 U).17 The recursivity of the representationalong with the fact that… is fully connected implies that if some �su ¤ 0, then forall s0 2 S , V.�; s0/ restricted to consumption streams is non-trivial.

Definition 7. %� has a greater preference for flexibility than % if f % ` impliesf %� ` for all ` 2 L and f 2 H .18

The comparison in the definition implies that % and %� have the same rankingover S-IHCSs, ie, ` % `0 if, and only if, ` %� `0. (This is lemma 34 in the appendixand also assumes Independence.) We require the following definition to comparebeliefs.

Definition 8. LetQ be a Markov kernel19 from U to itself. ThenQ is a dilation if itis expectation preserving, ie, if

RUu0Q.u; du0/ D u for each u 2 U. For probability

measures � and �� on U, �� is a dilation of � if there exists a dilation Q such that�� D Q�, ie, ��.du0/ WD

RQ.u; du0/ �.du/.

If �� is a dilation of �, then ��u D �u, because a dilation preserves expectations.In order to facilitate a comparison of two sets of measures, we shall say that aDPF representation

�.�s/s2S ;…; ı

�is canonical if k�suk2 D 1 for the smallest

s 2 S D f1; : : : ; ng such that �su ¤ 0. Obviously, % admits a canonical DPFrepresentation if, and only if, % satisfies Axioms 1–10.

Theorem 2. Let % and %� have canonical DPF representations�.�s/s2S ;…; ı

�and

�.��s /s2S ;…

�; ı��, respectively. Then, %� has a greater preference for flexibility

than % if, and only if,… D …�, ı D ı�, and ��s is a dilation of �s for each s 2 S .

The proof is in appendix F. Intuitively, DM* with preference %� has a greaterpreference for flexibility than DM with preference % precisely because he expectsmore uncertainty to resolve before making a choice, which increases the optionvalue of waiting to make a choice from the menu. Some asset pricing implicationsof Theorem 2 are explored in Krishna and Sadowski [2012].

(17) The “if” part of the claim is easy to see. To see the “only if” part, let �s have mean �su D0 for all s 2 S , and consider W 2 C.H � S/ such that W.`; s/ D 0 for all ` 2 L andW.f; s/ > 0 for all f 2 H . Then, with ˆ W C.H � S/ ! C.H � S/ given by ˆW.f; s/ WDPs02S ….s; s

0/�R

Umaxp2x

�u.pm/C ıW.ph; s/

�d�s.u/

�, we haveˆW.f; s/ > 0 for all f 2 H ,

with equality on L. Therefore, the unique fixed point of ˆ, namely the value function Vrepresenting %, must also have this property.

(18) The definition generalizes the definition of more averse to commitment in Higashi, Hyogoand Takeoka [2009] to apply to the domain of S-IHCPs. Their Theorem 4.2 is implied by ourTheorem 2.

(19) Let B be the Borel subsets of U, so that .U;B/ is a measurable space. Then,Q W U�B ! Œ0; 1�

is a Markov kernel from .U;B/ to itself if (i) for each u 2 U, Q.u; �/ is a probability measureon .U;B/, and (ii) for each D 2 B, Q.�;D/ is a measurable function defined on U.

21

The behavioral comparison in terms of beliefs provided by Theorem 2 is possibleonly because the beliefs .�s/ are identified up to scaling, and… and ı are identifieduniquely in our dynamic setting. In contrast, the notion of ‘greater preference forflexibility’ proposed in DLR for the static context cannot rely on beliefs becausethose are not identified in their model. Hence, instead of characterizing whether oneDM has a stronger preference for flexibility than another, DLR characterize whetherone DM has any preference for flexibility whenever the other does. Neither rankingis complete. While ours can only compare preferences that agree on the ranking ofsingletons, the ranking in DLR can only compare preferences with representationsfor which the support of the measure is ordered by set inclusion.We now compare DM’s strength of preference for flexibility across states.

Proposition 9. Suppose % has a DPF representation and satisfies Axiom 10. Then,for s; s0 2 S , %s exhibits a greater preference for flexibility than %s0 if, and only if,�s is a dilation of �s0 and….s; �/ D ….s0; �/.

Proposition 9 describes how DM trades off flexibility across states. For example,if �s is a dilation of �s0 and….s; �/ D ….s0; �/, and if � .s/ > � .s0/, then flexibilityis more valuable to DM in state s than in state s0. If all states are comparable inthe sense of greater preference for flexibility, then ….s; �/ is independent of s andobjective states evolve according to a process that is iid.

6. Conclusion

We provide foundations for a uniquely identified recursive representations of pref-erence for flexibility by relaxing the standard assumption of strategic rationalityto accommodate unobservable taste shocks. If taste shocks are transient, strategicrationality with respect to continuation problems should be satisfied unconditionally(CSR). If there are transient and persistent taste shocks, but only transient shocksare unobservable, then strategic rationality with respect to continuation problemsshould hold contingent on the objective state of the world (S-CSR).What if persistent taste shocks are also unobservable? Then CSR should be

satisfied only contingent on some history of past tastes and, in the simplest case,contingent on the current consumption taste. While this taste is by assumptionunobservable, CSR would also have to hold contingent on current consumptionchoice from a large enough menu. In a supplement, Krishna and Sadowski [2013],we pose an axiom called Choice Contingent CSR that captures this idea and derive arepresentation that can accommodate unobserved persistent taste shocks by allowingconsumption utilities themselves to follow a particular, and uniquely identifiedMarkov process. The proof builds on the proof for the DPF representation, wherethe collection of possible consumption rankings plays the role of the state space.While we suggest the three versions of CSR above—namely CSR along with

State Contingent and Choice Contingent CSR—as a natural starting point when

22

trying to understand how strategic rationality is violated, these could obviouslybe relaxed further. For example, CSR might only be satisfied contingent on finitehistories of consumption choices and states. Modelling the corresponding evolutionof beliefs as a Markov process would require a larger subjective state space, aswell as an assumption that ensures that preference for flexibility is sufficientlypersistent. Beyond that, our proofs of identification only require that the impliedMarkov process over preferences have a unique stationary distribution. Given thepersistence of preference for flexibility, such uniqueness is ensured by Stationarity(via an application of the Perron Theorem).Consequently, we view our work as illustrative of how to achieve a fully identified

recursive representation of choice when the assumption of strategic rationality isfurther weakened.

Appendices

In what follows, we will also consider the space of menus of consumption lotteries,K�P .M/

�, with typical members being a; b; c. By the recursive nature of H , continuation

problems are members ofH . Let A;B;C denote typical elements of the collection of menusof continuation lotteries, K

�P .H/

�. To ease notational burden, we will often write K for

K�P .M �H/

�, KM for K

�P .M/

�, and KH for K

�P .H/

�.

A. Abstract Representations

We shall first construct a general representation in the spirit of DLRS where the prize spaceis infinite. We then study the effect of assuming Strategic Rationality. We begin by definingand collecting some facts about support functions.Let Y be a compact metric space. Let P .Y / be the space of probability measures on

Y , endowed with the topology of weak convergence, which makes P .Y / compact andmetrizable, and let C.Y / denote the Banach space of uniformly continuous functions on Y .In what follows, for u 2 C.Y / and p 2 P .Y /, we will frequently denote

Ru dp by u.p/.

Fix p� 2 P .Y /, and let X WD fu 2 C.Y / W u.p�/ D 0g.Define UY WD fu 2 X W kuk1 D 1g, and for any weak* closed, convex G � P .Y /, let

�G W UY ! R given by �G.u/ WD maxp2G u.p/ be its support function. The extendedsupport function of such a set G is the unique extension of the support function �G to X bypositive homogeneity. A function defined on X is sublinear, norm continuous, and positivelyhomogeneous if, and only if, it is the extended support function of some weak* closed,convex subset of P .Y / [Theorem 5.102 and corollary 6.27 of Aliprantis and Border, 1999].We shall therefore call a function � W UY ! R a support function if its unique extension toX by positive homogeneity is sublinear and norm continuous.

Support functions have the following duality: For any weak* compact, convex subset Gof aff

�P .Y /

�, G�G

D G where G�GWD

¶p 2 aff

�P .Y /

�W u.p/ 6 �G.u/ for all u 2 UY

·.

Support functions also have the following useful properties20 for weak* compact, convexsubsets G;G0 of P .Y /: (i) G � G0 if, and only if, �G 6 �G0 , (ii) �tGC.1�t/G0 D t�G C .1�

(20) See, for instance, p 226 of Aliprantis and Border [1999].

23

t /�G0 for all t 2 .0; 1/, (iii) �G\G0 D �G ^ �G0 , and (iv) and �conv.G[G0/ D �G _ �G0 . (SinceG and G0 are convex, it follows from Lemma 5.14 of Aliprantis and Border [1999] thatconv.G [G0/ is also compact.) Notice also that �fp�g D 0.

A.1. Constructing a General Representation

Let Y be a compact metric space of prizes, so P .Y / is a compact metric space. Let K�P .Y /

�denote the space of compact subsets of P .Y /, and C

�P .Y /

�the space of closed, convex

subsets of P .Y /, both endowed with the Hausdorff metric. Then, C�P .Y /

�is a closed

subspace of K�P .Y /

�. We consider a preference % over K

�P .Y /

�. For notational ease, we

shall write K and C for K�P .Y /

�and C

�P .Y /

�respectively. Typical elements of K will

be denoted by G;G0 etc. As always, ˛G C .1 � ˛/G0 2K is the Minkowski sum.The space of all vN-M utility functions is simply C.Y /, the Banach space of uniformly

continuous functions on Y . In general, in a state dependent additive EU representation,the vN-M utility functions need only be identified up to positive affine transformation. Fixp� 2 P .Y / as above and, analogous to DLRS, let the subjective state space be given byUY WD fu 2 C.Y / W u.p

�/ D 0 and kuk1 D 1g, the space of all vN-M utility functions that(i) take the value 0 at p�, (ii) are nontrivial on P .Y /, and (iii) lie on the boundary of theunit ball of C.Y /.Let % be a preference on K. The axioms to be imposed on % are the same as those

imposed on %s, modulo the change in domain. Therefore, for expositional ease, we willsimply refer to the corresponding axioms for %s in the text.

We refer to the state spaceUY as the canonical state space. Let AUYbe the Borel algebra

of sets in UY , and � a normal21 charge on .UY ;AUY/. A pair .UY ; �/ is a finitely additive

EU representation of % if V.x/ DRUY

maxp2x u.p/ d�.u/ represents %. We say that afinitely additive EU representation .UY ; �/ is jointly identified if, given the state space UY ,the charge � is unique.22

Theorem 3. A preference, %, satisfies Non-triviality, Continuous Order, Monotonicity, andIndependence if, and only if, it admits a finitely additive EU representation.

Note the key differences between Theorem 3 and the additive EU representation theoremof DLR. They establish that given the canonical subjective state space, the measure is uniqueand countably additive, while the theorem above establishes neither. To see why, note firstthat our subjective state space is not compact. Therefore, the Riesz Representation Theoremonly guarantees that the measure is finitely additive. (Notice that this would remain the caseeven if one were able to establish uniqueness of the representation.) Second, we are unableto show that the finitely additive EU representation obtained is jointly identified. This isbecause when the subjective state space is finite dimensional, the span of the space of allsupport functions is dense in the space of all continuous bounded functions on the subjectivestate space. We are unable to establish this result in our infinite dimensional setting. Beforethe formal details are presented, we provide some intuition for the proof.

(21) A charge is outer regular if every set in AUYcan be approximated from without by open sets;

inner regular if it can be approximated from within by closed sets; and normal if it is both outerand inner regular – see also definition 10.2 in Aliprantis and Border [1999].

(22) This is the sense in which the representation in DLR is identified.

24

The proof of the representation naturally extends ideas in DLRS to the infinite dimensionalsetting. The first step is to show that each menu can be identified (isometrically) with itssupport function, and that support functions live in the space of twice normalized, non-trivialvN-M functions on the prize space Y . (This is exactly as in DLRS.) This allows us to definethe subjective state space as a space of all twice normalized, continuous, non-constant,non-trivial functions on Y . Instead of looking at the space of menus, we can look at thespace of support functions, a subset of all continuous bounded functions on the subjectivestate space.The second step of the proof shows that any linear functional on the space of menus

induces a continuous linear functional on the corresponding space of support functions.Moreover, since the preference % is monotone, this linear functional is Lipschitz, and cantherefore be extended to the space of all continuous bounded functions on the subjectivestate space. (This step uses the Hahn-Banach Theorem.) The final step uses the RieszRepresentation Theorem to show that any linear functional on the space of all continuousbounded functions can be written as an integral with respect to a normal finitely additivemeasure.

Proof of Theorem 3. By the continuity of %, and since % satisfies Independence (Axiom4), an adaptation of Lemmas 1 and 2 from DLR implies that G � conv.G/ for each G 2K .(In particular, we have conv

�tG C .1 � t /G0

�D t conv.G/C .1 � t / conv.G0/.) Following

DLR, it suffices to restrict attention to C , the space of all weak* compact, convex subsets ofP .Y /. Here, P .Y / is endowed with a metric inducing the weak* topology, and C is endowedwith the Hausdorff metric. Let K0 WD f� 2 Cb.UY / W � D �G for some G 2 Cg denote theaffine embedding of C in Cb.UY /, where Cb.UY / is the space of bounded and continuousfunctions onUY , endowed with the supremum norm, and recall that by construction, 0 2 K0.Define the induced preference %� on K0, so that G % G0 if, and only if, �G %� �G0 . It

is easily seen that %� is complete and transitive, and satisfies Continuity, Independence,and Monotonicity,23 since % has these properties. By the Mixture Space Theorem (see,for instance, Fishburn [1970] or Kreps [1988]), there exists a function ' W K0 ! R thatrepresents %�, is linear in the sense that for all �1; �2 2 K0 and ˛ 2 Œ0; 1�, '

�˛�1 C .1 �

˛/�2/�D ˛'.�1/C .1 � ˛/'.�2/, and is positive so that �1 > �2 implies '.�1/ > '.�2/.

Let K1 WDSr>0 rK0 be the cone generated by K0. It can be shown, following the

arguments in lemma S10 of the supplement to DLRS, that (i) K1 �K1 is a vector subspaceof Cb.UY /, and (ii) for any ! 2 K1 � K1, there exist �1; �2 2 K0 and r > 0 such that! D r.�1 � �2/. Extend ' to K1 �K1 by linearity and let ˆ0 denote this extension. This isachieved by settingˆ0.!/ WD r

�'.�1/�'.�2/

�, where for a given ! 2 K1�K1, �1; �2 2 K0

and r > 0 are such that ! D r.�1 � �2/. The arguments in lemma S11 of the supplement toDLRS allow us to establish that ˆ0 is positive, ie, !1 > !2 imply ˆ0.!1/ > ˆ0.!2/.As in DLRS, we claim that ˆ0 on K1 �K1 is Lipschitz. To see this, notice that for any

! 2 K1 �K1, we have ! 6 k!k11 (indeed, j!j 6 k!k11, where 1 is the constant functionthat maps to 1), so that ˆ0.!/ 6 k!k1ˆ0.1/, ie, ˆ0 is Lipschitz with constant ˆ0.1/.

Since ˆ0 is Lipschitz on a vector subspace K1 �K1, the Hahn-Banach Theorem allows

(23) Recall that G � G0 if, and only if, �G > �G0 . Monotonicity of % requires that if G � G0, thenG % G0, which implies that we must have �G %� �G0 . Thus, %� satisfies Monotonicity in thesense that for �1; �2 2 K0, �1 > �2 implies �1 %� �2.

25

us to extend it to Cb.UY /, with the extension being denoted by ˆ. The Riesz RepresentationTheorem [Theorem 13.9, Aliprantis and Border, 1999] allows us to represent the linearfunctional ˆ on Cb.UY / as an integral with respect to a normal charge � on AUY

, thealgebra generated by the open sets of UY , as desired.

A.2. Strategic Rationality

Let % be a preference on K .

Definition 10. A preference, %, is strategically rational if G % G0 implies G � G [G0.

In this section, we show that if % has a finitely additive EU representation .UY ; �/, it isstrategically rational if, and only if, � is carried by a singleton. We begin with a lemma.

Lemma 11. For any U0 � UY , the following are equivalent:

(a) jU0j D 1.

(b) For all p; q 2 P .Y / and for all u1;u2 2 U0, u1.p/ > u1.q/ if and only if u2.p/ > u2.q/.

Proof. It is clear that (a) implies (b). To see that (b) implies (a), suppose contrary to (a)we have u1;u2 2 U0, u1 ¤ u2. By the definition of UY , there exists p� 2 P .Y / such thatu.p�/ D 0 for all u 2 UY . Moreover, kuk1 D 1 for all u 2 UY . Therefore, u1 is not apositive affine transformation of u2.

At the same time, (b) implies via the expected utility theorem, that u1 is an affine trans-formation of u2, which is a contradiction.

Definition 12. The carrier of the charge� is the setU� WDTfN W N is closed,�.N c/ D 0g.

The carrier of the charge always exists, and is clearly well defined. If the charge is also ameasure, then the carrier is referred to as the support of the measure. Moreover, given thedefinition of �, we have U� � UY . For any p; q 2 P .Y /, define Up;q WD fu 2 UY W u.p/ >

u.q/g and Uıp;q WD fu 2 UY W u.p/ D u.q/g. Notice that Up;q is always open and Uıp;q isalways closed, since p is a continuous (linear) functional on UY , which is a closed set.

Lemma 13. If % on K is strategically rational, then minf�.Up;q/; �.Uq;p/g D 0 for allp; q 2 P .Y /.

Proof. Suppose to the contrary there exist p; q with minf�.Up;q/; �.Uq;p/g D �.Up;q/ > 0.It is clear that fp; qg � fpg; fqg, which violates Strategic Rationality.

Lemma 14. If for all p; q 2 P .Y /, minf�.Up;q/; �.Uq;p/g D 0, thenˇ̌U�

ˇ̌D 1.

Proof. If for all p; q 2 P .Y /, minf�.Up;q/; �.Uq;p/g D 0, then either (i)U� � Up;q[Uıp;q ,

or (ii) U� � Uq;p [Uıp;q , but not both (by the definition of U� and since Up;q and Uq;p areopen).

In other words, for all p; q 2 P .Y /, and for all u1;u2 2 UY , u1.p/ > u1.q/ if, and onlyif, u2.p/ > u2.q/. Lemma 11 now implies that

ˇ̌U�

ˇ̌D 1, as required.

We may now put all this together, as follows.

26

Proposition 15. Let% be a preference onK�P .Y /

�that has a finitely additive representation

.UY ; �/. Then, the following are equivalent:

(a) % is strategically rational.

(b) The carrier of the charge � is a singleton, ie,ˇ̌U�

ˇ̌D 1.

Proof. It is easy to see that (b) implies (a). Lemmas 13 and 14 show that (a) implies (b).

B. The Domain of SIHCPs

We now sketch the construction of State Contingent Infinite Horizon Consumption Problems(S-IHCPs) introduced in section 3.1. As in the text,M is a finite set of consumption prizesin any period, and S is a finite set of states.

LetH1 WD H�K�P .M/

��denote the set of acts that give a closed subset of P .M/ in each

state s 2 S . It is a compact metric space when endowed with the (product) Hausdorff metric.For each t > 1, inductively define Ht WD H

�K�P .M �Ht�1/

��, which is also a compact

metric space. Thus, each ft 2 Ht is an act that gives a closed set of probability measuresover M � Ht�1 in each state s 2 S . This allows us to define the space H� WD�1tD1Ht ,which is also a compact metric space.

Thus, an element in H� is a collection .ft /, where ft 2 Ht for each t > 1. Each ftcontains information about some of the f� 2 H� for all � < t since each pt 2 ft .s/ is aprobability measure overM �Ht�1, and each pt�1 2 ft�1.s/ is a probability measure overM �Ht�2, and so on. A sequence .ft / is consistent if, roughly speaking, the informationin ft about ft�2 is in accord with the information in ft�1 about ft�2.24 The space of allconsistent sequences, denoted byH � H�, is the space of S-IHCPs. A simple adaptation ofTheoremA1 of GP shows that there is a homeomorphism betweenH andH

�K�P .M�H/

��.

In fact, it can be shown that this homeomorphism is an affine or linear homeomorphism, sothat we can define a linear preference on H (ie, a preference that satisfies Independence)and study the naturally induced preference on H

�K�P .M �H/

��.

C. A Separable Representation

In the process of obtaining our representations, we will frequently find it useful to obtain anintermediate representation on one state space, and then transform the representation so it isdefined on another state space. The following lemma is an abstract version of this idea. Inwhat follows, we consider the prize spaceM � Y . If Y is compact,M � Y is also compact.This allows us to define the canonical state space UM�Y and with a typical state given byu 2 UM�Y .

Lemma 16 (Change of State Space). Let .UM�Y ; �/ be a finitely additive EU representationof %. A sufficient condition for .U0K�Y ; �

0/ to be another finitely additive EU representationof % is that there exist functions ‰ W UM�Y ! U0M�Y and .�; �/ W UM�Y ! RCC �R suchthat ‰ is a measurable bijection and .�; �/ are integrable, and that satisfy:

(24) See GP for a precise definition of consistency. It is easy to construct examples of sequences inH� that are not consistent.

27

(a) for all u0 in the image of ‰, u0.p/ D �.u/.‰u/.p/C �.u/ for all p 2 P .M � Y /, and

(b) the bijection ‰ is measure preserving, ie, for all measurable D0 � U0M�Y , �0.D0/ D

��‰�1D0

�, and for all measurable D � UK�Y , �.D/ D �0.‰D/.

If the finitely additive EU representation .UM�Y ; �/ is jointly identified (as in appendixA.1), then the condition is also necessary.

The proof of the sufficiency part of the lemma merely amounts to a change of variable,and is an instance of the nonuniqueness encountered in DLR. The necessary part of thelemma is also not difficult, and a statement and proof (albeit, in a slightly different setting)can be found in Schenone [2010].Separability (Axiom 5) says that if p; q 2 P .M � Y / are such that their marginals are

identical, ie, pm D qm and py D qy , then fp; qg � fpg. This gives us the following lemma.

Lemma 17. Let .UM�Y ; �/ be a finitely additive representation and satisfy Separability(Axiom 5). For p and q that induce the same marginals, ie, pm D qm and py D qy ,�fu 2 UM�Y W u.p/ > u.q/g D 0.

Proof. If the lemma were not true, we would have V.fp; qg/ > V.fqg/, which contradictsSeparability (Axiom 5).

Definition 18. Let .UM�Y ; �/ be a finitely additive EU representation. The representationis finitely additive, separable if for each u in the carrier of �, there exist u.u/ 2 U andv.u/ 2 C.Y / such that u.p/ WD u.pmIu/ C v.py Iu/ for each p 2 P .M � Y /, and if themapping u 7! .u; v/ is measurable. For ease of notation, we shall suppress the dependenceof u and v on u.

Lemma 19. A finitely additive EU representation satisfies Separability if, and only if, it isalso a finitely additive separable representation as in

V.G/ D

ZUM�Y

maxp2G

�u.pm/C v.py/

�d�.u/(C.1)

Proof. The ‘if’ part is clear. We now prove the ‘only if’ part. Fix Qm 2 M and Qy 2 Y .Since 1

2.m; y/C 1

2. Qm; Qy/ and 1

2.m; Qy/C 1

2. Qm; y/ have the same marginals, lemma 17 implies

that u�12.m; y/ C 1

2. Qm; Qy/

�D u

�12.m; Qy/ C 1

2. Qm; y/

�for all u in the carrier of �. Since

each u is linear in probabilities, u�.m; y/

�D u

�.m; Qy/

�C u

�. Qm; y/

�� u

�. Qm; Qy/

�must

be satisfied for each individual state u in the carrier of �.25 Following GP, we defineu.m/ WD u

�.m; Qy/

�and v.y/ WD u

�. Qm; y/

�� u

�. Qm; Qy/

�to find u

�.m; Qy/

�D u.k/ C v.y/.

This allows us to write V.G/ DRUM�Y

maxp2G�u.pm/Cv.py/

�d�.u/which is the desired

separable representation.

(25) In comparison, the weaker separability axiom in GP would only imply this condition for thepreference functional V that aggregates over all states.

28

C.1. Reduction of the State Space

Recall that U WD¶u 2 RM W

Pi ui D 0

·is the collection of all vN-M utility functions onM

modulo additive constants. Notice thatUM D¶r 2 U W krk2 D 1

·. SinceUM � U, we must

havePi ri D 0, so that r.p�m/ D 0 for all r 2 UM . We now use lemma 16 to show that for

any finitely additive separable representation, the state space can be viewed as (a subset of)UM � Œ0; 1��UY , with a typical vN-M utility function of the form r.pm/C .1� /w.py/.

Proposition 20. A preference % over the compact subsets of P .M � Y / with a finitelyadditive representation .UM�Y ; �/ satisfies Separability (Axiom 5) if, and only if, there is achange of state space as in lemma 16 that allows to write a finitely additive representationof % of the form Z

UM�Œ0;1��UY

maxp2G

� r.pm/C .1 � /v.py/

�d�0.r; ; v/

where �0 is a normal charge on UM � Œ0; 1� �UY .

Proof. The ‘if’ part is immediate, so we only prove the ‘only if’ part of the proposition.Let .UM�Y ; �/ be a finitely additive representation. (Recall that Theorem 3 guarantees theexistence of a finitely additive representation, though we have not ruled out the possibilitythat there could be many such representations.) By lemma 19, every such representationmust have the property that for every u in the carrier of �, u.p/ D u.pm/C v.py/, whereu 2 U and v 2 C.Y / are vN-M functions and the mapping u 7! .u; v/ is measurable.

Every separable utility function u.pm/C v.py/ is of the form ˛r.pm/C ˇw.py/, wherer 2 UM ,w 2 UY , and .˛; ˇ/ 2 R2Cn.0; 0/. LetX WD

�UM �RC

���UY �RC

�, and consider

UM�Y \X . A finitely additive separable representation is one where �.UM�Y \X/ D 1.An even smaller state space is UM � Œ0; 1� � UY , wherein the utility in state .r; ; v/ is

r.pm/C .1 � /w.py/.Define ‰ W UM�Y \X ! UM � Œ0; 1��UY as follows: ‰ W

�.r; ˛/; .w; ˇ/

�7!�r; ˛=.˛C

ˇ/;w�. It is clear that‰ is continuous, and hence measurable. Moreover,‰ is also a bijection.

To see this, suppose there are ˛; ˛0; ˇ; ˇ0 such that ˛=.˛ C ˇ/ D ˛0=.˛0 C ˇ0/. This implies˛0 D �˛ and ˇ0 D �ˇ for some � > 0, which is impossible since, as mentioned above,UM�Y \X only contains functions that are unique up to scaling.By lemma 16, we may define the charge �0.D/ WD �.‰�1D/ on UM � Œ0; 1� �UY , and

restrict attention to a separable representation on this state space. Notice that since ‰ iscontinuous, the charge �0 is normal (ie, both inner and outer regular).

C.2. Additively Separable Representation

We now show that in the presence of some additional assumptions, the normal probabilitycharge �0 in proposition 20 can be replaced by a regular probability measure, leading to anadditive, separable EU representation. Recall that�0 is a regular probability measure if it is(i) regular (ie, outer regular and tight), and (ii) a probability measure, ie, is countably additive,has �0.UM � Œ0; 1� �UY / D 1, and is defined on the Borel �-algebra of UM � Œ0; 1� �UY .

29

Definition 21. Let .UM � Œ0; 1� � UY ; �0/ be a finitely additive separable EU represen-

tation. The representation is Y -simple if the marginal charge of �0 on UY — given byRUM�Œ0;1��D

d�.r; ; w/ for anyD 2 AUY— has a finite carrier. It is Y -trivial if the carrier

of the marginal on UY is a singleton.

Lemma 22. Every finitely additive, separable representation that is Y -simple can be ex-tended uniquely to an additive, separable, Y -simple representation

V.G/ D

ZUM�Œ0;1��fw1;:::;wI g

maxp2G

� r.pm/C .1 � /w.py/

�d�.r; ; w/(C.2)

Proof. As �0 is Y -simple, there must exist an I 2 N with I > 0 such that the carrierof �0 is a closed subset of UM � Œ0; 1� � fw1; : : : ; wI g, where fw1; : : : ; wI g � UY . SinceUM � Œ0; 1� � fw1; : : : ; wI g is compact, it follows immediately that the carrier of �0 is alsocompact.The charge �0 is normal (see footnote 21), so for any Borel (algebra) measurable A �

UM � Œ0; 1� � fw1; : : : ; wI g, the inner regularity of �0 implies that �0.A/ D supf�.C/ WC � A and closedg. Since UM � Œ0; 1�� fw1; : : : ; wI g is compact, it follows that any closedC � A is also compact. Therefore, the charge �0 is ‘tight’ relative to a compact class of sets,namely the collection of all compact subsets of UM � Œ0; 1� � fw1; : : : ; wI g. By Theorem9.12 of Aliprantis and Border [1999], �0 is also countably additive (on the algebra of opensets).

By the Carathéodory Extension Procedure Theorem,�0 can be uniquely extended from thealgebra of open sets to the Borel �-algebra ofUM � Œ0; 1��fw1; : : : ; wI g. (The CarathéodoryTheorem is Theorem 9.22 of Aliprantis and Border [1999]. The extension of the measure isunique since �0 is a finite measure.) The unique extension of �0 will be written as �.Finally, Theorem 10.7 of Aliprantis and Border [1999] says that a measure on a Polish

space is regular, and because UM � Œ0; 1� � fw1; : : : ; wI g is a compact subset of Euclideanspace, it follows that � is regular.

C.3. Identification of the Representation

Recall that in the original, abstract EU representation theorem, Theorem 3, we are unable toestablish that � is a regular measure or that the representation is jointly identified. We haveseen identified additional assumptions, it is possible to show that � is a regular measure. Wenow show that under those same assumptions, the representation can be jointly identified.

Proposition 23. Suppose a continuous preference % has an additive, separable, Y -simplerepresentation as in equation (C.2). Then, the representation is jointly identified, ie, themeasure � is unique on the state space UM � Œ0; 1�� fw1; : : : ; wI g (where I > 0 is a naturalnumber).

Proof. Let % have a utility representation V as in (C.2), and let �0 D �. Suppose thereis another regular measure �1 (that need not be a probability measure) on UM � Œ0; 1� �

fwIC1; : : : ; wICJ g (where J > 0) such that

V.G/ D

ZUM�Œ0;1��fwIC1;:::;wICJ g

maxp2G

� r.pm/C .1 � /w.py/

�d�1.r; ; w/

30

It is without loss of generality to consider �0 and �1 as measures on UM � Œ0; 1� �

fw1; : : : ; wICJ g. We show that �1 D �0.Let Y0 � Y be a finite set such that for each i; j 2 1; : : : ; I where i ¤ j , there exists

yij 2 Y0 such that wi .yij / ¤ wj .yij /. The requirement is easily satisfied, since wi is acontinuous, non-constant function on Y and i ¤ j implies wi ¤ wj , which in turn impliesthat the two functions must disagree somewhere.

Now consider the set B WDM � Y0, and the domain K�P .B/

�. Then, each measure �j ,

j D 0; 1, induces the preference functional Wj on K�P .B/

�as follows:

Wj .G/ D

ZUM�Œ0;1��fw1;:::;wICJ g

maxp2G

� r.pm/C .1 � /w.py/

�d�j .r; ; w/

Define, as in DLR, UB WD¶r 2 RB W

Pi ri D 0;

Pr2i D 1

·. In that case, UM � Œ0; 1� �

fw1; : : : ; wICJ g is isomorphic to a subset of UB . Indeed, for any vN-M utility function onM � Y0 of the form ˛r.qm/C .1 � ˛/w.qy/, the two normalizations that map the functioninto UB are continuous.Thus, �0 and �1 are transformed into measures on UB in the obvious way, and have

supports on sets (in UB) that are isomorphic to UM � Œ0; 1� � fw1; : : : ; wICJ g. From thedefinition of the functionals W0 and W1, we know that for each menu G 2 K

�P .B/

�, we

have W0.G/ D W1.G/. The uniqueness part of the additive EU representation theorem inDLR now says that the two transformed measures agree on UB , and hence �0 D �1, asdesired.

It will be useful to denote Y -trivial, additive, separable EU representations of % by thecollection .U; v; �/ where v W Y ! R is a vN-M function, and � is a probability measureon (the Borel �-algebra of) U. It induces the preference functional

V.G/ D

ZU

maxp2G

�u.pm/C v.py/

�d�.u/(C.3)

that represents the preference % over menus. The transformation from the state space with.r; ˛/ 2 UM � Œ0; 1� to the state space with u 2 U is achieved by setting u WD ˛

.1�˛/r and by

applying the appropriate transformation to the measure, as in lemma 16.

D. Proofs from Section 3

D.1. Proof of Proposition 2

Let W 2 C.H � S/ and consider the function ˆW.f; s/, given by

ˆW.f; s/ WDXs02S

….s; s0/

�ZU

maxp2f .s0/

�u.pm/C ıW.ph; s

0/�d�s0.u/

�for all s 2 S . It is easy to see that ˆ is monotone, ie, W 6 W 0 implies ˆW 6 ˆW 0,and satisfies discounting, ie, ˆ.W C �/ 6 ˆW C ı� when � > 0. If we assume thatˆW 2 C.H � S/ for all W 2 C.H � S/, it follows that ˆ is a contraction mapping (withmodulus ı), and has a unique fixed point, which establishes the proposition. All that remainsis to show that ˆ is an operator on C.H � S/.

31

Before completing the proof, we establish a useful inequality:R

U kuk2 d�s < 1 foreach �s. To see that the inequality holds, let us define W W KM ! R as W.a/ WDR

U max˛2a u.˛/ d�.u/ for a 2 KM . Recall that p�m is the uniform lottery over M . Leta WD f˛ 2 P .M/ W k˛ � p�mk2 6 " for some " > 0g so that a 2KM . Then, for each u 2 U,max˛2a u.˛/ D " kuk2. Therefore, 0 6 W.a/ D "

RU kuk2 d�.u/. But W

�.P .M/

�D

W.M/ 6Pm � jumj <1 because � is nice, which implies that 0 6 W.a/ <1.

We will now show that if .fn/ 2 H1 is a sequence that converges to f 2 H , thenˆ.W /.fn; s

0/ ! ˆ.W /.f; s0/ whenever W 2 C.H � S/, which establishes that ˆW 2

C.H � S/. (Since S is finite, any convergent sequence in S must eventually be constant,which we take to be s0.)

For each x 2K , u 2 U,W 2 C.H �S/, and s 2 S , define '.x; u; s/ D maxp2x�u.pm/C

ıW.ph; s/�. Then, j'.x; u; s/j 6 maxp2x ju.pm/C ıW.ph; s/j 6 kuk2maxp2x

ˇ̌̌u.pm/kuk2

ˇ̌̌C

maxp2x ı jW.ph; s/j 6 kuk2K1 C K2;s where K1 WD maxx2K maxp2xˇ̌̌u.pm/kuk2

ˇ̌̌, K2;s > 0,

and the bounds follow from the definition of u 2 U, the compactness ofH , and the continuityof W .

AsW is continuous, the function u.pm/C ıW.ph; s/ 2 C.M �H/ is a continuous, linearfunctional on P .M �H/, when the latter is endowed with the topology of weak convergence(which is metrizable). Therefore, by the Maximum Theorem, for each u 2 U and s 2 S ,'.x; u; s/ is continuous in x.Consider the sequence .fn/ that converges to f . By the bounds established above,

j'.fn.s/; u; s/j 6 kuk2K1CK2;s , and kuk2K1CK2;s is �s-integrable since �s is nice (bythe inequality established above). Moreover,

limn!1

ˆW.fn; s0/ D lim

n!1

Xs

….s0; s/

�ZU

'.fn.s/; u; s/ d�s.u/�

D

Xs

….s0; s/

�ZU

limn!1

'.fn.s/; u; s/ d�s.u/�

D

Xs

….s0; s/

�ZU

'.f .s/; u; s/ d�s.u/�

D ˆW.f; s0/

As f and .fn/ are arbitrary, we conclude that ˆW 2 C.H � S/ whenever W 2 C.H � S/.The equalities above rely on the Dominated Convergence Theorem to interchange the orderof limits and integration, and the continuity of '.�; u; s/ for each u and s to establish thepointwise limit. This completes the proof.

D.2. Proof of Theorem 1

That the representation satisfies all the axioms is straightforward. Consider a preference% that satisfies Axioms 1–9. By Theorem 3, % has a finitely additive, EU representation.Separability (Axiom 5) implies, according to lemma 19 and proposition 20, that any suchfinitely additive EU representation also has a representation based on a regular countablyadditive measure.

32

In section D.2.1, we use S-CSR (Axiom 8) to establish in proposition 24 that for eachs 2 S , the preference %s has a H -trivial, separable, additive representation as in (C.3).This amounts to showing that the relevant state space is isomorphic to U. In section D.2.2,proposition 25 establishes that the parameter of the representation of%s , namely the measure�s on U, is uniquely identified up to a scaling. In section D.2.3, we invoke the remainder ofthe axioms and show in proposition 28 that % has a recursive representation, possibly withstate-dependent discount factors. In section D.2.4, we show that there exists an equivalentrepresentation of%with a state-independent discount factor ı 2 .0; 1/. We also show that thecollection of measures .�s/ in this representation is unique up to a common scaling, and thatthe Markov chain on S with transition probabilities… has a unique stationary distribution� (propositions 30, 31, and 32). This proves the theorem.

D.2.1. Separable Representation

It is easy to see that %s is continuous, and satisfies Independence, Monotonicity, andSeparability, so that by proposition 20, %s has a finitely additive separable representation.

Proposition 24. Suppose %s has a finitely additive separable representation. If %s alsosatisfies State Contingent CSR (Axiom 8), then the representation is H -trivial.

Proof. As in Proposition 20, we know that a separable, finitely additive separable represen-tation has the form

Us.x/ D

ZUM�Œ0;1��UH

maxp2x

� r.pm/C .1 � /v.ph/

�d�s.r; ; v/

For an arbitrary consumption prize m 2 M , let .m;A/ be a rectangular menu, definedas .m;A/ WD f.m; p´/ W p´ 2 A; A 2 KH g. Define a utility function Ws W KH ! R asWs.A/ D Us.m;A/. It follows from the separability of the representation that the choiceof m 2 M only affects Ws up to a constant. Let d��s .v/ D

’UM�Œ0;1�

d�s.r; ; v/ be themarginal of � on UH . Then, Ws.A/ D

RUH

maxph2A v.ph/ d��s .v/C constant.By proposition 15 above, State Contingent CSR (Axiom 8) implies that Ws.A/ D

maxph2A vs.ph/, where vs 2 C.H/ is given by vs.f / D Ws.ff g/. But this implies thatmaxph2A vs.ph/ D

Rmaxph2A v.ph/ d��.v/. Therefore, the carrier of �� must be a single-

ton.

It follows from lemma 22 that �s can be extended in a unique way to an additive measureon U. Abusing notation, this transformed measure is also denoted by �s . Thus, as mentionedat the end of section C.3, %s has a separable, additive, H -trivial EU representation .�s; vs/that can be written as

Us.x/ D

ZU

maxp2x

Œu.pm/C vs.ph/� d�s.u/(D.1)

We end with the observation that �s is nice. To see this, note that there exists p�h 2 P .H/

such that vs.p�h/ D 0. Now consider the menu .m; p�h/. It is easy to see that Us

�.m; p�

h/�D

�su.m/ WDR

U u.m/ d�s.u/. But Us�.m; p�

h/�is finite, which implies �su.m/ is finite for

every m 2M , which proves that �s is nice.

33

D.2.2. Identification

Section D.2.1 shows that the state space relevant for an additive, separable, H -trivial rep-resentation is U. Our goal in this section is to show that the measure �s on U and vs areunique up to a common scaling. We shall say that vs is non-degenerate if vs ¤ 0.

Proposition 25. Consider two separable, H -trivial, additive EU representations, .�s; vs/and .�0s; v0s/, of %s , wherein both vs and v0s are non-degenerate. There exists � > 0 such thatthe following properties hold:

(a) �s.�D/ D �0s.D/ for all measurable D � U,

(b) v0s D �vs C constant.

Proof. By lemma 16, our result on the change of state space, we know that there exists ameasurable bijection‰ W U! U, and integrable functions .�; �/ W U 7! .RCC;R/ such thatfor each u0 in the support of�0s , we have u0.pm/Cv0s.ph/ D �.u/

�.‰u/.pm/Cvs.ph/

�C�.u/.

Consider two lotteries p; q such that pm D qm but where vs.ph/ ¤ vs.qh/. Then,�u0.pm/C v

0s.ph/

���u0.qm/C v

0s.qh/

�D v0s.ph/ � v

0s.qh/

D �.u/Œvs.ph/ � vs.qh/�

But this implies �.u/ D v0s.ph/ � v0s.qh/

vs.ph/ � vs.qh/for all u, and hence �.u/ is constant, which proves

(b).Let U 0s be the functional induced by .v0s; �0s/. Then,

U 0s.x/ D

ZU

maxp2x

�u0.pm/C v

0s.ph/

�d�0s.u0/

D

ZU

maxp2x

��.‰u/.pm/C vs.ph/

�d Q�0s.u/C

Z��0s

D �

ZU

maxp2x

�Qu.pm/C vs.ph/

�d�00s . Qu/C constant

where Q�0s and �00s each obtain from a change of state space. Proposition 23 says that �00s D �s ,which implies that ‰ is the identity mapping. Thus, it must be that �0s.D/ D �s.�D/ for allmeasurable D � U. These observations prove part (a).

D.2.3. Recursive Representation

The preference % on H is continuous and satisfies Independence (Axiom 3). Therefore,there exists an additively separable representation with state dependent utilities as specifiedin (D.1), namely

W0.f / DXs

�.s/

�ZU

maxp2f .s/

�u.pm/C vs.ph/

�d�s.u/

�(D.2)

where � is the period 0 beliefs about the states in S . In the representation above, eachvs W H ! R is continuous. Let %�s be the preference relation on H induced by vs and recallthat H is convex.

34

Proposition 26. Suppose % has a representation as in (D.2) and satisfies Singleton Indiffer-ence to Timing (Axiom 7). Then, each %�s satisfies Independence.

Proof. Let f; g 2 H such that vs.f / > vs.g/ which implies Us�fp�m; f g

�> Us

�fp�m; gg

�.

Let f 0 2 H , and fix � 2 .0; 1�. By the linearity of Us, we have Us��fp�m; f g C .1 �

�/fp�m; f0g�> Us

��fp�m; gg C .1 � �/fp

�m; f

0g�. But %s satisfies Singleton Indifference to

Timing (Axiom 7), which implies that Us��fp�m; f g C .1 � �/fp

�m; f

0g�D Us

�fp�m; �f C

.1 � �/f 0g�, so that Us

�fp�m; �f C .1 � �/f

0g�> Us

�fp�m; �g C .1 � �/f

0g�, which in turn

implies that vs��f C .1 � �/f 0

�> vs

��g C .1 � �/f 0

�. We have just established that %�s

satisfies Independence (Axiom 3).

Proposition 27. Suppose % has a representation as in (D.2). If % satisfies Statewise Non-triviality (Axiom 1), History Independence (Axiom 9), and Aggregate Stationarity (Axiom6), then each %�s also satisfies Monotonicity and Statewise Non-triviality (Axiom 1).

Proof. By proposition 26, %�s satisfies Independence (Axiom 3), and hence vs is linear. Itfollows that there exist utility functions ws;s0 W K ! R for all s0 2 S such that vs.f / DPs0 ws;s0

�f .s0/

�. Notice that by History Independence (Axiom 9), we may assume that

ws;s0 D �s.s0/ws0 for all s; s0 2 S , where �s.s0/ > 0.

Notice that f xs % fys if, and only if, Us.x/ > Us.y/. By Aggregate Stationarity (Axiom

6), this is equivalent to requiring that f.m; f xs /g % f.m; fys /g, which holds if, and only

if,P0s �.s

0/vs0.fxs / >

P0s �.s

0/vs0.fys /. By the observations above, this is equivalent toP

s0 �.s0/�s0.s/ws.x/ >

Ps0 �.s

0/�s0.s/ws.y/. Let W 0s .x/ WDPs0 �.s

0/�s0.s/ws.x/. Then,W 0s is linear on K and W 0s .x/ > W 0s .y/ if, and only if, Us.x/ > Us.y/.Thus, %�s satisfies Monotonicity (Axiom 4) if, and only if, ws.x [ y/ > ws.x/ for all

x; y 2 K. But ws is monotone with respect to set inclusion if, and only if, Us is, whichproves that f x[ys0 %�s f

xs0 because x [ y %s x.

Finally, notice thatW 0QsDPs �.s/�s.Qs/wQs . ButW 0Qs is non-trivial because%Qs is non-trivial.

Therefore, it must be that wQs is non-trivial, from which it follows that %�Qssatisfies Statewise

Non-triviality (Axiom 1).

We now establish the existence of a recursive representation.

Proposition 28. Let % have a representation as in (D.2), and suppose % satisfies StatewiseNon-triviality (Axiom 1), History Independence (Axiom 9), and Singleton Indifference toTiming (Axiom 7). Then, there is a value function

V.f; s0/ DXs

….s0; s/

�ZU

maxp2f .s/

�u.pm/C ısV.ph; s/

�d�s.u/

�(D.3)

where… governs transition probabilities for a Markov process on S and….s0; s/ > 0 for alls0; s 2 S , such that

V0.f / DXs

�.s/

�ZU

maxp2f .s/

�u.pm/C ısV.ph; s/

�d�s.u/

�(D.4)

represents %.

35

Proof. Fix s0 2 S and consider the act f xs0 . Then,

W0.fxs0 / D �.s

0/

ZU

maxp2x

�u.pm/C vs0.ph/

�d�s0.u/C . � /

As noted above, vs.�/ induces a preference %�s on H such that (i) %�s is continuous, (ii) %�ssatisfies Independence (proposition 26), and (iii) %�s satisfies Monotonicity (proposition27). Then, by the Mixture Space Theorem and by Theorem 3, vs has a (state-separabale)finitely additive representation, which can be written as

vs.fxs0 / D ıs�s.s

0/

ZUM�H

maxp2x

u.p/ d Q�ss0.u/C ısXt¤s0

�s.t/

ZUM�H

maxp2f .t/

u.p/ d Q�st .u/

where ıs is a scaling factor chosen so that (i) �s 2 P .S/ is strictly positive and (ii) Q�st is aprobability charge for all t 2 S . Such a choice can be made as follows: If Q�st .UM�H / > 1,define � 0s.t/ WD Q�st .UM�H /, so that we can take Q�st to be a probability measure. Because� 0s.t/ > 0 for all t 2 S , it follows that we may let ıs D

Pt �0s.t/, so that the scaling factor

� 0s.t/ can be replaced by �s.t/ WD � 0s.t/=ıs .Singleton Indifference to Timing (Axiom 7) implies that �.s0/

RU maxp2x

�u.pm/ C

vs0.ph/�d�s0.u/ and ıs�s.s0/

RUM�H

maxp2x u.p/ d Q�ss0.u/ are (positive) affine transforma-tions of each other. By Statewise Non-triviality (Axiom 1) every state is non-null under � ,and by History Independence (Axiom 9), every state is also non-null under �s , ie, �s.s0/ > 0(this was established in proposition 27). Hence the measures f�; �s W s 2 Sg have fullsupport.

Recall that �s0 defined on U is the marginal of a measure on U�UH , where the marginalon UH has a singleton carrier. The transformations of the state space used in obtaining aseparable representation are an instance of those considered in lemma 16, so that we mayregard �s0 as a measure on UM�H .Since both Q�ss0 and �s0 are probability charges, we must have Q�ss0 D �s0 for all s 2 S .

Therefore, each vs can be written as

vs.f / D ısXs0

�s.s0/

ZU

maxp2f .s0/

�u.pm/C vs0.ph/

�d�s0.u/

Define the Markov transition probabilities on S to be….s; s0/ WD �s.s0/, and let V.f; s/ WDvs.f /=ıs so that

V.f; s/ DXs0

….s; s0/

ZU

maxp2f .s0/

�u.pm/C ıs0V.ph; s

0/�d�s0.u/

Finally, define

V0.f / WD W0.f / DXs

�.s/

ZU

maxp2f .s/

�u.pm/C ısV.ph; �s/

�d�s.u/

to attain the desired recursive representation.

It follows from the recursive representation and the fact that % satisfies Statewise Non-triviality (Axiom 1) that at least one �s must be non-trivial. To see this, suppose every �sis trivial, ie, �s.f0g/ for all s 2 S . Then, the recursive representation implies every V.�; s/must be identically 0, which violates Statewise Nontriviality (Axiom 1).

36

D.2.4. Unique Representation with a State-independent Discount Factor

Proposition 28 establishes the existence of a recursive value function as in (D.3), whichcan be described by the parameters

�.�s; �s; ıs/s2S

�. We will now show that there exists a

unique equivalent representation with a state-independent discount factor. Let � 2 RSCC

,and let h�; �i WD

Ps �.s/�.s/) for any � 2 P .S/.

Proposition 29. Let�.�s; �s; ıs/s2S

�and

�. O�s; O�s; Oıs/s2S

�be two recursive representa-

tions of % as in (D.3). Then, there exists � 2 RSCC

such that the parameters are relatedby the following transformations: O�.s/ WD �.s/�.s/= h�; �i, O�s.s0/ WD �s.s

0/�.s0/= h�s; �i,O�.D/ WD �

�D=�.s/

�, Oıs WD ıs h�s; �i =�.s/, and OV .�; O�s/ WD V.�; �s/= h�s; �i. Moreover, if�

.�s; �s; ıs/s2S�is a representation of%, then the transformed set of parameters

�. O�s; O�s; Oıs/s2S

�also represents %.

Proof. The last part of the proposition is immediate. By our identification result, proposition25, there exists � 2 RS

CCsuch that (i) O�s.D/ D �s

�D=�.s/

�for all s 2 S and for all

Borel measurable D � U, and (ii) Oıs OV .�; s/ WD ısV.�; s/=�.s/. By construction, OV .�; s/ DV.�; s/= h�s; �i, and hence we must have Oıs WD ıs h�s; �i =�.s/. Finally, the identificationfrom the Mixture Space Theorem implies that O�s.s0/ WD �s.s0/�.s0/= h�s; �i.

Proposition 30. There exists � 2 RSCC

such that Oıs is independent of s 2 S . Moreover, �is unique up to scaling, so that Oı is unique, and the corresponding measures . O�s/s2S areunique up to scaling.

Proof. A representation with a constant discount factor will obtain immediately if we canestablish that there exists a vector � � 0 and a number Oı > 0 such that Oı�.s/ D ıs h�s; �i forall s 2 S .

For S D f1; : : : ; ng, consider the stochastic matrix…, whose row s is �s . Define the n� ndiagonal matrix � as follows: �.i; j / D ıi if j D i , and 0 otherwise. In matrix notation,our problem amounts to finding a � � 0 and Oı > 0 such that Oı� D ��…. This amounts toshowing that (i) � is a (left) eigenvector of the matrix �…, and (ii) Oı is the correspondingeigenvalue.

We shall say that a matrix is positive if each of its entries is strictly positive. By proposition28 above, the matrix… is positive, so the matrix �… is also positive. Thus, by the Perrontheorem below, such � and Oı exist, � is unique up to scaling, and Oı is unique.By proposition 29, these are the only transformations that we need consider, which

establishes that the measures . O�s/s2S are unique up to a common scaling.

The Perron Theorem is standard and can be found, for instance, as Theorem 1 in chapter16 of Lax [2007]. For completeness, we state the relevant part of the theorem.

Theorem 4 (Perron). Every positive matrix A has a dominant eigenvalue denoted by Oıwhich has the following properties:

(a) Oı > 0 and the associated eigenvector � � 0.

(b) Oı is a simple eigenvalue, and hence has algebraic and geometric multiplicity one.

(c) A has no other eigenvector with nonnegative entries.

37

Thus, we have the established the existence of the following representation

V.f; s0/ DXs

….s0; s/

�ZU

maxp2f .s/

�u.pm/C ıV .ph; s/

�d�s.u/

�(D.5)

and

V0.f / DXs

�.s/

�ZUK

maxp2f .s/

�u.pm/C ıV .ph; s/

�d�s.u/

�(D.6)

where V0.�/ represents %.

Proposition 31. Let % have a recursive representation as in (D.6). If % satisfies AggregateStationarity (Axiom 6), then � is the unique stationary distribution of the Markov processwith transition matrix….

Proof. Recall that V0�f.m; f /g

�DPs �.s/

� Ru.m/C ıV .f; s/

�d�s.u/. Then, letting � WDP

s �.s/�su.m/, we see that

V0�f.m; f /g

�� �

D ıXs

�.s/

"Xs0

….s; s0/

�ZU

maxp2f .s0/

�u.pm/C ıV .ph; s

0/�d�s0.u/

�#

D ıXs0

hXs

�.s/….s; s0/i �Z

U

maxp2f x

s

�u.pm/C ıV .ph; s

0/�d�s0.u/

�

By Aggregate Stationarity (Axiom 6), V0�f.m; �/g

�and V0.�/ represent the same preference.

From propositions 26 and 28, we see that each V.�; s/ is linear onH . Therefore, V0�f.m; �/g

�is linear over H . By the Mixture Space Theorem, V0

�f.m; �/g

�and V0.�/ must be affine

transformations of each other. Therefore, �.s0/ DPs �.s/….s; s

0/ for all s0 2 S . In otherwords, � is a stationary distribution of…. Since… is positive, the stationary distribution isunique.

Proposition 32. If % has a recursive representation of the form in (D.6), then ı 2 .0; 1/.

Proof. It follows immediately from the non-triviality of % and from Aggregate Stationarity(Axiom 6) that ı > 0. We shall now show that ı < 1.

As � is the unique stationary distribution of …, we have for any f 2 H , V0.f / DPs �.s/V .f; s/. Now let a 2KM be the closed "-ball around the p�m, the uniform lottery over

M . It is clear that 0 <R

U maxp2a u.pm/ d�s.u/. But we also haveR

U maxp2a u.pm/ d�s.u/ 6RU maxp2P .M/ u.pm/ d�s.u/ D

RU maxm2M u.m/ d�s.u/ 6

Pm2M j�su.m/j <1, where

the last inequality holds because �s is nice. By the recursive construction of H , it fol-lows that there exists a unique act f � that gives the menu a in each period and in everystate, ie, f � ' .a; f �/. Letting � WD

Ps �.s/

�RU maxpm2a u.pm/ d�s.u/

�, we see that

V0.f�/ D � C ıV0.f

�/ D �P�>0 ı

� . Since � > 0 and because V0.f �/ is finite, we con-clude that ı < 1, as required.

38

E. Proofs from Section 4

Proof of corollary 4. Suppose first that % has a DPF representation�.�s/s2S ;…; ı

�and

that each %s jK.P .M�ff �g// is strategically rational. By proposition 15, strategic rationalityimplies that the support of each �s lies in f�us W � > 0g for some us 2 U.For the converse, suppose the support of �s lies in f�us W � > 0g for some us 2 U. It

is easy to see that for any a 2 K�P .M/

�,RRC

max˛2a �us.˛/ d�.�/ D max˛2a �su.˛/.Clearly, the functional a 7! max˛2a �su.˛/ represents %s jK.P .M�ff �g//, from which itfollows that the latter is strategically rational.

Proof of corollary 5. We shall define a sequence of extensions. Let H0 WD H�K�P .M �

L/��, and inductively defineHn WD H

�K�P .M �Hn�1/

��. It is easy to see thatHn�1 � Hn

and Hn " H .For each x 2 K

�P .M � L/

�, let '0.x/ WD fp W p & q 8q0 2 xg. Similarly, for each

f 2 H0, let 0.f /.s/ WD q, where q 2 '0.f .s//, and notice that 0.f / 2 L. Now define%0 on H0 as follows: f %0 g if, and only if, 0.f / & 0.g/.Now suppose %n�1 has been defined on Hn�1. For each x 2 K

�P .M � Hn�1/

�, let

'n�1.x/ WD fp W p %n�1 q 8q0 2 xg. Similarly, for each f 2 Hn�1, let n�1.f /.s/ WD q,where q 2 'n�1.f .s//, and notice that n�1.f / 2 Hn�1. Now define %n on Hn as follows:f %n g if, and only if, n�1.f / %n�1 n�1.g/.

Let %D limn!1 %n. Notice that this limit is unique because the extension at each stageis unique, and because each preference is continuous andH is compact. Notice also that% isa well defined preference relation on H and by construction satisfies Axioms 1–9 and each%s is strategically rational. This implies that % has a DPF representation

�.�s/s2S ;…; ı

�where by the strategic rationality of %s , each �s is a Dirac measure.

Now suppose, contra hypothesis, there exist two recursive Anscombe-Aumann represen-tations of & on L, namely

�.us/;…; ı

�, and

�.u�s /;…

�; ı��. Let �s be the Dirac measure

on U concentrated on us, and ��s the Dirac measure concentrated on u�s for each s. Then,�.�s/;…; ı

�and

�.��s /;…

�; ı��are two DPF representations that are identical on L.

It is easy to see that both�.us/;…; ı

�and

�.u�s /;…

�; ı��represent each %n on Hn for all

n > 0. Therefore, both represent % on H . But by the uniqueness of the DPF representation,this implies … D …�, ı D ı�, and .�s/s2S and .��s /s2S are identical up to a commonscaling. This establishes the uniqueness of the recursive Anscombe-Aumann representation,as required.

Proof of corollary 6. Let us suppose that �s is independent of s. Then, clearly, V.`; s/ isindependent of s for all ` 2 L0. But this implies that for all x 2 X ,

Rmaxp2x

�u.pm/ C

V.p`; s/�d�s.u/ is independent of s. This implies %s jX is independent of s, as claimed.

For the converse, let us assume %s jX is independent of s. Then, for any s; s0 2 S , it isthe case that for all m 2 M , `; `0 2 L0, .m; `/ %s .m; `0/ if, and only if, .m; `/ %s0 .m; `0/.This implies V.`; s/ > V.`0; s/ if, and only if, V.`; s0/ > V.`0; s0/. Therefore, we must haveVs D �s;s0Vs0 for some �s;s0 > 0. Let ` 2 L0 deliver lottery p in the first period and p�m inevery subsequent period. Then, V.`; s/ D

Ps �.s/ Nus.p/ D V.`; s0/ D �s;s0V.`; s/, which

implies �s;s0 D 1. Therefore, Vs is independent of s on L0.As %s jX is independent of s, we have UsjL0

/ Us0 jL0. But we have just established VsjL0

is independent of s, which implies UsjL0is also independent of s. Therefore, �s must be

39

independent of s, which completes the proof.

F. Proofs from Section 5

Lemma 33 is not part of the proof of a result in the text, but is referred to in section 5.1.

Lemma 33. Let�.us/s2S ;…; ı

�and

�.u0s/s2S ;…

0; ı0�be DPF representations of %, and

suppose Nu0s D �s Nus . Then, �s D �s0 for all s; s0 2 S .

Proof. Let us suppose �s is not independent of s. Let s�; s� 2 S be such that �s� 6 �s 6 �s�

for all s 2 S with �s� < �s� . Given that DM is in state s� today, let ˛ and ˇ be lotteries overM such that DM is indifferent between (i) receiving ˛ today and p�m for sure in all futureperiods, and (ii) receiving ˇ in state s tomorrow and p�m in all other states and periods. Sincep�m yields zero consumption utility and since the representations are separable and recursive,we have Nus�.˛/ D ı….s�; s/ Nus.ˇ/, and Nu0s�.˛/ D ı0…0 .s�; s/ Nu0s.ˇ/ (where we have used thefact that….s; s0/ > 0 for all s; s0 2 S and similarly for…0). With Nu0s D �s Nus and �s� > �s forall s, we obtain ı….s�; s/ 6 ı0…0 .s�; s/. As this holds for all s 2 S , and because….s�; �/ and…0 .s�; �/ are probability measures, we note that ı D ı

Ps….s

�; s/ < ı0Ps…0 .s�; s�/ D ı

0,where the inequality is strict because ı….s�; s�/ < ı0…0 .s�; s�/.

Similarly, for DM in state s� with �s� 6 �s for all s, we find that ı > ı0. Therefore, it mustbe that �s is independent of s 2 S , such that Nu0s D � Nus for all s 2 S and for some � 2 RC.This implies that for all s; s0 2 S , ı….s; s0/ D ı0…0.s; s0/. Because ….s; �/ and …0.s; �/

are probability measures, it follows that ı D ı0 and that ….s; �/ D …0.s; �/ for all s 2 S ,establishing the uniqueness claimed.

We now present some proofs concerning the behavioral comparison ‘greater preferencefor flexibility’ in section 5.3. We begin with some preliminary lemmas.

Lemma 34. Suppose %� has a greater preference for flexibility than % and both % and %�

satisfy Independence (Axiom 3). Then, for all `; `0 2 L, ` %� `0 if, and only if, ` % `0.

Proof. By hypothesis, ` % `0 implies ` %� `0, or equivalently, `0 �� ` implies `0 � `.Therefore, it suffices to show that `0 �� ` implies `0 � `. So, let us suppose `0 �� ` and,without loss of generality, assume that `0 � `. Let `� �� `0 �� ` (by Independence (Axiom3), it suffices to consider ` and `0 for which such an `� exists) so that for some sufficientlysmall � 2 .0; 1/, we have �`� C .1 � �/` �� `0 but `0 � �`� C .1 � �/`, which contradictsthe hypothesis, thereby completing the proof.

Lemma 35. Suppose %� has a greater preference for flexibility than %, and suppose % and%� have canonical DPF representations

�.�s/s2S ;…; ı

�and

�.��s /s2S ;…

�; ı��respectively.

Then, ı D ı�,… D …�, �su D ��su for all s, and V.`; s/ D V �.`; s/ for all ` 2 L and s 2 S .Moreover, V �.f; s/ > V.f; s/ for all f 2 H .

Proof. By lemma 34, we know that ` %� `0 if, and only if, ` % `0. Thus, % and %� representthe same preference on the restricted domain L. Let VL and V �L denote the value functionsfor the respective canonical DPF representations, restricted to L. As VL and V �L represent thesame preference, they are affine transformations of each other (in each state s). By corollary

40

5, �su D ���su for some � > 0, where � is independent of s. But both DPF representationsare canonical, which means that, in fact, � D 1.

Let `� denote the S-IHCS that gives the uniform lottery in every state in each period, soVL.`

�; s/ D V �L .`�; s/ D 0 for all s (and hence V0.`�/ D 0 D V �0 .`�/). For any probability

measure �s on U with k�suk2 ¤ 0, there exist m;m0 2 M such that �u.m/ > 0 >

�u.m0/. This implies that there exist `; `0 2 L such that ` � `� � `0 where `� is as above.Consequently, for any f 2 H , there exists � 2 .0; 1/ and `� 2 L such that �f C.1��/`� � `�,which means that �f C .1 � �/`� %� `� because %� has a greater preference for flexibilitythan %. This implies V �

��f C .1��/`�; s

�> V �.`�; s/ D V.`�; s/ D V

��f C .1��/`�; s

�,

from which it follows that V �.f; s/ > V.f; s/ for all f 2 H and s 2 S , as claimed.

It follows easily from lemma 34 that for each s, U �s > Us, where Us represents %s, andU �s represents %�s .

Let V be the value function that corresponds to the canonical DPF representation�.�s/;…; ı

�.

Notice that since there is no preference for flexibility with respect to continuation prob-lems in any state s, for the purpose of assigning utilities, we may restrict attention tothe domain H

�K�P .M � fhı; h

ıg/��, where V.hı/ 6 V.h/ 6 V.hı/ for all h 2 H , and

V.hı; s/ < V.hı; s/ for all s.

Lemma 36. Let 's W U! R be convex and D � U be compact, convex such that 'sjD isLipschitz. Then, there exists a compact x � affP .M �fhı; h

ıg/ such that maxp2x�u.pm/C

ıV .ph; s/�6 's.u/ with equality for all u 2 D.

Proof. Let u� 2 D, and let �u�.u/ W U! R be an affine function such that �u� 6 's withequality at u�. (The function �u� represents the hyperplane supporting 's at u�.) Then,there exist d 2 RM and d 0 2 R such that �u�.u/ D hd; ui C d 0. We shall now construct thecorresponding menu.Notice that �u�.u/ D

Pi diui C d

0 DPM�1iD1 ui .di � dM / C d

0 where we have usedthe fact that

Pi ui D 0 for all u 2 U. Let ˛ 2 affP .M/ be such that ˛i D di � dM for

i D 1; : : : ;M � 1, and ˛M D 1�PM�1iD1 ˛i . Also, let qh 2 affP .fhı; h

ıg/ be such that d 0 DıV .qh; s/ (where V has been extended to affP .fhı; h

ıg/ by linearity and hence uniquely).Now consider q.u�/ D .˛; qh/, the signed measure onM � fhı; hıg (with marginals ˛ andqh). Then, �u�.u/ D u.˛/C ıV .qh; s/.

Recall that 's restricted to D is Lipschitz. This implies that the set x WD convfq.u/ W u 2Dg is compact. (Intuitively, the set of ‘slopes’ and intercepts of the supporting hyperplanes of'sjD is compact.) It is clear that x � affP .M �fhı; h

ıg/. For each q 2 x, u.qm/C ıV .qh; s/is an affine function of u. Therefore, maxq2x

�u.qm/C ıV .qh; s/

�is a convex function of u.

By construction, this function is always dominated by 's and is equal to 's on D, whichcompletes the proof.

The proof of Theorem 2 relies on the following Theorem, which characterizes dilations.

Theorem 5. Let� and�� be probability measures onU. Then, the following are equivalent:

(a) �� is a dilation of �.

(b) ��' > �' for every continuous convex function ' W U! R that is �C �� integrable.

41

Theorem 5 is Theorem 7.2.17 in Torgersen [1991]. It was proved by Blackwell [1953] forthe case where the supports of � and �� are bounded.

Proof of Theorem 2. ‘Only if’. Suppose %� has greater preference for flexibility than %. ByTheorem 5, it suffices to show that for every continuous convex function 's W U! R that is�s C �

�s integrable, we have ��s 's > �s'

s .Fix such a function 's. We shall show that there exist continuous, convex functions

'sn W U ! R that are �s C ��s integrable, such that (i) 'sn 6 'snC1, (ii) 'sn " 's, and(iii) �s'sn 6 ��s '

sn. Since 's1 is (�s C ��s ) integrable, it follows that 's � 's1 is integrable.

Therefore, 0 6 's � 'sn 6 's � 's1, so the Dominated Convergence Theorem and (iii) aboveimply �s's 6 ��s '

s .Let .Dn/ be an increasing sequence of compact subsets of U such that

SnDn covers

the effective domain of 's. By standard arguments (because the effective domain has arelative interior in our finite dimensional setting), we may assume that for all n, 'sjDn

is Lipschitz. Then, by lemma 36, there exists xn � affP .M � fhı; hıg/ so that 'sn.u/ WD

maxq2xn

�u.qm/C ıV .qh; s/

�satisfies 'sn 6 's , and 'snjDn

D 'sjDn. It follows immediately

from the construction (in lemma 36) that 'sn 6 'snC1 (since xn � xnC1). All that remains isto show that for all n, �s'sn 6 ��s '

sn.

For a fixed s, letWs andW �s represent the restrictions of Vs and V �s toK�P .M�fhı; h

ıg/�.

Thus, for any x 2K�P .M �H

�, there exists x0 2K

�P .M � fhı; h

ıg/�such that Vs.x/ D

Ws.x0/. Moreover, two DPF representations differ if, and only if, they differ on the domain

K�P .M�fhı; h

ıg/�. Abusing notation again, letWs andW �s denote the respective extensions

to all compact subsets of affP .M � fhı; hıg/.

Recall that by lemma 35, V �s .y/ > Vs.y/ for all y 2 K�P .M �H

�and Vs.hı/ > 0 >

Vs.hı/. Then, by linearity, we must have W �s .x/ > Ws.x/ for all compact x � affP .M �

fhı; hıg/. But Ws.xn/ D

RU 'sn.u/ d�s.u/ (which implies, in particular, that each 'sn is

integrable, since xn is compact and maxfjWs.xn/j ; jW �s .xn/jg <1), so that we have�s'sn 6��s '

sn as required.

‘If’. Consider the operators ˆ;ˆ� W C.H � S/! C.H � S/ defined as follows:

ˆW.f; s/ WDXs0

….s; s0/

ZU

maxp2x

�u.pm/C ıW.p´/

�d�s.u/ and

ˆ�W.f; s/ WDXs0

….s; s0/

ZU

maxp2x

�u.pm/C ıW.p´/

�d��.u/

As observed in the proof of Proposition 2, both ˆ and ˆ� are monotone and also satisfydiscounting, and are therefore contractions. For each x 2K

�P .M �H/

�, maxp2x

�u.pm/C

ıW.ph; s/�is a continuous and convex function of u. Therefore, by Theorem 5, for any W 2

C.H � S/, ˆW 6 ˆ�W . Let ˆn and ˆ�n denote the n-th iterates of ˆ and ˆ� respectively.We claim that ˆnW 6 ˆ�nW for all W 2 C.H � S/ and for all n > 1. We have alreadyestablished this for n D 1. Suppose this is true for n � 1, ie, ˆn�1W 6 ˆ�.n�1/W . Then,ˆ.ˆn�1W / 6 ˆ�.ˆn�1W / 6 ˆ�.ˆ�.n�1/W /, ie, ˆnW 6 ˆ�nW , as claimed. Finally, letV and V � be the unique fixed points of ˆ and ˆ� respectively, so that V 6 V �, whichcompletes the proof.

Proof of Proposition 9. Let Us and Us0 respectively represent %s and %s0 over K where,as before, Us.x/ D

RU maxp2x

�u.pm/C ıV .ph; s/

�d�s.u/ and similarly for s0. Following

42

the arguments in lemma 35, we see that when restricted to consumption streams, UsjL is apositive affine transformation of Us0 jL. It is easy to see that the constant term must be zero,so let us suppose UsjL D �Us0 jL for some � > 0. Then, it must be that �s and �s0 differ bya scaling of � so that �su D ��s0u. We also have VL.�; s/ D �VL.�; s0/.Now consider the S-IHCS `mt that delivers in each period, and in every state, p�m, the

uniform lottery over M , except at time t C 1, where it delivers the prize m 2 M . Then,Us.`

mt /=Us0.`

mt / D � > 0. Define �s WD .0; : : : ; 1; : : : ; 0/, where the 1 is the s-th entry. The

probability distribution over states S at time t C 1, conditional on being in state s in date 1 is�s…

t . Therefore,Us.`mt / D ıtPQs �s…

t .Qs/�Qsum, which implies that � D Us.`mt /=Us0.`mt / D�PQs �s…

t .Qs/�Qsum��P

Qs �s0…t .Qs/�Qsum

� (or at least, such an m 2M can be chosen because �su ¤ 0 for some

s 2 S). But… is fully connected and has a unique stationary distribution � , which implieslimt!1 �s…

t D � D limt!1 �s0…t , which means we must have � D 1, ie, UsjL D Us0 jL

and, in particular, �su D �s0u.Consider now, the state Qs 2 S , and let ` 2 L be such that V.`; Qs/ ¤ 0. Such an ` exists

because V.�; Qs/ is non-trivial over consumption streams. Let `� be the S-IHCS that givesthe uniform lottery p�m in each state and in every period. Now, consider the consumptionstream `� that gives the consumption stream .p�m; `/ in state Qs and the S-IHCS `� in everyother state. Then, V.`�; s/ D ….s; Qs/V .`; Qs/ D ….s0; Qs/V .`; Qs/ D V.`�; s0/, which implies….s; Qs/ D ….s0; Qs/ because V.`; Qs/ ¤ 0.

Adapting the arguments from the proof of Theorem 2, we can now show the equivalenceclaimed, which proves the proposition.

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