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Modeling Nonmonotone Preferences:

The Case of Utility Smoothing

Katsutoshi Wakai June 2010

Abstract

We propose a model of intertemporal choice in which a strong dislike of volatility involved in a utility sequence causes preferences to be nonmonotone.

In particular, this notion of utility smoothing allows us to axiomatize a rep- resentation that captures an extreme dislike of losses. When applied to a consumption-saving problem, the nonmonotone preferences induced by our model never suggest a monotonically decreasing consumption pro…le. Further- more, an optimal consumption sequence need not be monotonically increasing.

Our model may suggest spreading large and small consumption allocations over time if the volatility involved in a utility sequence is su¢ ciently low.

Keywords: discount factor, gain/loss asymmetry, nonmonotone preferences, recursive utility, reference point, utility smoothing

JEL Classi…cation Numbers: D90, D91

Graduate School of Economics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606- 8501, Japan; E-mail: wakai@econ.kyoto-u.ac.jp

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Acknowledgements

I am grateful for valuable comments from Eddie Dekel, Larry Epstein, Itzhak Gilboa, Takashi Hayashi, Youichiro Higashi, Kazuya Kamiya, Takao Kobayashi, Hitoshi Matsushima, Eiichi Miyagawa, Stephen Morris, Ben Polak, and seminar participants at Keio University, Kobe University, University of Tokyo, Washington University, the RUD 2004 Conference, and the Summer Workshop on Economic The- ory in Hokkaido 2009. I am also thankful for helpful comments and suggestions from an associate editor and anonymous referees. Financial support from the Japanese government in the form of research grant, Grant-in-Aid for Scienti…c Research (C) (20530146), is gratefully acknowledged.

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1 Introduction

Conventionally, intertemporal choice has been analyzed via Samuelson’s (1937) dis- counted utility model: at each time t, the conditional utility of a consumption sequence c= (c0; :::; cT) is represented by

Vt(c) XT

=t

tU(c ); (1)

where 0 < <1 is a single-period discount factor and U is an instantaneous util- ity function. When the consumption good is a desirable one, this model assumes monotonicity, that is, an increase in consumptionc at any period increases over- all utility Vt(c). In addition, the model implies utility independence: each utility sequence (U(ct); :::; U(cT))is weighted by the same series of discount factors.

The assumption of monotonicity seems innocuous, but it may be inconsistent with an aversion to volatility involved in a consumption sequence. For example, consider a popular version of a habit-formation model, where the habit level is a function of past consumption. It is well known that in this model, a departure from a past standard of living is so costly that a decision maker (DM) may dislike an increase in consumption at a particular period as it increases volatility of the given consumption sequence. In a recent paper, Rozen (2009) provides an axiomatic foundation for this version of a habit-formation model and proves that it is consistent only with a weaker version of monotonicity.

Habit-formation models capture nonmonotone preferences by making an instan- taneous utility function U history dependent, while maintaining the assumption of utility independence. On the contrary, in a descriptive approach, Loewenstein and Prelec (1993) propose a model that captures a desire for smoothing a distribution of instantaneous utility over time, where as in (1), the instantaneous utility function is assumed to be independent of the consumption history. In particular, this notion of utility smoothing is shown to be consistent with the experimental result called a preference for spread, that is, the DM prefers spreading good and bad outcomes

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evenly over time, which is a preference that is di¢ cult to incorporate into habit- formation models.1 Moreover, their model suggests that when the DM strongly dislikes volatility in a utility sequence, the DM’s preferences violate monotonicity.

In a recent paper, Wakai (2008) provides an axiomatic foundation for a notion of utility smoothing similar to the one de…ned in Loewenstein and Prelec (1993), but his analysis assumes monotonicity so that it fails to characterize a type of utility smoothing that violates monotonicity.

The purpose of this paper is to …ll the gap left in Wakai’s (2008) analysis by providing an axiomatic foundation for a notion of utility smoothing that may not satisfy monotonicity. In particular, we extend his model of utility smoothing by in- troducing a weaker version of monotonicity. Formally, at each timet, the conditional utility of a consumption sequence cis expressed in the recursive form

Vt(c) min

t+12[t+1; t+1]

[(1 t+1)U(ct) + t+1Vt+1(c)]; (2) where t+1and t+1are the upper and lower bounds of single-period discount factors, respectively, and satisfy0 < t+1 <1 and t+1 t+1. We provide an axiomatiza- tion for this representation by adapting a method developed in a di¤erent context by Gilboa and Schmeidler (1989) and Epstein and Schneider (2003).

The representation (2) is based on recursive gain/loss asymmetry characterized by the following two properties: (i) current utility U(ct) becomes a reference point to evaluate future utility Vt+1(c), where Vt+1(c) is de…ned as the average utility of future periods, and (ii) the di¤erence between future utility Vt+1(c) and current utility U(ct) de…nes a gain or a loss, and gains are discounted more than losses.

Hence, as similar to Kahneman and Tversky’s (1979) prospect theory, representation (2) is a class of reference-based preferences that also incorporate a loss-aversion type of behavior, which captures a dislike of the volatility involved in a sequence of the average utility of future periods. Furthermore, a key departure of representation

1See Loewenstein and Prelec (1991).

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(2) from Wakai (2008) is the range of discount factors: in (2), the upper bound t+1

can be more than one, whereas in Wakai (2008), it must be less than one. If the upper bound t+1 is greater than one, the weight(1 t+1)becomes negative, that is, monotonicity is violated at time t. Thus, if future utility is less than current utility, the incentive to avoid a loss is so strong that the DM would rather replace current consumption with that of lower utility. On the contrary, if future utility is greater than current utility, monotonicityalways holds. Therefore, in our model, a strong desire for utility smoothing is expressed as an extreme dislike of losses.

The technical contribution of this paper is the introduction of a weaker version of monotonicity that is consistent with a stronger notion of utility smoothing. In- tuitively, this axiom says that for any consumption sequence c = (c0; c1; c2) that satis…esct2 fG; Bg;

(G; G; G) (c0; c1; c2);

with a strict ranking if at least one of ct is B, where G and B represent “good”

and “bad”consumption, respectively. This assumption excludes the possibility that U(ct) is always assigned with a negative weight because the reduction in instan- taneous utility from a constant utility sequence always makes the DM worse o¤.

Moreover, the assumption doesnot necessarily imply (c0; c1; c2) (B; B; B);

for a consumption sequence (c0; c1; c2) having ct =G for some t. This means that the DM may prefer a constant sequence ofBto a sequence that involves bothGand B. Therefore, this axiom allows us to model a strong dislike of volatility in utility sequences.

We also apply representation (2) to a three-period consumption-saving problem, where bond prices are given exogenously. A key observation is that if representation (2) violates monotonicity (that is, 1; 2>1), the DM never chooses a consumption sequence in which Vt(c) is decreasing over time because nonmonotone preferences

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induce a strong dislike of losses. This feature implies the following two distinct results. First, the nonmonotone preferences induced by our model never suggest a monotonically decreasing consumption pro…le. In particular, when bond prices are high, our model suggests complete consumption smoothing, which contrasts with a decreasing consumption sequence implied by the discounted utility model. Second, our model can accommodate a preference for spread as long as the volatility involved in a utility sequence is low enough to makeVt(c) (weakly) increasing. This happens when bond prices are consistent with attaining an optimal consumption sequence satisfying c0 > c1 and c1 < c2. Moreover, the ratio between optimal consumption allocations assigned at any two periods is far more stable than the ratio implied by an optimal consumption sequence derived from the discounted utility model. Thus, in our model, consumption goods are spread in a well-controlled manner.

The remainder of the paper proceeds as follows. Section 2 axiomatizes repre- sentation (2). Section 3 compares the notion of weak monotonicity introduced in this model with other notions used in the related literature. Section 4 studies a consumption-saving problem to gain insights into our model. Section 5 concludes the paper. All proofs are presented in the appendices.

2 Representation

The objective of this section is to derive a representation for preferences de…ned on the set of deterministic consumption sequences. However, to model the preference for utility smoothing based on simple and intuitive axioms, we follow Wakai (2008) and enrich the domain by adopting the Anscombe–Aumann (1963) framework with a temporal interpretation: preferences are de…ned on the set of sequences whose outcome at any period is a lottery de…ned over a consumption set. The domain we are interested in, the set of deterministic consumption sequences, is identi…ed with a subset of this enriched domain, where an element of this subset is a sequence of degenerate lotteries. As shown later, we assume that each lottery is evaluated

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independently. Therefore, although the payment is a lottery, we are not interested in modeling attitudes toward temporal risk together with the preference for utility smoothing.2 Such a model requires a more involved setting, and we investigate it in a di¤erent paper.3

We consider a discrete-time model in a …nite horizon setting, where time varies overT =f0;1; :::; Tg withT satisfying 0< T <1. For eacht2 T, a continuation of time is denoted byTt ft; t+ 1; :::; Tg. A nonempty consumption set isX, which can be a convex set in R or a …nite set of objects. We denote by M the set of all probability distributions over X with …nite support. An act is a functionl :T

! M, where l = (l0; l1; :::; lT). A constant act p is a function l : T ! M such that lt = p 2 M for all t 2 T; p is also identi…ed with p 2 M. Let L = MT be the collection of all acts, and let C be a collection of all constant acts. A mixture operation + on L is de…ned by a period-wise application of a probability mixture operation + on M, that is, ( l+ (1 )l0)t = lt+ (1 )l0t for 2 [0;1] and l; l02 L.

The primitive of the model is the collection of complete and transitive prefer- ence orderings f tg f t j t 2 T g, whose elements are de…ned on L. For each t2 T, assume that a conditional ordering t is Archimedean, nondegenerate, and independent of the payo¤ history(l0; l1; :::; lt 1). A conditional ordering onM, also denoted by t, is de…ned as follows: for p; q 2 M, p t q if and only if p t q.

Furthermore, for expositional purposes, we assume the following.

2Alternatively, by adapting the method developed by Casadesus-Masanell, Klibano¤, and Oz- denoren (2000) in a context of ambiguity aversion in Savage’s (1954) setting, we can de…ne utility smoothing directly on a set of deterministic consumption sequences. However, such axiomatization is deeply involved and less intuitive, so we do not adapt their method.

3The representation that takes into account both the preference for utility smoothing and atti- tudes toward temporal risk is a class of recursive representations suggested by Kreps and Porteus (1978), where an aggregator function incorporates utility smoothing as shown in representation (2).

See Wakai (2010).

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Assumption 1: For each t 2 T, there exists an a¢ ne function Ut : M ! R that represents t on M.

As we show later, the axiom discussed below implies Assumption 1.

Formally, a standard model of intertemporal choice, such as the discounted utility model, assumes the following property.

Strict monotonicity (SMT): For each t2 T and for all l; l0 2 L,if l tl0 for all 2 T,then l tl0. In addition,if for some t,l tl0,then l tl0.

To model utility smoothing, we state SMT based on instantaneous utility rather than on consumption, where the latter is a special case of the former.

The objective of this paper is to model intertemporal choice that may not satisfy SMT. To this end, we need to replace SMT with a weaker condition. In particular, we follow Shalev (1997) by adopting the condition below.4

Axiom 1 –Utility equivalence (UE):For each t2 T and for all l; l02 L,if l 'tl0 for all 2 T,then l'tl0.

UE assumes that each lottery is evaluated independently, that is, two acts are equally preferred as long as lotteries yield the same utility at each period. Thus, for each t2 T, we de…ne a collection of utility sequencesUt by

Ut f(Ut(lt); Ut(lt+1); :::; Ut(lT))jfor all l2 Lg:

We ignore the utility of past payo¤s because preference ordering t is independent of the payo¤ history. Then UE induces a conditional ordering on a set of utility sequences Ut, also denoted by t, as follows: for u; u0 2 Ut, u t u0 if and only if l t l0, where u = (Ut(lt); Ut(lt+1); :::; Ut(lT)) and u0 = (Ut(lt0); Ut(l0t+1); :::; Ut(l0T)).

Because the existence of this induced ordering allows us to de…ne a preference for

4In Shalev (1997), Axiom 1 is called substitutability.

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utility smoothing, UE is a crucial property of SMT that we need to preserve. More- over, for l; l0 2 L, let u and u0 be the utility sequences implied by l and l0, respec- tively. Then the mixture operation on L, l+ (1 )l0, yields a utility sequence of u+(1 )u0, which is a weighted summation of the corresponding utility sequences.

Next, we maintain the following two axioms assumed in Wakai (2008).5 The

…rst axiom weakens the assumption underlying the discounted utility model.

Axiom 2 –Constant independence (CI):For eacht2 T and for all l; l0 2 L, p2 C,and for all 2(0;1),l tl0 if and only if l+ (1 )p t l0+ (1 )p.

The discounted utility model is based on a version of the independence axiom, where the preference orderingl tl0 is preserved under the mixture operation with any l00 2 L. This assumption implies that all utility sequences must be discounted by the same series of discount factors. However, CI preserves the preference ordering l tl0 only under the mixture operation with a constant act p 2 C. In particular, p induces a constant utility sequence, so that a utility sequence implied by l and a utility sequence implied by l+ (1 )p must have a similar pattern. Thus, it is now possible that a di¤erent series of discount factors can be applied to a di¤erent pattern of utility sequences. Moreover, CI implies Assumption 1, that is, there exists the a¢ ne function Ut:M !Rthat represents ton M.

The second axiom de…nes the notion of the aversion to volatility involved in utility sequences.

Axiom 3 – Time-variability aversion (TVA): For each t 2 T and for all l; l02 L,and for all 2(0;1),l'tl0 implies l+ (1 )l0 tl.

In terms of the induced ordering onUt, TVA means that u'tu0 implies u+ (1 )u0 tu;

5In Gilboa and Schmeidler (1989), Axiom 2 is called certainty independence and Axiom 3 is called uncertainty aversion.

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whereuandu0 are utility sequences implied bylandl0, respectively. This condition makes an indi¤erence curve convex in Ut regardless of the functional form of Ut. Thus, TVA assumesutility smoothing.

The following proposition is a modi…ed version of Gilboa and Schmeidler’s (1989) multiple priors model with a temporal interpretation, where discount factors can be negative at certain utility sequences (for the proof, see Appendix A).

Proposition 1: The following statements are equivalent: (i) f tg satisfy Axioms 1–3.

(ii) At each t, there exists an a¢ ne function Ut:M !R and a nonempty, closed, and convex set Dt RT+1 t; each element of which, t 2 Dt, is a discount vector

t= ( tt; :::; tT)satisfying PT

=t t = 1 such that t is represented by Vt, where Vt(l) min t2Dt

XT

=t

tUt(l ): (3)

Moreover, Dt is uniquely de…ned and Ut is unique up to a positive a¢ ne transfor- mation.

The proof follows essentially from Gilboa and Schmeidler (1989); however, to take into account that SMT no longer holds, we modify their proof accordingly.

In representation (3), the discount factor of period , t, can take a negative value at all utility sequences. If this happens, decreasing the utility of consumption at period always increases overall utility. However, such preferences are not con- sistent with the notion of utility smoothing. Hence, to avoid the above problem, we reintroduce the following property implied by SMT as a new axiom.

Axiom 4 –Monotonicity for strong utility smoothing (MSUS):For each t2 T with t <T and for any p; p0 2M satisfying p tp0,and for any nonempty proper subset A of Tt,if l =pfor all 2A and l 0 =p0 for all 0 2 TtnA,then p

tl.

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The intuition behind MSUS is explained by the following example. Consider three utility sequences in Ut, u = (2;2;2), u0 = (1;1;1), and u00 = (2;1;2). Then SMT implies that u t u00 t u0, where t denotes an induced ordering on a set of utility sequences Ut. However, MSUS implies that u t u0 and u t u00, but it does not necessarily imply u00 tu0. Thus, MSUS can exhibit a dislike of volatility involved in utility sequences more strongly than SMT can. Moreover, MSUS never allows a situation where the discount factor of period , t, takes a negative value at all utility sequences because, from a constant utility sequence, reducing utility at any period always makes the DM worse o¤.

Finally, following Wakai (2008), we specify the relationship between conditional orderings.

Axiom 5 –Dynamic consistency (DC):For each t2 T with t < T and for all l; l0 2 L, if l =l0 for all t and if l t+1 l0, then l t l0; the latter ranking is strict if the former is strict.

As assumed in many other models of intertemporal choice, we adopt DC to avoid a con‡ict between di¤erence selves.

The following proposition encapsulates the main results of this paper (for the proof, see Appendix B).

Proposition 2: The following statements are equivalent: (i) f tg satisfy Axioms 1–5.

(ii) There exists an a¢ ne function U: M ! Rand a collection of sets of discount factors f[ t; t]g1 t T satisfying 0 < t <1 and t t for all t 2 f1; :::; Tg such that: for each t2 T, t is represented by Vt(:); where fVt(l)g0 t T are recursively de…ned by

Vt(l) min

t+12[ t+1; t+1]

[(1 t+1)U(lt) + t+1Vt+1(l)]; (4) and VT(l) U(lT). Moreover, [ t; t] is uniquely de…ned for all t 2 f1; :::; Tg, and U is unique up to a positive a¢ ne transformation.

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The proof proceeds as follows. First, MSUS and DC imply that an a¢ ne function Ut is time invariant. Second, MSUS and DC are su¢ cient to adapt Wakai’s (2008) proof, which derives representation (4) with a set of discount factors [ t+1; t+1], where t+1 t+1. Third, MSUS requires that the weight on current utility U(lt) is positive for some utility sequence. This condition implies t+1 < 1. Fourth, for preferences to satisfy DC, the weight for future utilityVt+1(l) must be positive for all utility sequences. This condition results in 0< t+1.

Finally in this section, we want to consider a continuous-time analogue of (4).

For this analysis, de…ne a discount-factor processB :T !R++ by B0 1 and by Bt+1 (1 rt)Bt, wherertis the rate of depreciation ofBt de…ned byrt 1 t+1 with t+1 2 [ t+1; t+1]. Let B be a collection of all discount-factor processes B = (B0; :::; BT). Then, for a consumption sequencec, at eacht < T, (4) can be rewritten as

Vt(c) min

B2B

(BT

BtU(cT) +

T 1

X

=t

B

BtF(c ; r ) )

; (5)

whereF(c ; r ) r U(c ). Similarly, de…ne a bounded Riemann integrable discount- factor processBb : [0; T]!R++byBb0 1and byBbt exp( Rt

k=0rkdk), where the rate of depreciation of Bbt is the discount ratert satisfying rt rt rt and 0< rt. LetBbbe a collection of all bounded Riemann integrable discount-factor processesB.b Furthermore, assume thatX is a nonempty, convex, and compact subset ofR+ and U is continuous. Then, for a bounded Riemann integrable consumption sequencec, at each t < T, a continuous-time analogue of (5) is written as

Vt(c) min

B2Bb

(BbT

BbtU(cT) + Z T

t

Bb

BbtF(c ; r )d )

; (6)

where F(c ; r ) r U(c ).6 In both (5) and (6), for a constant consumption se- quencec such thatct=x for all t,Vt(c) =U(x).

6As shown in Wakai (2008), if (i)T =1, (ii)rt =r andrt =rfor all t, and (iii) 0< r < r, then (6) is a special form ofvariational utilityas introduced by Geo¤ard (1996).

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To see how monotonicity can be violated in (6), suppose that rt=r and rt=r for all t, where r <0< r. Consider a consumption sequence c such that ct=x for 0 t t1 and ct=y fort1 < t T, where0< t1 < T. Then, at t= 0, the utility of this consumption sequence cbecomes

V0(c) =U(x)(1 Bbt1)+U(y)(Bbt1 BbT)+U(y)BbT =U(x)+fU(y) U(x)gBbt1: (7) Equation (7) essentially replicates the utility of a two-period consumption sequence (x; y) of a discrete-time model, which incorporates gain/loss asymmetry. When a consumption sequence is increasing (x < y), the minimizing discount rate is rt = r > 0 for all 0 < t t1. This means that the smaller consumption assigned on [0; t1]receives the maximal weight (1 Bbt1) =Rt1

0 r B db = 1 exp( rt1), which is positive. Thus, a di¤erentiable increase in x ory increases V0(c). On the contrary, when a consumption sequence is decreasing (x > y), the minimizing discount rate is rt = r <0 for all 0< t t1. This means that the larger consumption assigned on [0; t1] receives the minimal weight (1 Bbt1) = Rt1

0 r B db = 1 exp( rt1), which is negative. Therefore, although a di¤erentiable increase iny increasesV0(c), a di¤erentiable increase in x decreases V0(c), that is, the monotonicity is violated.

Note that for this consumption sequencec, the minimizing discount raterton(t1; T] can take any value in[r; r]because for any change in a discount rate rt on(t1; T], a utility change in the second term in (6) is o¤set by a utility change in the …rst term in (6).

3 Comparison with Other Notions of Weak Monotonic- ity

In an in…nite horizon model, Rozen (2009) provides an axiomatic foundation for the version of a habit-formation model where a habit level is a function of past consumption. She shows that the model is consistent with a weaker version of the monotonicity axiom, which states that by adding the same amount of consumption

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to today’s and future consumption, the DM always prefers a new consumption sequence rather than the original. A …nite horizon analogue of her axiom is as follows.

Gain Monotonicity: For all t 2 T, for all > 0, and for all consumption sequence c,

(c0; ::; ct 1; ct+ ; ct+1+ ; ::cT + ) t(c0; ::; ct 1; ct; ct+1; ::cT):

Clearly, gain monotonicity does not imply MSUS, and MSUS does not imply gain monotonicity even under an increasing instantaneous utility function. Further- more, gain monotonicity is based on consumption smoothing rather than on utility smoothing. Thus, our model (4) does not necessarily satisfy gain monotonicity (see Appendix C). Conversely, MSUS assumes the existence of an instantaneous util- ity function that is independent of a consumption history. This means that the habit-formation model characterized by Rozen (2009) does not necessarily satisfy an in…nite horizon version of MSUS (see Appendix C).

Utility smoothing can be de…ned di¤erently to our model. For example, to capture an attitude toward the utility variations between adjacent periods, Gilboa (1989) axiomatizes the following representation for the ex ante preference ordering

0:

V0(c) 0U(c0) + XT

t=1

f tU(ct) + tjU(ct) U(ct 1)jg;

where t j tj+j t+1j for allt 2 T with 0 0 and T+1 0, whose conditions ensure that preferences satisfy monotonicity. To incorporate a notion similar to Kah- neman and Tversky’s (1979) loss aversion into intertemporal choice, Shalev (1997) further extends Gilboa’s (1989) model by relaxing the assumption of monotonicity.7

7Waegenaere and Wakker (2001) model nonmonotone intertemporal preferences by generalizing the regular Choquet integral. They show that Shalev (1997) is a special case of their representation.

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Formally, Shalev (1997) axiomatizes the following representation for the ex ante preference ordering 0:8

V0(c) =U(c0) + XT

1 +

t max[U(ct) U(ct 1);0] + t min[U(ct) U(ct 1);0] : (8) Moreover, he additionally imposes the axioms that derive

t +

t >0 for all t, (9)

whose conditions imply loss aversion in intertemporal choice, that is, the di¤erence in utility between adjacent periods is de…ned as a gain or loss, and gains are discounted more than losses. In particular, if the DM strongly dislikes a loss, the DM has an incentive to lower the reference consumption level (that is, consumption in the previous period), which may cause preferences to be nonmonotone.

Gilboa (1989) and Shalev (1997) de…ne preferences on the same domain as used in our model, which is the Anscombe–Aumann (1963) framework with a temporal interpretation. Hence, the axioms are comparable. For example, to derive (8), Shalev (1997) assumes UE, which has been adopted in this paper. In addition, to model loss aversion, Shalev (1997) assumes two more axioms, one of which is a weaker version of monotonicity de…ned as follows.

Constant-Tail Monotonicity: For all l; l0 2 Land for all t2 T,if (i)l =l0 for all < t, (ii) l =l 0 and l0 =l00 for all ; 0 t, (iii)l 0 l0 for some t, then l 0l0.

Constant-tail monotonicity states that if the payment is constant after a certain period of time, the DM always prefers increasing the constant payment. This axiom ensures that t ; t+>0for allt. Making t larger than +t as shown in (9) requires

8Gilboa (1989) and Shalev (1997) do not model conditional preference orderings. Moreover, the decisions based on Gilboa’s (1989) or Shalev’s (1997) model are dynamically inconsistent if we assume that each conditional preference ordering is independent of the consumption history.

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one more technical axiom.9 Furthermore, constant-tail monotonicity does not imply MSUS because (8) does not necessarily satisfy MSUS under the condition of t ; t+>

0 for all t (see Appendix C). Similarly, MSUS alone does not imply constant-tail monotonicity. However, as shown in Appendix B, MSUS and DC together imply constant-tail monotonicity.

Our notion of utility smoothing is most closely related to the notion introduced by Loewenstein and Prelec (1993). Their model is based on the idea that the DM prefers a utility sequence under which at any point in time, the average utility of future periods is close to the average utility of the whole sequence. However, Loewenstein and Prelec (1993) do not o¤er an axiomatic foundation for their repre- sentation. Hence, it is di¢ cult to deduce a notion of weak monotonicity that their representation satis…es. To this end, we can only show an example in which their representation does not satisfy MSUS (see Appendix C).

4 Application

Consider a three-period consumption-saving problem, where time varies over T = f0;1;2g. At each period, a single perishable consumption good is available, and its price is assumed to be one throughout the model. In addition, at time 0, a short- term bond that pays one consumption good at time 1 is issued, and its price is denoted byq0;1. Similarly, at time 1, a short-term bond that pays one consumption good at time 2 is issued, and its price is denoted byq1;2. Assume that q0;1 and q1;2

are exogenously given and announced at time 0. Furthermore, bothq0;1 andq1;2 are positive and …xed throughout the analysis.

At each period, a DM engages in trade and obtains an optimal consumption sequence c = (c0; c1; c2) from an initial wealth W. We examine the key features of an optimal consumption sequence implied by representation (4) that violates

9This axiom is called loss aversion in Shalev (1997), which is a modi…ed version of variation aversion introduced by Gilboa (1989).

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monotonicity (USNM). We also compare its key features with those of an optimal consumption sequence implied by the following three models: the discounted utility model (DU), representation (4) that maintains monotonicity (USM), and Shalev (1997).10 For comparability with Shalev (1997), who models only ex ante prefer- ences, we assume that the DM submits all trades at time 0, which will be executed in subsequent periods.

To focus on investigating the e¤ects caused by a di¤erence in the structure of discount factors, we use the same instantaneous utility function U : X ! R for all four models, which is assumed to be di¤erentiable and strictly concave on X = R+. Furthermore, we specify the structure of the discount factors of each model as follows. In DU, the conditional utility at timetis expressed by

Vt(c) X2

=t

tU(c ); (10)

where the discount factor satis…es0< <1.

As for representation (4), we assume

1(1 2)

(1 1) = and 2

(1 2) = . (11)

These parameter values imply that for an increasing consumption sequence c, rep- resentation (4) at time t <2is rewritten as

Vt(c) = (1 t+1) X2

=t

tU(c );

which is ordinally equivalent to (10).

For USM, we additionally assume

1(1 2)

(1 1) = , 2

(1 2) = , and > : (12)

1 0The discounted utility model with 0 < < 1 is not a special case of Loewenstein and Pr- elec’s (1993) model. This feature makes it hard to compare the implications of their model in a consumption-saving problem with those of related models, so we do not provide such an analysis.

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These parameter values imply that for a decreasing consumption sequencec, repre- sentation (4) at timet <2 is rewritten as

Vt(c) = (1 t+1) X2

=t

tU(c ): (13)

However, for USNM, we assume

1 >1 and 2 >1; (14)

under which the DM’s preferences violate monotonicity for a decreasing consumption sequence.

In terms of Shalev (1997), to be compatible with USNM, we assume that 1> +1 > 2+>0;

and

+

1 +

2

1 +1 = and

+ 2

1 1+ = 2. (15)

These parameter values imply that for an increasing consumption sequence c, the representation at time 0 is rewritten as

V0(c) = (1 1+) X2

=0

tU(c );

which is ordinally equivalent to (10). For a decreasing sequence c, we assume that monotonicity is violated, that is,

2 > 1 >1: (16)

We now investigate an optimal consumption sequence. First, the DM whose preferences follow USNM never chooses a consumption sequence in which Vt(c) is decreasing over time because nonmonotone preferences induce a strong dislike of losses. This implies that an optimal consumption sequence c must satisfy one of the following conditions: (i) c is constant, (ii) c is (weakly) increasing, or (iii) c shows a pattern of c0 > c1 < c2 (see Appendix D). Condition (i) happens when

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bond prices are high, whereas condition (ii) happens when bond prices are low.

Condition (iii) captures a preference for spread, and it happens when q0;1 is high butq1;2 is low. In particular, condition (iii) implies that USNM allows the optimal consumption sequence to ‡uctuate over time as long as the volatility involved in a utility sequence is low enough to make Vt(c) (weakly) increasing.

Among nonmonotone preferences, condition (iii) is the distinct feature of USNM because nonmonotone preferences alone do not necessarily induce condition (iii). For example, in Shalev (1997), a utility variation between adjacent periods is de…ned as a gain or loss. Thus, if Shalev’s representation violates monotonicity, the DM never chooses a consumption sequence in which U(ct)is decreasing over time. This means that irrespective of asset prices, an optimal consumption sequencecis either constant or (weakly) increasing but never exhibits a pattern of condition (iii) (see Appendix D).

On the contrary, the DM whose preferences follow DU or USM chooses a con- sumption sequence that does not necessarily satisfy any of the conditions mentioned above (see Appendix D). In particular, if bothq0;1 andq1;2 are su¢ ciently high, the DM chooses a consumption sequence that is decreasing over time. This implies that to avoid a strictly decreasing consumption pro…le, utility smoothing alone is not su¢ cient; preferences also need to be nonmonotone. Furthermore, in DU and USM, an optimal consumption sequence can exhibit a pattern of condition (iii). Thus, to gain a better understanding of condition (iii), we derive an optimal consumption sequence based on a particular instantaneous utility function.

Let U(x) =p

x. As shown in Appendix D, for each of DU, USM, and USNM, condition (iii) happens when q0;1 > and q1;2 < . For this range of asset prices,

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an optimal consumption sequence cimplied by USNM is as follows.

c= q20;1q1;22

4 ;q1;22

2 ;1

!

c2 if < q0;1<(1 2) + 2

2

q1;2

; and (17)

c= n

(1 2)q1;2

+ 2 o2

;q1;22

2 ;1

!

c2 ifq0;1 (1 2) + 2

2

q1;2; (18) where c2 should be solved from the budget constraint. Furthermore, (17) corre- sponds to the case where an optimal consumption sequencecsatis…esU(c0)< V1(c), whereas (18) corresponds to the case wherec satis…esU(c0) =V1(c).

The crucial result is that when U(c0) = V1(c), the ratio between optimal con- sumption allocations assigned at any two periods is independent of q0;1 (the level indicated byc2 depends onq0;1). Hence, however expensiveq0;1 becomes, consump- tion goods are distributed over time in a well-controlled manner. This contrasts with the optimal consumption sequence implied by DU, which is indeed identical to the one shown in (17) regardless of the levels of asset prices. Evidently,c0 increases relative toc1 and c2 asq0;1 increases because future consumption becomes more ex- pensive relative to current consumption. Thus,c0 can be disproportionately higher thanc1 and c2.

To see whether or not the above result is induced by utility smoothing alone, we examine the optimal consumption sequence of USM as follows.

c= q20;1q1;22

4 ;q1;22

2 ;1

!

c2 if < q0;1<(1 2) + 2

2

q1;2

;

c= n

(1 2)q1;2

+ 2o2

;q1;22

2 ;1

!

c2 (19)

if(1 2) + 2

2

q1;2

q0;1 (1 2) + 2

q1;2

1(1 2) (1 1) ; and c= q0;1 (1 1)

1(1 2)

2 q1;22

2 ;q21;2

2 ;1

!

c2 ifq0;1 > (1 2) + 2 q1;2

1(1 2) (1 1) ; where c2 should be solved from the budget constraint. The ratio between optimal consumption allocations is independent of q0;1 only at (19), so that a strong dislike

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of losses induced by nonmonotone preferences is necessary to derive the result ifq0;1

is su¢ ciently high.

5 Conclusion

We have shown that the DM’s preferences may become nonmonotone if they ex- hibit a strong dislike of volatility in a utility sequence. In particular, nonmonotone preferences are captured via an extreme dislike of losses, where a loss is a situa- tion in which the average utility of future periods is less than current utility. We also study a consumption-saving problem when our model induces nonmonotone preferences. The resulting optimal consumption sequence is never monotonically decreasing. Furthermore, it need not be monotonically increasing because one of the consumption allocations in future periods can be less than current consumption as long as the average utility of future periods is more than or equal to current utility. This feature implies that there is a range of asset prices in which our model suggests spreading large and small consumption allocations over time. Moreover, in this situation, consumption goods are distributed over time in a well-controlled manner because the ratio between optimal consumption allocations assigned at any two periods is far more stable than the ratio implied by an optimal consumption sequence derived from the discounted utility model.

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Appendix A: Proof of Proposition 1

Necessity of the axioms is routine. The proof of su¢ ciency is based on Lemmas A.1–

A.4, which correspond to Lemmas 3.1, 3.2, 3.3, and 3.5 in Gilboa and Schmeidler (1989), respectively.

Lemma A.1: Assume Axioms 1–2. Then for each t2 T,there exists an a¢ ne function Ut:Y !Rsuch that for all p; p0 2 C,p tp0 if and only if Ut(p) Ut(p0).

Furthermore, Ut is unique up to a positive a¢ ne transformation.

Proof. The proof follows from Gilboa and Schmeidler (1989, Lemma 3.1).

By nondegeneracy, we use a speci…cUtsuch that[ 1;1] Ut(M). Letp[ 1]; p[0]; p[1] 2M satisfyUt(p[ 1]) = 1,Ut(p[0]) = 0, andUt(p[1]) = 1, respectively. Because strict monotonicity may not hold, the following Lemmas A.2–A.4 are more involved than Lemmas 3.2, 3.3, and 3.5 in Gilboa and Schmeidler (1989).

Lemma A.2: Assume Axioms 1–2. For eacht2 T,let Utbe an a¢ ne function derived in Lemma A.1. Then there exists a function Jt:L !R such that:

(i) For all l; l02 L, l tl0 if and only if Jt(l) Jt(l0).

(ii) For all p2 C, J is uniquely determined by Jt(p) =Ut(p).

For all l2 L,Jt is determined in the following way.

(iii) For all l 2 L such that p[1] t l t p[0], Jt(l) = l, where l uniquely satis…es lp[1]+ (1 l)p[0] 'tl.

(iv) For all l2 Lsuch that p[0] tl 0 p[ 1], Jt(l) = l 1, where l uniquely satis…es lp[0]+ (1 l)p[ 1] 'tl.

(v) For all l 2 L such that l t p[1], Jt(l) = 1

l

, where l uniquely satis…es

ll+ (1 l)p[0]'tp[1].

(vi) For all l 2 L such that p[ 1] t l, Jt(f) = 1

l 1, where l uniquely satis…es lp[0]+ (1 l)l'tp[ 1].

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Proof. Given Lemma A.1 and the assumption on Ut, (ii)–(iv) follow from Gilboa and Schmeidler (1989, Lemma 3.2).

(v) This is shown by proving the followings …ve steps.

(Step 1) For all ; 0 2(0;1)such thatp[1] t l+(1 )p[0]; 0l+(1 0)p[0] t p[0], > 0 if and only if l+ (1 ) p[0] t 0l+ (1 0)p[0].

Assume that > 0. By the Archimedean property and nondegeneracy, there exists a unique 2(0;1)such that

p[1]+ (1 )p[0] 't l+ (1 )p[0]: (A.1) De…ne 0 2(0; ) by 0

0

. It follows from CI that

0h

l+ (1 )p[0]i

+ (1 0)p[0] 't 0 h

p[1]+ (1 )p[0]i

+ (1 0)p[0]:

Because the left-hand side is 0l+ (1 0)p[0] and the right-hand side is 0p[1]+ (1

0)p[0],

0l+ (1 0)p[0] 't 0p[1]+ (1 0)p[0]: (A.2) Furthermore, it follows from CI that

p[1]+ (1 )p[0] t 0p[1]+ (1 0)p[0]: (A.3) Hence, (A.1) to (A.3) imply that l+ (1 )p[0] t 0l+ (1 0)p[0]. The converse follows from a similar argument.

(Step 2) For all 2 (0;1), l t l+ (1 ) p[0] t p[0]. Moreover, for all

; 0 2(0;1), > 0 if and only if l+ (1 ) p[0] t 0l+ (1 0)p[0].

By CI, for all 2 (0;1), l+ (1 ) p[0] t p[0]. We shall show that l t l+ (1 )p[0]. Suppose, by way of contradiction, that there exists 2(0;1)such that l+(1 )p[0] tl. It follows from the Archimedean property and CI that there exists 2(0;1)such that p[1] t [ l+ (1 )p[0] ] + (1 )p[0] t l+ (1 )p[0], which contradicts (Step 1). The second claim follows from a similar proof.

(Step 3) There exists a unique l 2(0;1)such that ll+ (1 l) p[0] 'tp[1].

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It follows from (Step 1), (Step 2), and the Archimedean property.

(Step 4) For all l; l0 2 Lsatisfying l; l0 tp[1],l tl0 if and only if l0 > l. Assume thatl tl0 tp[1]. Then by CI, l0l+(1 l0)p[0] t l0l0+(1 l0)p[0] 't p[1] tp[0]. It follows from (Step 3) that there exists a unique 2 (0;1) such that [ l0l+(1 l0)p[0]]+(1 )p[0]'tp[1]so that l= l0 < l0. Conversely, if l0 > l, (Step 2) implies that l0l+ (1 l0)p[0] t ll+ (1 l)p[0] 't l0l0+ (1 l0)p[0]. By CI, this implies thatl tl0.

(Step 5) For all l2 L satisfying l tp[1],Jt(l) = 1

l

represents t. By (Step 3) and (Step 4), 1

l

represents t on l 2 L satisfying l t p[1]. We now show that this Jt is consistent withUt, that is, given (ii), Jt is the unique way to represent t for l 2 L satisfying l t p[1] if there exists p 2 C such that p 't l.

To prove this claim, it su¢ ces to show that Jt(p) =Ut(p) for allp2 C that satisfy p tp[1]. If suchpexist, it follows from (Step 3) that there exists a unique l2(0;1) such that lp+ (1 l)p[0] 'tp[1] (here, l p). Then by (ii),

1 =Jt(p[1]) =Ut( lp+ (1 l)p[0]) = lUt(p) + (1 l)Ut(p[0]) = lUt(p):

This implies thatUt(p) = 1

l

=Jt(p). Finally, ifl'tp[1], CI implies l 1. Hence, Jt(l) = 1, which is consistent with (iii).

(vi) This is shown as follows. By adapting an argument similar to that used in the proof of (v), we can easily show that for anyl 2 L satisfying p[ 1] tl, (a) for any 2 (0;1), p[0] t p[0]+ (1 )l t l, (b) for all ; 0 2(0;1), > 0 if and only if p[0]+ (1 )l t 0p[0]+ (1 0)l, and (c) there exists a unique l2(0;1) such that lp[0]+ (1 l)l 't p[ 1]. These results and CI also imply that for all l; l0 2 L such thatp[ 1] t l and p[ 1] t l0, l t l0 if and only if l0 > l. Hence,

1

l 1 represents t on l 2 L satisfying p[ 1] t l. We now show that this Jt is consistent withUt, that is, given (ii),Jtis the unique way to represent t forl2 L satisfying p[ 1] t l if there exists p 2 C such that p 't l. To prove this claim, it su¢ ces to show that Jt(p) = Ut(p) for all p 2 C that satisfy p[ 1] t p. If such p

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exist, there exists a unique l 2 (0;1) such that lp[0] + (1 l)p 't p[ 1] (here, l p). Then by (ii),

1 =Jt(p[ 1]) =Ut( lp[0]+ (1 l)p) = lUt(p[0]) + (1 l)Ut(p) = (1 l)Ut(p):

This implies that Ut(p) = 1

l 1 = Jt(p). Finally, if l't p[ 1], CI implies l 0.

Hence,Jt(l) = 1, which is consistent with (iv).

Let Bt be a collection of all bounded and real-valued functions on Tt. De…ne a functionUt (:)t:L !BtbyUt (l)t =Ut(l )for each 2 Tt. LetUt fUt (l)tjl2 Lg. For 2R, de…ne by the function onTt, where = for each 2 Tt.

Lemma A.3: Assume Axioms 1–3. Then for each t 2 T, there exists a functional It:Bt!R such that

(i) For all l2 L; It(Ut (l)t) =Jt(l).

(ii) It is homogeneous (of degree 1).

(iii) Itis C-independent: for any a2Btand 2R,It(a+ ) =It(a)+It( ).

(iv) It is superadditive.

Proof. We de…neIt on Ut by (i); given Lemma A.2 and UE,It is well de…ned onUt. Because tsatis…es TVA, ifItis homogeneous andC-independent, it follows from Gilboa and Schmeidler (1989, Lemma 3.3) thatIt is superadditive. Hence, it su¢ ces to show that It is homogeneous andC-independent.

First, we claim that It is homogeneous of degree 1. To prove this claim, it is enough to show that for any a; b 2 Ut with a = b for 2 (0;1], It(a) = It(b).

There are three cases, as follows.

(Case 1) For b2Ut such that 1 It(b) 1:

The conclusion follows from Gilboa and Schmeidler (1989, Lemma 3.3).

(Case 2) For b2Ut such that It(b)>1:

Letlbe an act inLsuch thatUt (l)t=b(this meansl tp[1]). Then, by Lemma A.2,Jt(l) = 1

l

, where l uniquely satis…es ll+ (1 l)p[0] 't p[1]. Furthermore,

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de…nel0 2 Lby l0 l+ (1 )p[0]. By Lemma A.1, Ut ( l+ (1 )p[0])t= b, and by CI, l+ (1 ) p[0] t p[0]. Hence, It( b) = It(a) = Jt(l0) >0. Moreover, (Step 2) of Lemma A.2 implies Jt(l)> Jt(l0).

If 0 < It(a) 1, by Lemma A.2, Jt(l0) = l0, where l0 uniquely satis…es

l0p[1]+(1 l0)p[0] 'tl0. Then,Jt(l) = 1

l

andJt(l0) = l0 implyJt(l0) = l0 lJt(l).

In addition, by Lemma A.2 and CI,

l0 't l0p[1]+ (1 l0)p[0]

't l0( ll+ (1 l)p[0]) + (1 l0)p[0]

't l0 ll+ (1 l0 l)p[0];

which implies that l0 l= becausel0 = l+ (1 )p[0]. Hence, It( b) =It(a) = Jt(l0) = l0 lJt(l) = Jt(l) = It(b).

IfIt(a)>1, by Lemma A.2, Jt(l0) = 1

l0

, where l0 uniquely satis…es l0l0+ (1

l0)p[0] 't p[1]. Then, Jt(l) = 1

l

and Jt(l0) = 1

l0

imply Jt(l0) = l

l0

Jt(l), where

l0 > l. In addition, by Lemma A.2 and CI,

l0l0+ (1 l0)p[0] 't ll+ (1 l)p[0] 't l0[ l

l0

l+ (1 l

l0

)p[0]] + (1 l0)p[0];

which implies that l

l0

= because l0 = l+ (1 )p[0]. Hence, It( b) = It(a) = Jt(l0) = l

l0

Jt(l) = Jt(l) = It(b).

(Case 3) For b2Ut such that It(b)< 1:

Let l be an act in L such that Ut (l)t = b (this means p[ 1] t l). Then, by Lemma A.2, Jt(l) = 1

l 1, where l uniquely satis…es lp[0] + (1 l)l 't p[ 1]. Furthermore, de…ne l0 2 L by l0 l+ (1 )p[0]. By Lemma A.1, Ut ( l+ (1

)p[0])t= b, and by CI,p[0] t l+ (1 ) p[0]. Hence,It( b) =It(a) =Jt(l0)<0.

Moreover, the proof of Lemma A.2-(vi) implies Jt(l0)> Jt(l).

If 1 It(a) <0, by Lemma A.2, Jt(l0) = l0 1, where l0 uniquely satis…es

l0p[0] + (1 l0)p[ 1] 't l0. Then, Jt(l) = 1

l 1 and Jt(l0) = l0 1 imply

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