An alternative axiomatization of intertemporal utility smoothing

Economics Letters 119 (2013) 224–227

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Economics Letters journal homepage: www.elsevier.com/locate/ecolet

An alternative axiomatization of intertemporal utility smoothing Katsutoshi Wakai ∗ Graduate School of Economics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan

highlights • We axiomatize a model of intertemporal utility smoothing. • The axiomatization does not require introducing auxiliary consumption risk. • We derive a particular form of the aggregator function in the recursive utility.

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Article history: Received 15 October 2012 Received in revised form 13 February 2013 Accepted 22 February 2013 Available online 5 March 2013

abstract We propose an alternative axiomatization of the model of intertemporal utility smoothing suggested by Wakai (2008) without introducing auxiliary consumption risk. © 2013 Elsevier B.V. All rights reserved.

JEL classification: D80 D90 Keywords: Discount factor Utility smoothing Recursive utility

1. Introduction Intertemporal choices are often analyzed using the discounted utility model, but experimental studies have generated results that seem to contradict this model. Wakai (2008) proposes the following refinement of the discounted utility model that is consistent with some of those experimental findings. Formally, the utility of an infinite consumption sequence (c0 , c1 , . . .) is expressed in a recursive form: V (c0 , c1 , . . .) = min {(1 − δ)u(c0 ) + δ V (c1 , c2 , . . .)} , δ∈[δ,δ]

(1)

where u is the instantaneous utility function and δ and δ are the upper and lower bounds of discount factors satisfying 0 < δ ≤ δ < 0. Representation (1) captures the notion of intertemporal utility smoothing – that is, a desire to lower volatility in a utility sequence – via the following form of recursive gain/loss asymmetry: (i) current utility u(c0 ) becomes a reference point to evaluate future utility V (c1 , c2 , . . .), and (ii) the difference between future utility V (c1 , c2 , . . .) and current utility u(c0 ) defines a gain or a loss, and

∗

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0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.02.025

gains are discounted more than losses. In particular, if δ = δ , representation (1) coincides with the discounted utility model. On theoretical grounds, Koopmans (1960) provides an axiomatization for the discounted utility model on the preference domain that consists of deterministic consumption sequences. In particular, he derives the instantaneous utility function u based on timeseparability of preferences. This means that the technique used in Koopmans (1960) cannot be adopted to derive the instantaneous utility function u in representation (1) because representation (1) is non-time-separable. Therefore, Wakai (2008) introduces auxiliary consumption risk because such devices allow us to derive the instantaneous utility function u in a simple axiomatic system.1 However, the introduction of auxiliary randomization devices comes with a cost: he must impose an assumption that is inconsistent with the way intertemporal consumption risk resolves over time. Given the above problem, the objective of this paper is to axiomatize representation (1) on the set of deterministic consumption sequences without introducing auxiliary consumption risk. 1 To model a dislike of utility variations between adjacent periods, Gilboa (1989) and Shalev (1997) adopted the same domain as Wakai (2008).

K. Wakai / Economics Letters 119 (2013) 224–227

We achieve this goal by deriving a particular form of the aggregator function in the framework of Koopmans (1960) recursive utility. More specifically, we adopt the notion of biseparable preferences as developed by Ghirardato and Marinacci (2001) in the context of choice under uncertainty on a Savage (1954) domain by reinterpreting time periods as possible states of the world. Biseparable preferences describe the behavior of two-dimensional choice problems, which is represented by the function that is a weighted average of cardinal vNM utility indices under a rank-dependent weighting function. In Koopmans (1960) recursive utility, the stationarity axiom reduces the intertemporal choice problem to a two-dimensional choice problem with the dimensions being utility today versus continuation utility from tomorrow onward. Thus, the adoption of biseparable preferences leads to a representation that is a weighted average of today’s utility and continuation utility under a rank-dependent weighting function. We then impose an axiom that captures the notion of intertemporal utility smoothing and derive a particular form of weighting function as shown in representation (1). Finally, we solve the recursive equation (1) and derive the explicit functional form of representation V . The remainder of the paper presents sets of axioms and representations. We also provide proofs that show the equivalence between these axioms and representations.

225

In terms of the relationship between the ordering on X and the ordering on Y , much of the literature assumes the following form of monotonicity. Monotonicity. For any ⟨ct ⟩ , ct′ ∈ Y , if ct ≽ ct′ for all t ∈ N, then

 

 ′

⟨ct ⟩ ≽ ct . The latter ranking is strict if the former ranking is strict for some t ∈ N. The following result can be easily derived so we state it without a proof. Lemma 1. Given continuity, Axioms 1 and 2 imply monotonicity. Because X is connected and compact, it follows from Lemma 1 and continuity that, for each ⟨ct ⟩ ∈ Y , there exists ⟨x⟩∗ ∈ C such that ⟨x⟩∗ ≃ ⟨ct ⟩. We call this x a constant equivalent of ⟨ct ⟩ and refer to it as ce(⟨ct ⟩). To model the recursive gain/loss asymmetry, we must first derive asymmetric weights for gains versus losses as well as the instantaneous utility function u. Thus, we consider a binary sequence ⟨x : y⟩, which is a consumption sequence ⟨ct ⟩ ∈ Y such that ct = x ∈ X for t = 0 and ct = y ∈ X for t ≥ 1. Let Yb be the collection of all binary sequences, each element of which, as shown  above, is either increasing or decreasing. Furthermore, for ⟨ct ⟩ , ct′ ∈ Yb ,

 

the mixture of ⟨ct ⟩ and ct′ is the binary sequence in Yb , denoted by

2. Representation We consider an infinite-horizon, discrete-time model, where time varies over {0, 1, 2, . . .} = N. The axiomatization exhibited below can be easily adapted to a finite-horizon, discrete-time model with a minor modification. A decision maker (DM) consumes a single perishable good at each period t ∈ N from a connected and compact set X = [x, x] ⊂ R++ , where x > x. We denote a set of deterministic consumption sequences by Y ≡ {(c0 , c1 , . . .) ∈ R∞ |ct ∈ X for each t ∈ N}, which is endowed with the product topology. Let ⟨ct ⟩ be a generic element of Y , where ⟨ct ⟩ = (c0 , c1 , . . .). Let C be the set of all constant deterministic consumption sequences, where a generic element of C is denoted by ⟨c ⟩∗ = (c , c , c , . . .). The DM faces the same choice set Y at each time t, and the DM’s preference ordering on Y , denoted by ≽, is assumed to be complete, transitive, continuous, nondegenerate, and independent of time and the payoff history. We first assume the following axioms that characterize the recursive utility. Axiom 1 (Atemporal Preference (AP)). For all ⟨ct ⟩ , ct′

 

∈ Y and  

x, x′ ∈ X , (x, ⟨ct ⟩) ≽ (x′ , ⟨ct ⟩) if and only if (x, ct′ ) ≽ (x′ , ct′ ).

 

Axiom 2 (Stationarity (ST)). For all (x, ⟨ct ⟩) and (x, ct′ ) ∈ X × Y ,

 

(x, ⟨ct ⟩) ≽ (x, ct ) if and only if ⟨ct ⟩ ≽ ct .  ′

     ⟨ct ⟩ : ct , such that, for each τ ∈ N, ⟨ct ⟩ : ct′ τ = ce( cτ : cτ′ ).   ′  Thus, by monotonicity, for each τ ∈ N, cτ ≽ ⟨ct ⟩ : ct τ ≽ cτ′ if    cτ ≽ cτ′ , and cτ′ ≽ ⟨ct ⟩ : ct′ τ ≽ cτ if cτ′ ≽ cτ . Moreover, we also  ′ state that ⟨ct ⟩ and ct are comonotonic if there are no τ , τ ′ ∈ N such that cτ ≻ cτ ′ and cτ′ ′ ≻ cτ′ . 

 ′

Koopmans (1960) introduced these axioms, which are also a part of assumptions that characterize the discounted utility model (AP is Postulate (3a) and ST is a combination of Postulate (3b) and Postulate (4)). In particular, AP induces the ordering on X , which is independent of a continuation payoff ⟨ct ⟩: for x, x′ ∈ X , x ≽ x′ if and only if (x, ⟨ct ⟩) ≽ (x′ , ⟨ct ⟩), where ⟨ct ⟩ is any element in Y .2 Furthermore, ST assumes that the passage of time does not alter the preference ordering, which induces a dynamically consistent decision process. 2 The literature regarding risk and uncertainty often defines the induced ordering on the consumption set X as follows: for x, y ∈ X , x ≽ y if and only if ⟨x⟩∗ ≽ ⟨y⟩∗ , where ⟨x⟩∗ and ⟨y⟩∗ are acts that pay x and y at every state, respectively. This definition lacks a behavioral foundation in an intertemporal setting because consuming x at each period is not identical to consuming x in a single period.

 ′ 

Recursive gain/loss asymmetry implies that comonotonic consumption sequences in Yb are evaluated under the same decision weight. To capture this idea, we adopt the following version of the independence axiom on Yb from Ghirardato and Marinacci (2001), which is suitably modified to fit our framework, where {x, y} ≽ z stands for x ≽ z and y ≽ z. Axiom 3 (Comonotonic Independence Consumption Se   for Binary  quences (CI)). For all ⟨x : y⟩ , x′ : y′ , x′′ : y′′ ∈ Yb that are pairwise ′ ′′ ′′ ′ comonotonic, if {x, x′ } ≽ x′′ and  {y,y } ≽ y (or x ≽ {x, x }and y′′ ≽ {y, y′ }), then ⟨x : y⟩ ≽ x′ : y′ implies ⟨x : y⟩ : x′′ : y′′ ≽



 











 



x′ : y′ : x′′ : y′′ and x′′ : y′′ : ⟨x : y⟩ ≽ x′′ : y′′ : x′ : y′ .3

CI states that among the comonotonic consumption sequences in Yb satisfying the stated condition, the mixture operation does not alter the preference ordering. The required condition is that the mixture must be taken with a dominated (or dominating) consumption  sequence because such an operation   guarantees  that ⟨x : y⟩, x′ : y′ , and a mixture of ⟨x : y⟩ or x′ : y′ with x′′ : y′′ are all pairwise comonotonic. The above axiom leads to the following lemma. Lemma 2. Assume that ≽ satisfies Axioms 1 and 2. Then the following statements are equivalent. (i) ≽ satisfies Axiom 3. (ii) There exists a continuous and nontrivial function u : X → R, real numbers δ and δ satisfying 0 < δ, δ < 1 such that ≽ on Yb is represented by F : Yb → R, where F (⟨x : y⟩) ≡



(1 − δ)u(x) + δ u(y) (1 − δ)u(x) + δ u(y)

if x ≽ y if x ≼ y.

(2)

Moreover, δ and δ are unique, and u is unique up to a positive affine transformation. 3 This is a simplified version of the Binary Comonotonic Act Independence axiom used in Ghirardato and Marinacci (2001).

226

K. Wakai / Economics Letters 119 (2013) 224–227

Note that (2) does not define a relationship between δ and δ . Proof. Let Σ = {∅, {0}, N \ {0}, N}. In Ghirardato and Marinacci ⟨x : y⟩ corresponds to the bet on A, and a (2001), abinary sequence  mixture ⟨ct ⟩ : ct′ corresponds to the statewise A-mixture of ⟨ct ⟩

 

and ct′ , where A ∈ {{0}, N \ {0}}. Furthermore, because monotonicity holds on a strict ordering, all nonempty subsets in Σ satisfy their definitions of essential events. Then (2) follows from Theorem 11 of Ghirardato and Marinacci (2001), which is a class of the Choquet expected utility defined on Σ with a unique set function ρ : Σ → [0, 1] satisfying ρ(∅) = 0, ρ(N) = 1, ρ({0}) = (1 − δ), and ρ(N \ {0}) = δ .  Now, given the cardinal utility defined by (2), for x, y ∈ X , consider z ∈ X that satisfies 1

1

u(x) + u(y). (3) 2 2 The existence of such z is guaranteed because X is connected and u is continuous. Furthermore, Ghirardato et al. (2003) show that u(z ) =



= z ′ ∈ X | ⟨x : y⟩ ≃ ce x : z 

 

 ′

: ce z ′ : y 



⟨ct ⟩ ⊕

1 2

 ′ 

∈ Yb by   1  1 1 1 ⟨ct ⟩ ⊕ ct′ ≡ cτ ⊕ cτ′ 2

2

ct

2

2

τ

2

2

⟨ct ⟩ ⊕

1 2

F (⟨ct ⟩) = max [(1 − δ)u(c0 ) + δ u(c1 )], δ∈[δ,δ]

which is inconsistent with TVA.  Given (6), define V : Y → R by V (⟨ct ⟩) ≡ u(ce(⟨ct ⟩)).

(7)

Then, it is clear that V (⟨ct ⟩) = F (⟨ct ⟩) for ⟨ct ⟩ ∈ Yb .

(8)

δ∈[δ,δ]

By ST, ⟨ct ⟩ ≃ ⟨c0 : ce((c1 , c2 , . . .))⟩. Then (6) implies that V (⟨ct ⟩) = F (⟨c0 : ce((c1 , c2 , . . .))⟩)

The conclusion follows from (7).

 

1

Suppose, by way of contradiction, that δ > δ . Then representation (2) is rewritten as

δ∈[δ,δ]

for all τ ∈ N.

Axiom 4 (Time-Variability Aversion (TVA)). For any ⟨ct ⟩ , ct′ ∈ Yb ,

 

(6)

δ∈[δ,δ]

= min [(1 − δ)u(c0 ) + δ u(ce(c1 , c2 , . . .))].

The next axiom assumes that the DM is averse to the volatility involved in utility sequences.

if ⟨ct ⟩ ≃ ct′ , then

F (⟨ct ⟩) = min [(1 − δ)u(c0 ) + δ u(c1 )].

V (⟨ct ⟩) = min [(1 − δ)u(c0 ) + δ V ((c1 , c2 , . . .))].

.

Thus, we can elicit the equivalent class of z in X , denoted by E (x, y), without referring to the utility function u.  We  denote by 1 x ⊕ 12 y ∈ X an element in E (x, y), and for ⟨ct ⟩ , ct′ ∈ Yb , define 2

1

(Step 1): The real numbers δ, δ derived in Lemma 2 satisfy 0 < δ ≤ δ < 1 and (2) is rewritten as follows: for all ⟨ct ⟩ ∈ Yb

(Step 2): For all ⟨ct ⟩ ∈ Y ,

E (x, y) ≡ z ′ ∈ X |z ′ ≃ z , where z satisfies (3)



Proof. Necessity of the axioms is routine. The proof of sufficiency is divided into three steps.

 ′ 

(Step 3): For all ⟨ct ⟩ ∈ Y , V (⟨ct ⟩) = min

 ∞ 

b∈D

≽ ⟨ct ⟩.

ct



 bt u(ct ) .

(9)

t =0

TVA makes an indifference curve convex in the utility domain, regardless of the functional form of u. Thus, TVA defines utility smoothing, which leads to δ ≤ δ . The following is a stationary and infinite-horizon version of Wakai’s (2008) model of utility smoothing.

Moreover, V is continuous on Y . First, for all ⟨ct ⟩ ∈ Yb , it follows from the construction of D in Wakai (2008) that (9) attains the samevalue ∞as (8). For ⟨ct ⟩ ∈ Y ,  n  ∞ consider two sequences, c t 1 and c nt 1 , such that for each n≥1

Proposition 1. The following statements are equivalent. (i) ≽ satisfies Axioms 1–4. (ii) There exists a continuous and nontrivial function u : X → R, a set [δ, δ] ⊂ R satisfying 0 < δ ≤ δ < 1, and a nonempty, weak*closed, and convex set D, each element of which, b ∈  D, is a strictly ∞ positive discount function b : N → R++ , satisfying t =0 b t = 1 such that: ≽ on Y is represented by V : Y → R, where

c t = ct

V (c0 , c1 , . . .) ≡ min

 ∞ 

b∈D

 bt u(ct )

= min {(1 − δ)u(c0 ) + δ V (c1 , c2 , . . .)} . δ∈[δ,δ]

(4)

t =0

(5)

Moreover, δ , δ , and D are unique, and u is unique up to a positive affine transformation. Furthermore, V is continuous on Y , and D is recursively constructed from [δ, δ], as shown in Wakai (2008). Note that representation (4) is a version of the multiple-priors utility as introduced by Gilboa and Schmeidler (1989), where time is interpreted as states. As for the proof, Wakai (2008) first imposes a temporal version of the multiple-priors axiom on the preference ordering ≽ and derives representation (4). Then he shows that stationarity induces a particular structure on D, under which representation (4) satisfies representation (5). In contrast, we first impose a temporal version of the multiple-priors axiom on Yb and derive representation (5). Then we show that, given continuity, the solution of recursive representation (5) is representation (4).

n

c nt

for t ≤ n

and c t = xu for t > n, n

= ct for t ≤ n and

and

= x for t > n,

c nt

l

where xu ∈ arg maxx∈X u(x) and xl ∈ arg minx∈X u(x). By monotonicity, for each n ≥ 1 V

 n  ct

  ≥ V (⟨ct ⟩) ≥ V c nt .

(10)

 n Furthermore, it follows from monotonicity that V ( c t ) is weakly  n decreasing and V ( c t ) is weakly increasing. As both sequences are bounded, each sequence converges. Let V and V be the limits of  n  V ( c t ) and V ( c nt ), respectively. Moreover, it follows from the construction of D in Wakai (2008) that, for each n ≥ 1, V

 n  ct

= min b∈D

 b t u( ) n ct

t =0

≥ min b∈D

 ∞   ∞ 

 ≥ min b∈D

∞ 

 bt u(ct )

t =0

 b t u( ) c nt

  = V c nt .

(11)

t =0

Now, we claim that V = V so that (10) and (11) imply (9). For each n ≥ 1, let bn be an element in D such that n

b ∈ arg min b∈D

 ∞  t =0

 bt u( ) . c nt

K. Wakai / Economics Letters 119 (2013) 224–227

It follows from the construction of D in Wakai (2008) that, for each b ∈ D and each t ≥ 0, ∞  τ =t

t

∞ 

bnτ u(xu ) − u(xl ) < ε





 n  ct

≤

∞ 

bnt u(c t ). n

Acknowledgments

(12)

I am grateful to the editor and the referee for their comments and suggestions. Financial support from the Japanese government in the form of a research grant, Grant-in-Aid for Scientific Research (C) (23530219), is gratefully acknowledged.

(13)

References

for any n ≥ 1. Moreover, for each n ≥ 1, V

Finally, because X is bounded and Y is adopted with the product topology, it can be shown by a standard argument that V , defined by (9), is continuous on Y . 

bτ ≤ δ .

Thus, for any ε > 0, there exists T > 0 such that for each t ≥ T

τ =t

227

t =0

Then, given (12) and (13), (11) implies that, for any ε > 0, there exists T > 0 such that, for all n ≥ T ,

  n    V c − V c n  ≤ t t =

  ∞   n   n n b u(c t ) − V c t    t =0 t  ∞  τ =n +1

bnτ u(xu ) − u(xl ) < ε,





which proves the claim. Note that the boundedness of X is crucial for this conclusion.

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