Characterization of stationary preferences in a continuous time framework

Journal of Mathematical Economics 63 (2016) 34–43

Contents lists available at ScienceDirect

Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Characterization of stationary preferences in a continuous time framework✩ Kazuhiro Hara Department of Economics, New York University, 19th West 4th Street, New York, NY 10012, United States

article

info

Article history: Received 26 February 2015 Received in revised form 22 November 2015 Accepted 29 November 2015 Available online 29 December 2015 Keywords: Stationary preference Exponential discounting Continuous time Skorohod metric

abstract We characterize preference relations on continuous time consumption paths which admit an exponential discounting representation. We provide two theorems as such, one in the cardinal framework and another in the ordinal framework. Our characterizations parallel the known characterizations in discrete time framework. In the cardinal framework, we adopt the axioms of Epstein (1983), which characterize a stationary preference relation in discrete time, and obtain the exponential discounting model as a special case of the discounting model proposed by Uzawa (1968). In the ordinal framework, we adopt the axioms of Bleichrodt et al. (2008) which were proposed to generalize Koopmans’ classical characterization of stationary preferences. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In continuous time economic models, it is commonplace to assume that an agent has a preference relation that is represented by a utility function U which is of the following exponential discounting form: U (x) =

∞



e−λt u(x(t )) dt ,

(1)

0

where, loosely speaking, u is an instantaneous utility function, λ is a discount rate, and x is any continuous time consumption path. The main goal of this paper is to characterize those preference relations that admit this type of representation both in the cardinal and ordinal frameworks. In the cardinal framework, we characterize a preference relation that admits a general discounting representation proposed by Uzawa (1968) and pin down the functional form in (1) as a special case. Specifically, we characterize a preference relation on lotteries over consumption paths that admits an expected utility representation where the corresponding von-Neumann Morgenstern (vNM) utility function is of the following form: U (x) =

∞



e−

t 0

β(x(s))ds

where β is a function that determines a discount rate for each level of consumption. This model has been applied to growth theory and international economics.1 Moreover, it can accommodate anomalies of the exponential discounting model such as the sign effect and magnitude effect.2 In the ordinal framework, on the other hand, we characterize a preference relation that admits a utility representation precisely of the form (1). Axiomatic foundations for the exponential discounting in the discrete time setup are well established. In particular, in the cardinal framework, Epstein (1983) has characterized a preference relation on lotteries over consumption streams that has an expected utility representation with a vNM utility function of the form: U (x) =

∞ 

δ t u(xt ),

(2)

t =0

where δ is a discount factor and x is any discrete time consumption stream. In the ordinal framework, Koopmans (1960, 1972) has characterized the preference relations that admit a utility representation by functions of the form (2). There are also many

u(x(t )) dt ,

0

✩ I am grateful to Efe Ok for his extensive suggestions and encouragement. I thank Edward Green, Fabio Maccheroni, Hiroo Sasaki, Igor Kopylov, Nobusumi Sagara, and Peter Wakker for useful comments. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.jmateco.2015.11.005 0304-4068/© 2015 Elsevier B.V. All rights reserved.

1 See for instance Calvo and Findlay (1978), Epstein (1987a), Epstein and Hynes (1983), Nairay (1984), and Obstfeld (1981). 2 The sign effect refers that monetary losses are discounted at a lower rate than gains. The magnitude effect refers that smaller outcomes are discounted at a higher rate than large outcomes. See Frederick et al. (2002) for an excellent survey on time preference including details of these anomalies.

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43

generalizations of Koopmans’ characterization.3 In particular, the recent work by Bleichrodt et al. (2008) has provided such a characterization in the most general (but still discrete time) framework. It is natural to ask whether these characterizations continue to hold in the continuous time setup. Given the importance of continuous time models in macroeconomics and finance, this problem seems to have practical importance beyond mathematical consistency. And, in fact, such problems have received attention in the previous literature. This literature includes Epstein (1987b), Harvey and Østerdal (2012), and Sagara (2013), which characterize general classes of preference relations (on continuous time consumption paths) that include the exponential discounting model as a special case. None of these works pin down a utility function of the form (1), however. Moreover, they assume that the range of the consumption paths has a Euclidean structure, and uses axioms that exploit this structure, while no such need arises in the discrete time setup. Kopylov (2010), in contrast, takes the time axis as a state space and uses a Savage type representation theorem to pin down the utility functions that are of form (1) with no structural assumptions on the consumption space. Yet, Kopylov’s axiomatic system has no discrete time counterpart and derives a utility function which is not necessarily continuous. (More on the relation between our approach and that of Kopylov shortly.) We wish to provide here a characterization of the exponential discounting model in continuous time in a way that is parallel to the known characterizations in the discrete time setup. We provide a characterization both in the cardinal and the ordinal frameworks with axioms that are adopted from the corresponding characterizations in the discrete time setup. Specifically, we adopt the axioms of Epstein (1983) and Bleichrodt et al. (2008), in a context where each consumption path carries the form of a cadlag function (i.e. a right-continuous function with left limits) that takes its values in a separable metric space. Clearly, to obtain a representation with a continuous utility function, one needs a metric or a topology on the set of all consumption paths. Choice of topological assumptions requires care since it is well-known that the continuity property in an infinite dimensional framework can have behavioral implications. (See Epstein (1990).) Immediate candidates for this purpose are the product topology and the sup distance. However, the exponential discounting is not continuous with respect to the product topology in general. On the other hand, the sup distance yields a nonseparable space, and on such spaces the expected utility theorem fails. To deal with such difficulties, we endow the set of consumption paths with the Skorohod metric in this paper. While this metric is quite well-behaved, and is used extensively in literature on stochastic processes,4 it has not been used (to the best of our knowledge) in the literature on decision theory before. We will see below that the Skorohod metric furnishes a separable topology which is ideal for studying preference relations in a continuous time framework. In what follows, we first investigate the problem in the cardinal framework (Section 2). Our first theorem is a characterization of preference relations that admit a representation studied by Uzawa (1968), which contains the exponential discounting as a special case. We then pin down the exponential discounting model in our second theorem. In Section 3, we provide characterizations in the ordinal framework. Section 4 contains the proofs of our main theorems and several properties of the Skorohod metric that are used in the proofs. Finally, Appendix contains the definition of the Skorohod metric and some auxiliary proofs.

3 See, for instance, Hübner and Suck (1993) and Dolmas (1995). 4 See, for example, Billingsley (1999), Ethier and Kurtz (1986), and Pollard (1984).

35

2. Cardinal framework 2.1. Preliminaries Throughout this section, (X , dX ) stands for a separable metric space. A mapping f : [0, ∞) → X is an X -valued cadlag function if it is right-continuous and has left limits. That is, f is cadlag iff f (t +) = f (t ) and f (t −) exists for every t ∈ [0, ∞). The set of all cadlag functions is denoted by D∞ . The set D∞ contains plenty of consumption paths that are relevant for economic analysis. For example, it contains all continuous paths. Moreover, sample paths of most of stochastic processes (such as Lévy processes) belong to this set. We denote the set of all consumption paths by C and in what follows, assume C is the set of all X -valued cadlag functions whose range is relatively compact.5 (In particular, if X is compact, then C = D∞ , and if X is a finite-dimensional Euclidean space, C consists of all bounded X -valued cadlag functions.) As we seek a continuous utility representation, we need a topology on the set of all consumption paths. To do this, we endow D∞ with the Skorohod metric d∞ .6 The Skorohod metric has several nice properties and is used extensively in the literature on stochastic processes. Its most relevant feature for our purposes is that when X is a separable metric space, D∞ is a separable metric space relative to the Skorohod metric. This ensures that C is a separable metric space. We denote the set of all Borel probability measures on a given metric space S by △(S ). As usual, we endow △(S ) with the topology of weak convergence. A primitive of our analysis is a complete preorder % on △(C ).7 We impose the standard axioms of expected utility theory on %. Independence. For any p, q, r ∈ △(C ) and any λ ∈ (0, 1), p % q implies λp + (1 − λ)r % λq + (1 − λ)r . Continuity. For any p ∈ △(C ), the sets {q | q % p} and {q | p % q} are closed in △(C ). It is well-known that a complete preorder % satisfies these two axioms if and only if it admits an expected utility representation. (See, for instance, Theorem 3 of Grandmont (1972).) That is, there is a continuous and bounded U : C → R such that for each p, q ∈ △(C ),



 U dp ≥

p % q if and only if C

U dq.

(3)

C

(We note that the separability of C is essential for the validity of this statement.) We need to impose additional axioms to pin down the shape of U in accordance with the exponential discounting model. To this end, we introduce some notation. For each T ∈ [0, ∞), let DT be the set of all cadlag functions restricted to [0, T ). We endow the set DT with the Skorohod metric dT .8 For each x ∈ DT and y ∈ DT ′ where T , T ′ ∈ [0, ∞], define the real map xT y on [0, T + T ′ ) by xT y(t ) :=

x(t ) y(t − T )



if 0 ≤ t < T if T ≤ t < T + T ′

for every t ∈ [0, T + T ′ ). In words, xT y is the path on the time interval [0, T +T ′ ) which follows x up to time T and from T onwards

5 That is, x ∈ C iff x is an X -valued cadlag function such that the closure of x([0, ∞)) is compact in X . 6 See Appendix for the definition of this metric. 7 The asymmetric part of % is denoted by ≻, while its symmetric part is denoted by ∼. 8 See Appendix for details.

36

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43

continues according to the path t → y(t − T ).9 For any T > 0, x ∈ DT and p ∈ △(C ), we define the Borel probability measure xT p on C by xT p(S ) = p(x S )

(4)

where S is any Borel subset of C and x S := {y ∈ C | xT y ∈ S }. (This measure is well-defined; see Remark 3.) The following axiom imposes stationarity of preference over time. Stationarity. For any T > 0, x ∈ DT and p, q ∈ △(C ), xT p % xT q if and only if

p % q.

This axiom states that if two lotteries yield the same path up to time T , then the path up to that time is irrelevant for ranking of the lotteries. In order to state our final axiom, we introduce one more notation. For any T > 0, p ∈ △(DT ), and y ∈ C , define a Borel probability measure pT y on C by pT y(S ) = p(Sy )

(5)

where S is any Borel subset of C and Sy = {v ∈ DT | vT y ∈ S }. Risk separability. There is T > 0 such that for any p, q ∈ △(DT ) and y, y′ ∈ C , pT y % qT y if and only if

pT y′ % qT y′ .

This axiom states that there is a point T in time such that if two lotteries yield the same path after that time, then the path after that time is irrelevant for ranking of the lotteries. Our axiomatic system is parallel to that of Epstein’s (1983). Assumption 1 in Epstein (1983) requires ⊁= ∅, but we do not need to impose this here. His Assumption 4 corresponds to the independence and continuity axioms. On the other hand, Epstein’s Assumptions 2 and 3 imply that if two lotteries yield the same consumption paths up to time T , then the path up to T is irrelevant for ranking of the lotteries. Our stationarity axiom is a continuous time version of this property. Likewise, our risk separability axiom corresponds to Epstein’s Assumption 5. 2.2. Characterization theorems Our first result is an axiomatic characterization of a preference model proposed by Uzawa (1968). Theorem 1. A complete preorder % on △(C ) satisfies the independence, continuity, and stationarity axioms if and only if there are continuous functions u : X → R and β : X → R++ such that βu is bounded and (3) holds for each p, q ∈ △(C ) with U (x) =

u′ = α u + γ β. By adding the risk separability axiom to the set of axioms in Theorem 1, we obtain a characterization of the exponential discounting model as follows. Theorem 3. A complete preorder % on △(C ) satisfies the independence, continuity, stationarity, and risk separability axioms if and only if there is a continuous and bounded u : X → R and λ > 0 such that (3) holds for each p, q ∈ △(C ) with U (x) =

∞



e−λt u(x(t ))dt ,

(8)

0

for every x ∈ C . Remark 1. In Epstein (1983), each discrete time consumption path is assumed to take values in a compact and connected metric space.10 By contrast, we assume neither the compactness nor connectedness in Theorems 1 and 3. Instead, we impose the weaker assumption that X is a separable metric space and the image of each consumption path is relatively compact. At any rate, connectedness is not invoked in the proof of the first theorem in Epstein (1983). Moreover, steps of his proof that use compactness still go through even if the compactness is replaced by the assumption that the image of each path is relatively compact. Therefore, the characterization in his first theorem still holds under our assumptions alone. In fact, our proof adopts the patent of his proof, except that we need to modify parts of the argument to accommodate the continuous nature of the domain. The connectedness assumption, on the other hand, is used in a crucial manner in the proof of the second theorem of Epstein (1983) in which the exponential discounting in discrete time is characterized. There is, however, an alternative way to prove that theorem which does away with connectedness, that theorem is in fact valid under the weaker assumption that the image of each consumption path is relatively compact. Our proof of Theorem 3, which is fundamentally different from that of Epstein, is based on this alternative approach. 3. Ordinal framework 3.1. Preliminaries

∞



Proposition 2. Let % be a complete preorder on △(C ) such that p ≻ q for some p and q in △(X ). Then the two pairs (β, u) and (β ′ , u′ ) both represent % in the sense of Theorem 1 if and only if β = β ′ and there are two real numbers α > 0 and γ such that

ϕ(x, t )u(x(t ))dt ,

(6)

0

for every x ∈ C , where ϕ is the real map on C × [0, ∞) defined by

ϕ(x, t ) := e−

t 0

β(x(s))ds

.

(7)

u

Moreover, β is not constant, provided that p ≻ q holds for some p, q ∈ △(C ). Epstein (1983) provides a characterization of Uzawa’s time preference in discrete time. Theorem 1 above may thus be seen as a continuous time version of the first theorem of Epstein (1983). The nature of the uniqueness of the representation given in Theorem 1 is identified as follows.

9 Note that x y depends not only on T but also on T ′ , but our notation does not T make this explicit.

Throughout this section (X , dX ) stands for a separable and connected metric space, and D∞ for the set of all X -valued cadlag functions. For any T ∈ [0, ∞] and c ∈ X , cT is a map defined on [0, T ) that takes constant value c. When there is no risk of confusion, we drop the subscript and denote the constant map by c. For any T ≥ 0, let CT be the set of all cadlag functions that are constant after time T . That is, CT := {x ∈ D∞ | x(t ) = x(T −) for all t ≥ T }. We denote the set of all consumption paths by C and in what follows, assume C is any subset of D∞ that satisfies the following conditions: (1) CT ⊂ C for any T ≥ 0, and (2) xT y is in C for any x, y in C and T > 0.

10 The consumption space is assumed to be a closed interval in his paper. However, the proof works for arbitrary connected and compact metric space.

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43

37

The primitive in this section is a complete preorder % on C. The following axioms are imposed on %.

3.3. An alternative characterization

O1. There are c, c ′ in X , x in C, and T > 0 such that cT x ≻ c ′ T x. O2. For any T > 0 and any convergent sequences (xn ) and (yn ) in CT ,

An alternative characterization of the exponential discounting model in the continuous time framework is obtained by Kopylov (2010) through a measure theoretic approach. In this approach, the time axis is taken as a state space and a representation is obtained through a Savage type representation theorem. The advantage of this approach, which is also briefly discussed in Wakker (1993), is that it allows for arbitrary consumption spaces. It also has some disadvantages, however. For example, it lacks a discrete time counterpart and ignores the topological structure of the time space [0, ∞), which is intrinsic to the intertemporal choice model. By contrast, we impose a continuity property that respects the topological structure of the time space, and as a consequence obtain a continuous representation. As such, our approach parallels Koopmans’ classical approach that was originally carried out in the discrete time framework. Nevertheless, the exponential discounting model with a continuous instantaneous utility function can be characterized with Kopylov’s axioms if continuity is assumed in addition. To see this, for any x, y ∈ D∞ and 0 ≤ t < s ≤ ∞, let x[t ,s) y denote the path defined by x[t ,s) y(t ′ ) = x(t ′ ) for t ′ ∈ [t , s), and x[t ,s) y(t ′ ) = y(t ′ ) otherwise. The following axioms were introduced by Kopylov (2010).

xn % yn

for every n implies lim xn % lim yn .

O3. For any x in C, there is c in X such that x ∼ c. O4. For any x in C and c in X , if x ≻ c (c ≻ x) then there is T > 0 such that xt c ≻ c (resp. c ≻ xt c) for every t ≥ T . O5. For any x, y, z in C and T > 0, xT y % xT y if and only if xT z % xT z . O6. For any c in X , x, y in C, and T > 0, cT x % cT y if and only if x % y. Axioms O1–O6 can be seen as continuous time versions of properties used in Bleichrodt et al. (2008). O1 states that preferences are sensitive on the time interval up to some time T and excludes preferences that are determined by the tail behavior of consumption paths. O2 is a continuity property imposed on ultimately constant paths. O3 guarantees that for each path, there is a constant path that is indifferent to it. O4 states, in words, that a far future does not matter. O5 states that if two paths coincide after time T , the path beyond that time is irrelevant for the evaluation of the paths. O6 is a stationarity condition. It posits that if two paths yield the same constant consumption up to time T , then the path up to time T is irrelevant for ranking of the paths.

Theorem 4. A complete preorder % on C satisfies O1–O6 if and only if there is a nonconstant continuous function u : X → R and λ > 0 such that for each x and y in C,

 if and only if

∞

e−λt u(y(t )) dt .

implies x[t ,s) z ′ % y[t ,s) z ′ .

implies a′ [t ,s) b′ % a′ [t ′ ,s′ ) b′ .

K4. For all x, y ∈ D∞ , c ∈ X , increasing (tn ), and decreasing (sn ) such that sn < tn for all n and lim tn = lim sn , c[tn ,sn ) x % y for all n implies x % y.

K5. For all a, b ∈ X , t < s, t ′ < s′ , and l ≥ 0,

∞

≥

a[t ,s) b % a[t ′ ,s′ ) b

x % c[tn ,sn ) y or

e−λt u(x(t )) dt

0



x[t ,s) z % y[t ,s) z

K2. For all x, y ∈ D∞ , x ≥∗ y implies x % y.11 K3. For all t < s, t ′ < s′ , and a, a′ , b, b′ in X with a ≻ b and a′ ≻ b′ ,

3.2. Characterization theorem

x%y

K1. For all x, y, z , z ′ ∈ D∞ and t < s,

(9)

a[t ,s) b % a[t ′ ,s′ ) b

if and only if a[t +l,s+l) b % a[t ′ +l,s′ +l) b.

0

Moreover, λ is unique while u is unique up to a positive affine transformation. Remark 2. By Observation 3 in Bleichrodt et al. (2008), some of the axioms can be relaxed or replaced on particular domains. When C consists of ultimately constant paths, O3 can be dropped. For any consumption path x, y, we write x ≥∗ y iff x(t ) % y(t ) for all t. A consumption path x is bounded if there are c , c ′ ∈ X such that c % x(t ) % c ′ for all t. When C contains only bounded consumption paths, O4 can be replaced by the following monotonicity condition. A preference relation satisfies monotonicity if x ≥∗ y implies x % y and x >∗ y implies x ≻ y. In Theorem 4, u and λ are derived so that the discounted utility is well-defined for every x ∈ C. In particular, u is bounded when C contains all cadlag functions. In that case, the model satisfies the following standard continuity condition. Continuity. For any convergent sequences (xn ) and (yn ) in C, xn % yn

for every n implies lim xn % lim yn .

Notice that the axioms O2 and O4 are implications of this condition. The following corollary provides a characterization in the special case when C contains all cadlag functions using continuity instead of O2, O3, and O4. Corollary 5. A complete preorder % on D∞ satisfies O1, O5, O6, and continuity if and only if there is a nonconstant continuous and bounded function u : X → R and λ > 0 such that (9) holds for each x and y in D∞ .

K1 states that the ranking of two consumption paths is independent of common past and common future consumption. K2 is a natural monotonicity condition. K3 requires that the preference for consuming a on [t , s) rather than on [t ′ , s′ ) while consuming b outside of these intervals should be preserved when a and b are replaced by a′ and b′ such that a′ ≻ b′ . K4 states that consumption over [tn , sn ) becomes irrelevant as the intervals collapse to a point in finite time or go to infinity. K5 is a version of stationarity condition. These axioms together with continuity characterize those complete preorders on the space of all cadlag functions which are of the exponential discounting model: Theorem 6. A complete preorder % on D∞ satisfies K1–K5, and continuity if and only if there is a continuous and bounded function u : X → R and λ > 0 such that (9) holds for each x and y in D∞ .12 Moreover, when ⊁= ∅, λ is unique while u is unique up to a positive affine transformation. 4. Proofs In this section, we provide the proofs of the results in Sections 2 and 3. A map x in DT is simple, if there are 0 = T0 < T1 < · · · <

11 See Remark 2. 12 I thank a referee of the journal for this theorem and its proof.

38

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43

Tk = T , and c 1 , . . . , c k in X such that x(t ) = c

i

whenever Ti ≤ t < Ti+1 for i = 0, . . . , k − 1.

(10)

For each n ∈ N, let En := {x ∈ D∞ | x is constant on [ , for all m ∈ Z+ }. For any n ∈ N, define σn : D∞ → D∞ by m n

σn (x)(t ) :=

   x k n

x ( n)

if

k n

≤t<

k+1 n

m+1 n

)

, 0 ≤ k < n2 − 1

if n ≤ t .



Recall that for each T ∈ [0, ∞], the Skorohod metric on DT is denoted by dT . We write convergence with respect to this metric dT

on DT by →. The following lemma summarizes properties of the Skorohod metric that are relevant for the proofs of our main results. We provide the proof of the lemma in the Appendix. Lemma 7. The Skorohod metric satisfies the following properties. (a) (D∞ , d∞ ) is separable when (X , dX ) is separable. d∞

(b) xn → x implies xn (t ) → x(t ) all but countably many points. d∞

T

T

(c) xn → x implies 0 f (xn (t ))dt → 0 f (x(t ))dt for every continuous f : X → R and T > 0. (d) Simple functions are dense in DT for T ∈ [0, ∞]. dT

d∞

d∞

(e) xn → x and yn → y imply xnT yn → xT y. d∞

(f) Tn → ∞ implies xTn y → x for every x, y ∈ D∞ . d∞

(g) If (xn ) is a sequence in En , then xn → x iff xn (t ) → x(t ) for all t. d∞

(h) σn (x) → x for every x ∈ D∞ . The properties (a)–(f) are relevant for the results in the cardinal framework. On the other hand, the properties (b), (c), (g), and (h) are used in the ordinal framework. 4.1. Cardinal framework

Since cl(x[0, ∞)) is compact, there is ϵ > 0 such that β ◦ x > ϵ . T Hence, e− 0 β(x(s))ds → 0 as T → ∞. By letting T → ∞ in (12), we obtain (11).  Proof of Theorem 1. Let u : X → R and β → R++ be continuous functions such that βu is bounded and (3) holds for every p and q in △(C ) where U is defined by (6). We wish to show that % satisfies the independence, continuity, and stationarity axioms. To this end, we first show that U is continuous and bounded. Take M > 0 such that | βu | ≤ M. Observe that for every x ∈ C , ∞



|ϕ(x, t )u(x(t ))| dt ≤ M

where the last equality follows from Lemma 8. This implies that U is bounded and |U | ≤ M in particular. For each x ∈ C and T T > 0, let UT (x) := 0 ϕ(x, t )u(x(t ))dt and T x(t ) := x(t + T ) for all t ∈ [0, ∞). Then, U (x) = UT (x) + ϕ(x, T )U (T x). Let (xn ) be a sequence in C that converges to x. By (c) in Lemma 7, T T ϕ(xn , t ) → ϕ(x, t ) for all t and 0 |u(xn (t ))|dt → 0 |u(x(t ))|dt. Then, by (b) in Lemma 7, ϕ(xn , t )u(xn (t )) → ϕ(x, t )u(x(t )) almost everywhere. On the other hand, ϕ(xn , t )u(xn (t )) ≤ |u(xn (t ))| for all t. By Young’s Theorem,15 therefore, UT (xn ) → UT (x). For ϵ . any ϵ > 0, take T > 0 large enough so that ϕ(x, T ) < 2M There is N such that n ≥ N implies |UT (xn ) − UT (x)| < 2ϵ and ϵ ϕ(xn , T ) < 2M . Then, |U (xn ) − U (x)| ≤ |UT (xn ) − UT (x)| + |ϕ(xn , T )U (T xn ) − ϕ(x, T )U (T x)| < ϵ . Thus, U is continuous and bounded. By the expected utility theorem,16 therefore, % satisfies the independence and continuity axioms. To see that % satisfies the stationarity axiom, let T > 0. Define the map ρ : DT × C → C by ρ(x, y) = xT y. By (e) in Lemma 7, ρ is continuous. Observe that for any x, x′ ∈ DT and p, q ∈ △(C ),

Lemma 8. Let β : X → R++ be a continuous function. Then, for every x ∈ C , ∞

β(x(t ))e−

t 0

β(x(s))ds

dt = 1.

(11)

0

Proof. Since every cadlag function is continuous almost everywhere,14 β ◦ x is continuous almost everywhere. Therefore, the  map t → 1 − e−

t 0

β(x(s))ds is differentiable almost everywhere with  − 0t β(x(s))ds

the derivative β(x(t ))e . Moreover, since x[0, ∞) is relatively compact, the map is absolutely continuous. Therefore, for every T > 0,

C

β(x(t ))e−

t 0

β(x(s))ds

dt = 1 − e−

T 0

β(x(s))ds

.

(12)

0

13 x y is in C . This is because cl(x([0, T ))) is compact. (See page 122 of Billingsley, T 1999.) Then, cl(xT y([0, ∞))) is also compact. 14 See Lemma 5.1 in Chapter 3 of Ethier and Kurtz (1986).

U ◦ ρ(x, ·) dp ≥ C



U ◦ ρ(x, ·) dq C



⇔ UT (x) + ϕ(x, T ) U dp ≥ UT (x) + ϕ(x, T ) C   ⇔ U dp ≥ U dq. C

 U dq C

C

In view of (3), % satisfies the stationarity axiom. Conversely, let % be a complete preorder on △(C ) that satisfies the independence, continuity, and stationarity axioms. If p % q holds for all p, q ∈ △(C ), then take any constant functions u : X → R and β : X → R++ and define U by (6) to obtain a desired representation. In what follows, we assume p ≻ q for some p and q in △(C ). By (a) in Lemma 7, (D∞ , d∞ ) is a separable metric space. Therefore, C is separable. By the expected utility theorem, (see footnote 16) there is a nonconstant, continuous, and bounded function U : X → R that satisfies (3) for each p, q ∈ △(C ). Observe that for any T > 0, x ∈ DT , and p ∈ △(C ),



 U dxT p = C

U ◦ ρ(x, ·) dp.

C

By stationarity, therefore,



U ◦ ρ(x, ·) dp ≥

p%q⇔ C

T



 U dxT q ⇔

C

The following lemma is used in the proof of Theorem 1.



 U dxT p ≥

We begin with the following remark.

is measurable. Therefore, p ◦ σy−1 is well-defined.

ϕ(x, t )β(x(t ))dt = M , 0

0



Remark 3. Probability measures xT p and pT y defined in (4) and (5) respectively are well-defined. To see this, define ρx : C → C by ρx (y) = xT y13 and observe that xT p = p ◦ ρx−1 . By (e) in Lemma 7, ρx is continuous. Hence, it is measurable. Therefore, p ◦ρx−1 is welldefined. Similarly, define σy : DT → C by σy (v) = vT y and observe that pT y = p ◦ σy−1 . By (e) in Lemma 7, σy is continuous. Hence, it

∞





U ◦ ρ(x, ·) dq

C

for each p, q ∈ △(C ). Thus, U and U ◦ ρ(x, ·) are two vNM utility functions for %. By the uniqueness of the expected utility

15 See Theorem 2.8.8 in Bogachev (2007). 16 See Theorem 3 of Grandmont (1972).

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43

representation, there are two real numbers a(x) > 0 and b(x) such that for every y ∈ C , U ◦ ρ(x, y) = b(x) + a(x)U (y).

(13)

Since T and x ∈ DT are arbitrary, a and b define the maps a :   T >0 DT → R++ and b : T >0 DT → R respectively. In what follows, y, y are two elements in C such that U (y) > U (y). We establish that U takes the form (6) through the following claims. Claim 1. For any T > 0, a and b are continuous on DT . Proof of Claim 1. Let (xn ) be a convergent sequence in DT and x := lim xn . For every n, by (13), U (xnT y) − U (xnT y) = a(xn )(U (y) − U (y)).

(14)

39

Claim 5. For any T > 0 and x ∈ DT , a(x) = e−

T 0

β(x(s))ds

.

Proof of Claim 5. First, consider the case when x = cT for some c ∈ X . If T is an integer, then by Claim 3, a(cT ) = a(c1 )T = e−β(c )T . If T is a rational number, then mT is an integer for some integer m. Then, a(cmT ) = e−β(c )mT . On the other hand, a(cmT ) = a(cT )m by Claim 3. Thus, a(cT ) = e−β(c )T . If T is a real number, then there is an increasing sequence of rational numbers (Tm ) that converges to T . Then, a(cTm ) = e−β(c )Tm converges to e−β(c )T , while by Claim 4, a(cT −Tm ) converges to 1 as m → ∞. Since a(cT ) = a(cTm )a(cT −Tm ) for every m, a(cT ) = e−β(c )T . In light of Claim 3, one can easily show that the claim holds when x is a simple function of the form (10). To complete the proof, let x be an arbitrary element of DT . By (d) in Lemma 7, there is a sequence of simple functions (xn ) that converges to x. Since the T

Similarly, U (xT y) − U (xT y) = a(x)(U (y) − U (y)).

(15)

By continuity of U and (e) in Lemma 7, the left hand side of (14) converges to that of (15). This implies that a is continuous. In view of (13), b is also continuous on DT . 

claim holds for simple functions, we have a(xn ) = e− 0 β(xn (s))ds for every n. By Claim 1, a(xn ) converges to a(x). Therefore, we are T T done if 0 β(xn (s))ds converges to 0 β(x(s))ds. To this end, for each n define x˜ n by x˜ n (t ) = xn (t ) for t ∈ [0, T ) and x˜ n (t ) = x(T −) d∞

otherwise. Define x˜ similarly. Then, by (e) in Lemma 7, x˜ n → x˜ . By T T T (c) in Lemma 7, 0 β(xn (s))ds = 0 β(˜xn (s))ds → 0 β(˜x(s))ds =

Claim 2. For any T > 0 and x ∈ DT , 0 < a(x) < 1.

T

Proof of Claim 2. By construction, a > 0. To see a < 1, suppose there are T > 0 and x ∈ DT such that a(x) ≥ 1. Define z by

Define u : X → R by u(c ) := β(c )U (c∞ ). By continuity of β and U, u is continuous. Since U is bounded, βu is bounded.

0

z (t ) := x(t − kT ) whenever kT ≤ t < (k + 1)T for k = 0, 1, . . . for each t ∈ [0, ∞). By (13), U (zTn y) − U (zTn y) = a(x)

n −1

(U (y) − U (y))

(16)

holds for every n where Tn = nT . Since U is bounded, the left hand side of (16) is bounded uniformly for every n. If a(x) > 1, the right hand side of (16) diverges to infinity as n → ∞. Therefore, a(x) = 1 is the case. However, in that case, (13) yields U (z ) = b(x) + U (z ), which implies b(x) = 0. Then, for every y ∈ C and n, U (zTn y) = U (y). By (f) in Lemma 7, zTn y converges to z. Since U is continuous, U (z ) = U (y) for every y ∈ C . That is, U is constant. This is a contradiction.  Claim 3. For any T < T ′ , x ∈ DT , and x′ ∈ DT ′ −T , a(xT x′ ) = a(x)a(x′ ) and b(xT x′ ) = b(x) + a(x)b(x′ ). Proof of Claim 3. By (13), b(xT x ) + a(xT x )U (y) = b(x) + a(x)b(x ) + a(x)a(x )U (y), ′

′

′

′

for every y ∈ C . If a(xT x′ ) ̸= a(x)a(x′ ), then U must be constant as y ∈ C is arbitrary. Thus, a(xT x′ ) = a(x)a(x′ ) holds. This in turn implies b(xT x′ ) = b(x) + a(x)b(x′ ).  For any c ∈ X and T ∈ [0, ∞], cT stands for the path defined on [0, T ) that takes constant value c. Claim 4. For any c ∈ X and any sequence (Tn ) in [0, ∞) that converges to 0, lim a(cTn ) = 1. Proof of Claim 4. By Claim 3, a(c1 ) = a(c 1 ) for every n. Since 1

a(c1 ) > 0, a(c 1 ) = a(c1 ) n → 1 as n → ∞. For any n, take m large n

enough so that Tm <

1 . n

Then, a(cTm ) > a(c 1 ) by Claims 2 and 3.

Therefore, a(cTm ) converges to 1 as m → ∞.

n



Let β(c ) := − log a(c1 ) for each c ∈ X . By Claim 1, β : X → R is continuous.17 Moreover, β > 0 by Claim 2.

dT 17 It is immediate from the definition of d that c → c iff c → cT . T n nT



Claim 6. For any T > 0 and x ∈ DT , b(x) =

T 0

e−

t 0

β(x(s))ds

u(x(t ))dt.

Proof of Claim 6. First, consider the case when x = cT for some c ∈ X . By (13), b(cT ) = (1 − a(cT ))U (c∞ ). In view of Claim 5, 1 − a(cT )

b(cT ) =

β(c )

u( c ) =

T



e−β(c )t u(c )dt .

0

Hence, the claim is true for this case. In light of Claim 3, one can easily show that the claim holds when x is a simple function of the form (10). To complete the proof, let x be an arbitrary element of DT . By (d) in Lemma 7, there is a sequence of simple functions (xn ) that converges to x. Since the claim holds for simple functions, we have T



b(xn ) =

e−

t 0

β(xn (s))ds

u(xn (t ))dt ,

0

for every n. By Claim 1, b(xn ) converges to b(x). Therefore, we  T t  T t

are done if 0 e− 0 β(xn (s))ds u(xn (t ))dt converges to 0 e− 0 β(x(s))ds u(x(t ))dt. Define x˜ n for each n and x˜ as in Claim 5. Then, by (e) in d∞

Lemma 7, x˜ n → x˜ . By (b) of Lemma 7, u(˜xn (t )) → u(˜x(t )) almost everywhere. Then, by (c) in Lemma 7, e−  e

t 0

−

β(˜x(s))ds

t 0

β(˜xn (s))ds

u(˜xn (t )) →

u(˜x(t )) almost everywhere. On the other hand, by (c)  t

in Lemma 7, 0 |u(xn (t ))|dt → 0 |u(x(t ))|dt. Since e− 0 β(˜xn (s))ds u(˜xn (t )) ≤ |u(xn (t ))| everywhere for each n, we can invoke Young’s Theorem to obtain

T

T



e−

n

n

β(x(s))ds.

t 0

β(˜xn (s))ds

T

u(˜xn (t ))dt →

0

Since

T



e−

t 0

β(˜x(s))ds

u(˜x(t ))dt .

0

T 0

t

 T −  t β(x (s))ds e 0 n u(xn (t ))dt for 0  T −  t β(x(s))ds u(˜x(t ))dt = 0 e 0 u(x(t ))dt, we

β(˜xn (s))ds u(˜xn (t ))dt  T −  t β(˜x(s))ds 0

e−

0

each n and 0 e are done. 

=

We are ready to obtain an expression in (6). Define ϕ by (7). For any x ∈ C and T > 0, let T x be an element of C defined by T x(t ) = x(t + T ) for every t ∈ [0, ∞). By (13), Claims 5, and 6, U (x) =

 0

T

ϕ(x, t )u(x(t ))dt + ϕ(x, T )U (T x).

(17)

40

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43

Since cl(x([0, ∞))) is compact and β is continuous, there is ϵ > 0 such that β(x(t )) ≥ ϵ for every t. Therefore, ϕ(x, T ) ≤ e−ϵ T . Since U is bounded, the second term of (17) converges to 0 as T → ∞. Therefore, the first term converges to U (x) as T → ∞. That is, U (x) =

∞



ϕ(x, t )u(x(t ))dt .

the independence, continuity, stationarity and risk separability axioms. In light of Theorem 1, it is enough to check that % satisfies the risk separability axiom. For any y ∈ C , define σy : DT → C by σy (v) = vT y. By (e) in Lemma 7, σy is continuous. Moreover, for any p ∈ △(DT ), pT y = p ◦ σy−1 . Observe that



0



We conclude the proof by the following claim.

DT

C

U (y) = lim b(y|[0,T ) ) = K (1 − lim a(y|[0,T ) )) T →∞

T →∞

= K. This contradicts with the fact that U is not constant.





Proof of Proposition 2. We continue using notations introduced in the proof of Theorem 1. First, let β ′ = β and u′ = α u + γ β for some α > 0 and γ . Let U be the function defined by (6) associated with (β, u). Similarly, let U ′ be the function defined by (6) associated with (β ′ , u′ ). Observe that for any y ∈ C , ∞

e−

t

e−

t

0

β(y(s))ds

u(y(t ))dt

DT

∞

+γ

β(y(s))ds

0

where the second equality is due to Lemma 8. Thus, U ′ = α U + γ . Therefore, U and U ′ are two vNM functions of the same relation. Conversely, let U and U ′ be two vNM utility functions for the same relation of the form (6) associated with (β, u) and (β ′ , u′ ) respectively. Then, there are α > 0 and γ such that U ′ = α U + γ . For any c ∈ X and y ∈ C , U (c1 y) = b (c1 ) + e ′

−β ′ (c )

(α U (y) + γ ),

−γ + b′ (c1 ) + γ e−β (c ) ′ + e−β (c ) U (y). α ′

Plug U (c1 y) = b(c1 ) + e

−γ + b′ (c1 ) + γ e−β (c ) . (18) α ′

(e−β(c ) − e−β (c ) )U (y) = −b(c1 ) + ′

Eq. (18) holds for any y ∈ C . The difference of (18) for y and y yields −β ′ (c )

(e−β(c ) − e

u′ (c )

u′ (c )

u(c )

U (c∞ ) = β(c ) and U ′ (c∞ ) = β ′ (c ) . Therefore, β ′ (c ) = α β(c ) + γ . Since β = β ′ , this implies u′ (c ) = α u(c ) + γ β(c ). Thus, u′ = αu + γ β .  Proof of Theorem 3. We continue using notations introduced in the proof of Theorem 1. First, let U be a vNM utility function for % with the expression in (8). We show that % satisfies

dq

Hence, the risk separability axiom is satisfied. Conversely, let % be a complete preorder that satisfies the independence, continuity, stationarity, and risk separability axioms. If ≻= ∅, any λ > 0 and any constant function u represent %. Let ⊁= ∅. By Theorem 1, there are a continuous and bounded function U : X → R and continuous functions u : X → R and β : X → R++ that satisfy (3) and (6). We wish to show that β is constant. To this end, let T > 0 be the time with which risk separability holds. That is, for any p, q ∈ △(X ) and y, y′ ∈ C , pT y % qT y iff pT y′ % qT y′ . Fix y, y′ ∈ C such that U (y) > U (y′ ). Define a binary relation D on △(DT ) as follows; for each p, q ∈ △(DT ), U ◦ σy dp ≥



U ◦ σy dq.

DT

By definition, U ◦ σy is a vNM utility function of D. On the other hand, by risk separability,





pDq⇔

U dpT y ≥

C ⇔ C

U dpT y′ ≥

U dqT y

C

U dqT y′

C

U ◦ σy′ dp ≥



U ◦ σy′ dq. DT

That is, U ◦σy′ is another vNM utility function for D. Therefore, there are two constants γ > 0 and η such that U ◦ σy′ = γ U ◦ σy + η. Then, for any v, w ∈ DT ′ where T ′ := T2 , and

U (wT ′ (vT ′ y )) = γ U (wT ′ (vT ′ y)) + η. ′

In view of (13), these two equations become b(v) + a(v)b(w) + a(v)a(w)U (y′ )

= γ (b(v) + a(v)b(w) + a(v)a(w)U (y)) + η, b(w) + a(w)b(v) + a(w)a(v)U (y )

)(U (y) − U (y)) = 0,



0

′

which implies β(c ) = β ′ (c ) for all c ∈ X . Finally, for any c ∈ X , u(c )

e−λt u(v(t ))dt

⇔ pT y′ % qT y′ .

U (vT ′ (wT ′ y′ )) = γ U (vT ′ (wT ′ y)) + η,

U (y) and rearrange to obtain,

T



 dp ≥ DT

DT



−β(c )



0

DT

⇔

′ 1 where b′ (c1 ) = 0 e−β (c ) u′ (c )dt. On the other hand, U ′ (c1 y) = α U (c1 y) + γ . Then,

U (c1 y) =

e−λt u(v(t ))dt

DT

0

′

T



 ⇔



= α U (y) + γ ,

dp + e−λT U (y).

0

pT y % qT y

p D q iff

β(y(t ))dt



Therefore, for any p, q ∈ △(DT ) and y, y′ ∈ C ,

0



e−λt u(v(t ))dt

=

Proof of Claim 7. Suppose u(c )β(c )−1 = K for some K for every b(c ) c ∈ X . Then, for any T > 0, K = U (c∞ ) = 1−a(Tc ) . Thus, T b(cT ) = K (1 − a(cT )) for every c ∈ X . Let x ∈ DT be any simple function of the form (10). Then, by Claim 3, b(x) = K (1 − a(x)) holds. Since a and b are continuous and step functions are dense by (d) in Lemma 7, b(x) = K (1 − a(x)) holds for all x ∈ DT . Then, for any y ∈ C ,



T





Claim 7. βu is not constant.

U ′ (y) = α

U ◦ σy dp

U dpT y =

= γ (b(w) + a(w)b(v) + a(w)a(v)U (y)) + η, which can be rewritten as,

η = (1 − γ )b(v) + (1 − γ )a(v)b(w) + (U (y′ ) − γ U (y))a(v)a(w), and η = (1 − γ )b(w) + (1 − γ )a(w)b(v) + (U (y′ ) − γ U (y))a(w)a(v).

and

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43 d∞

The difference of the two equations yields,

(1 − γ )(b(v) − b(w)) + (1 − γ )(a(v)b(w) − a(w)b(v)) = 0. (19) Suppose γ ̸= 1. Then, (19) implies b(v)(1 − a(v))−1 = b(w)(1 − a(w))−1 . Since v, w ∈ DT ′ are arbitrary, this implies u(c )β(c )−1 = u(c ′ )β(c ′ )−1 for every c , c ′ ∈ X . This contradicts with Theorem 1. Thus γ = 1. Then, η = (U (y′ ) − U (y))a(v)a(w) for every v, w ∈ DT ′ . In particular, by letting v = w = cT ′ for any c ∈ X , η = (U (y′ ) − U (y))e−β(c )T . Since U (y′ ) ̸= U (y) and η is constant, β(c ) = β(c ′ ) for every c , c ′ ∈ X . Hence, β is constant.  4.2. Ordinal framework Proof of Theorem 4. Let λ > 0 and u be a nonconstant continuous function such that (9) holds for every x, y ∈ C. We show that % satisfies O2. It is fairly easy to show that % satisfies other conditions. T For any x ∈ C and T ∈ [0, ∞], let UT (x) := 0 e−λt u(x(t )) dt. It is enough to show that for any convergent sequence (xn ) in CT , U∞ (xn ) → U∞ (x) where x := lim xn . By (b) in Lemma 7, u(xn (t )) → u(x(t )) almost everywhere. By (c) in Lemma 7, T T |u(xn (t ))|dt → 0 |u(x(t ))|dt. For each n, e−λt u(xn (t )) ≤ 0 |u(xn (t ))| for every t. By Young’s Theorem, UT (xn ) → UT (x). On the hand, xn (T ) →  ∞ other  ∞x(T ) by (b) in Lemma 7. This implies e−λt u(xn (T )) dt → T e−λt u(x(T )) dt. Therefore, U∞ (xn ) → T U∞ (x). Conversely, let % be a complete preorder on C that satisfies O1–O6. By O1, there are c and c ′ in X , x in C and T ∗ > 0 such that cT ∗ x ≻ c ′ T ∗ x. Without  loss of generality, we assume T ∗ = 1. For each n ∈ N, let Zn := ( T >0 CT ) ∩ En . Define a map Fn : Zn → X N ) for every m ∈ N. By (g) in Lemma 7, Fn is a by Fn (x)m := x( m n homeomorphism between Zn and Fn (Zn ) where Fn (Zn ) is endowed with the product topology. Define a binary relation %n on Fn (Zn ) by (xm ) %n (ym ) if and only if Fn−1 ((xm )) % Fn−1 ((ym )). In light of O1–O6, Fn (Zn ) and %n satisfy the conditions in Theorem 2 of Bleichrodt et al. (2008). Therefore, there are δn ∈ (0, 1) and a nonconstant continuous function un : X → R such that for every x, y in Zn , x % y if and only if

∞ 

δn un (xt ) ≥

t =0

t

∞ 

δn un (yt ), t

(20)

t =0

where xt = x( ) for t ∈ Z+ and similarly for y. Observe that Z1 is a subsetof Zn for every  n. Therefore, upon normalization if necessary, t n

u1 =

n−1 t =0

δn t un and δ1 = δn n for every n. Let λ := − log δ1 .

λ u1 (c ) for every c ∈ X . Let Z := Define u : X → R by u(c ) := 1−δ 1   Z and define a function U : Z → R by U (x) = ∞ n∈N n t =0 δn un (xt ) where x is in Zn and (xt ) = Fn (x). Then U is well defined. Moreover,

U ( x) =

∞



41

e−λt u(x(t )) dt

(21)

0

holds for every x in Z . In view of (20), % admits a desired representation on Z .18 Now, define U (x) by (21) for every x ∈ C. We show that U (x) is well defined and for each x, y ∈ C, x % y if and only if U (x) ≥ U (y). To this end, it is enough to show the following claim. Claim. U (x) = U (c ) for every x in C and c in X with x ∼ c. First, consider the case where x is in CT for some T > 0. In this case, cl(x[0, T )) is compact and cl(x[0, T )) = cl(x[0, ∞)). Therefore, u ◦ x is bounded. Hence, U (x) is well defined. By (h)

18 This conclusion is obtained without using O3. See Remark 2.

in Lemma 7, xn := σn (x) → x. Observe that xn and x are in CT +1 . By (b) in Lemma 7, u(xn (t )) → u(x(t )) almost everywhere.

 T +1

 T +1

By (c) in Lemma 7, 0 |u(xn (t ))|dt → 0 |u(x(t ))|dt. For each n, e−λt u(xn (t )) ≤ |u(xn (t ))| for every t. By Young’s The T +1 −λt  T +1 −λt orem, 0 e u(xn (t ))dt → 0 e u(x(t ))dt. On the other hand, x ( T + 1 ) → x ( T + 1 ) by (b) in Lemma 7. This implies n  ∞ −λt ∞ e u(xn (T + 1)) dt → T +1 e−λt u(x(T + 1)) dt. Therefore, T +1

U (xn ) → U (x). Now, let c be an element of X such that x ∼ c.19 Here, we only treat the case when a ≻ x ≻ b for some a and b in X as other cases can be treated similarly. Take sequences (an ) and (bn ) in X such that U (an ) > U (c ) > U (bn ) for every n and both U (an ) and U (bn ) converge to U (c ).20 By O2, for every n, U (an ) > U (xm ) > U (bn ) for large m and hence U (an ) ≥ U (x) ≥ U (bn ). We obtain the conclusion by letting n → ∞. Next, consider the case when x is an arbitrary element in C. Let c be an element of X such that x ∼ c. We only treat the case when a ≻ x ≻ b for some a and b in X as other cases can be treated similarly. Take sequences (an ) and (bn ) in X such that U (an ) > U (c ) > U (bn ) for every n and both U (an ) and U (bn ) converge to U (c ). Take any sequence (Tn ) in [0, ∞) such that Tn → ∞. By O4, for each n, there is M such that m ≥ M implies an ≻ xTm an % xTm b. Therefore, for m ≥ M, U (an ) >



Tm

e−λt u(x(t )) dt +

0



∞

e−λt u(b) dt .

Tm

Since the second term of the right hand side vanishes as m → ∞, by letting n → ∞, U (c ) ≥ lim sup

Tm



m→∞

e−λt u(x(t )) dt .

0

By the similar argument, one can show U (c ) ≤ lim inf

Tm



m→∞

e−λt u(x(t )) dt .

0

Therefore, U (c ) = lim

m→∞

Tm



e−λt u(x(t )) dt .

0

Since (Tm ) is arbitrary, U (x) is well defined and the claim holds. Finally, we establish a uniqueness result for u and λ. It is clear that if (u, λ) represents % in the sense of (9), then so does (α u + γ , λ) for any α > 0 and γ . Conversely, let (u, λ) and (u′ , λ′ ) be two representations of % in the sense of (9). Define U from (u, λ) by (21). For each (a, b) ∈ X 2 , let f1 (a) := (1 − e−λ )u(a), f2 (a) := e−λ u(a) and f (a, b) := f1 (a)+f2 (b). Observe that f (a, b) = λU (a1 b) for each (a, b) ∈ X 2 . Define U ′ , f1′ , f2′ and f ′ similarly from (u′ , λ′ ). Then, for each (a, b), (a′ , b′ ) ∈ X 2 , f (a, b) ≥ f (a′ , b′ ) iff U (a1 b) ≥ U (a′1 b′ ) iff U ′ (a1 b) ≥ U ′ (a′1 b′ ) iff f ′ (a, b) ≥ f ′ (a′ , b′ ). Thus, f and f ′ are two additive representations of the same binary relation on X 2 . Therefore, by the uniqueness result of additive representations, there are α > 0 and γ1 , γ2 ∈ R such that f1′ = α f1 + γ1 and f2′ = α f2 + γ2 . (See, for example, Theorem 5.4 in Fishburn, 1970.) In particular, for any a ∈ X , u′ (a) = f ′ (a, a) = α f (a, a) + γ1 + γ2 = α u(a) + γ1 + γ2 . Hence, u′ is a positive affine transformation of u. Given the fact that u is not constant, λ = λ′ follows from f1′ = α f1 + γ1 . 

19 Note that the fact U (x) is finite and X is connected together with O2 imply existence of such c. Hence, O3 is not necessary up to this point. 20 Such sequences exist because X is connected and u is continuous.

42

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43

Proof of Corollary 5. Let λ > 0, u be a nonconstant bounded continuous function and (9) holds for every x, y ∈ D∞ . By Theorem 4, % satisfies O1, O5, and O6. It is easy to see continuity and boundedness of u implies that % satisfies continuity. Conversely, let % be a complete preorder that satisfies O1, O5, O6, and continuity. By Theorem 4, there are a nonconstant continuous  u : X → R and λ > 0 such that (9) holds for each x, y ∈  T >0 CT . Now, let D = {x ∈ D∞ | x ∼ c for some c ∈ X }. Then, T >0 CT ⊂ D. Moreover, by the same argument as the claim in Theorem 4, the desired representation holds on D. We wish to show that D = D∞ . To see this, suppose D ̸= D∞ and let x ∈ D \ D∞ . Then, by continuity and connectedness of X , x ≻ c for all c ∈ X or c ≻ x for all c ∈ X . Take a, b, and c in X such that a ≻ b ≻ c. Notice that a1 x is not in D. Because otherwise the discounted utility for a1 x is well-defined and finite which in turn implies that of x is finite. But that implies x ∈ D which is a contradiction. Similarly, b1 x and c1 x are not in D. Since a ≻ b1 a ≻ c1 a, a1 x ≻ b1 x ≻ c1 x holds by O5. Then, a1 x ≻ c ′ and b1 x ≻ c ′ for all c ′ ∈ X , or c ′ ≻ b1 x and c ′ ≻ c1 x for all c ′ ∈ X . Consider the case when a1 x ≻ c ′ and b1 x ≻ c ′ for all c ′ ∈ X . By continuity and (h) in Lemma 7, σn (a1 x) ≻ b1 x for large n. But σn (a1 x) is in D, hence σn (a1 x) ∼ c ′ ≻ b1 x for some c ′ ∈ X . This is a contradiction. We obtain the similar contradiction when c ′ ≻ b1 x and c ′ ≻ c1 x for all c ′ ∈ X . Thus D = D∞ must hold. Finally, note that u must be bounded because otherwise one can construct a path in D∞ such that the discounted utility along the path diverges.  Proof of Theorem 6. It is easy to see that the exponential discounting model satisfies K1–K5. By continuity and boundedness of u, continuity is also satisfied. Let % be a complete preorder that satisfies K1–K5 and continuity. Let ⊁= ∅ as there is nothing to prove otherwise. We use the set Z introduced in the proof of Theorem 4. Let S be the set of all simple cadlag functions in D∞ . By Corollary 4 in Kopylov (2010), there is u : X → R and λ > 0 such that (9) holds for each x, y ∈ S . Continuity of u follows from the continuity condition. Since Z is a subset of S , (9) holds for each x, y ∈ Z . Then, following the proof of the claim in Theorem 4, one can show that the representation holds on T >0 CT . The rest of the proof is same as Corollary 5 except the last step. In the last step, we invoke K1 instead of O5 to obtain the same conclusion. The uniqueness result can be shown by the same argument as in Theorem 4.  Appendix. Skorohod metric In this appendix, we define the Skorohod metric and provide the proof of Lemma 7. To this end, let T ∈ [0, ∞] and ΛT be the set of all increasing bijective selfmaps on [0, T ). Define γT : ΛT → R ∪ {∞} by

   µ(t ) − µ(s)   γT (µ) := sup log . t −s 0≤s




dT (x, y) := inf max γT (µ), sup dX (x(t ), y(µ(t ))) . µ∈ΛT

0≤t
For any x, y ∈ D∞ , µ ∈ Λ∞ , and t ≥ 0, let

v(x, y, µ, t ) = sup min{1, dX (x(s ∧ t ), y(µ(s) ∧ t ))}, 0≤s

where a∧b := min{a, b}. The Skorohod metric d∞ on D∞ is defined by



d∞ (x, y) := inf max γ∞ (µ), µ∈Λ∞

∞



Proof of Lemma 7. (a) See Theorem 5.6 in Chapter 3 of Ethier and Kurtz (1986). (b) See Lemma 5.1 and Proposition 5.2 in Chapter 3 of Ethier and Kurtz (1986). (c) Let DR be the set of all real valued cadlag functions. We endow DR with the Skorohod metric. Then, the map x → f ◦ x is a continuous map defined on D∞ . (See page 151 of Ethier and Kurtz, 1986.) For any h in DR , let I (h) be the cadlag function defined by T I (h)(T ) = 0 h(t ) dt for every T ∈ [0, ∞). Then, the map h → I (h) is a continuous map defined on DR . (See page 153 of Ethier and Kurtz, 1986.) Since xn converges to x, I (f ◦ xn ) converges to I (f ◦ x). Therefore, I (f ◦ xn )(T ) converges to I (f ◦ x)(T ) for all continuity point T of I (f ◦ x). (See Proposition 5.2 in Chapter 3 of Ethier and Kurtz, 1986.) Notice that I (f ◦ x) is continuous and thus I (f ◦ xn )(T ) T converges to I (f ◦ x)(T ) for all T . That is, 0 f ◦ xn dt converges to

T

f ◦ x dt for all T . 0 (d) For T = ∞, see Theorem 5.6 in Chapter 3 of Ethier and Kurtz (1986). For T < ∞, see Theorem 12.2 and page 131 of Billingsley (1999). To prove (e), (f) and (g), we use the following observation. For dT

any T ∈ (0, ∞), xn → x iff there is a sequence (µn ) in ΛT such that sup |µn (t ) − t | → 0, and

(22)

sup dX (xn (t ), x(µn (t ))) → 0,

(23)

t ∈[0,T ) t ∈[0,T )

as n → ∞. This is a direct consequence of the definition of dT . d∞

Similarly, xn → x iff there is a sequence (µn ) in Λ∞ such that (22) and (23) hold for all T > 0.21 dT

d∞

(e) Let xn → x and yn → y. There is a sequence (µn ) in ΛT that satisfies (22) and (23). There is a sequence (νn ) in Λ∞ that satisfies (22) and (23) for all T > 0 for (yn ) and y. For each n, define τn by τn (t ) = µn (t ) for t < T and τn (t ) = νn (t − T ) + T for t ≥ T . Then, (τn ) is a sequence in Λ∞ that satisfies (22) and (23) for all T > 0 d∞

for (xnT yn ) and xT y. Hence, xnT yn → xT y. (f) For each n, define µn by µn (t ) = t for every t. Then, (µn ) d∞

satisfies (22) and (23) for all T > 0 for (xTn y) and x. Thus, xTn y → x. d∞

(g) Let (xn ) be a sequence in En . If xn → x, by (b), xn (t ) → x(t ) almost everywhere. Since each xn is constant over intervals, xn (t ) → x(t ) holds for all t. Conversely, if xn (t ) → x(t ) holds for all t, take a sequence (µn ) where µn (t ) = t for all n and t. Then, (µn ) satisfies (22) and (23) for all T > 0 for (xn ) and x. Hence, d∞

xn → x. (h) See page 151 of Ethier and Kurtz (1986).



References Billingsley, P., 1999. Convergence of Probability Measures. John Wiley & Sons, New York. Bleichrodt, H., Rohde, K.I.M., Wakker, P.P., 2008. Koopmans’ constant discounting for intertemporal choice: A simplification and a generalization. J. Math. Psych. 52, 341–347. Bogachev, V.I., 2007. Measure Theory Vol. I. Springer-Verlag, Berlin. Calvo, G.A., Findlay, R., 1978. On the optimal acquisition of foreign capital through investment of oil export revenues. J. Int. Econ. 8, 513–524. Dolmas, J., 1995. Time-additive representations of preferences when consumption grows without bound. Econom. Lett. 47, 317–325. Epstein, L.G., 1983. Stationary cardinal utility and optimal growth under uncertainty. J. Econom. Theory 31, 133–152. Epstein, L.G., 1987a. A simple dynamic general equilibrium model. J. Econom. Theory 41, 68–95.



e−t v(x, y, µ, t )dt . 0

We provide the proof of Lemma 7 as follows.

21 See Proposition 5.3 in Chapter 3 of Ethier and Kurtz (1986) for the proof of this statement.

K. Hara / Journal of Mathematical Economics 63 (2016) 34–43 Epstein, L.G., 1987b. The global stability of efficient intertemporal allocations. Econometrica 55, 329–355. Epstein, L.G., 1990. Impatience. In: Eatwell, J., Milgate, M., Newman, P. (Eds.), Utility and Probability. W. W. Norton & Company, Inc., New York, London. Epstein, L.G., Hynes, J.A., 1983. The rate of time preference and dynamic economic analysis. J. Polit. Econ. 91, 611–635. Ethier, S.N., Kurtz, T.G., 1986. Markov Processes: Characterization and Convergence. John Wiley & Sons, New York. Fishburn, P., 1970. Utility Theory for Decision Making. Wiley, New York. Frederick, S., Loewenstein, G., O’Donoghue, T., 2002. Time discounting and time preference: A critical review. J. Econom. Lit. 40, 351–401. Grandmont, J.M., 1972. Continuity properties of a von Neumann–Morgenstern utility. J. Econom. Theory 4, 45–57. Harvey, C.M., Østerdal, L.P., 2012. Discounting models for outcomes over continuous time. J. Math. Econom. 48, 284–294. Hübner, R., Suck, R., 1993. Algebraic representation of additive structures with an infinite number of components. J. Math. Psych. 37, 629–639. Koopmans, T.C., 1960. Stationary ordinal utility and impatience. Econometrica 28, 287–309.

43

Koopmans, T.C., 1972. Representation of preference orderings over time. In: McGuire, C.B., Radner, Roy (Eds.), Decision and Organization. NorthHolland, Amsterdam. Kopylov, I., 2010. Simple axioms for countably additive subjective probability. J. Math. Econom. 46, 867–876. Nairay, A., 1984. Asymptotic behavior and optimal properties of a consumption–investment model with variable time preference. J. Econom. Dynam. Control 7, 283–313. Obstfeld, M., 1981. Macroeconomic policy, exchange-rate dynamics, and optimal asset accumulation. J. Polit. Econ. 89, 1142–1161. Pollard, D., 1984. Convergence of Stochastic Processes. Springer, New York. Sagara, N., 2013. Representation of preference orderings with an infinite horizon: Time-additive separable utility in continuous time. J. Int. Econ. Stud. 27, 3–22. Uzawa, H., 1968. Time preference, the consumption function, and optimum asset holdings. In: Wolfe, J.N. (Ed.), Value, Capital and Growth. Edinburgh University Press, Edinburgh. Wakker, P., 1993. Unbounded utility for Savage’s ‘‘Foundations of Statistics,’’ and other models. Math. Oper. Res. 18, 446–485.