数理解析研究所講究録
第 1557 巻 2007 年 59-70

59

Representation of Preference Orderings on$Lp$ -spaces
by Integral Functionals: Myopia,
Continuity and TAS Utility*
Nobusumi Sagara (佐柄信純)
Faculty of Economics, Hosei University (法政大学経済学部)
4342, Aihara, Machida, Tokyo, 194-0298, Japane-mail: nsagara@hosei.ac.jp

March 31, 2007
Key Words: Preference orderings;$L^{P}$-space; TAS utility; Recursive utility;

Integral representation.

1

Introduction

Time additive separable (TAS) utility functions have been used in the analysis of intertemporal optimal behaviors and equilibria over time in various
fields. The first axiomatization of TAS utility with an infinite horizon wasprovided by Koopmans (1972b) in a discrete time framework. Koopmans
employed a truncation method to embed preference orderings with an infinite horizon into finite dimensional preference orderings with an additive
separable representation by using the result of Debreu (1960), and then extended the preference orderings with a finite horizon to those with an infinite
horizon by a kind of limiting argument.
While the result of Koopmans was restricted to bounded programs, Dolmas (1995) generalized it to unbounded programs. Epstein (1986) obtained
the TAS representation under the hypothesis of constancy of marginal rates of’This research is part of the “International Research Project on Aging (Japan, China
and Korea)“ of the Hosei Institute on Aging, Hosei University, and supported by theSpecial Assistance of the Ministry of Education, Culture, Sports, Science and Technology
and by a Grant-in-Aid for Scientific Research (No. 18610003) from the Japan Society for
the Promotion of Science.

60
intertemporal substitution. However, the above works require strong assumptions and it is difficult to apply these results to a continuous time framework.
In particular, Epstein (1986) requires that preference ordering is represented
by differentiable utility functions and the truncation method of Koopmans(1972b) and Dolmas (1995) do not work because program spaces in continuous time are infinite dimensional even if time horizons are fixed to be

finite.
The purpose of this paper is to present an axiomatic aPproach in acontinuous time framework for representing preference orderings on -spaces
in terms of integral functionals. We show that if preference orderings on
spaces $satis9^{r}$ continuity, separability, sensitivity, substitutability, additivity and lower boundedness, then there exists autility function representing
the preference orderings such that the utility function is an integral functional with an upper semicontinuous integrtd satisfying the growth condition. Moreover, if the preference orderings $satis\Psi$ the continuity with respect
to the weak topology of -spaces, then the integrand is aconcave integrand.
As aresult, TAS utility functions with constant discount rates are obtained.
$L^{P}$$L^{P_{-}}$$L^{p}$

Finitely Additive Representation

2ofbe a measure space with. a countably generated, complete and nonatomic measure of . For each, and a$1\leq p<\infty$ , let
be the set of measurable function from,endowed with the If-normtowith. Sinceis countably generated,
whereis the Euclidean norm of
is a separable Banach space (see Billingsley 1995, Theorem
19.2).
be a charis called a trajectory. Let
An element in$t\inA$$\chi_{A}(t)=0$
and
that is, $\chi_{A}(t)=1$ if
acteristic function ofdenotes a trajectory taking its valotherwise. If is a trajectory, then. Thus, if and are trajectories andues $x(t)$ on and zero on$A\cap B=\emptyset$
is a “patched” trajectory taking its values $x(t)$, then.
a.e. $t\in A$ and $y(t)$ a.e. $t\in B$ , and vanishing on

Let
a set

$\sigma- field$$\mathscr{J}$$(\Omega, \mathscr{J}, \mu)$$\sigma- finite$$\Omega$$\mu$$\mathscr{J}$$\Vert f\Vert_{p}=(\int_{\Omega}|f|^{p}d\mu)^{1/p}$$\int_{\Omega}|f|^{p}d\mu<\infty$$\mathbb{R}^{n}$$\Omega$$f$$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$\mathscr{J}$$\mathbb{R}^{n}$$|\cdot|$$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$L^{p}(\Omega, \mathscr{J}, \mu)\mathbb{R}^{n})$$\chi_{A}$$A\in\not\subset \mathscr{J}$$x$$x\chi_{A}$

$A$

$x$

$\Omega\backslash A$$y$$x\chi_{A}+y\chi_{B}$$\Omega\backslash (A\cup B)$

ofDefinition 2.1. A subset
conditions are satisfied: (i)$\chi$$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$0\in\ovalbox{\tt\small REJECT}^{-}$$x\chi_{A}+y\chi_{B}\in\chi$; (ii) $x,$

is admissible if the following
and $A\cap B=\emptyset$ imply$y\in\ovalbox{\tt\small REJECT}$.

be an admissible set of trajectories. Then $x\chi_{A}\in\chi$ for every$:=\{x\chi_{A}|x\in X\}$ is contained in. A, and henceand$p_{7}efe7$ ence $relation\succ on\sim\chi$ is a complete transitive binary relation on ,S2‘.
Let$x\in\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}^{-}$$A\in \mathscr{J}$$X_{A}$$\chi$

61
We introduce the following axioms on the preference relation.
$\bullet$, the upper contour set
Continuity; For every
are closed inand the lower contour set.
$\{y\in\ovalbox{\tt\small REJECT}|$$x\in\ovalbox{\tt\small REJECT}$$\{y\in\ovalbox{\tt\small REJECT}|x\sim\succ y\}$$y\sim\succ x\}$$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$\bullet$Sensitivity: For every A.
such that$\in \mathscr{J}$with $\mu(A)>0$ , there exist $x,$$y\in\ovalbox{\tt\small REJECT}$$x\chi_{A}\succ y\chi_{A}$$\bullet$with $A\cap B=\emptyset,$for every $z\in\chi$ .

Separability: For every $A,$
$x\chi_{A}+z\chi_{B}\sim\succ y\chi_{A}+z\chi_{B}$$B\in \mathscr{J}$$x\chi_{A}\sim\succ y\chi_{A}$implies

The continuity axiom is a standard condition of the continuity of preference relations on topological spaces. The sensitivity of rules out the
witb $\mu(A)>0$ issituation in which the induced preference relation on
is indifferent. The separabihty of“degenerate” in that every element infor every
if and only if
implies thatby
induces a preference relation onwith $A\cap B=\emptyset$ . Thus,.
Let $I=\{1, \ldots, m\}$ be a finite set of natural numbers andhas a positive measure. Define
be a partition of such that each
is identified with thefor each $i\in I$ . Since every trajectoryand every elementin the product spaceelement, it
is identified with its algebraic sum
is the algebraic sum,S37, where
follows that.
of$\sim\succ$$\chi_{A}$$\ovalbox{\tt\small REJECT}_{A}^{\sim}$$x\chi_{A}+z\chi_{B}\sim\succ y\chi_{A}+z\chi_{B}$$x\chi_{A}\sim\succ y\chi_{A}$$\sim\succ$$\ovalbox{\tt\small REJECT}_{A}$$z\in\ovalbox{\tt\small REJECT}^{-}$$\sim\succ$$restricting\succ to\ovalbox{\tt\small REJECT}_{A}\sim$$\{\Omega_{1}, \ldots, \Omega_{m}\}$$\Omega$$\chi_{i}=$$\Omega_{i}$$x\in\ovalbox{\tt\small REJECT}$$\mathscr{X}_{\Omega_{i}}$$(x\chi_{\Omega_{1}}, \ldots , x\chi_{\Omega_{m}})$$\prod_{i\in I}\ovalbox{\tt\small REJECT}_{i}$$\sum_{i\in I}x_{i}\in\ovalbox{\tt\small REJECT}$$(x_{1}, \ldots , x_{m})\in\prod_{i\in I}\ovalbox{\tt\small REJECT}_{i}$$\ovalbox{\tt\small REJECT}=\prod_{i\in I}\chi_{i}=\sum_{i\in I}$$\ovalbox{\tt\small REJECT}_{1}^{\sim},$$\sum_{i\in I}\ovalbox{\tt\small REJECT}_{i}$$\ldots\ovalbox{\tt\small REJECT}_{m}$is a sepLemma 2.1. Let ,S2‘ be a admissible set of trajectories. Thenis connected and separable
is connected, thenarable metric space. If
for each $i\in I$ .
$\ovalbox{\tt\small REJECT}$$\chi$$\ovalbox{\tt\small REJECT}_{i}$,is a subset of the separable Banach space If
be the projection
is connected. Let
it is also separable. Suppose that, it follows that
. Sinceis continuous and
into
from
is a connected set as the image of the connected set by the continuous
arbitrarily. Notechoosemapping. To show the separability ofis admissible. Then there exists a sequenceis a subset of ,SIY sincethat. Therefore, is theby the separability ofinsuch thatin view ofincluster point of the sequence. Since$P_{7}\cdot oof$$(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$\ovalbox{\tt\small REJECT}$$\chi$$\chi_{i}$$\ovalbox{\tt\small REJECT}$$pr_{i}$$\ovalbox{\tt\small REJECT}_{i}=pr_{i}(\ovalbox{\tt\small REJECT})$$pr_{i}$$\ovalbox{\tt\small REJECT}_{i}$$x\in\ovalbox{\tt\small REJECT}_{i}^{\sim}$$\chi_{i}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}_{i}$$\{x^{\nu}\}$$\ovalbox{\tt\small REJECT}$$x^{\nu}arrow x$$\ovalbox{\tt\small REJECT}^{\sim}$$\{pr_{i}(x^{\nu})\}$

$x$

$x$

$\ovalbox{\tt\small REJECT}_{i}$$pr_{i}(x^{\nu})arrow pr_{i}(x)\square =$.

Suppose that $m\geq 3$ . Let $N$ be an arbitrary subset of I. $Since\succ satisfies\sim$,S27 a preference relation
separability, $\sim\succ induces$ on the product space$\prod_{i\in N}$

62$\sim\succ_{N}$by$(x_{i})_{i\in NN}\succ\sim(y_{i})_{i\in N}\Leftrightarrow^{def}[(x_{i})_{i\in N}, (z_{i})_{i\in I\backslash N}]_{\sim}\succ[(x_{i})_{i\in I}, (z_{i})_{i\in I\backslash N}]\forall z\in\ovalbox{\tt\small REJECT}^{\sim}$

.

(2.1)

of , the preference rela. Thus for every subset. By theis independent of any
onfor each $i\in I$ .
such that
there exist
sensitivity
By Lemma 2.1, we can apply the theorem of Debreu-Gorman (Debreu 1960;
Gorman 1968) to obtain an additive separable utility function representing

We denote$tion\sim\succ_{N}$$I$$(z_{i})_{i\in I\backslash N} \in\prod_{i\in I\backslash N}\ovalbox{\tt\small REJECT}_{i}$$\prod_{i\in N}\ovalbox{\tt\small REJECT}_{i}$$of\succ\sim$$\sim\succ$by$\sim\{i\}\succ$$N$$\sim\succ_{i}$$x_{i},$’$x_{i}\succ iy_{i}$$y_{i}\in\ovalbox{\tt\small REJECT}_{i}^{\wedge}$.

ev

be a connected admissible set of trajectories.
Theorem 2.1. Let
satisfies continuity, separability and sensitivity, then for each $i\in I$ , there
such thatonexists a continuous function$If\succ\sim$$U_{i}$$\ovalbox{\tt\small REJECT}_{i}$$x \succ\sim y\Leftrightarrow\sum_{i\in I}U_{i}(x_{i})\geq\sum_{i\in I}U_{i}(y_{i})$This representation.

$of\succ\sim is$.

unique up to increasing linear tmnsfo rmation of$\sum_{i\in I}U_{i}$Remark 2.1. The general result of Debreu (1960) on the additive separable representation of preference relations on product topological spaces wereextended by Gorman (1968), who demonstrated that the separability axiom(2.1) can be replaced with the weaker condition. The terminologies for theabove axioms are different from those of Debreu (1960) and Gorman (1968).
We follow the usage of the expositive article by Koopmans (1972a). Notethat the requirement $m\geq 3$ is crucial for the additive separable representation. Koopmans (1972a) gave a counter example such that for $m=2$ , every
that
preference relation on a connected separable topological spacesatisfies continuity, separability and sensitivity cannot be represented by anadditive separable utility function !
$\chi_{1}\cross\chi_{2}$Integral Representation

3

We introduce the following axioms on the preference relation.
Substitutability: For every
exists some $y\in\chi$ such that$x\in\ovalbox{\tt\small REJECT}$$\bullet$and$x\sim y\chi_{A}$.

$A\in \mathscr{J}$with $\mu(A)>0$ , there

satisfying $A\cap B=$
Additivity: For every $x,$ $y\in X$ and $A,$ $B,$ $E,$
$E\cap F=\emptyset,$
imply $x\chi_{A}+x\chi_{B}\sim y\chi_{E}+y\chi_{F}$ .
and$F\in \mathscr{J}$$\bullet$$x\chi_{A}\sim y\chi_{E}$$x\chi_{B}\sim y\chi_{F}$63$\bullet$$Lowe7^{\cdot}$

everyboundedness: There exists some$x\in\ovalbox{\tt\small REJECT}$

such that$x_{0}\in\ovalbox{\tt\small REJECT}$$x\sim\succ x_{0}$for.

When maximal elements with respectsubstitutability becomesa somewhat strong requirement because it necessarily implies the existence ofmultiple maximal elements. In particular, if
is convex, then substitutability excludes the strict convexity’ which guarantees a unique maximal element. However, we do not as sume the compactness of , and hence substitutability is not a strong restriction when maximal elements are nonexistent.
The lower boundedness $of\succ excludes\sim$ that a utility function representing
is identically equal to-oo, which is an innocuous requirement.
In essence, additivity implies separability; More precisely, additivity implies the following weaker form of the separability:$to\succ exist\sim\rangle$$\ovalbox{\tt\small REJECT}$$of\succ\sim$$\ovalbox{\tt\small REJECT}$$\sim\succ$$\bullet$Indifferent sepambility: For every$y\chi_{A}$with $A\cap B=\emptyset,$.
implies $x\chi_{A}+z\chi_{B}\sim y\chi_{A}+z\chi_{B}$ for every$A,$ $B\in \mathscr{J}$$x\chi_{A}.\sim$$z\in\ovalbox{\tt\small REJECT}$To demonstrate this claim, let $x,y,$and $A\cap B=\emptyset$ . Suppose that. Define $v=x\chi_{A}+z\chi_{B}$ and $w=y\chi_{A}+z\chi_{B}$ . Since $v\chi_{A}=x\chi_{A}$ ,$y\chi_{A}=w\chi_{A}$ and $v\chi_{B}=w\chi_{B}$ by construction, we haveand
. The additivity $of\succ\sim impliesv\chi_{A}+v\chi_{B}\sim w\chi_{A}+w\chi_{B}$ , whichis equivalent to $x\chi_{A}+z\chi_{B}\sim y\chi_{A}+z\chi_{B}$ , from which indifferent separability
follows.
$z\in\ovalbox{\tt\small REJECT}^{-}$$x\chi_{A}\sim y\chi_{A}$$v\chi_{A}\sim w\chi_{A}$$v\chi_{B}\sim w\chi_{B}$Theorem 3.1. Let
be an admissible set of tmjectones that is connected
and closed in If.
satisfies continuity, sepambility, sensitivity, substitutability, additivity and lower boundedness, then there vists
a unique extended real-valued func tion :with thefollowing properties;$\chi$$(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$If\succ\sim$$e$$\Omega\cross \mathbb{R}^{n}arrow \mathbb{R}\cup\{-\infty\}$$f$(i) $f(t, \cdot)$ is upper semicontinuous on
surable on for every.
$\Omega$.

$t\in\Omega$(iii) For every.

$t\in\Omega$

and $f(\cdot, v)$ is mea-$v\in \mathbb{R}^{n}$(ii) There exist some$\beta|v|^{p}a.e$$\mathbb{R}^{n}a.e$$\alpha\in L^{1}(\Omega, \mathscr{J}, \mu)$for every$v\in \mathbb{R}^{n}$andsuch that $f(t, v)\leq\alpha(t)+$.

if and only if$A\in \mathscr{J}_{\rangle}x\chi_{A}\sim\succ y\chi_{A}$$\int_{A}f(t, y(t))d\mu(t)$$\beta\geq 0$.

$\int_{A}f(t, x(t))d\mu(t)\geq$

A function :is a normal integmnd $if-g$ satisfies condition (i) of Theorem 3.1. Thus condition (i) states that $-f$ is anormal integrand, which we say that is upper semicontinuous integrandin the sequel. Condition (ii) is called growth condition in optimal control$g$$\Omega\cross \mathbb{R}^{n}arrow \mathbb{R}\cup\{+\infty\}$

$f$

64
theory. The meaning of the uniqueness of is as follows: If is another upper semicontinuous integrand satisfying the conditions of Theorem 3.1, then.
$g(t, v)=f(t, v)$ a.e.
for every$f$$t\in\Omega$$g$$v\in \mathbb{R}^{n}$

Proof of Theorem 3.1. By virtue of Theorem 2.1, there exists acontinu-

with the form $U(x)=$which representsonous utility function. Without loss of generality one may assume that $U_{i}(0)=0$, that is,for each $i\in I.$ We shall show that is disjointly additive on$U$$\ovalbox{\tt\small REJECT}$$\sim\succ$$\sum_{i\in I}U_{i}(x_{i})$$U$$\ovalbox{\tt\small REJECT}$

imply $U(x\chi_{A}+y\chi_{B})=U(x\chi_{A})+U(y\chi_{B})$ .
with $A\cap B=\emptyset$ . Let
and $A,$To this end, take any$E,$for some partition
andbe such that$\{J, K\}$ of $N$ , and let $E$ and $F$ have positive measlre. Then $E$ and $F$ aredisjoint. By the substitutability’there exist $v\in\chi$ such thatis admissible,. Define $y=u\chi_{E}+v\chi_{F}$ . Since
and. Note that $y\chi_{E}=u\chi_{E}$ and $y\chi_{F}=v\chi_{F}$ . We thus have
we have. By the additivity
and’we have $x\chi_{A}+x\chi_{B}\sim$for each $i\in N$ . Then $E\cup F$
and. Defineis decomposed into an -tuple of pairwise disjoints sets
andfor $j\in J.$ Since
for $k\in K$ andwith. Thus, we, and similarly, we havewith $y\chi_{E_{k}}=0$ for $k\in K$ and.
have
and $y\chi_{F_{j}}=0$ for $j\in J.$ Therefore,$A\cap B=\emptyset$

and

$x,$

$y\in\ovalbox{\tt\small REJECT}$$B\in \mathscr{J}$$x\in\ovalbox{\tt\small REJECT}$$F\in \mathscr{J}$$F \subset\bigcup_{k\in K}\Omega_{k}$$E \subset\bigcup_{j\in J}\Omega_{j}$$of\succ\sim$$u,$$x\chi_{A}\sim$$\ovalbox{\tt\small REJECT}$$x\chi_{B}\sim v\chi_{F}$$u\chi_{E}$$y\in\ovalbox{\tt\small REJECT}$$x\chi_{A}\sim y\chi_{E}$$of\succ\sim$$x\chi_{B}\sim y\chi_{F}$$F_{i}=F\cap\Omega_{i}$$E_{i}=E\cap\Omega_{i}$$y\chi_{E}+y\chi_{F}$$\{(E_{j})_{i\in J}, (F_{k})_{k\in K}\}$$n$$E_{k}=\emptyset$$F_{j}=\emptyset$$y\chi_{E}:=(x\chi_{E})\chi_{\Omega_{i}}$$y\chi_{E}\in\ovalbox{\tt\small REJECT}$$y\chi_{F_{1}}\in\ovalbox{\tt\small REJECT}_{i}$$y\chi_{E_{l}}\in\ovalbox{\tt\small REJECT}_{i}$$y \chi_{E}=(y\chi_{E_{1}}, \ldots, y\chi_{E_{n}})\in\prod_{i\in N}\ovalbox{\tt\small REJECT}_{i}$$y \chi_{F}=(y\chi_{F_{1}}, \ldots, y\chi_{F_{n}})\in\prod_{i\in N}\ovalbox{\tt\small REJECT}_{i}$$U(x \chi_{A})=U(y\chi_{E})=\sum_{j\in J}U_{j}(y\chi_{E_{j}}),$ $U(x \chi_{B})=U(y\chi_{F})=\sum_{k\in K}U_{k}(y\chi_{F_{k}})$and $U(x \chi_{A}+y\chi_{B})=U(y\chi_{E}+y\chi_{F})=\sum_{j\in J}U_{j}(y\chi_{E_{j}})+\sum_{k\in K}U_{k}(y\chi_{F_{k}})$ ,and hence $U(x\chi_{A}+x\chi_{B})=U(x\chi_{A})+U(x\chi_{B})$ . Rom this condition, we
can derive the disjoint additivity of U. To demonstrate this, let
sinceand $A\cap B=\emptyset$ . Define $z=x\chi_{A}+y\chi_{B}$ . We then haveis admissible, and $z\chi_{A}+z\chi_{B}=x\chi_{A}+y\chi_{B}$ by construction. Thus,$x,$$y\in\ovalbox{\tt\small REJECT}$$z\in\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$U(x\chi_{A}+y\chi_{B})=U(z\chi_{A}+z\chi_{B})=U(z\chi_{A})+U(z\chi_{B})=U(x\chi_{A})+U(y\chi_{B})$Define the functional$\Phi$:$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})\cross \mathscr{J}arrow \mathbb{R}\cup\{-\infty\}$.

by$\Phi(x, A)=\{\begin{array}{ll}U(x\chi_{A}) if x\in\chi-\infty otherwise.\end{array}$

By construction,$\bullet$$\bullet$$\bullet$$\Phi(\cdot, \Omega)$$\Phi$

satisfies the following properties:

is upper semicontinuous on

is finitely additive on$L^{p}(\Omega, .\mathscr{J}, \mu;\mathbb{R}^{n})$, that is, $A,$$\Phi(x, A\cup B)=\Phi(x, A)+\Phi(x, B)$ for every$\Phi$$\Phi$$\mathscr{J}$is local on ,9, that is, $x,$$\Phi(x, A)=\Phi(y, A)$.

$B\in \mathscr{J}$.

and $A\cap B=\emptyset$ imply$x\in L^{p}(\Omega, ff \mu;\mathbb{R}^{n})$$y\in L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$.

and $x\chi_{A}=y\chi_{A}$ imply

65$-\infty<\Phi(x_{0}, A)$$\bullet$for every$A\in \mathscr{J}$.

Then by the representation tbeorem of Buttazzo and Dal Maso (1983), there
exists a unique upper semicontinuous integrand :with the following properties:$\Omega\cross \mathbb{R}^{n}arrow \mathbb{R}\cup\{-\infty\}$$f$(a) There exist somea.e.
for every$\alpha\in L^{1}(\Omega, \mathscr{J}, \mu)$$t\in\Omega$$\beta|v|^{p}$$v\in \mathbb{R}^{n}$and$\beta\geq 0$such that $f(t, v)\leq\alpha(t)+$.

(b) $\Phi(x, A)=\int_{A}f(t, x(t))d\mu(t)+\Phi(x_{0}, A)$ for every$A\in \mathscr{J}$$x\in L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$.

andConditions (i) and (ii) of the theorem follows from this result. Since an ad$hom$ condiditive constant does not affect the representation’ it follows
tion (b) that
if and only if $\int_{A}f(t, x(t))d\mu(t)\geq\int_{A}f(t, y(t))d\mu(t)$ ,which shows condition (iii) in the above theorem.
$of\succ\sim$$x\chi_{A}\sim\succ y\chi_{A}$$\square$Example 3.1. Suppose that the admissible set ,92“ is a positive cone ofgiven by$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$\ovalbox{\tt\small REJECT}=${$x\in L^{p}(\Omega,$ $\mathscr{J},$ $\mu;\mathbb{R}^{n})|x(t)\geq 0$a.e.

$t\in\Omega$}.

Let be a continuous linear functional on the Banach spacesuch thatand ker $x^{*}=\{0\}$ , where the duality
for each. Suppose
relation is denoted by $x^{*}(x)=\langle x,$ ) for each
if and, that is,thattois represented by the restriction of
only if. It is evident $that\succ satisfies\sim$ continuity, separability. Since $x\neq 0$
is the origin of
and additivity. The lower bound of
implies, for every
with positive measure, it follows thatby choosingwith $x(t)>0$ on . Thus, ): satisfies
and withsensitivity. To show the substitutability
take any
positive measure. Let $y\in\chi$ be such that $y(t)>0$ on . We then have) defined\langle) $>0$ . Consider the continuous increasing function onsuch that. Then there exists someby. This, we have. Sinceis a positive cone and.
demonstrates the substitutability
Therefore, by Theorem 3.1, there exists a unique upper semicontinuousfor every $x\in$function $f(t, )$ on
such that. On the other hand, the Riesz representation theorem implies that there
such thatwithexists a uniqueis the inner product, wherefor every
for everyof . By the uniqueness of , we obtain$x^{*}$$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$\langle x, x^{*}\rangle\geq 0$$x\in\ovalbox{\tt\small REJECT}^{-}$$x\in L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$x^{*}$$x^{*}$$\sim\succ$$x\sim\succ y$$\ovalbox{\tt\small REJECT}$$\langle x, x^{*}\rangle\geq\langle y, x^{*}\rangle$$\ovalbox{\tt\small REJECT}$$\sim\succ$$A\in \mathscr{J}$$\langle x, x^{*}\rangle>0$$\langle x\chi_{A}, x^{*}\rangle>0$$A$$x\in\ovalbox{\tt\small REJECT}$$of\succ\sim$$A$$x\in\ovalbox{\tt\small REJECT}$’$A$$y\chi_{A},$$[0, \infty$$x^{*}$$\lambda\geq 0$$\lambda\mapsto\langle\lambda y\chi_{A}, x^{*}\rangle$$y\chi_{A}\in\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\langle x, x^{*}\rangle$$\langle\lambda y\chi_{A}, x^{*}\rangle=$$\lambda y\chi_{A}\in\ovalbox{\tt\small REJECT}$$of\succ\sim$$\langle x, x^{*}\rangle=\int_{\Omega}f(t, x(t))d\mu(t)$$\mathbb{R}^{n}$$\ovalbox{\tt\small REJECT}$$\varphi\in L^{q}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$\int_{\Omega}\langle x(t), \varphi(t)\rangle d\mu(t)$$f$$\mathbb{R}^{n}$a.e.

$t\in\Omega$$x\in\ovalbox{\tt\small REJECT}$.

$\frac{1}{p}+\frac{1}{q}=1$$\langle x, x^{*}\rangle=$$\langle x(t), \varphi(t)\rangle$$f(t, v)=\langle v, \varphi(t)\rangle$$v\in \mathbb{R}^{n}$

66
Convexity of Preferences
We introduce the convexity axiom of the preferences,
Convexity; Letbe a convex admissible set. For every
is convex.
upper contour set$\ovalbox{\tt\small REJECT}$$\bullet$$x\in\ovalbox{\tt\small REJECT},$$\cdot the$$\{y\in\ovalbox{\tt\small REJECT}|y\sim\succ x\}$Theorem 3.2. Suppose
satisfies the axioms in Theo 7 $em3.1$ replacing the stmng continuity with the weak continuity of the weak topology of. Then the integmnd in Theorem 3.1 is a concave integmnd,
and $hence\succ\sim is$ convex.
$that\succ\sim$$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$that the preference relation is represented by a weakly continuous utility function. Thus, the functional
defined in the proof of Theorem 3.1 is weakly upper semicontinuous on. The representation theorem of Buttazzo and Dal Maso
(1983) guarantees the concavity of the integrand $f(t, \cdot)$ .

Proof. The weak continuity$of\sim\succ implies$$\Phi$$L^{p}(\Omega, \mathscr{J}, \mu;\mathbb{R}^{n})$$\square$not assumed explicitly, the weak continuity
Even if the convexity$of\sim\succ necessarily$ implies the convexity$of\succ is\sim$$of\succ!\sim$

Stationarity of Preferences
Let $X$ be a subset ofeach $v\in X$ and$\mathbb{R}^{n}$$A\in \mathscr{J}$such that $x(t)\in X$ for every
with $\mu(A)>0$ , we say that$x\in\ovalbox{\tt\small REJECT}$$v\chi_{A}$a.e.

$t\in\Omega$. For

is a locally constanttrajectory in $X$ .

be an admissible set that contains every locally
Stationarity: Letwith $A\cap B=\emptyset$ ,constant trajectory in $X$ . For every $A,$for every $v\in X$ .
$\mu(A)=\mu(B)$ implies$\ovalbox{\tt\small REJECT}$$\bullet$$B\in \mathscr{J}$$v\chi_{A}\sim v\chi_{B}$be the Borel
Theorem 3.3. Let
of $\Omega=[0, \infty$ ) and be a regular
Borel measure. Suppose
satisfies the arioms in Theorem 3.1. Fur$thermo7e,$satisfies stationa , then the integrand is independent of$t\in\Omega$that is, there exists a unique upper semicontinuous functionon$g:Xarrow \mathbb{R}\cup\{-\infty\}$ such that $f(t,v)=g(v)a.e$ .
for $even/v\in X$ .
$\sigma- field$$\mathscr{J}$$\mu$$that\succ\sim$$if\succ\sim$$f$$r\dot{\tau}ty$$X_{f}$$t\in\Omega$with $s<t$ be arbitrary, and let
and $I_{\epsilon’}(t)=(t-\epsilon’, t+\epsilon’)$ be disjoint open intervals with
we have. By the stationarity
for every $v\in X$ and hence. Let$P_{7}oof$$s,$$I_{\epsilon}(s)=(s-\epsilon, s+\epsilon)\cap(O, \infty)$$t\in\Omega$$\mu(I_{\epsilon}(s))=\mu(I_{\epsilon’}(t))$$of\sim\succ$$\epsilon,\epsilon’>0$

and,$v\chi_{I_{\epsilon}(s)}\sim v\chi_{I_{\epsilon(t)}}$$\int_{I_{e}(s)}f(\tau, v)d\mu(\tau)=\int_{I_{e},(\epsilon)}f(\tau, v)d\mu(\tau)$

for every

67
Thus, by the Lebesgue-Besicovitch differentiation theorem (Evans
and Gariepy, 1992, Theorem 1.7.1), we have$v\in X$ .

$f(t, v)= \lim_{\epsilonarrow 0}\frac{1}{\mu(I_{\epsilon}(s))}\int_{I_{\epsilon}(s)}f(\tau, v)d\mu(\tau)$$= \lim_{\epsilonarrow 0}\frac{1}{\mu(I_{\epsilon(t)})}\int_{I_{e’}(t)}f(\tau, v)d\mu(\tau)=f(s, v)$

Therefore, $f(t, v)$ is constant a.e.

4$t\in\Omega$.

for arbitrarily fixed $v\in X$ .

口TAS Representation with Myopia

of . Let be a Lebesgue
Let $\Omega=[0, \infty$ ) and ,9 be the Borel
be awith positive values and let
integrable continuous function on
given by $\mu_{\rho}(A)=$nonatomic finite measure of a measurable space.
for$\sigma- field$$\Omega$$\rho$$\Omega$$\mu_{\rho}$$(\Omega, \mathscr{J})$$A\in \mathscr{J}$$\int_{A}\rho(t)dt$

Recursive Utility
with
Suppose that the admissible set of trajectories is$1\leq p<\infty$ . A preferenceis given by the following recursive
integral functional$\ovalbox{\tt\small REJECT}=L^{p}(\Omega, \mathscr{J}, \mu_{\rho};\mathbb{R}_{+}^{n})$$relation\succ on\ovalbox{\tt\small REJECT}\sim$$\forall x,$$y\in\ovalbox{\tt\small REJECT}$:$x \succ\sim y\Leftrightarrow\int_{\Omega}f(t, x(t))F(t,$$\int_{0}^{s}r(s, x(s))ds)dt$(4.1)$\geq\int_{\Omega}f(t, y(t))F(t,$ $\int_{0}^{s}r(s, y(s))ds)dt$

where andfunction on
$f$

$r$

are measurable functions on.

$\Omega\cross \mathbb{R}_{+}^{n}$

and

$F$

,

is a measurable$\Omega\cross \mathbb{R}$Assumption 4.1. (i) $f(t, \cdot)$ is continuous on.
is measurable on for every$\Omega$(ii) There exist some.

$t\in\Omega$

and $f(\cdot, v)$$v\in \mathbb{R}_{+}^{n}$$\alpha\in L^{1}(\Omega, \mathscr{J}, \mu_{\rho})$$|f(t, v)|\leq\alpha(t)+a|v|^{p}$(iii) $F(t, \cdot)$ is continuous on.
for every$\mathbb{R}_{+}^{n}a.e$$\mathbb{R}$a.e.

and $a>0$ such thatfor every$t\in\Omega$$(t, v)\in\Omega\cross \mathbb{R}_{+}^{n}$.

and $F(\cdot, z)$ is measurable on$\Omega$

is measurable on$\Omega$$z\in \mathbb{R}$(iv) $r(t, \cdot)$ is continuous on.
for every$v\in \mathbb{R}_{+}^{n}$$\mathbb{R}_{+}^{n}a.e$.

$t\in\Omega$and$r(\cdot, v)$

68(v) There exists somesuch that$\beta\in L_{1oc}^{1}(\Omega, \mathscr{J}, \mu_{\rho})$a.e.

$|r(t, v)|\leq\beta(t)$for every$t\in\Omega$$v\in \mathbb{R}_{+}^{n}$

and

a.e.

$|F(t,$ $\int_{0}^{t}\beta(s)ds)|\leq\rho(t)$(vi) $f(t, O)F(t, \int_{0}^{t}r(s, O)ds)=0$ a.e.

Assumption 4.2.
(ii)

$a.e.t\in F(t,z)\geq 0\Omega$a.e.

$t\in\Omega$(i) $f(t, x)\geq 0$ a.e.
$t\in\Omega$for every$t\in\Omega$.

.

for every$t\in\Omega$$z\in \mathbb{R}$$x\in \mathbb{R}_{+}^{n}$.

and $F(t, \cdot)$ is decreasing on$\mathbb{R}$.

(iii) $f(t, \cdot)F(t, \cdot)$ is concave on(iv) $r(t, \cdot)$ is concave on$\mathbb{R}_{+}^{n}\cross \mathbb{R}$$\mathbb{R}_{+}^{n}a.e$.

$t\in\Omega$a.e.

$t\in\Omega$.

.

It is easy to verify that by growth conditions (ii) and (v) of Assumption
4.1, we have$|f(t, x(t))F(t,$ $\int_{0}^{t}r(s, x(s))ds)|\leq(\alpha(t)+a|x(t)|^{p})\rho(t)$and the right-hand side of the above inequality isa.e.
for every. Thus, the preference relation
Lebesgue integrable over for every
given above is well defined.
By the similar argument developed by Sagara (2007), under Assumption
4.1, one can show the continuity of the recursive integral functional$x\in\ovalbox{\tt\small REJECT}$$t\in\Omega$$\Omega$$x\in\ovalbox{\tt\small REJECT}$$x rightarrow\int_{\Omega}f(t, x(t))F(t,$ $\int_{0}^{t}r(s, x(s))ds)dt$

satisfied. It is easy to verify
on , and hence the continuity axiom
that separability, additivity, indifferent separability are satisfied. If, in ad$of\succ is\sim$$\ovalbox{\tt\small REJECT}$

dition, Assumption 4.2 is satisfied, then the recursive integral functional isconcave on$\ovalbox{\tt\small REJECT}$.

Theorem 4.1 (Sagara). $Let\succ\sim be$ a refe 7 ence 7 elation on
defined by(4.1). Suppose that Assumption 4.1 is satisfied. Then there exists a uniquesuch thatupper semicontinuous integmnd on$\ovalbox{\tt\small REJECT}$

$p$

$g$

$\forall x,$$\Omega\cross \mathbb{R}^{n}$$y\in X$ : $x \succ\sim y\Leftrightarrow\int_{\Omega}g(t, x(t))\rho(t)dt\geq\int_{\Omega}g(t, y(t))\rho(t)dt$ .

If, $rnoreove7^{\cdot}$, Assumption 4.2 is satisfied, then$g$is a concave integrand.

There is a degree of freedom for the choice of . By choosing $\rho(t)=$$\exp(-\rho t)$ , one obtains a TAS utility function with exponential discounting.
$\rho$69

TAS Utility
We denote by
the set of essentially bounded functions on
towith respect to the Lebesgue measure. In view of the inclusion$L^{\infty}(\Omega, \mathscr{J};\mathbb{R}^{n})$$\Omega$$\mathbb{R}^{n}$$L^{\infty}(\Omega, \mathscr{J};\mathbb{R}^{n})\subset L^{\infty}(\Omega, \mathscr{J}, \mu_{\rho};\mathbb{R}^{n})\subset L^{p}(\Omega, \mathscr{J}, \mu_{\rho};\mathbb{R}^{n})$for $p\geq 1$ ,it is legitimate to endow
with the relative If-norm topology, instead of the essential $sup(ess. \sup)$ norm topolfrom
ogy of. By changing the $ess$ . sup norm of
to thenorm, we can deal withas an admissible set of trajectories in.
The following main result of this paper strengthens Theorem 4.1 under
the alternative hypotheses on the preference relation.
$L^{\infty}(\Omega, \mathscr{J};\mathbb{R}^{n})$$L^{p}(\Omega, \mathscr{J}, \mu_{\rho};\mathbb{R}^{n})$$L^{\infty}$$L^{p_{-}}$$L^{\infty}(\Omega, \mathscr{J};\mathbb{R}^{n})$$L^{\infty}(\Omega, \mathscr{J};\mathbb{R}^{n})$$L^{p}(\Omega, \mathscr{J}, \mu_{\rho};\mathbb{R}^{n})$be an admissible set of tmjectot ies closed and convex
satisfies continuity, sensiti vity, sepam bility, substitutability, additivity, lower boundedness, $stationari\cdot ty$ , then there exists a
unique upper semicontinuous integmnd onsuch that

Theorem 4.2. Letin.

$\ovalbox{\tt\small REJECT}$$L^{p}(\Omega, \mathscr{J}, \mu_{\rho};\mathbb{R}^{n})$$If\succ\sim$$g$$\forall x,$$y\in\ovalbox{\tt\small REJECT}$If, moreover,$\sim\succ$:$\mathbb{R}^{n}$$x_{\sim} \succ y\Leftrightarrow\int_{\Omega}g(x(t))\rho(t)dt\geq\int_{\Omega}g(y(t))\rho(t)dt$

satisfies convexity, then$g$.

is a concave integrand.

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