Technical Memorandum Number 724F

Discounting and Risk Neutrality

by

Matthew J. Sobel

Revised September 2006

Department of Operations Weatherhead School of Management

Case Western Reserve University 330 Peter B Lewis Building

Cleveland, Ohio 44106

DISCOUNTING AND RISK NEUTRALITY‡

Matthew J. Sobel**

September 1990; Revision September 15, 2006

Abstract

Let be a binary relation in which is a real vector space of vector-valued discrete-timeÐ ¤ ß Z Ñ Z

stochastic processes with as the sequence of zero vectors. If satisfies four axioms, then! Ð ¤ ß Z Ñ

there are unique discount factors such that preferences regarding stochastic processes induce prefences

among present value random vectors. Three of the axioms are familiar: weak ordering, continuity, and

non-triviality. The fourth axiom, , is implies where isdecomposition \ ] ¤ Ð £ Ñ \ ¤ Ð £ Ñ ]! !

the sequence of zero vectors. Also, if preferences satisfy the four axioms then the following properties

are equivalent: the converse of decomposition, the existence of a felicity function on the set of random

vectors, and risk neutrality. In this limited sense, discounting implies risk neutrality.

JEL

D81, D99, G12

*In honor of Martin Shubik and in memory of Robert Rosenthal.**Department of Operations, Weatherhead School of Management, Case Western Reserve University,Cleveland, OH 44106-7235.

Samuelson (1937) proposed the discounted utility model in a deterministic framework which was

subsequently axiomatized by Koopmans (1960), Williams and Nasser (1966), and others. An

axiomatic foundation for discounting in a stochastic setting, the subject of this paper, addresses the

basic properties of intertemporal preferences which yield utility functions having particular structures.

Since the discounted utility model is additively separable, the literature on separability is relevant to

this issue (Blackorby, Primont, and Russell (1998)). In spite of this considerable attention,

inconsistencies abound in professional practice and in research literatures.

Under a strong form of Koopmans' axioms, there exists an utility function and aintraperiod ?

discount factor such that, for all deterministic scalar sequences ( , , ...) and ( ," − Ð!ß "Ñ B œ B B C œ C" # "

C#, ...),

(1) is weakly preferred to ( ) ( )B C Í ? B   ? C! !> >

>" >"> >" "

If and were replaced by stochastic processes ( , , ...) and ( , , ...), then each sideB C \ œ \ \ ] œ ] ]" # " #

of the inequality would be a random variable. Would a combination of Koopmans' axioms and those

from utility theory von Neumann and Morgenstern (1944) for example, imply the existence of anß

interperiod utility function such that@

(2) is weakly preferred to [ ( )] [ )]\ ] Í I @ ? \   I @ ?Ð]Š ‹ Š ‹! !> >

>" >"> >" "

(where denotes expected value)?I

In the axiomatization of preferences among scalar sequences by Williams and Nassar (1966), is?

the identity function. So

(3) is weakly preferred to B C Í B   C! !> >

>" >"> >" "

Would a combination of axioms from Williams and Nassar and utility theory lead to

(4) is weakly preferred to [ ( )] [ ( )]\ ] Í I @ \   I @ ]! !> >

>" >"> >" "

The preference model in numerous research fields is

(5) is weakly preferred to ) )\ ] Í IÒ ?Ð\ Ó   IÒ ?Ð] Ó! !> >

>" >"> > " "

2

for an intraperiod utility function . Of course this is (2) with risk neutrality and risk sensitivity,?

respectively, concerning interperiod utility and intraperiod utility. An elementary ordering that is

widely used in professional practice and several research fields is

(6) is weakly preferred to ) )\ ] Í IÐ \   IÐ ]! !> >

>" >"> > " "

Of course this is (2) with risk-neutral interperiod and intraperiod utility functions, and it is (5) with a

risk-neutral intraperiod utility function.

Alternative decision rules are compared via (5) and (6) in diverse areas such as accounting

[Moriarity and Allen (1984)], advertising [Monahan (1983)], banking [Shubik and Sobel (1992)],

biology [McNamara (1990), Mendelssohn (1979)], capital accumulation [Sethi (1998)], energy

[Murphy, Toman and Weiss (1989)], engineering [Yeh (1985)], finance [Altug and Labadie (1994),

Cochrane (2000), Duffie (2001), Sharpe (1985), Ho and Lee (1986)], game theory [Shapley (1953)],

production [Arrow, Karlin and Scarf (1962)], replacement [Jorgenson, McCall and Radner (1967)],

social psychology [Shubik (1970)], and technological change [Balcer and Lippman (1984)].

Can (2), (4), (5), and (6) be reconciled? Do successively stronger assumptions underlie each of (2),

(4), and (6) and (2), (5), and (6)? Researchers utilize (5) instead of (6) to investigate the effects of risk

sensitivity. So it is not interesting that (2), (4), (5), and (6) are equivalent if the decision maker is risk

neutral.

There is a fundamental difficulty with using (5) to investigate the effects of risk sensitivity. When

axioms that imply the existence of discount factors (Theorem 1 in §2) are augmented with the

existence of an period utility function, then that function is linear (Theorem 2 in §3). Professorintra

James E. Smith, Duke University, notes that (4) implies that an interperiod utility function is an

intraperiod utility function in a single period mdel. So it too must be linear. In this limited sense,

discounting implies risk neutrality.

Miyamoto and Wakker (1996) unify much of the literature on multiattribute preference orderings

and utility functions that is reviewed by Dyer and Sarin (1979), Farquhar (1977), Fishburn (1978), and

Keeney and Raiffa (1976). Wakker (1993) has axioms implying that there are positive numbers ""ß

"#ß ÞÞÞ <Ð † Ñ \ ]and a real-valued function of random variables such that is weakly preferred to if and

3

only if where . Let be a degenerate random variable that takesYÐ\Ñ   YÐ] Ñ YÐ\Ñ œ <Ð\ Ñ HD "> > > +

the value with probability one, and let be the real-valued function on the reals such that+ A

AÐ Ñ œ <ÐH Ñ <a . The linearity of (in probabilities in the sense of (11) in §3) implies+

Y Ð\Ñ œ IÒ AÐ\ ÑÓÞ AÐ † ÑD "> > > This paper shows that is linear ( , risk neutral) under weak sufficienti.e.

conditions for additive separability. This result diminishes the justification for (5).

Literatures which depend on (5), such as asset pricing and, more generally, theoretical financial

economics, are left in an ambiguous position. However, individuals exhibit sensitivity to riskdo

(gambling, insurance, etc.); so this paper casts further doubt on the descriptive validity of conclusions

predicated on expected present value. Experimental work challenges expected utility theory and

comparisons of deterministic temporal sequences via present values [Frederick, Loewenstein, and

O'Donoghue (2003)].

A relatively small body of work provides axiomatic justifications for comparing deterministic

sequences via their present values. Koopmans (1960) and Koopmans, Diamond, and Williamson

(1964) study orderings of countable sequences of vectors of real numbers. They postulate the

existence of an interperiod utility function such that deterministic sequence is@Ð † Ñ B œ B >

weakly preferred to if and only if ( ) ( ). They obtain sufficient conditions for theC œ C @ B   @ C>

existence of ( ) and an intraperiod utility function such that" ‘ ‘− !ß " ? À ÄQ

@ B œ ?ÐB Ñ( ) . Koopmans (1972) specifies conditions on a binary relation which imply the!_>œ"

>">"

existence of with the properties assumed in the earlier studies.@Ð † Ñ

Williams and Nassar (1966) obtain sufficient conditions for a weak ordering on a real vector_¤

space of finite sequences ( ,..., ) of real numbers to have the following property: there existsB œ B B" X

" "− " B ¤ C B C  (0, ) such that if and only if ( ) 0._ !X>œ"

>"> >

Lancaster (1963), Fishburn (1970), and others cited by Fishburn and Rubinstein (1982) and

Frederick, Loewenstein, and O'Donogue (2003) investigate closely related deterministic models of time

preference and impatience. Epstein (1983) studies a stochastic model that relates time preference and

risk preference. He specifies necessary and sufficient conditions for a binary relation on the set of_¤

real-valued discrete-time stochastic processes to have the following property. Let ( , ,...) andB œ B B" #

4

C œ C C( , ,...) be degenerate ( constant with probability one) real-valued stochastic processes." # i.e.,

There exists (0, ) and such that if and only if _" ‘ ‘ "− " ? À Ä B ¤ C ?ÐB Ñ!_>"

>">

  ?ÐC Ñ ?!_>"

>">" . Technical differences aside, this paper presents conditions under which is affine.

The axioms in this paper are briefly compared in §1 with the assumptions made by Williams and

Nassar, Koopmans, Epstein, and Miyamoto and Wakker.

Meyer (1976) uses the same rationale that leads from (1) to (2). He develops a cardinal comparison

of alternative sample paths which are outcomes in the sample space, say of ( , , ...) andB œ B B" #

C œ C C( , , ...), and then, observing that the cardinal measures are random variables, he applies" #

expected utility theory. He implicitly assumes orthogonality of the axioms underlying time preference

and risk preference. The present paper answers a question addressed to the author by Robert

Rosenthal (1987) who wondered if the two axiom systems might be related. They are indeed related.

It is not clear how the previously cited studies and this paper are are related to investigations of

consistency in intertemporal choice preference structures. See Kreps and Porteus (1979), Johnson and

Donaldson (1985), Machina (1989), and their references.

Section 1 presents notation and discusses axioms. Section 2 has sufficient conditions for

discounting, , four axioms that imply that comparisons between vector-valued stochastic processesi.e.

correspond to comparisons between their present values (which are random vectors) A further axiomÞ

yields a vector of familiar geometric discount factors. Three of the axioms are familiar: weak ordering,

continuity, and non-triviality. Instead of an independence or Archimedean axiom, the fourth is

decomposition, namely implies where is the sequence of zero\ ] ¤ Ð £ Ñ \ ¤ Ð £ Ñ ]! !

vectors. Section 3 shows that if preferences satisfy the four axioms then the following properties are

equivalent: the converse of decomposition, the existence of a felicity function, and risk neutrality.

In §4, a combination of results in earlier sections with some in Koopmans (1972) yields sufficient

conditions for a scalar discount factor to generate vectors of geometric discount factors. Section 5

briefly examines relationships among alternative continuity axioms and definitions of impatience.

Section 6 sketches the implications of §3 for attribute decomposition in random vectors.

5

1. Notation and Axioms

Fix a probability space ( , ,P). Let and be respective sets of positive integers and countableH Y M Z

stochastic sequences ,... defined on ( , ,P) with for all .\ œ Ð\ ß\ Ñ \ Ð Ñ − Ð>ß Ñ − M ‚" # >QH Y = ‘ = H

Let denote the - component of . If , , and , let and\ 7 \ \ − Z ] − Z , − \ ] − Z7> >th ‘

,\ − Z Ð>ß Ñ − M ‚ \ ] ,\ have values at given by ( ) ( ) and ( ). Let be the zero vector in= H = = = )> > >

‘ ) )Q , and let ( , ,...) . So is a real vector space with zero element . ! !œ − Z Z

Let be the - unit vector in and let be the sequence of vectors except that the/ 7 / − Z7 7>Qth ‘ ) !

t- vector is . For ( ) and ( ) let denote .th / ? œ ? − œ − ? † ?7 3 3 3 3Q Q Q

3œ"‘ 8 8 ‘ 8 8! Let be a binary relation on , write if and , and write if ¤ Z \ µ ] \ ¤ ] ] ¤ \ \ ¢ ] \ ¤ ]

but not . The following notations are written interchangeably: and , and] ¤ \ \ ¤ ] ] £ \ \ µ ]

] µ \ \ ¢ ] ] ¡ \, and and . A (sometimes called a ) is a completeweak ordering strong ordering

transitive binary relation.

Let be the set of -valued random vectors. For , let ( , ) denote ( , , ,...). ThenW G − W G − Z G‘ ) )Q !

( , ) induces the following binary relation on : if and only if ( , ) ( , ). Let_ _¤ Z ¦ W E ¦ F E ¤ F! !

[ ¸ ^ [ ¦ ^ ^ ¦ [ [ ¦ ^ [ ¦ ^ ^ ¦ [ denote and and let denote but not . Notations_ _ _ _

employed interchangeably include and , and , and and_ _[ ¦ ^ ^ ¥ [ [ ¸ ^ ^ ¸ [ [ ¦ ^

^ ¥ [ E − W IÐEÑ −. For , let denote the vector of expectations of components (when the‘Q

expectations exist).

Axioms

In the following axioms, , , and\ œ Ð\ ß\ ß ÞÞÞÑ − Z ] œ Ð] ß ] ß ÞÞÞÑ − Z ß > − 7 œ "ß ÞÞÞßQ" # " # Mß

- 0.

(A1) Transitivity and rationality: weakly orders¤ Z

(A2) Decomposition: implies\ ] ¤ Ð £ Ñ \ ¤ Ð £ Ñ ]!

(A3) { : } is closedContinuity: ! !\ ] ¤ Ð £ Ñ !

(A4) More is better: / ¢7> !

(A5) ( , ,...) ( , , ,...) Stationarity: implies\ \ µ \ \ µ" # " #! !)

(A6) ( )Sooner is better: \ / / ¢ \- 7> 7ß>"

6

The first four axioms lead to a present value formula. The fifth axiom yields the usual geometric

form of discount factors; the sixth axiom implies that discount factors are less than one. A loose

interpretation of (A5) is that indifference between receiving a cash flow and not receiving it implies

indifference for the one-period delay.

Reasonable alternative versions of the continuity axiom (A3) include the following statements:

(A3 ) 0 0w \ ¢ Ð ¡ Ñ \ ¡ Ð ¢ Ñ\ ! implies that there exists such that if! ! ! !* *

(A3 ) ww \ ¤ Ð £ Ñ Ê ,\ ¤ Ð £ Ñ a ,   !! !

However, (A3 ) and (A3 ) are equivalent to each other and to the restriction of (A3) in which w ww ] œ !

(Proposition 5 in §7).

Decomposition axiom and its converse

Axiom (A2) is particularly important and may be contrasted with its converse:

(A2 ) ( ; )c \ ¤ Ð £ Ñ ] Ê \ ] ¤ Ð £ Ñ \ − Z ] − Z!

The present value formula in §2 invokes (A2) but not (A2 ).-

Proposition 1: Suppose (A1) is valid.

(a) (A2) [ , , ]Í [ ¤ Ð £ µ Ñ Ê [ ^ ¤ Ð £ µ Ñ^ a ^ − Z!

(b) (A2 ) [ , , ]c Í [ ^ ¤ Ð £ µ Ñ^ Ê [ ¤ Ð £ µ Ñ !

Proof: For sufficiency in (a), use and . For necessity in (a), use and\ œ [ ^ ] œ ^ [ œ \ ]

^ œ ] \ œ [ ^ ] œ ^. For (b), let and .

Thus, with (A2), is as good as the status quo only if incrementing with is as good as .[ ^ [ ^any

With (A2 ), is not as good as the status quo if there is which is better than augmented by- [ ^ ^any

[ . The combination of (A2) and (A2 ) is a version of the “independence" assumptions that recur (andc

whose descriptive validity is questioned) in axiomatic theories of decision making. Also, this

combination corresponds to “persistence," “monotonicity," and “consistent choice" in dynamic

programming [Denardo (1967), Sobel (1975, 1980), and Blair (1984)]. Two applications of part (a)

imply

[ ] [ ]\ ] µ \ µ ] Í [ µ Ê [ ^ µ ^ a ^ − Z! !Ê

7

The counterpart to in Miyamoto and Wakker (1996) can be construed as a set of sequences ofZ

time-indexed marginal probability distributions. Allowing for difference in the models, their axioms

include (A1) and (A3) and, instead of (A2), assumptions of outcome monotonicity and the Thomsen

condition. Define outcome monotonicity here as and implyE ¤ Ð £ ÑF [ ¤ Ð £ Ñ^ E [ ¤

Ð £ ÑF ^ . The following result shows that (A2) alone is less restrictive than outcome monotonicity

because the former is equivalent to (A2) together with (A2 ).c

Proposition 2: If satisfies (A1) then it satisfies outcome monotonicity if and only if it satisfiesÐ ¤ ß Z Ñ

both (A2) and (A2 ).c

Proof: (A2) and (A2 ) outcome monotonicity:c Ê

Using (A2 ), and imply and So Lemma 1(b) in §2 impliesc E ¤ F [ ¤ ^ EF ¤ [ ^ ¤! !.

E[ F ^ ¤ E[ ¤ F ^! yielding by (A2).

:Outcome monotonicity (A2)Ê

Let , , and . Thus, and are and . SoE œ \ ] F œ [ œ ^ œ ] E ¤ F [ ¤ ^ \ ] ¤ ] ¤ ]! !

outcome monotonicity, , is .E[ ¤ F ^ \ ¤ ]

:Outcome monotonicity (A2 )Ê c

Let , , and . Thus, and are and . SoE œ \ F œ ] [ œ ^ œ ] E ¤ F [ ¤ ^ \ ¤ ] ] ¤ ]

outcome monotonicity is . \ ] ¤ !

Preferences are if implies for every that isconstant risk averse \ ¤ ] \ ¤ ] − Z% % %

constant with probability one. However, and (A2 ) imply \ ¤ ] \ ] ¤ \ ¤ ] c ! Ê % %

from (A2). Preferences are if implies for all . However,relative risk averse \ ¤ ] +\ ¤ +] + !

\ ¤ ] \ ] ¤ Í +Ð\ ] Ñ ¤ +   !, (A2 ), and Lemma 2(a) in §2 imply for all . Then (A2)c ! !

yields . Since axioms (A1), (A2), (A2 ), and (A3) imply both constant and relative risk+\ ¤ +] c

aversion, one should anticipate risk neutrality (cf. Theorems 1 and 2 in Miyamoto and Wakker).

Axiom sets

The first four axioms imply that there are discount factor vectors such that a vector-valued stochastic

process is weakly preferred to another if and only if there is a corresponding weak preference

concerning the present value random vectors. Conversely, axioms (A1) and (A3) are satisfied by any

8

preference relation among stochastic processes that is determined by preference among present value

random vectors. Also, (A4) is satisfied if the preference among random vectors has the property that

more is preferred to less. However, (A2) is necessarily satisfied only if and are independent. So\ ]

the key axiom is decomposition, namely (A2).

The following example satisfies (A1), (A3), and (A4) but neither (A2) nor (A2 . Let ,-Ñ Q œ X œ "

let denote preferences among random variables corresponding to variance minus expected value,¦

H œ Ö+ß ,× + , \Ð+Ñ œ !ß \Ð,Ñ œ "ß ] Ð+Ñ œ "ß with probabilities 3/4 for and 1/4 for , and and

] Ð,Ñ œ ! \ ] ¦ ] ¦ \ \ ] ¢ ] ¢ \. Then and , so and .) !

A loose interpretation of (A5) is that indifference between receiving a cash flow and not receiving it

implies the same indifference for the one-period delay. In §4, the converse of (A5) implies that a

scalar discount factor generates a vector-valued discounte factor.

Consider restrictions of the assumptions to finitely long deterministic sequences of scalars. The

axioms in Williams and Nasser (1966) correspond to (A1) through (A6) (with (A2 ) and a continuityc

assumption different from (A3)). The postulates in Koopmans (1972) roughly correspond to (A1)

through (A5), (A2 ) and the converse of (A5). It is surprising that “sooner is better" is implied by hisc

postulates (although he does not directly assume the counterpart of (A6)). The preference ordering in

Epstein (1983) is defined on the space of probability measures on the Borel field of countable

sequences of real numbers. Epstein's assumptions, a restatement of Koopmans' in a stochastic

framework, correspond roughly to (A1) through (A5), (A2 ), and the converse of (A5). The continuityc

assumptions in Williams and Nassar (1966), Koopmans (1972), and Epstein (1983) differ from each

other and from (A3).

2. Discounting Theorem

This section owes much to Williams and Nassar (1966). Its main results, Theorem 1 and

Corollaries 1 and 2, are stated and then proved with several lemmas.

It is convenient to define for each , ( ,..., ) for each , and for to write" " " "7" > "> Q>œ " 7 œ > \ − Z

" "> > 7> 7>\ 7 \ ¤ Z for the random vector with component . Recall that ( , ) induces the binaryth

9

relation ( , ) on the set of random -vectors. The following result gives sufficient conditions for_¦ W Q

preferences in finitely long processes to be consistent with a stochastic discounting formula. The proof

would be simplified by replacing the assumption of (A2) with the stronger assumption of outcome

monotonicity.

Theorem 1: (A1), (A2), (A3), (A4) 0 ( Axioms and imply the unique existence of for each"7> 7

and such that>)

(7) _\ ¤ ] Í \ ¦ ]! !X X>œ" >œ"> > > >" "

for all and with ,..., , ) ,..., , ) .\ œ Ð\ \ ] œ Ð] ] X − M" X " X0 0

The proof of Theorem 1 uses the property that , , implies ] µ 4 œ "ß ÞÞÞß N ] µ( ) ( )4 4N4œ"! !!

(Lemma 1(d)). The following countable analog is one of several ways to extend the theorem:

(8) If for all and then .] µ 4 ] − Z ] µ( ) ( ) ( )4 4 4_ _4œ" 4œ"! !! !

Corollary 1. (A1), (A2), (A3), (A4) (8)Axioms and and imply

_\ ¤ ] Í \ ¦ ]! !_ _>œ" >œ"> > > >" "

if and exist and have finite components with probability one .! !_ _>œ" >œ"> > > >" "\ ]

Now the stationarity axiom (A5) yields geometric discount factors.

Corollary 2. (A1), (A2), (A3), (A4), (A5) Axioms and imply for each there uniquely7 œ "ß ÞÞÞßQ

exists such that for all If also is valid then ( ) . (A6) ." " " "7 7> 7 7>" ! œ > − M "

Proof of Theorem 1

Lemma 1. Assuming (A1) and (A2):

(a) ( , ) ( , )\ ¢ ¤ µ Í \ ¡ £ µ! !

(b) , , ,\ ¤ Ð £ µ Ñ ] ¤ Ð £ µ Ñ Ê \ ] ¤ Ð £ µ Ñ! ! !and

(c) \ ¤ Ð £ Ñ ] ¢ Ð ¡ Ñ Ê \ ] ¢ Ð ¡ Ñ! ! !and

(d) \ µ ] µ Ê \ ] µ! ! !and

(e) \ ] µ Ê \ µ ]!

Proof. (a): due to (A2). So .\ £ Í \ £ Ê \ ¤ \ ¤ Í \ £! ! ! ! ! !

Now and and \ µ Í \ ¤ \ £ \ £ \ £ Í \ µ Þ! ! ! Í ! ! !

10

Now and not So not Also, So\ ¢ Í \ ¤ \ µ \ µ Þ \ ¤ \ £! ! ! ! ! Í !. .

\ ¡ Þ!

(b) and (c): The contrapositive of (c) is implies , , or both. From (A2),\ ] £ \ ¡ ] £! ! !

\ ] £ \ £ ] \ ¤ ] ¤ £ ] ¢! ! ! ! ! implies . So (A1) and imply and (a) yields Y . If

then and (a) imply . Using (c), (b) follows from [ ]\ £ ] \ ¡ \ ¤ ] µ Ê \ ] ¤! ! ! !and

due to Proposition 1(a) and (A1).

(d): Two applications of (b).

(e): Two applications of (A2).

Lemma 2. Assuming (A1), (A2), (A3):and

(a) 0.\ ¤ Ð £ Ñ Í -\ ¤ Ð £ Ñ -  ! ! for all

(b) and .! £ Ð ¤ Ñ\ ! Ÿ - - Ê -\ £ Ð ¤ Ñ Ð- Ñ\# #

(c) .\ µ -\ µ - −! !Ê for all ‘

(d) 0 - Ê\ µ - Á \ µ Þ! !and

Proof. (a): is trivial ( ) and is trivial if . For when and (positiveÉ - œ " Ê - œ ! Ê \ ¤ - − M!

integers), Proposition 1(a) implies so . Inductively, and\ £ #\ £ #\ 8\ £ Ð8 "Ñ\!

! !£ Ð8 "Ñ\ -\ ¤ - − M - −. So . Let where is the set of positive rational numbers.for all g g

So with and , and . Therefore, if If - œ 7Î8 7 − M 8 − M - œ \Î8 -\ ¤ \Î8 ¤ \Î8 ¡!"

7

! !Þ !

then Proposition 1(a) with yields ; so (A1) implies [ œ ^ œ \Î8 \Î8 ¤ #\Î8 #\Î8 ¡ !.

Inductively, and for In particular, SoÐ5 "Ñ\Î8 £ 5\Î8 5\Î8 ¡ 5 − MÞ \ œ 8\Î8 ¡ Þ! !

\Î8 ¤ \ ¤ Þ -\ ¤ - −! ! !because Therefore, for all .g

If , let where ( ] and { }. Hence, and there exists a-  - œ 8 0 0 − !ß " 8 − M ! 8\ ¤g !

sequence with as . So for all , (A3), and Lemma 1(b) imply 0 − 0 Ä 0 3 Ä _ 0 \ ¤ 3 0\ ¤3 3 3g ! !

and -\ œ 8\ 0\ ¤ Þ!

(b): , (a), and Proposition 1(a).! ! !£ \ £ -\ £ \, , #

(c) [ and ] (by Lemma 1(a) and (a)) for all \ µ Í \ ¤ \ £ Í -\ ¤ - − d! ! ! !

(d) Use Lemma 1(a) and replace with in (a) replace with and with .- ,ß \ ,\ - ,"

11

The proof of Lemma 3(a) uses an argument due to Professor James C. Alexander, Department of

Mathematics, Case Western Reserve University.

Lemma 3. (A1), (A2), (A3) (A4):Assuming and

(a) For all and there uniquely exists such that ... 0 > œ #ß $ß 7 œ "ß ÞÞÞßQ "7>

.! µ / /7" 7>"7>"

(b) Also assuming (A6) implies ." "7ß>" 7> "

Proof. (a): If and with and thenUniqueness - ! - ! / -/ µ / - / µw w7" 7> 7" 7>! !

/ - / µ Ð- - Ñ/ µ7" 7> 7>w w! ! from Lemma 1(a), so from Lemma 1(b). Uniqueness follows from

(A4) and Lemma 2(d).

Let . If then for all large enoughExistence E œ Ö À / / ¢ × E œ g / Ð"Î Ñ/ £! ! !7> 7" 7> 7"! !

! (Lemma 2(a)), so (A3) implies . But so ./ £ / ¢ E Á g7> 7>! !

Let . If then so ((A2) and (A4)). Now,- œ 380Ö − E× − E / / ¢ / ¤ / ¢! ! ! !7> 7" 7> 7"! !

! is neither zero ((A1)) nor negative (Lemma 2(a) and (A4)). So .-   !

From (A3), For any , so from (A3).-/ / ¤ Þ - / / £ -/ / £7> 7" 7> 7" 7> 7"! ! !! !

Therefore, . If then so (Lemma 1(a)); but ((A4)).-/ / µ - œ ! / µ / µ / ¢7> 7" 7" 7" 7"! ! ! !

So . Let .- ! œ "Î-"7>

(b): (A6) with and . - œ " \ œ !

If (A6) were replaced with ( ) for , then Lemma 3(b) would become\ / / ¢ \ >  - 77> 7ß>"

" " 77ß>" 7> " >   for .

Conclusion of proof of Theorem 1. From Lemma 3, . So Lemma 2(c)! µ < œ / /7> 7" 7>"7>"

implies . Therefore,! µ / /"7> 7" 7>

( )! µ \ / /! !Q X7œ" >œ" 7> 7> 7> 7""

œ \ \ /! !Q X7œ" >œ" 7> 7> 7""

( , ) ( , ).œ \ \ / œ \ \! ! !Q X X7œ" >œ" >œ"7> 7> 7 > >" "! !

Thus Lemma 2(a) yields ( , ) and\ µ \!X>œ" > >" !

( , ) ( , ) . _\ ¤ ] Í \ ¤ ] Í \ ¦ ]! ! ! !X X X X>œ" >œ" >œ" >œ"> > > > > > > >" " " "! !

12

Proof of Corollary 2. From the proof of Lemma 3, for each< œ / / µ7> 7" 7>"7>" !

7 œ "ß ÞÞÞßQ > œ #ß $ß µ / / ¸ 1 and ..... From (A5), and the theorem implies ! 7# 7ß>""7>" )

where ( ,..., , / , ,..., ). Then for each and from (A4); so1 œ ! ! ! ! 1 œ 7 >" " " )7# 7ß>" 7>

" "7ß>" 7#>œ ( ) .

Let , . If (A5) is replaced with] œ > − M> )

[( ,..., , , ,...) ( ,..., , , , ,...) ]] ] \ \ µ Ê ] ] \ \ µ" X " # " X " #! !)

then ( ) for all , where / ." " " " " "7> 7X 7 7 7 X" 7X>Xœ > X œ ,

3. Risk Neutrality

Recall that denotes the set of -valued random vectors. A is order-W ? À W Ä‘ ‘Q felicity function

preserving and linear, i.e., it satisfies (9) and (10) for all [ , ], , and :- − ! " E − W F − W

(9) _E ¦ F ?ÐEÑ   ?ÐFÑif and only if

(10) )?Ò E Ð" ÑFÓ œ ?ÐEÑ Ð" ?ÐFÑ- - - -

Note that a felicity function is unique up to a positive affine transformation; that is, if and are? ? A

felicity functions, then there are and such that ., ! + − A œ + ,?‘

The following result axiomatizes the comparison of expected present values of cash flows, namely

(6). Let be the subset of whose present value vectors, using any discount factor withZ Z* "

components , are absolutely bounded with probability one.! Ÿ ""7

Theorem 2: (A1), (A2), (A3), (A4), then the following properties areIf satisfies andÐ ¤ ß Z Ñ

equivalent:

(a) (A2 );Ð ¤ ß Z Ñ satisfies -

(b) There exists a felicity function;

(c) , , There exists and such that¯ ‘ " ‘− − > − MQ Q>

(12) X Y ) )¤ Í † IÐ \   † IÐ ]¯ " ¯ "! !X X>œ" >œ"> > > >

for all with for for some , .\ − Z ] − Z \ œ ] œ > X X − M* *> > )

Proof. Since (a) and (b) are immediate consequences of (c) with the felicity function ?ÐEÑ œ

¯ "† IÐ EÑ" , the proof shows that (b) implies (c), and that (a) implies (b). Uniqueness of a felicity

13

function up to a positive affine transformation implies that there is no loss of generality from assuming

?Ð Ñ œ !) .

Lemma 4: (A1), (A2), (A3) ( ) imply:Axioms and and the existence of a felicity function with? ? œ !)

(a) , , , , .?ÐEÑ œ ?Ð EÑ E − W @ÐBÑ œ @Ð BÑ B − @Ð Ñ œ !and with‘ )Q

(b) , , , .?ÐE FÑ œ ?ÐEÑ ?ÐFÑ EßF − W @Ð+ ,Ñ œ @Ð+Ñ @Ð,Ñ +ß , −and ‘Q

Proof: (a): For ,E − W

(Lemma 2(c))E ¸ E Ê E Ð EÑ ¸ Ê Ð"Î#ÑE Ð"Î#ÑÐ EÑ ¸) )

[ ((9))Ê ! œ ? Ð"Î#ÑE Ð"Î#ÑÐ EÑÓ œ Ð"Î#Ò?ÐEÑ ?Ð EÑÓ

.Ê ?ÐEÑ œ ?Ð EÑ

Hence, for all and ( ) .@ÐBÑ œ @Ð BÑ B − @ œ !‘ )Q

(b): From (10),

Ð"Î#ÑE Ð"Î#ÑF ¸ Ð"Î#ÑÐE FÑ Ð"Î#Ñ)

] ]Ê Ð"Î#Ñ?ÐEÑ Ð"Î#Ñ?ÐFÑ œ ?ÒÐ"Î#ÑE Ð"Î#ÑF œ ?ÒÐ"Î#ÑÐE FÑ Ð"Î#Ñ)

œ Ð"Î#Ñ?ÐE FÑ

Let partially order according to first-order stochastic dominance; that is, ifŸ W E Ÿ F

TÖE Á F ! TÖE Ÿ D   TÖF Ÿ D× D − B œ ÐB Ñ −} and } for all . Hence, for and‘ ‘Q Q3

C œ ÐC − B Ÿ C B Á C B Ÿ C 3 œ "ß ÞÞÞßQÞ W § W3 3 3Q) , if and for Let be the set of absolutely‘ *

bounded random vectors: : | | , for some .W œ E − W TÖ E O 7 œ "ß ÞÞÞßQ× œ " O _* ˜ ™7 E E

Lemma 5: (A1), (A2), (A3) Axioms and and the existence of an intraperiod utility function with?

? œ !( ) imply:)

(a) , .@ÐBÑ œ † B B −¯ ‘Q

(b) .?ÐEÑ œ † IÐEÑ E − W¯ for all *

Proof: (a): (i) : From Lemma 4(b), .B − M @ÐBÑ œ @Ð B / Ñ œ @ÐB / Ñ œ † BQ Q Q7œ" 7œ"7 7 7 7! ! ¯

(ii) : For each , let / with , . So Lemma 4(b) impliesB − 7 B œ ; = ; = − Mg Q7 7 7 7 7

@ÐB / Ñ œ @Ð; / = Ñ œ ; @Ð/ = @Ð/ Ñ œ @Ð= / = = @ / =7 7 7 7 7 7 7 7 7 7 7 7 7 7 7/ / ). Also, / )= ( / ). Therefore,

@ / = œ @ / = @ B / œ ; @ / = œ B @ÐBÑ œ † B( / ) ( )/ ; so ( ) ( )/ and .7 7 7 7 7 7 7 7 7 7 78 8

14

(iii) : Let with 0 for each . Then Lemma 2(a) andC Ÿ B Ê @ÐCÑ Ÿ @ÐBÑ D œ B C D   77

¯7 7 7 @ÐD / Ñ   @ÐB CÑ œ @ÐDÑ   @ÐCÑ Ÿ @ÐBÑ0 imply 0. So 0. Therefore, (A2) implies .

(iv) (0, ) : Let , with for each =1,2,... with bothB − _ B B − B Ÿ B Ÿ B 3Q 3 3 Q 3 3+ +g

sequences converging to . From (b) and (c), ) . Letting iB † B œ @ÐB Ñ Ÿ @ÐBÑ@ÐB œ † B Ä _¯ ¯3 3 3 3+ +

implies .@ÐBÑ œ † B¯

(v) : If 0, (d) and Lemma 4(a) imply B − B @ÐB / Ñ œ @Ð B / Ñ‘Q7 7 7 7 7

œ Ð B Ñ@Ð/ Ñ œ B B œ ! @ÐB / Ñ œ @ œ œ B7 7 7 7 7 7 7 7 7¯ ) ¯. If then ( ) 0 .

(b): Part (a) and (11) imply for all . For any , there is a sequence?ÐEÑ œ † IÐEÑ E − W E − W¯ w *

( , ) with , , so ) ) and ) ),E E E E − W E Ÿ E Ÿ E ?ÐE Ÿ ?ÐEÑ Ÿ ?ÐE IÐE Ÿ IÐEÑ Ÿ IÐE3 3 3 3 33 3 w 3 3 3

3 − M IÐE Ä IÐEÑ IÐE Ä IÐE 3 † IÐE Ÿ YÐEÑ Ÿ † IÐE, with ) and ) ). For each , ) ).33 3 3¯ ¯

Convergence of the expectations to implies . IÐEÑ YÐEÑ œ † IÐEѯ

Lemma 6: (A1), (A2), (A2 ), (A3) Axioms and imply the existence of a felicity function.c

Proof. From Herstein and Milnor (1953), (9) and (10) are implied by (13), (14), and (15) (for all

EßFßG − W) which follow:

(13) E ¸ F EÎ# GÎ# ¸ FÎ# GÎ#implies

(14) ;_¦ Wweakly orders

(15) { [ ]: ( } _ _! ! !− !ß " E " ÑF ¦ Ð ¥ ÑG is a closed set

In order to prove (15), let , ) and , ). Then\ œ ÐE F ] œ ÐG F! !

{ [ ]: ( } { [ ]: 0} = { [ ]: } which is_ ~! ! ! ! ! ! !− !ß " E " ÑF ¦ G œ − !ß " \ ] ¤ − !ß " \ ¤ ]

closed due to (A3). Similarly, { [ ]: } is closed._! ! !− !ß " E Ð" ÑF ¥ G

Axiom (A1) implies (14).

In order to prove (13), use (A2), (A2 ), and parts (d) and (e) of Lemma to obtainc

( , ( )( , )_E ¦ F Í EF ¤ Ê "Î# E F ¤!Ñ ! ! !

( )( , ) + ( )( , ) ( )( , Ê "Î# E F "Î# G ¤ "Î# G! ! !Ñ

(( )( ), ) (( )( ), )Ê "Î# E G ¤ "Î# F G! !

( )( ) ( )( )_Ê "Î# E G ¦ "Î# F G

Similarly, ( )( ) ( )( ). _ _ _E ¥ F Í F ¦ E Ê "Î# F G ¦ "Î# E G

15

This completes the proof of Theorem 2.

Extensions of Theorem 2

It is an open question whether there are conditions weaker than (A1) through (A5) and (A2 ) thatc

would axiomatize (4) and the optimization of the “expected utility of the present value" of a time

stream of rewards. See surveys by White (1988) and Whittle (1990).

The remainder of this section concerns several routes that can be followed to extend Lemma 5(b)

beyond (hence Theorem 2 beyond ). Let be the subset of whose elements take only finitelyW Z W W* ‡ w

many values (with probability one). Recall that and denote the zero vector and the - unit) / 77 th

vector in . Suppose that is a felicity function with , for let be a random‘ ) ‘Q QB?Ð † Ñ ?Ð Ñ œ ! B − H

variable with , , , , and . Then (A4)TÖH œ B× œ " @ÐBÑ œ ?ÐH Ñ œ @Ð/ Ñ 7 œ "ß ÞÞÞßQ œ Ð ÑB B 7 7 7¯ ¯ 8

and imply for each . The expected utility formula, a consequence of (10), is@Ð Ñ œ ! ! 7) ¯7

(11) , ?ÐEÑ œ TÖE œ B ×@ÐB Ñ œ IÒ@ÐEÑÓ E − W!3 3 3

w

First, at the expense of additional assumptions and complexity, versions of (11) are valid for larger

collections than [Fishburn (1982)]. These versions would expand the set of random vectors that canWw

be approximated in the proof of Theorem 2. Second, additional assumptions would induce continuity

of in the topology of weak convergence, and that would permit approximating with?Ð † Ñ E − W W*

a sequence in . For a third route, let be the set of random vectors whose components haveW W § W* o

finite expectations. The next result assumes essentially that is continuous at .¦ )

Lemma 7: ( ) Suppose that a felicity function satisfies and? ? œ !)

(16) For all sequences in with as and for allØG Ù W TÖG œ × Ä " 3 Ä _ H − W3 3o o)

, with there is a such that implies andH ¦ 4 3   4 H ¥ G G ¥ H) 3 3

Then and imply for all (A1), (A2), (A3) .?ÐEÑ œ † IÐEÑ E − W¯ o

Proof: (a) From Lemma 5(b), for all there exists with . Let be a sequence# # ! H − W YÐHÑ œ G*3

in with } as . So (16) implies that there exists j such that | | for allW TÖG œ Ä " 3 Ä _ YÐG Ñ o3 3) #

3   4 YÐG Ñ Ä ! 3 Ä _. Therefore, as .3

(b) For , let have component max { ,0}, . ThenE − W E − W 7 E 7 œ "ß ÞÞÞßQo + o th7

E œ E E YÐEÑ œ YÐE YÒ E Ó E Ò E Ó+ + + + + +( ) and ) ( ) with each component of ( )

16

bounded below (above) by 0. So it suffices to prove the assertion for those elements in whoseWo

components are all either bounded above by zero or bounded below by zero.

(c) Let with and 0. Define and if for ;E − W E   D H œ E H œ E Ÿ D 7 œ "ß ÞÞÞßQo ) )D 7D

otherwise, and . Hence, for each and } as .H œ H œ E E œ H H D TÖH œ Ä " D Ä _D DD D D) )

So part (a) and Lemmas 4(b) and 5(b) imply ) ) ) )YÐEÑ œ YÐH YÐH œ † IÐH YÐHD DD D8

Ä † IÐEÑ D Ä _ E Ÿ8 ) as . A similar argument suffices for .

Corollary 1 and Lemma 7 extend Theorem 2 to stochastic processes with countable time indices.

COROLLARY 3: Assumptions (8) and (16) imply that Theorem 2 is valid with (12) replaced by (17)

if both expectations exist with finite components:

(17) X Y ¤ Í † IÐ \ Ñ   † IÐ ] Ñ \ − Z ß ] − Z8 " 8 "! !_ _>œ" >œ"> > > >

Corollary 2 augments (A1) through (A4) with (A5) to obtain ( ) ; including (A6) yields" "7> 7>"œ

"7 1. So Corollaries 2 and 3 give conditions which imply that preferences are consistent with the

maximization of . The algorithm in Feinberg and Schwartz (1994) computes an¯ "† IÐ \ Ñ!_>œ"

>">

optimal policy for this criterion in a Markov decision process with and finitely many states andQ "

actions. Furukawa (1980) and Henig (1983) study Pareto optimization of the vector IÐ \ Ñ!_>œ"

>">"

when ." "" Qœ † † † œ

4. Scalar Discount Factor

This section combines Corollary 2, Theorem 2, and Koopmans (1972) to specify conditions which

imply . Let be the sequences that are constant with probability one, and for" " ƒ" Qœ † † † œ § Z

B − ² B ² œ =?: 7+B B © B B œ > Xƒ ƒ ƒ ) let | |. Let contain the sequences for which if .> 7 7> X >

Let and with for each and let and withB − C − B ¤ C 8 − B − C −( ) ( ) ( ) ( )8 8 8 8ƒ ƒ \ ƒ ƒ

² B B ² Ä ² C C ² Ä 8 Ä _ ¤( ) ( )8 80 and 0 as . Say that is continuous on with respect toƒ

² † ² B ¤ C if . There are two additional assumptions:

(A3 ) * ¤ ² † ²is continuous on with respect toƒ

(A5 ) ( , , ,...) ( , ,...)c ) \ \ µ \ \ µ" # " #! !implies

Axiom (A5 ) is the converse of (A5).c

17

Proposition 4: (A1) (A5), (A3 ), (A5 )Axioms through and and the existence of an intraperiod utility* c

function imply ." "" Qœ † † † œ

Proof: The assumptions satisfy Postulates 1 through 4 in Koopmans (1972). So in ƒT

(18) B µ C Í ?ÐB Ñ œ ?ÐC Ñ! !X X>œ" >œ"

>" >"> >! !

where is unique and generates a class that is unique up to a positive affine! ‘ ‘− Ð!ß "Ñ ? À ÄQ

transformation. Let and for all ; from Theorem 2, generates a?Ð Ñ œ ! B œ C œ > " ?ÐAÑ œ † A) ) ¯> >

class that is unique up to a positive linear transformation.

Let and ; so from Lemma 3 and (A2). From (18), if\ œ / ] œ / \ µ ] / µ /7> 7> 7" 7> 7> 7"" "

and only if . Thus, . ! ¯ " 8 " !>" >"7 7> 7 7>œ œ

5. Continuity Axioms and Impatience

Reasonable alternative versions of the continuity axiom

(A3) { : } is closed ! !\ ] ¤ Ð £ Ñ !

include these statements:

(A3 ) such that ( , )w \ ¢ Ð ¡ Ñ Ê b ! \ ¡ Ð ¢ Ñ\ a − !! ! ! ! !* *

(A3 ) ww \ ¤ Ð £ Ñ Ê ,\ ¤ Ð £ Ñ a ,   !! !

Let (A3 ) denote the restriction of (A3) in which .o ] œ !

Proposition 5. (A1), (A2), (A2 ). (A3 ), (A3 ), (A3 ) .Assume and Then and are equivalentc o w ww

Proof. Let (3 ), (3 ), and (3 ) denote the respective unions of {(A1),(A2),(A2 )} with {(A3 )},o c ow ww

{(A3 )}, and {(A3 )}.w ww

(3 ) (3 ): Lemma 2(a) and its proof.o Ê ww

(3 ) (3 ): For , let be a sequence in that converges to from below. To initiate aw ww3Ê , ! Ø, Ù ,g

contrapositive proof, if then (due to (A2 )). So,\ ¡ ,\ ¢! ! c

,\ ¢ , , \ , , 3( ) for sufficiently near (due to (A3 )). Therefore, there exists such that3 3w *

3   3 ¡ ,\ , , \ œ , \ , \ ¡ , \ ¤ 3* implies ( ) . So contrary to for all because! ! !3 3 3 3

\ ¤ , !! and is rational.3

(3 ) (3 ): To initiate a contrapositive proof,ww wÊ

18

(( )( )\ £ \ Í \ \ ¤ Í " \ ¤! ! !! !

Let and ; so . Now (3 ) implies ( )( ) for all . If! ! ! ! !* * " ! " ! , " \ ¤ ,   !ww !

, œ " ( ) then! "

( )( ) ! ! !£ , " \ Í ¤ \ Í \ £!

(3 ) (3 ): Let with for all . If then ((A1)). If ( ) ww3 3Ê Ä \ ¤ 3 œ ! \ ¤ !o ! ! ! ! ! !! !

then without loss of generality ( ) for all . So (3 ) implies ( ) for all .! !3 "ww ! 3 , \ ¤ £ ,   !!

Let / to obtain . , œ \ ¤! ! !" !

Impatience

For any and -dimensional random vector , let ( , ) denote the sequence in with first\ − Z Q ^ ^ \ Z

component and component for all . Let be the sequence in with component^ > \ >   # \ Z >th th>" #

\ −>" for all t . Versions of impatience that may appear to differ from\

(A6) ( )Sooner is better: \ / / ¢ \- 7> 7ß>"

include the following statements:

(19) ( ) ( ) \ ¢ ] Í \ ] ¢ ^ß\ ^ß ] a ^

(20) ( , ) ( , , ) ( , )\ ¢ ^ \ Í \ ^ \ ¢ ^ \# " #

To interpret (19), which is implicit in Koopmans (1960), suppose that is preferred to . Then \ ] \

is preferred to with an "intensity" which is greater than the intensity with which a delay of is] \

preferred to the same delay of . For an interpretation of (20) that corresponds to the formal definition]

of impatience in Koopmans (1960), suppose that is preferred to the sequence in which is\ \"

replaced by . Then ( , , , ,...) is preferred to ( , , , ,...).^ \ ^ \ \ ^ \ \ \" # $ " # $

Proposition 6. (A1) (A6) (A2 ) (19) (20).Axioms through and imply andc

Proof. To prove the claim regarding (19), let ( ) ( ) have components whose existence is" "œ − !ß "7Q

asserted in Corollary 2, let be the vector with component ( ) , and let be the -vector of" "> >77 Qth 1

ones. Then

( ) ( , ) [ ( )] [ ( )]\ ] ¢ ^ß\ ^ ] Í \ ^ß\ ] ^ß ] ¢ !

( )( ) ( ) ( )Í ¥ \ ] œ \ ]) " " " "! !X X>œ" >œ"

>" > >"> > > >1

19

( ) .Í ¥ \ ] Í \ ¢ ]) "!X>œ"

>"> >

For the claim concerning (20),

( , ) ( )\ ¢ ^ \ Í \ ¦ ^ Í \ ¦ ^# " "1 "

Í \ ^ \ ¦ ^ ^ \" > >X X>œ# >œ#

> >" " " "! ! ( , , ) ( , ) Í \ ^ \ ¢ ^ \" #

6. Attribute Decomposition in Random Vectors

Several of the previous results have interpretations for preference orderings of -dimensionalQ

random vectors. This section is confined to the special case of Theorem 2 when is a binary relation¤

on sequences ( , for all . If axiom (A2 ) were invoked, the restriction would be equivalent toE Ñ E − W! c

fixing and considering the sequences ( ) for all . This yields the following\ − Z Eß\ − Z E − W

version of Theorem 2 in which is regarded as a primitive binary relation on (rather than a binary_¦ W

relation that is induced by on ). Recall that denotes the set of absolutely bounded random¤ Z W*

vectors and consists of vectors whose components have finite expectations.W § Wo

Corollary 4: , ( ), and_If is weakly ordered,Ð ¦ WÑ / ¦ 7 œ "ß ÞÞÞßQ7 )

(21) ( , )_ _EF ¦ Ð Ñ E ¦ Ð ÑF E − W F − W¥ ¥) implies

( , )_Ö − Ò!ß "Ó À E ¦ Ð ÑF× E − W F − W! ! ¥ is closed

then the following properties are equivalent:

(a) , (A2 );_Ð ¦ WÑ satisfies -

(b) There exists a felicity function;

(c) There exists with positive components such that¯ ‘− Q

, _E ¦ F † IÐEÑ   † IÐFÑ ÐE − W F − W Ñif and only if ¯ ¯ * *

The assertion above is valid with replaced by if also W W* o

as for all sequences in with and for allØG Ù W TÖG œ × Ä " 3 Ä _ H − W3 3 )

, . with there exists such that implies andH ¦ 4 3   4 H ¥ G G ¥ H) 3 3

20

Assumption (21) corresponds to the decomposition axiom (A2) and is a strong form of utility

independence which is investigated in references cited in §1 and Blackorby, Primont, and Russell

(1978).

7. Conclusion

Assumptions that imply discounting are orthogonal to assumptions that yield an intraperiodnot

utility function. Weak axioms that are consistent with discounting in a deterministic environment carry

the seeds of risk neutrality in a stochastic environment. The following assumptions imply that

preferences among stochastic processes correspond to comparisons of random present value vectors:

weak ordering, decomposition, continuity, and more is better. If there exists an intraperiod utility

function, the same assumptions imply risk neutrality. An axiomatic justification of expected present

value of intraperiod utility (with nonlinear intraperiod utility) seems to require a justification of

discounting in a stochastic environment ( , (7)) with an assumption that is weaker thani.e.

decomposition (A2).

Acknowledgements

The author is grateful to Professor James C. Alexander for comments and the argument on which

the proof of Lemma 3(a) is based, to Professor Eugene Feinberg and Dr. Charu Sinha for comments,

and to an anonymous commentator on an earlier version of the paper who noted that confining to ] !

in (A3) permitted a counterexample to a lemma.

21

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