Utility Functions for Infinite-Period
Planning
Harvey, C.M.
IIASA Working Paper
WP-86-082
November 1986
Harvey, C.M. (1986) Utility Functions for Infinite-Period Planning. IIASA Working Paper. IIASA, Laxenburg, Austria, WP86-082 Copyright © 1986 by the author(s). http://pure.iiasa.ac.at/2779/
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WITHOUT THE PERMISSION
OF THE AUTHOR
UTILITY FUNCTIONS FOR --PERIOD
PIANNING
Charles M. Harvey
November 1 9 8 6
WP-86-82
Working Papers are interim r e p o r t s on work of t h e International
Institute f o r Applied Systems Analysis and have received only Limited
review. Views or opinions expressed herein d o not necessarily
r e p r e s e n t those of t h e Institute or of i t s National Member
Organizations.
INTERNATlONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
A-2361 Laxenburg, Austria
This p a p e r p r e s e n t s a systematic discussion of decision analysis models f o r attitudes toward r i s k when t h e effects of a public policy choice extend into t h e distant or unbounded future. S e v e r a l issues of social r i s k attitudes are identified and
discussed. Conditions on p r e f e r e n c e s are presented by which value judgments concerning these issues can b e included in a formal model f o r a public policy evaluation.
Alexander B . Kurzhanski
Chairman
System and Decision Sciences Program
- iii -
UTILITY FUNCTIONS FOR lNFlWTE-PERIOD PLANNING*
C h a r l e s M. H a r v e y
1. Introduction
There are a variety of methods f o r treating t h e uncertainty of f u t u r e outcomes in a multiperiod planning model. Cost-benefit studies often: (1) add a "risk
premium" t o t h e discount rate (e.g., Sugden and Williams 1978, p. 60), (2) r e p o r t a
probability distribution of net p r e s e n t values (e.g., Mishan 1976, p. 372), (3) use
t h e net p r e s e n t value function as a utility function, o r (4) s u b t r a c t a multiple of
t h e variance of t h e net r e t u r n from t h e expected net r e t u r n (Markowiti 1959).
These methods do not a d d r e s s p r e f e r e n c e issues regarding a n institution's o r
society's attitude toward r i s k .
Decision analysis models t h a t do a d d r e s s t h e s e issues have been developed by
a number of authors.
Fishburn (1965a), (1970) h a s introduced conditions on
p r e f e r e n c e s t h a t imply a utility function having an additive form. Pollak (1967)
has introduced p r e f e r e n c e conditions t h a t imply a utility function having a n additive form o r a log-additive form. Meyer (1970), (1972) h a s shown t h a t weakened
assumptions of mutual utility independence imply t h a t t h e utility function h a s a n additive form, a positive multiplicative form, o r a negative multiplicative form (see
Keeney 1968, 1974 f o r similar r e s u l t s in a multiattribute setting). Richard (1975)
h a s introduced a condition of s t r i c t multivariate risk aversion t h a t t o g e t h e r with
mutual utility independence implies a negative multiplicative form. P r e f e r e n c e
conditions concerning utility dependence, e.g., t h e dependence of t h e decision
maker's r i s k attitude f o r one period on t h e outcomes in o t h e r periods, have been
introduced by Bell (1977). Fishburn (1965b), and Meyer (1977). Related work includes t h a t of B a r r a g e r (1980), Bell (1974), Bodily (1981), and Fishburn and Rubinstein (1982).
This p a p e r i s a systematic study of p r e f e r e n c e issues regarding attitudes toward r i s k in multiperiod planning.
A number of decision analysis models a r e
presented t h a t when a p p r o p r i a t e can b e used t o formulate social p r e f e r e n c e s in a
risk management study o r t o formulate c o r p o r a t e p r e f e r e n c e s in a long r a n g e
planning study by a p r i v a t e form.
*This research i s supported in part b y the National Science Foundation under Grant no. SES 8410665.
The models in t h i s p a p e r are c o n c e r n e d with infinite-period outcomes, i.e.,
with a n u n b ~ u n d e d ~ p l a n n i nhorizon.
g
Corresponding r e s u l t s f o r a bounded planning
horizon c a n b e deduced by assuming t h a t t h e amounts in t h e p e r i o d s beyond t h e
horizon are a specified, s t a n d a r d amount.
A g e n e r a l t y p e of planning model is p r e s e n t e d in Section 2. I t s primary
c h a r a c t e r i s t i c s are t h a t t r a d e o f f s between d i f f e r e n t p e r i o d s satisfy conditions of
p r e f e r e n t i a l independence and t h a t r i s k a t t i t u d e s satisfy conditions of e x p e c t e d
utility. Then, in Section 3, f o u r p r e f e r e n c e issues are identified t h a t c a n b e included in such a planning model (and six p r e f e r e n c e issues are identified t h a t cannot b e included).
Sections 4-8 are c o n c e r n e d with s p e c i a l conditions on social o r c o r p o r a t e
p r e f e r e n c e s , a n d with t h e implications of t h e s e p r e f e r e n c e conditions f o r t h e form
of a utility function in a g e n e r a l t y p e of planning model. Sections 4 and 5 discuss
a t t i t u d e s toward multiperiod r i s k ; Section 6 discusses c o n c e r n f o r i n t e r t e m p o r a l
equity; and Section 7 discusses a t t i t u d e s toward p r e s e n t r i s k . Then, in Section 8, a
number of planning models are p r e s e n t e d which use a combination of p r e f e r e n c e
conditions r e g a r d i n g t h e s e issues.
The planning models discussed in t h i s p a p e r are intended t o b e p r e s c r i p t i v e
r a t h e r t h a n d e s c r i p t i v e o r normative; t h a t is, t h e y are intended t o c l a r i f y t h e implications from value judgments concerning relatively simple c h o i c e s t o p r e f e r e n c e s between t h e r e l a t i v e l y complex a c t u a l alternatives.
The utility functions
discussed are intended t o b e a n inobtrusive technical means by which t h e analyst
can e v a l u a t e t h e s e implications.
A utility function c a n b e determined by t h e following s t e p s : (1)identify which
p r e f e r e n c e issues are important, (2) v e r i f y p r e f e r e n c e conditions r e l a t e d t o t h e s e
issues, and (3) f o r e a c h p r e f e r e n c e issue t h a t is deemed important, make a specific
p r e f e r e n c e assessment. S u c h a p r o c e d u r e h a s s e v e r a l advantages: i t is possible
t o choose where t o simplify t h e planning model, t h e number of value judgments
needed i s minimized, a n d a sensitivity analysis of t h e e f f e c t s of t h e value judgments
i s possible.
2. Expected-Utility Planning Models
This section p r e s e n t s a n expected-utility model f o r infinite-period planning
t h a t is based on t h e additive-value model presented in Harvey (1986a). The r e a d e r
is r e f e r r e d t o t h a t p a p e r f o r a more detailed discussion of the definitions and
results concerning value tradeoffs that a r e needed f o r t h e present discussion of
risk attitudes. Moreover, the r e a d e r is r e f e r r e d to Appendix A of this p a p e r f o r
much of t h e technical material (definitions and proofs) used in this section.
Suppose t h a t a unit of time is chosen, e.g., a y e a r , and t h a t the future i s divide d into a sequence of periods ( t -1, t ] , t
during a period, t
= 1 3 ,..., of
unit duration. The outcomes
= 1 2 ,..., a r e t o b e described by a single, readily interpretable
variable zt t h a t may involve t h e pricing out of nonmonetary a t t r i b u t e s o r the summing of t h e costs, risks, and benefits t h a t a c c r u e t o t h e individuals in a specific
~ , of
group. The consequences o v e r t h e f u t u r e are described by sequences ( z ~ , z ...)
t h e single-period
iwnite-period
amounts zt , t = 1 , 2 ,.... These consequences will b e called
consequences.
Uncertain events having a finite number of
infinite-period consequences as possible outcomes will be called i w n i t e - p e r i o d
lotteries.
Definition 1. A model having t h e following s t r u c t u r e will b e called a p l a n n i n g
model: (a) z* denotes a specified amount of t h e variables zt to b e used f o r normalization purposes, (b) I denotes a common interval on which t h e variables zt are
defined, (c) C denotes t h e s e t of all infinite-period consequences having amounts
zt in I, (d) L denotes t h e set of all infinite-period lotteries having consequences in
C,and (e) 2 denotes a p r e f e r e n c e relation defined on t h e s e t L of lotteries.
The amount zt =z* will b e called a s t a n d a r d amount. For technical reasons, i t will b e assumed t h a t t h e interval I has one of the forms ( z - , z + ) , [ z S , z + ) ,
and (z-,z*] where each of t h e amounts z*, z - , z + can be e i t h e r finite or infinite.
Here, t h e standard amount z* will b e called intermediate, least prqferred, and
most prqfetred respectively. If z* i s a l e a s t p r e f e r r e d o r m o s t p r e f e r r e d amount
in I, then L i s to b e r e s t r i c t e d t o those lotteries not having the extreme consequence (z* ,z* ,...) as a possible consequence.
In t e r m s of t h e above notation, a n ordinal-value model i s defined in Harvey
(1986a) by t h e p r e f e r e n c e conditions (A)-(I?,)r e s t a t e d in Appendix A. The primary
r e s u l t i s as follows.
Lemma 1. The conditions (A)-(E) on tradeoffs are satisfied if and only if p r e f e r ences among t h e infinite-period consequences in the set C* (defined in Appendix A)
are r e p r e s e n t e d by a n infinite-series value function
such that: V(zl,z2, ...) converges if and only if (z1,z2,...) i s in t h e set C* ; t h e timi n g w e i g h t s at , t
= 1.2, ..., are positive; and t h e tradeofls & n c t i o n v i s continu-
ous, s t r i c t l y increasing, and normalized s o t h a t v (z* )
= 0.
In this p a p e r , t h e p r e f e r e n c e conditions (A)-(E) are augmented by two conditions as follows:
(F) For any consequence (z1,z2, ...) in C*, t h e r e exists a n indifferent conse-
quence ( ~ 1 ~
i * 2 , ~ ...)
* 3 in
, C* f o r some amount ~i in I.
(G) The p r e f e r e n c e relation 2 satisfies t h e conditions f o r expected utility f o r
t h e set L* of l o t t e r i e s in L having consequences in C*.
(See, e.g.. Herstein and
Milnor 1953). Moreover, any l o t t e r y in L* has a n indifferent consequence in C*.
(This condition on t h e existence of c e r t a i n t y equivalents i s included to e n s u r e t h a t
t h e utility function i s continuous.)
Theorem 1. A planning model satisfies t h e p r e f e r e n c e conditions (A)-(G) if and
only if t h e p r e f e r e n c e relation 2 r e s t r i c t e d t o t h e set L* of infinite-period lott e r i e s i s r e p r e s e n t e d by a utility function having t h e forms:
Here, t h e value function V(x1,x2,..,)
= v (zl)
+ a 2 v (z2) + - - .
is as described in
Lemma 1 ; t h e m u l t i p e r i o d r i s k f i n c t i o n f is a continuous, s t r i c t l y increasing
function defined on t h e r a n g e of V ; and t h e first-period u t i t i t y f u n c t i o n u i s a
s t a n d a r d conditional utility function f o r t h e f i r s t period, i.e., a utility function on
consequences of t h e form ( x i , z$,z;
,...).
The p u r p o s e of Theorem 1 i s t o provide t h e g e n e r a l forms (2)-(4) of utility
functions f o r modeling t h e p r e f e r e n c e issues (i)-(iv) discussed in t h e n e x t section.
The timing weights at model t h e issue (i) of t h e importance of t h e f u t u r e ; t h e
t r a d e o f f s function v models t h e issue (ii) of i n t e r t e m p o r a l t r a d e o f f s ; t h e firstp e r i o d utility function u models t h e issue (iii) of r i s k in t h e p r e s e n t ; and t h e multiperiod r i s k function f models t h e issue (iv) of multiperiod r i s k .
9. Preference Issues
This section f i r s t identifies a number of issues concerning individual and soc i a l p r e f e r e n c e s t h a t c a n b e modeled by a utility function of t h e g e n e r a l forms
(2)-(4) d e s c r i b e d in Theorem 1. I t then identifies o t h e r issues t h a t are excluded by
t h e assumptions of Theorem 1. R e f e r e n c e s are provided t o more detailed discussions of e a c h issue.
(i) The importance of f u t u r e periods.
This issue i s c o n c e r n e d with t h e
dependence of t r a d e o f f s between two f u t u r e periods s a n d t on t h e f u t u r i t y of t h e
p e r i o d s s and t
.
In m o s t cost-benefit and r i s k analysis studies, p r e f e r e n c e s r e g a r d i n g t h i s iss u e are modeled by p r e s e n t value discounting, t h a t is, by assuming t h a t t h e timing
weights at form a geometric sequence, l , a , a 2 ,..., f o r some constant 0
< a < 1.
Timing weights of t h i s form are implied by s e v e r a l p r e f e r e n c e conditions, e.g., t h e
s t a t i o n a r i t y condition in Koopmans (1960), (1972) and t h e pairwise invariance condition in Keeney a n d Raiffa (1976, p. 480) which states t h a t t r a d e o f f s between two
p e r i o d s s a n d t depend only o n t h e (absolute) difference t -s between t h e periods.
More g e n e r a l timing weights are considered, f o r example, in Williams and Nassar
(1966) a n d Weibull (1986).
Another p r e f e r e n c e condition, called r e l a t i v e timing p r e f e r e n c e s , i s introduced in Harvey (1986a). This condition states t h a t t r a d e o f f s between t w o periods
s a n d t depend only on t h e r e l a t i v e difference t / s between t h e periods. I t implies
t h a t t h e timing weights at form a n arithmetic sequence, 1 , ( 1 / 2)',(1/
some constant r
> 0. The t e r m
3
)
'
,
..., f o r
r e l a t i v e v a l u e d i s c o u n t i n g i s used f o r a model of
this type.
Relative value discounting models a c c o r d more importance t o t h e distant fut u r e , and hence t o f u t u r e generations, than do p r e s e n t value discounting models.
This distinction between t h e t w o types of models o c c u r s because any arithmetic sequence
at-l,
( l t ) r 0
decays
more
slowly
than
any
geometric
sequence
o < a <I.
(ii) Concern f o r i n t e r t e m p o r a l e q u i t y . This issue has two equivalent forms.
The f i r s t form i s a concern f o r equity in t h e amounts zt in different periods; typically, t h e r e i s a n aversion toward l a r g e differences in t h e amounts zt
.
The second
f o r m i s a dependence of t h e tradeoffs between two periods s and t on t h e base
amount, e.g., asset position or social wealth, in one of t h e periods; f o r example,
t h e r e typically i s a willingness t o incur g r e a t e r costs in a period s f o r a benefit in
a period t if t h e base costs in period s are low than if t h e base costs in period s
are a l r e a d y high.
This issue i s not included in most cost-benefit studies.
P r e f e r e n c e s are
modeled by assuming t h a t t h e tradeoffs function v i s linear, i.e., v ( z t ) = zt. This
form of v is implied by a n attitude of neutrality toward intertemporal inequity o r ,
equivalently, by t h e condition t h a t tradeoffs between t w o periods are independent
of t h e base amount in one of the periods. P r e f e r e n c e conditions t h a t d o model
aversion toward intertemporal inequity and t h a t imply a parametric form f o r t h e
tradeoffs function v are introduced in Harvey (1986a).
(iii) A t t i t u d e t o w a r d r i s k in t h e p r e s e n t .
Suppose t h a t ( z t ) denotes a n
infinite-period consequence having t h e amount zt in period t and standard amounts
z* in e v e r y o t h e r period. Then, this issue can be described as a concern f o r r i s k
in l o t t e r i e s having consequences of t h e form (xi), i.e., l o t t e r i e s such t h a t all of
t h e uncertainty i s in t h e f i r s t period, t
= 1.
This issue and t h a t of (iv) below are two components of t h e overall issue of
risk in infinite-period planning. Most cost-benefit studies e i t h e r d o not include t h e
r i s k issues (iii), (iv) o r model t h e overall issue of r i s k by one of t h e methods
described in the introduction.
An attitude of neutrality toward risk in t h e p r e s e n t implies t h a t t h e firstperiod utility function u in linear, i.e., u (x
=x
A number of a u t h o r s have in-
troduced p r e f e r e n c e conditions t h a t model a n attitude of aversion toward r i s k in
t h e p r e s e n t and t h a t imply a parametric form f o r t h e function u . References are
provided in Sections 4 and 5.
(iv) A t t i t u d e t o w a r d m u l t t p e r i o d r i s k . Suppose t h a t (x,,xt) denotes a n
infinite-period consequence having t h e amounts
I, and xt
in t w o periods s , t and
having s t a n d a r d amounts x* in e v e r y o t h e r period. One form of t h e multiperiod
r i s k issue i s a c o n c e r n f o r joint probabilities as w e l l as marginal probabilities in
assessing p r e f e r e n c e s between infinite-period lotteries.
that
< (x,,xt),
For example, suppose
( x i ,xi ) > denotes a l o t t e r y having t h e equally likely consequences
(I, , x t ), ( x i ,xi ) and
< (x,
, x i ), (x;,xt)
> denotes a
lottery having t h e equally like-
ly consequences (I, ,xi ),( x i ,xt ). Even though these l o t t e r i e s have equal marginal
distributions f o r t h e period s and f o r t h e period t , if x,
< x,'
and xt
< xi , then
t h e second l o t t e r y may be p r e f e r r e d t o t h e f i r s t because of a n aversion to "catastrophe" if t h e consequence (x,.xt) occurs. References f o r t h i s issue are provided
in Sections 4 and 5.
An attitude of neutrality toward multiperiod r i s k (for example, indifference
between any two l o t t e r i e s described above) implies t h a t t h e multiperiod r i s k function f i s linear, i.e., f (V)= V. F o r such a model, t h e infinite-series value function
V(xl ,la,...) in Theorem 1i s also a utility function.
In c o n t r a s t t o t h e issues (i)-(iv) discussed above, t h e following issues (v)-(x)
are excluded by t h e p r e f e r e n c e conditions in Theorem 1 and hence cannot be
r e p r e s e n t e d by t h e models to b e discussed in this p a p e r .
(v) Dependence of t h e tradeoffs between two periods on t h e amounts in t h e intervening periods. This issue is excluded by condition (C) in Appendix A. Such
p r e f e r e n t i a l complementarity i s discussed, for example, in Gorman (1968), and
models are proposed in Bordley (1985) and Meyer (1977).
(vi) Dependence of t h e tradeoffs within a period on t h e base amounts in t h a t
period.
For a multiattribute decision problem, t h e r e may be, f o r example, a
dependence of t h e pricing-out amounts f o r t h e non-monetary a t t r i b u t e s on t h e base
amount of t h e monetary a t t r i b u t e . F o r a group decision problem, t h e r e may be a
concern f o r equity in t h e consequences t o t h e affected individuals. These issues
are excluded by t h e requirement t h a t the outcomes in a period t c a n b e described
by a single, readily i n t e r p r e t a b l e variable, i.e., t h a t multiple a t t r i b u t e s c a n b e reduced to a single a t t r i b u t e by "willingness-to-pay" methods, and group consequences can b e aggregated by a "sum-of-benefits" method. Multiattribute tradeoffs
dependence is discussed, f o r example, in Harvey (1985a) and Kirkwood and S a r i n
(1980); interpersonal equity in consequences is discussed, f o r example, in Atkinson
(1970), B a r r a g e r (1980), Fleming (1952), and Harvey (1985b).
(vii) Dependence of tradeoffs comparisons on t h e period. This issue i s excluded by condition (E) in Appendix A, which requires t h a t f o r any two periods s ,t
and any two p a i r s of amounts z1 < z2and z3 < z 4 , if t h e r e is a willingness t o tradeoff more t o increase z, from z1 t o z2 than to increase z, from z3 t o z 4 , then t h e r e
is a willingness t o tradeoff more t o increase zt from z1 t o z 2 than t o increase zt
from z 3 t o z4. A general planning model t h a t does not assume condition (E) is
presented in Harvey (1986a); however, t h e r e are no known p r e f e r e n c e conditions
on dependence of tradeoffs comparisons t h a t could b e used t o s t r u c t u r e such a
model.
(viii) Attitude toward intertemporal inequity in risks. This issue must b e distinguished from t h e issue (ii) of concern f o r intertemporal equity in amounts. The
present issue can be described as follows. Consider two p a i r s of amounts z,
and zt
< z;
such t h a t (z, )
-
<zJ
(zt ) and (2,' )- (2; ). Then, t h e consequences (z, ,z; )
and (zJ ,zt ) a r e indifferent and have t h e same inequity in amounts. By t h e substitution principle, i t would follow t h a t t h e lottery
< (2,
,z; ), ( z i , z t ) > is indifferent
t o e i t h e r of the consequences (z, , ~), j (2,' ,zt ). However, t h e lottery might b e
p r e f e r r e d to, f o r example, t h e consequence (z, ,z; ) since t h e lottery presents
equal r i s k in both periods whereas t h e consequence (z, , z j ) is unfair t o t h e period
s (and thus t o t h e affected individuals in period s ) .
Such p r e f e r e n c e s are excluded by t h e expected utility conditions (G). Inequit y in r i s k f o r group decision problems is discussed, f o r example, in Broome (1982),
Harvey (1985c), Keeney (1980), Keeney and Winkler (1985), Kirkwood (1979), and
S a r i n (1985).
(ix) Regret, Allais paradoxes, and framing effects. These issues a r e excluded
by condition (G). Discussions of these issues may be found, f o r example, in Allais
and Hogen (1979), Bell (1983), (1985), Kahneman and Tversky
(1979), and
Schoemaker (1982). Models t h a t do not assume condition (G) are presented in Chew
(1983), Chew and MacCrimmon (19'79). Fishburn (1982), (1983), and Machina (1982).
(x) Time resolution of uncertainty.
This g r o u p of issues i s discussed by
Mayer (in Keeney and Raiffa 1976, Chapter 9) where s e v e r a l issues are identified,
including: (1) changes in personal or social p r e f e r e n c e s o v e r time, and (2) t h e
time at which uncertainty i s resolved.
The p r e f e r e n c e s of a f u t u r e society a r e included in a utility function of
Theorem 1only in t h a t the p r e f e r e n c e s of c u r r e n t society depend in p a r t on a conc e r n f o r such f u t u r e p r e f e r e n c e s ; t h e planning models in this p a p e r do not include
a probability distribution of f u t u r e preferences. A s e p a r a t e issue i s whether t h e
social p r e f e r e n c e s r e p r e s e n t e d in a specified utility function change with time.
For a p r e s e n t value discounting model, t h e tradeoffs between t h e y e a r s 2050 and
2051 are t h e same whether t h e model r e p r e s e n t s t h e p r e f e r e n c e s of c u r r e n t society or t h e p r e f e r e n c e s of society in t h e y e a r 2050.
For a r e l a t i v e value
discounting model, however, t h e r e i s more indifference between t h e y e a r s 2050 and
2051 if t h e model r e p r e s e n t s c u r r e n t p r e f e r e n c e s than if t h e model r e p r e s e n t s
p r e f e r e n c e s in t h e y e a r 2050.
The time at which uncertainty is resolved c a n be included in a planning model
by t h e usual backward induction of decision-tree analysis. However, t h e issue of
anxiety caused by prolonged uncertainty i s not included in t h e planning models in
this p a p e r . This issue of "anxiety along the way" i s discussed in t h e work of Meyer
cited above.
4. A b s o l u t e H u l t i p e r i o d Risk
This section and t h e next discuss p r e f e r e n c e conditions regarding t h e issue of
multiperiod risk. General conditions are introduced in Sections 4.1 and 5.1 to define t h e attitudes of neutrality, aversion, and proneness toward multiperiod r i s k ,
and special conditions are introduced in Section 4.2 and 5.2.
Each g e n e r a l or special p r e f e r e n c e condition i s shown t o be equivalent t o a
condition on t h e f o r m of t h e multiperiod risk function f in Theorem 1. In this
sense, t h e function f c a n be r e g a r d e d as t h a t p a r t of a utility function U(x1,x2, ...)
which encodes t h e multiperiod risk attitude f o r p r e f e r e n c e s among infinite-period
lotteries. Table 1 below lists all of t h e special conditions on multiattribute risk
and t h e corresponding special forms of t h e utility function U. The p r o p e r t i e s
within e a c h of p a r t s (I)-(IV) are equivalent.
Table 1: Conditions on Multiperiod Risk and Forms of U(z1,z2, ...)
(11)
(1)
(Absolute) Risk Neutrality (a)
Relative Risk Neutrality (a)
(Absolute) Risk Neutrality (b)
Relative Risk Neutrality (b)
U l i n e a r in v
U logarithmic in v
U additive in u
U log-exp in u
(111
(IV)
Absolute Risk Constancy
Relative Risk Constancy
Utility Independence
Timing Independence
U l i n e a r or exponential in v
U logarithmic or power in v
U additive or multiplicative in u
U log-exp or power-root in u
Both single-period r i s k attitudes and multiperiod r i s k attitudes c a n b e defined
by specifying a period t and considering p r e f e r e n c e s between those l o t t e r i e s having consequences (zt ) in which t h e amounts in t h e periods o t h e r than period t are
s t a n d a r d amounts I*.
For such l o t t e r i e s , t h e r e i s no uncertainty e x c e p t f o r t h e
outcome in period t .
For a single-period r i s k attitude, t h e consequences ( x t ) are measured directly in terms of t h e amounts zt in period t . In p a r t i c u l a r , t h e tradeoffs between zt
and t h e amounts in o t h e r periods are not considered. For a multiperiod r i s k attitude, however, t h e consequences ( z t ) are measured in terms of t h e amounts v ( z t )
by considering t h e t r a d e o f f s between t h e amounts zt in period t and t h e amounts in
o t h e r periods; then, changes in the period t will b e r e f e r r e d t o a s changes in
value.
4.1 Absolute multiperiod risk neutrality and aversion
The multiperiod risk attitudes of neutrality, aversion, and proneness may b e
defined by a number of equivalent conditions concerning changes in value. Two
such conditions are specified in this subsection. The following concepts will b e
needed f o r t h e f i r s t of t h e s e conditions.
Definition 2. P a i r s of amounts xt ,xi and yt ,yj in a period t will be said t o have
equal (absolute) changes in v a l u e provided t h a t
f o r a common p a i r of amounts z , z f in a n o t h e r period. In particular, a n amount
will be called t h e (absolute) t r a d e o n . mmidvalue of two amounts xt
that t h e p a i r s of amounts xt , z t and
zt ,xi
< x;
zt
provided
have equal changes in value.
A s a type of example, suppose t h a t p r e f e r e n c e s f o r intertemporal equity are
not included in the model (as is typically t h e case f o r cost-benefit models). Then,
two p a i r s of amounts xt ,xi and y t ,y ; have equal changes in value provided t h a t
xi = x t + h , y i
amounts xt
< x;
=
yt
+h
f o r t h e same change h , and t h e tradeoffs midvalue of two
is t h e ordinary average of xt and x;
.
Lemma 2. For a planning model of type (A)-@), any two amounts xt < xi in any
period t have a unique tradeoffs midvalue
zt. Any two p a i r s of
amounts x ,x ' and
y , y f in t h e interval I have equal changes in value when they are in a period t if
and only if they have equal changes in value when they a r e in any o t h e r period.
Thus, an amount E is t h e tradeoffs midvalue of a p a i r of amounts x
z , 5 , x f a r e in a period t if and only if
x ,G,x ' are in any o t h e r period.
z i s t h e tradeoffs
midvalue of x
< x'
< x'
when
when
Suppose t h a t a single period 1 h a s been chosen, and t h e tradeoffs midvalue
of a p a i r of amounts xt
< xi
zt
has been assessed. Multiperiod r i s k attitudes can be
defined as in p a r t s (a) of Theorem 2 below by comparing t h e "average" consequence (<)with t h e "risky" lottery < ( z t ) , ( x ;)>. For example, muLtiperiod r i s k
a v e r s i o n can b e defined as t h e condition t h a t ( S t ) i s p r e f e r r e d to < ( x t ) , ( x i)>. In
applications, i t may often b e convenient t o choose t h e period 1 as t h e f i r s t period,
1 =I.
These definitions are similar to t h e definitions in Dyer and S a r i n (1982) comparing s t r e n g t h of p r e f e r e n c e midvalues and r i s k midvalues (i.e.,
certainty
equivalents). The difference t o be emphasized i s t h a t t h e above definitions do not
r e q u i r e a p r e c i s e assessment of t h e c e r t a i n t y equivalent of <(zt), (2; ) >; instead,
(Zt ) and <(zt) , ( x i )> are compared directly.
Multiperiod r i s k attitudes can also b e defined by considering consequences of
t h e form ( z , ,zt) f o r two periods s and 1. Suppose t h a t t h e periods s and 1 have
been chosen, and two p a i r s of amounts z , , x ;
and z t , z ; have been assessed such
that
z,
< z ; , z t < z;
and
( z t1, ( 2 ; I N ( 2 ; )
(2,)
.
(5)
These indifference relations imply t h a t t h e two infinite-period consequences
(z,,z;)and
(2;
, z t )are indifferent.
A s one type of example, if s and t are chosen to b e adjacent periods in t h e
distant f u t u r e and t h e f u t u r e i s valued in a c c o r d with t h e conditions of relative
timing p r e f e r e n c e s (so t h a t a, / at a I ) , then xt and
imately zt = z, and
zi
c a n b e assessed as approx-
zi = z; .
Multiperiod r i s k attitudes c a n b e defined as in p a r t s (b) of Theorem 2 by comparing t h e "average" consequences
< ( z , ,zt) , ( x i
(z,,zi ),
) >. For example, muLtiperiod r i s k a v e r s i o n c a n be defined as
t h e condition t h a t t h e consequences ( z ,,xi ),
< ( x , ,zt) , ( z d
(z; , z t ) with t h e "risky" lottery
>.
(zi , z t )are p r e f e r r e d
to the lottery
These definitions are similar t o t h e definitions in R i c h a r d (1975) of multivari-
ate r i s k n e u t r a l , s t r i c t l y multivariate r i s k a v e r s e , and s t r i c t l y multivariate r i s k
seeking. The d i f f e r e n c e i s t h a t in Richard's definitions t h e consequences (z, , ~), j
( x i , z t ) are not a s s e s s e d t o b e indifferent, and t h e l o t t e r y
compared with
< (z,
, z t ) , (z,'
< (z,
, z j ), (z,'
,zt ) > is
, x i ) >. In t h e terminology of F a r q u h a r (1984), t h e
comparisons in R i c h a r d ' s definitions are evaluated by a paired-gamble method
while t h e comparisons in t h e above definitions are evaluated by a standard-gamble
method.
Theorem 2. F o r a planning model of t y p e (A)-(G), t h e p r o p e r t i e s within e a c h of t h e
following p a r t s (I)-(111) are equivalent.
I. MuLtiperiod r i s k n e u t r a l i t y :
(a) T h e r e e x i s t s a p e r i o d t such t h a t f o r a n y amounts zt
( z t ) i s indifferent t o t h e l o t t e r y
< z;
t h e consequence
< ( z t ),( z j ) >.
(b) T h e r e e x i s t two p e r i o d s s a n d t such t h a t f o r a n y amounts as in (5) t h e
consequences (z, , z j )
-
(2: , z t ) are indifferent t o t h e l o t t e r y
< (z,
, s t ), (2,' , z j ) >.
(c) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (2) in which t h e mulj i s l i n e a r , t h a t is,
tiperiod r i s k function '
where t h e t r a d e o f f s function v i s to be assessed. H e r e , v i s normalized so t h a t
v (z*) = 0 .
(d) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (4) of t h e form
where t h e first-period utility function u i s t o be assessed. H e r e , u i s normalized
so t h a t u (z* ) = 0.
11. MuLtiperiod r i s k a v e r s i o n :
< z;
(a) T h e r e e x i s t s a p e r i o d t such t h a t f o r any amounts zt
(Zt ) i s p r e f e r r e d t o t h e l o t t e r y
t h e consequence
< ( z t ),(Zi) >.
(b) There e x i s t two p e r i o d s s and t such t h a t f o r any amounts as in (5) t h e
consequences (z, , z i )- (zd,zt ) are p r e f e r r e d t o t h e l o t t e r y
< (z,
,zt ),(zd , z i ) >.
(c) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (2) o r (4) in which multiperiod r i s k function 'j i s s t r i c t l y concave.
ITI. Multiperiod r i s k proneness:
P r o p e r t i e s analogous to 11. (a)-(c) c a n also be specified.
The salient f e a t u r e of t h e t w o sets (a), (b) of conditions on multiperiod r i s k attitudes discussed in t h i s subsection i s t h a t t h e y r e q u i r e t r a d e o f f s assessments.
P a r t of t h e assessment task i s t h e r e b y shifted from r i s k assessments t o tradeoffs
assessments.
4.2 A b s o l u t e m u l t i p e r i o d risk c o n s t a n c y
This section discusses t w o equivalent conditions o n multiperiod r i s k attitudes
t h a t c o r r e s p o n d to t h e condition on single-period risk a t t i t u d e s of constant r i s k
aversion (Arrow 1971, Pfanzagl1959, and P r a t t 1964).
D e f i n i t i o n 3. P r e f e r e n c e s will b e said to b e absolute multiperiod r i s k constant
f o r a p e r i o d t provided t h a t f o r any t h r e e p a i r s of amounts wt
yt
< yi
< w; , zt < zj
, and
t h a t have equal absolute changes in value, if ( z t ) i s t h e c e r t a i n t y
equivalent of <(wt ), (yt )>, t h e n ( z j ) i s t h e c e r t a i n t y equivalent of <(wi ), ( y i
One p r o c e d u r e f o r verifying this condition i s t o s e l e c t amounts wt
< yt
)>.
and
assess t h e c e r t a i n t y equivalent (zt ) of <(wt ),(yt )>. Then, consider a n amount
z;
yt
> zt ,
< y;
assess two amounts w i
,yi
s u c h t h a t t h e p a i r s wt
< w i , zt < z; ,
and
have equal absolute changes in value, and a s k whether (Zj ) i s t h e certain-
t y equivalent of <(w/ ),( y i ) >.
A second condition equivalent to t h a t of absolute multiperiod r i s k constancy is
t h e condition of utility independence introduced by Pollak (1967) f o r multiperiod
models and by Keeney (1968), (1974) f o r multiattribute models. Consider those lot-
t e r i e s having consequences of t h e form ( x t , z ) where z i s a n amount in a fixed
period s o t h e r than period t . Period t i s said to be u t i l i t y i n d e p e n d e n t of t h e
period
s
provided
that
(xt , z )
< (wt , z ' ) , ( y t ,z') > f o r any amount
2 < (wt , z ) , (yt , z ) >
implies
that
(xt , z ') ;?
z ' in period s, t h a t is, t h e attitude toward risk
f o r period t i s independent of t h e fixed amount z in t h e period s.
For a finite-period model or a multiattribute model, t h e equivalence of prop e r t i e s similar to (b) and (c) below i s discussed by Meyer and P r a t t (in Keeney and
Raiffa 1976, p. 330) and in Pollak (1967), and t h e equivalence of p r o p e r t i e s similar
to (b) and (d) below i s discussed in Keeney (1968), (1974) and Meyer (1970).
Theorem 3. For a planning model of type (A)-(G), t h e following p r o p e r t i e s are
equivalent:
(a) There e x i s t s a period t f o r which p r e f e r e n c e s are absolute multiperiod
r i s k constant.
(b) There e x i s t periods s and t t h a t are utility independent.
(c) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (2) in which t h e function
f i s linear or exponential, t h a t is,
where t h e tradeoffs function v and t h e constant r are to be assessed. Here, v i s
normalized so t h a t v (x* )
= 0.
(d) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (4) t h a t is additive or
multiplicative, t h a t i s
where t h e first-period utility function u and t h e constant a
sessed. Here, u i s normalized so that: u (x*)
>0
are t o be as-
= 1 in t h e f i r s t case, u ( x * ) = 0 in
the
second
case,
and
u(z*) = -1
in
the
third
case.
Moreover,
a ( u ( z ) - 1 ) + I > O f o r a l l z i n 1 i n t h e f i r s t case, and a ( u ( z ) + 1 ) -1 < O f o r a l l
z in I in the t h i r d case.
Note: A s is w e l l known, a utility function in p a r t s (c), (d) is multiperiod risk
a v e r s e in t h e third cases and i s multiperiod risk prone in t h e f i r s t cases.
5. Relative Mdtiperiod Risk
This section discusses p r e f e r e n c e conditions on relative ( o r proportional)
changes in one o r more periods. These p r e f e r e n c e conditions are analogous t o the
p r e f e r e n c e conditions discussed in the preceding section on absolute changes.
P r e f e r e n c e conditions on relative changes a p p e a r t o b e more realistic than
p r e f e r e n c e conditions on absolute changes f o r many problems, in particular, f o r
those problems in which t h e degree of risk aversion f o r one period d e c r e a s e s as
the amounts in t h e o t h e r periods become more favorable.
Consider a planning model of type (A)-(G) having a least p r e f e r r e d standard
amount z*. Relative changes will b e measured with r e s p e c t t o z* as t h e amount
having "zero value." For example, a n amount z:
will b e said t o have "one-half the
value" of an amount z-t2
provided t h a t z: is t h e tradeoffs midvalue of z * and
~ zt2 .
Typically, z* will b e less than any amount of zt t h a t has an appreciable
chance of occurring. A s a n illustration, suppose t h a t zt describes t h e amount of
consumption of a non-renewable r e s o u r c e (e.g., helium) during period t . Then, z*
might b e chosen as the outcome of no consumption o r of a specified minimal level of
consumption of t h e r e s o u r c e being studied.
5.1 Relative mdtiperiod risk neutrality and aversion
This subsection discusses risk attitudes involving relative changes in value
t h a t are analogous to t h e risk attitudes of neutrality, aversion, and proneness discussed in Section 4.1.
Relative changes in value can be defined by means of t h e concept of a "stand a r d sequence." Krantz et al. (1971) contains an extensive discussion of this idea
in a more general setting. An increasing sequence of amounts
in a period t will be called a s t a n d a r d sequence provided t h a t the pairs
zf -I, z f , k
= 1 ,...,n , have equal absolute changes in value (Definition 2 ) .
For a planning model of type (A)-@'),a sequence of amounts is a standard sequence if and only if v ( z f )
- v (zf -I) is constant f o r k
= 1,...,n . Thus, a sequence
of amounts is a standard sequence in a period t if and only if i t is a standard sequence in any o t h e r period.
Definition 4. P a i r s of amounts zt , z ; and y t ,y ; not equal t o z* will b e said t o
have equaL reLative changes i n v a l u e provided t h a t f o r any two standard sequences
zp<z;<...<zT
with t h e initial amounts zp
then zj
< z? if
and y p < y ; < . . . < y p
= y p = z * , if
and only if yj
< yp.
zt
=
zr
and yt = y r f o r some m
<n,
In particular, an amount Zt will be called the
reLative tradeofls midvaLue of two amounts zt
< z;
provided t h a t t h e p a i r s of
amounts zt ,Zt and Zt ,zi .have equal relative changes in value.
Lemma 3. For a planning model of type (A)-@'),
yt , y j
two p a i r s of amounts zt.z; and
have equal relative changes in value if and only if v (2; )
v ( y ; ) = p v ( y t ) f o r t h e s a m e constant p
two amounts zt
f o r t h e same p
> 0.
=pv
(zt) and
The relative tradeoffs midvalue of
< zj is t h e amount zt such t h a t v (St ) = p v (zt) and v (2; ) = p v (zt)
> 0. The statements of Lemma 2 are t r u e with the adjective "rela-
tive" added t o the terms "equal changes in value" and "tradeoffs midvalue."
Suppose t h a t a period t has been chosen and t h e relative tradeoffs midvalue
St of two amounts zt
< zj
has been assessed. Relative multiperiod risk attitude
can b e defined as in p a r t s (a) of Theorem 4 below by comparing t h e "average"
consequence
(st)with
the l o t t e r y
< (zt),(zj ) >.
The situation is analogous to t h a t
f o r absolute multiperiod r i s k attitudes. In particular, the discussion in Section 4.1
on standard-gamble methods of verification is equally pertinent in t h e present context.
The condition (a) below of relative multiperiod risk neutrality corresponds t o
t h e single-period risk attitude of logarithmic utility (Harvey 1981, 1986b).
Relative multiperiod r i s k attitudes can also b e defined by considering t h e
same amount in s e v e r a l different periods. For technical reasons, we will consider
not only actual periods t but also imaginary periods t having a r b i t r a r y timing
weights, 0
< at < 1. With
al timing weights, at , t
t h e non-restrictive assumption t h a t t h e sequence of actu-
= 1 3 ,..., decreases t o z e r o as t increases, these imaginary
periods correspond t o actual periods f o r a different choice of t h e unit of time.
For a period t
> 1 (actual o r
imaginary), a period m where 1 < m
<t
will be
called t h e temporal m i d v a l u e of the f i r s t period and period t provided t h a t t h e
tradeoffs between periods 1 and m coincide with the tradeoffs between periods m
and t (Harvey 1986a). For example, in a present value discounting model t h e temporal midvalue is t h e arithmetic mean, i.e., t h a t period m such t h a t m
t =m
+ h f o r t h e same increase h .
In a relative value discounting model t h e t e m -
poral midvalue is the geometric mean, i.e., t h a t period m such that m
t =k
.m
= 1 + h and
= k . 1 and
f o r the same proportional increase k .
Consider a n amount z in one of t h e t h r e e periods 1 < m
temporal midvalue of 1and t
.
<t
where m is the
Relative multiperiod risk attitudes can be defined as
in p a r t s (b) of Theorem 4 by comparing t h e "average" consequence (2,)
with the
"risky"1ottery < ( z l ) , ( z t ) > w h e r e z = z l =z, = z t .
Theorem 4. For a planning model of type (A)-(G) having a least p r e f e r r e d standard amount, t h e p r o p e r t i e s within each of t h e following p a r t s (1)-011) a r e
equivalent.
I. ReLative m u t t i p e r i o d r i s k n e u t r a l i t y :
< zi
) >.
(a) There exists a period t such t h a t f o r any amounts zt
t h e consequence (St) is indifferent t o t h e lottery
< (zt),(z;
not equal t o z*
(b) For any period t
z * , t h e consequence (2,)
> 1 (actual
o r imaginary) and any amount z not equal t o
i s indifferent t o the l o t t e r y
temporal midvalue of 1and t , and z
< ( z l ) , ( z t ) > where
m is the
= z l = z, = zt.
(c) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (2) in which t h e function
f i s logarithmic, t h a t is,
where t h e tradeoffs function v i s t o be assessed. Here, v is normalized s o t h a t
v(z*)
= 0.
(d) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (4) of t h e form
where t h e first-period utility function u and t h e constant c
Here, u (z*) =
> 0 are t o be assessed.
-- and u (z+) = +-.
11. ReLative muLtiperiod r i s k a v e r s i o n :
(a) There exists a period t such t h a t f o r any amounts zt
t h e consequence (St) i s p r e f e r r e d t o t h e lottery
(b) For any period t
> 1 (actual
not equal t o z *
< ( z t ) , ( z i ) >.
o r imaginary) and any amount z not equal t o
z * , t h e consequence (z, ) i s p r e f e r r e d t o t h e lottery
temporal midvalue of 1and t , and z
< z;
< (zl), (zt ) > where
m is the
= zl = z, = zt .
(c) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (2) o r (4) such t h a t the
composite function f oexp i s s t r i c t l y concave.
111. ReLatiue muLtiperiod r i s k proneness:
P r o p e r t i e s analogous t o 11. (a)-(c) can also be specified.
5.2 R e l a t i v e m u l t i p e r i o d r i s k constancy
This subsection discusses two equivalent conditions on multiperiod risk attitudes t h a t correspond t o t h e single-period risk attitude of linear r i s k aversion
(Harvey 1981, 1986b) and constant proportional risk aversion ( P r a t t 1964).
Definition 5. P r e f e r e n c e s will be said t o be r e l a t i v e m u l t i p e r i o d r i s k c o n s t a n t
f o r a period t provided t h a t f o r any t h r e e p a i r s of amounts wt
yt
< yi
< w; , xt < xi
, and
t h a t have equal r e l a t i v e changes in value, if ( x t ) i s t h e c e r t a i n t y
equivalent of
< (wt ), (yt ) >, then
(x; ) is t h e c e r t a i n t y equivalent of
< (w;
),(y; ) >.
This p r e f e r e n c e condition i s similar t o t h e multiattribute r i s k condition of
proportional utility dependence defined in Harvey (1984). I t may be verified by
means of t h e procedure t h a t i s described in Section 4.2 f o r t h e condition of absolute multiperiod r i s k constancy; t h e only modification i s t h a t h e r e t h e p a i r s
wt
< w j , xt < x i
and yt
< yi
have equal relative changes in value r a t h e r than
equal absolute changes in value.
A second condition equivalent t o t h a t of r e l a t i v e multiperiod r i s k constancy
c a n b e defined as follows. For e a c h period, t
= 1.2, ..., t h e p r e f e r e n c e relation 2
on infinite-period l o t t e r i e s induces a p r e f e r e n c e relation
on those l o t t e r i e s
having consequences of t h e type (xt ). For t h e discussion of utility independence in
Section 4.2, t h e issue was whether t h e p r e f e r e n c e relation
3
i s t h e same as a
p r e f e r e n c e relation on l o t t e r i e s having consequences (xt , z ) where z # z* i s a
fixed amount in a n o t h e r period. Now, consider t h e issue of whether t h e p r e f e r ence relation
& i s t h e s a m e as t h e p r e f e r e n c e relation
as f o r a n o t h e r period
s.
I t will b e said t h a t t h e periods a r e t i m i n g i n d e p e n d e n t provided t h a t f o r any two
periods s and t (actual o r imaginary) t h e p r e f e r e n c e relations
at and 2,
are t h e
same.
For a multiattribute model, t h e equivalence of p r o p e r t i e s similar to (a) and (d)
below i s discussed in Harvey (1984); f o r a group decision model, t h e equivalence of
p r o p e r t i e s similar t o (b) and (d) i s discussed in Harvey ( 1 9 8 5 ~ ) .
Theorem 5. For a planning model of type (A)-(G) having a l e a s t p r e f e r r e d stand a r d amount, t h e following p r o p e r t i e s are equivalent:
(a) There exists a period t f o r which p r e f e r e n c e s are relative multiperiod
risk constant.
(b) The periods are timing independent.
(c) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (2) in which t h e function
j
' i s logarithmic o r power, t h a t is,
where t h e t r a d e o f f s function v a n d t h e constant r are t o b e assessed. H e r e , v i s
normalized so t h a t v (z* )
= 0.
(d) P r e f e r e n c e s are r e p r e s e n t e d by a utility function (4) t h a t i s logexponential or power-root,that is,
where t h e first-period utility function u and t h e constants c
assessed. Here, u i s normalized s o t h a t u (z*)
> 0 and r are
t o be
= 0 in t h e f i r s t c a s e , and u (Z +) = 0
in t h e t h i r d case.
Note: I t follows from Theorem 2 and 4 t h a t a utility function in p a r t s (c), (d) is
multiperiod r i s k a v e r s e if r
< 1and i s r e l a t i v e
multiperiod r i s k a v e r s e if r
< 0.
6. Hultiperiod Tradeoffs
This section d e s c r i b e s f o u r p r e f e r e n c e conditions on multiperiod tradeoffs.
Each condition i s equivalent t o a s p e c i a l form f o r t h e t r a d e o f f s function v and
hence to s p e c i a l forms f o r the utility functions (2) and (3) t h a t contain v . The spec i a l forms f o r (2) are immediate in t e r m s of v whereas t h o s e f o r (3) r e q u i r e t h e
calculation of v
These r e s u l t s follow by combining r e s u l t s in Harvey (1985a).
(1986a), and hence are only summarized h e r e . The terminology is chosen t o b e
consistent with t h a t in Sections 4, 5 and t h a t in Harvey (1986b).
Consider a p a i r of base amounts z , and zt in two periods s , t where z , is
fixed and t h e e f f e c t s of changes in z t a r e t o b e examined. For a change w , in
period s , t h e corresponding change wt in period t such t h a t
is called the
"t
tradeons amount
f o r the change w , ,
and i s denoted by
=S(ws;s,t,zt).
If w t does not depend on t h e base amount z t , t h a t is, w t = W ;
f o r any
W;
= f ( w , ; s , t ,zi ), then t h e r e is said t o be absolute tradeons independence. If
wt
-W;
depends only on the difference zt
- z ; and on the
tradeoffs amount w t ,
then t h e r e is said t o be absolute tradeofls constancy.
P r e f e r e n c e s are in accord with the condition of absolute tradeoffs independence if and only if t h e r e i s a tradeoffs function of the form
where z* is finite. Then, the utility function ( 3 ) has t h e special form
P r e f e r e n c e s are in a c c o r d with the condition of absolute tradeoffs constancy
if and only if t h e r e is a tradeoffs function of the form
where z* can b e -= if q
is ( 6 ) if q
= 0 o r if
q
> 0 and can
be
+= if
q
< 0. Then, t h e
utility function ( 3 )
# 0 i s of t h e special form
I t is also possible t o consider relative changes in the amounts z t , o r more
generally relative changes in amounts yt
= zt + c > 0 where c is a constant t o be
specified o r evaluated. The amounts yt can be interpreted e i t h e r as sums, i.e.,
zt
+ z 0 where
z0=
-C
z0 = c i s a n "asset position," or as differences, i.e., zt
- z0 where
i s a "point of ruin."
Suppose t h a t f o r a b a s e amount yt = zt
+c
in p e r i o d t and a change w , in
period s t h e r e i s a p r o p o r t i o n pt such t h a t
(z,, Zt + p t y t ) " ( 2 ,
+ w,, z t ) .
Then, pt i s called t h e tradeofls proportion f o r t h e change w,, and i s denoted by
Pt = g ( w , ; s , t , z t ) .
If pt does n o t depend on t h e b a s e amount z t , t h a t i s , pt = p i f o r any
pi
= g (w,; s ,t ,zi
pt
- pi
), t h e n t h e r e i s said t o b e relative tradeofls independence. If
depends only on t h e r a t i o yt / y i and on t h e t r a d e o f f s p r o p o r t i o n pt , then
t h e r e i s said to b e relative tradeofls constancy.
P r e f e r e n c e s are in a c c o r d with t h e condition of r e l a t i v e t r a d e o f f s independ e n c e if and only if t h e r e i s a t r a d e o f f s function of t h e form
where -c
< z * < +a. Then, t h e utility
function ( 3 ) h a s t h e s p e c i a l form
P r e f e r e n c e s are in a c c o r d with t h e condition of r e l a t i v e t r a d e o f f s constancy
if and only if t h e r e i s a t r a d e o f f s function of t h e form
(zt + c ) q
- (z* + c)q ,
q
>0
v ( z t )= ' log((zt + c ) / ( z * + c ) ) , q = O
-(zt
+ c)q + ( z * + c)q
,
q
<0
I
where z* c a n b e -c if q
i s ( 8 ) if q = 0 or if q
> 0 and
can b e
+a
if q
# 0 i s of t h e s p e c i a l form
< 0. Then,
t h e utility function ( 3 )
E a c h of t h e a b o v e conditions on multiperiod t r a d e o f f s i s equivalent t o a condition on a t t i t u d e toward i n t e r t e m p o r a l equity.
F o r example, t h e r e i s absolute
t r a d e o f f s independence if a n d only if t h e r e i s a n a t t i t u d e of intertemporal i n e q u i -
t y n e u t r a l i t y , t h a t i s , a n y two indifferent consequences (2: , z t ) and (z, ,z; ) are
1
indifferent t o t h e "equable" consequence (?(I,
+ z;
). +(zt
2
+ zi )).
7. First-Period Risk
This s e c t i o n briefly d e s c r i b e s f o u r p r e f e r e n c e conditions on r i s k a t t i t u d e s
f o r consequences ( z l ) having a n amount z l in t h e f i r s t p e r i o d a n d s t a n d a r d
amounts z * in t h e o t h e r periods. These conditions are r e s t a t e m e n t s (with new terminology) of t h e conditions on single-attribute r i s k a t t i t u d e s t h a t are r e f e r e n c e d
in Section 4.2 a n d 5.2. E a c h condition i s equivalent t o a s p e c i a l form f o r t h e
first-period utility function u , a n d h e n c e t o s p e c i a l forms f o r t h e utility functions
(3) and (4).
F o r two amounts zl and
p o i n t of z l and
zi
t h e r i s k midpoint
Zi
defined by
sl of z l , z i
in t h e f i r s t period, l e t
-
(si) < ( z l ) , ( z i )
>.
sl denote t h e r i s k
mid-
If f o r any amounts z l and
zi
i s t h e a v e r a g e of z a n d z i , t h e n t h e r e i s said t o b e
first-period r i s k n e u t r a l i t y . If f o r any (absolute) c h a n g e h , t h e r i s k midpoint of
zl
+h
and
Zi +h
is
sl + h , t h e n t h e r e i s said t o b e absolute first-period
risk
constancy.
T h e r e i s first-period r i s k n e u t r a l i t y if a n d only if u ( z l )
= z i s a first-period
utility function. T h e r e i s first-period r i s k constancy if a n d only if t h e r e i s a
first-period utility function of t h e form
f o r some amount of t h e p a r a m e t e r r .
Conditions on first-period r i s k a t t i t u d e s c a n a l s o b e defined b y considering
p r o p o r t i o n a l c h a n g e s in t h e amounts y l = zl
+ c > 0 where
specified o r evaluated. If f o r any amounts zl and z;
c i s a constant t o b e
t h e r i s k midpoint of zl and
z; i s t h a t amount
such t h a t
+ c = k ( z l + c ) and z; + c = k(G1 + c ) f o r t h e
same proportion k , then t h e r e i s said t o be relative first-period r i s k n e u t r a l i t y .
If f o r any relative change k t h e risk midpoint of z
is
+ k ( z + c ) and z i + k ( z i + c )
sl + k ( z l + c ) , then t h e r e i s said t o be r e l a t i v e f i r s t - p e r i o d
There i s relative
log(zl
+ c)
first-period
r i s k neutrality
r i s k constancy.
if and only if
u (zl) =
i s a first-period utility function. There is relative first-period risk
constancy if and only if t h e r e is a first-period utility function of t h e form
f o r some amount of t h e p a r a m e t e r r
8. Examples of Planning Models
This section specifies a number of planning models t h a t include t h e following
p r e f e r e n c e issues: (i) t h e importance of t h e f u t u r e , (ii) intertemporal equity, (iii)
first-period r i s k , and (iv) multiperiod risk. P r e f e r e n c e conditions from Sections
4-7 are assumed concerning t h e issues (ii)-(iv). For each of t h e resulting models,
e i t h e r t h e p r e f e r e n c e condition of p r e s e n t value discounting o r t h a t of relative
discounting can be used. Thus, i t is not necessary t o discuss h e r e t h e issue (i) of
t h e importance of t h e future, even though this issue i s included in t h e models.
One planning model i s t h a t in which t h e r e is absolute tradeoffs independence,
first-period risk neutrality, and multiperiod risk neutrality. Such p r e f e r e n c e s
are r e p r e s e n t e d by t h e utility function
t h a t i s often used in cost-benefit studies and risk analysis studies.
This model may be r e g a r d e d e i t h e r as excluding all t h r e e of t h e p r e f e r e n c e
issues (ii)-(iv) o r as formulating these issues in t h e simplest possible manner.
From e i t h e r viewpoint, t h e model i s as simple as possible.
The next simplest models would b e those t h a t assume a p r e f e r e n c e condition
o t h e r than t h e above f o r only one of t h e issues (ii)-(iv). However, such models are
inconsistent according to t h e following result.
Theorem 6. For a planning model of type (A)-(G), any two of t h e p r e f e r e n c e conditions: absolute tradeoffs independence, first-period r i s k neutrality, and multiperiod r i s k neutrality, imply t h e third.
Thus, any model t h a t assumes a n o t h e r condition f o r one of t h e issues (ii)-(iv)
must assume a n o t h e r condition f o r at l e a s t one o t h e r of these issues. First, consid-
e r those models which include t w o of t h e issues (ii)-(iv) in t h e sense t h a t only one
of the conditions in Theorem 6 i s assumed.
Theorem 7. For a planning model of type (A)-(G):
(a) If t h e r e i s multiperiod risk neutrality, then t h e attitude toward intertemporal equity coincides with t h e attitude toward first-period r i s k , t h a t is, v
= u.
(b) If t h e r e i s absolute tradeoffs independence, then t h e attitude toward multiperiod r i s k coincides with t h e attitude toward first-period r i s k , t h a t is, f
=u.
(c) If t h e r e i s first-period r i s k neutrality, then t h e attitudes of intertemporal
inequity aversion and multiperiod risk aversion are inconsistent.
As a n example of p a r t (a), consider a planning model having a n intermediate
s t a n d a r d amount. The p r e f e r e n c e conditions of: (1) multiperiod r i s k neutrality and
(2) e i t h e r relative tradeoffs independence or relative first-period risk neutrality,
imply t h a t p r e f e r e n c e s are r e p r e s e n t e d by t h e utility function
U(x1,z2, ...)
= log-
zl+c
x* +c
+
a 21og
z2+2
z
+...
(11)
A s a n example of p a r t (b), consider a planning model having a least p r e f e r r e d
s t a n d a r d amount. The p r e f e r e n c e conditions of: (1) absolute tradeoffs independence and (2) e i t h e r relative multiperiod r i s k neutrality or relative first-period
r i s k neutrality with r e s p e c t to y t
= xt
- z * imply t h a t p r e f e r e n c e s a r e r e p r e s e n t -
e d by t h e utility function
U(z1,z2, ...) = log( ( z l
-z*) + a2(z2-z*) + - - . 1
.
Next consider those planning models which include all t h r e e of t h e p r e f e r e n c e
issues (ii)-(iv) in t h e sense t h a t none of the t h r e e p r e f e r e n c e conditions in
Theorem 6 a r e assumed.
A s a n example of such a planning model having a n intermediate standard
amount, consider t h e following conditions:
(1) There is absolute multiperiod risk constancy ( o r utility independence), and
t h e r e i s multiperiod risk aversion.
(2) There is relative tradeoffs independence with r e s p e c t t o yt
= zt + c .
(3) There is relative first-period risk constancy with r e s p e c t t o yt
= zt + c ,
and t h e r e is relative first-period risk aversion.
Theorem 8. Any two of t h e above conditions (1)-(3) imply t h e third. P r e f e r e n c e s
satisfy t h e conditions (1)-(3) if and only if p r e f e r e n c e s a r e r e p r e s e n t e d f o r some
r
< 0 by
a utility function of t h e form
U ( z l , z 2,...)
= - e x p ( r [log-
Z1+C
z * +C
+a210g-
Note: Here, t h e attitude toward risk in a period
function u t ( z t )
= -(zt
Z 2+C
z* +C
+...I)
.
(13)
t is represented by t h e utility
+ ~ ) ' ~ ' , r< 0. Thus, the d e g r e e of
risk aversion is less f o r
periods in the more distant future.
A s a n example of a planning model having a least p r e f e r r e d standard amount
such t h a t all t h r e e of the issues (ii)-(iv) a r e included, consider the following conditions:
(1) There is relative multiperiod r i s k neutrality.
(2) There is relative tradeoffs constancy with r e s p e c t t o yt
= zt
- z*
(3) There is relative first-period risk neutrality.
Theorem 9. Any two of t h e above conditions (1)-(3) imply t h e third. P r e f e r e n c e s
satisfy t h e conditions (1)-(3) if and only if preferences a r e r e p r e s e n t e d f o r some
q
> 0 by
a utility function of t h e form
Note:
Here,
u (zt ) = log(zt
the
-z*)
periods are timing
independent
(Section 5.2),
and
i s a utility function f o r any period t conditional on z,
= z*
f o r all s # t . The d e g r e e of r i s k aversion in a period t is less f o r fixed amounts in
the o t h e r periods t h a t are more p r e f e r r e d . There i s a n attitude of intertemporal
inequity aversion if and only if 0
< q < 1.
A p p e n d i x A: G e n e r a l Planning M o d e l s
This appendix lists t h e p r e f e r e n c e conditions f o r t h e expected-utility planning model in Theorem 1 and provides a proof of t h a t result. The definition of t h e
set C* and t h e conditions (A)-(E) are as in Harvey (1986a). Conditions (F), (G) are
added in t h e p r e s e n t paper.
The p r e f e r e n c e relation 2 defined on the set L of infinite-period l o t t e r i e s i s
assumed t o be a p a r t i a l o r d e r , i.e., reflexive and transitive, and t o b e s t r i c t l y increasing in t h a t l a r g e r amounts zt in e a c h period, t
= 1.2, ..., are p r e f e r r e d .
The following definition specifies t h a t subset of infinite-period consequences
on which tradeoffs between t h e periods are considered.
Definition A l .
c
The set C* consists of those infinite-period
consequences
= (z1,z2, ...) in C such that:
(a) For any z i > z l, t h e r e exists an N such t h a t
(b) For any
-
cn
<zl,
t h e r e exists a n N such t h a t
= ( ~ , zi2,...,2, , z * ~+ l , ~ * n+2 s...) 2 c f o r all n
2N .
The tradeoffs conditions (A)-(E) below are used t o establish t h e value function
(1) f o r infinite-period consequences in the set C* .
(A) The p r e f e r e n c e relation
,> i s a complete ordering on t h e set C*.
(B) The p r e f e r e n c e relation 2 i s continuous on P in e a c h variable.
(C) Each p a i r of adjacent variables zt , z t +1, t
dependent on C* of t h e o t h e r variables.
= 1.2, ..., i s preferentially in-
(D) Each consequence c = ( z l , z 2...)
, in C t h a t s a t i s f i e s t h e following conditions i s in t h e set C*.
(a) F o r any z ; > z
C,
t h e r e exists a n N such that
+ 2 C , = ( Z ,..., z , ,z*, + l r ~ *+2,...)
n
f o r all m ,n
(b) F o r a n y
2N .
z i < z l , t h e r e exists a n N such that
-
c , )?c,
= ( z l,...,z n , z * n + l , z * n + 2,...) f o r a l l m , n > = N .
( E ) Any two p e r i o d s s , t have equal t r a d e o f f s comparisons, i.e., f o r a n y two
p a i r s of amounts z 1 < z 2 and z 3 < z 4 in I , if s o c i e t y i s willing t o tradeoff more t o
i n c r e a s e z, from z 1 t o z 2 t h a n t o i n c r e a s e z, from z 3 t o z 4 in period s , t h e n soc i e t y i s willing t o tradeoff more t o i n c r e a s e zt from z 1 t o z 2 t h a n to i n c r e a s e zt
from z t o z in p e r i o d t .
Lemma A l . A planning model satisfies t h e conditions ( A ) - @ ) on t r a d e o f f s if and
only if t h e p r e f e r e n c e r e l a t i o n 2 r e s t r i c t e d t o t h e set C* of consequences is
r e p r e s e n t e d by a value function ( 1 ) as d e s c r i b e d in Lemma 1 where f o r a n intermediate amount z * t h e r a n g e of t h e t r a d e o f f s function v i s (--,-),
f o r a least pre-
f e r r e d amount z * t h e r a n g e of v i s [ O , - ) , a n d f o r a most p r e f e r r e d amount z * t h e
r a n g e of v i s (--,O].
Proof. F o r a model having a n intermediate s t a n d a r d amount, it i s shown in Harvey
(1986a) t h a t conditions (A)-(E)are equivalent t o t h e e x i s t e n c e of a value function
as d e s c r i b e d in Lemma 1. F o r a model having a n e x t r e m e s t a n d a r d amount, a simil a r argument c a n b e used to establish t h i s equivalence. Thus, f o r e a c h t y p e of
planning model, i t remains t o show t h a t condition (F) i s equivalent t o t h e p r o p e r t y
t h a t t h e r a n g e of v i s t h e a p p r o p r i a t e i n t e r v a l
(-a,=),
[O,-),
or (--,O].
F o r a model having a n intermediate s t a n d a r d amount, t h e r e e x i s t s amounts
z
> z*
and z
<z*
in t h e open i n t e r v a l I. Condition (F) implies t h a t f o r any conse-
quence c = ( zl , z z , z * 3,z* 4 , . . . ) t h e r e e x i s t s a n z i in I s u c h t h a t ( zi ) -- c , t h a t is,
v
( z i )= v ( z l )+ a2v( z 2 ) .Choosing
a fixed z z < z * o r z z > z * , i t follows t h a t t h e
r a n g e of v i s unbounded a b o v e a n d below. Thus, since v i s continuous t h e r a n g e of
v is
c
(--,a).Conversely,
if t h e r a n g e of v i s
(--,a),t h e n
f o r a n y consequence
= (z1,z2
...), with a c o n v e r g e n t value function V ( c ) t h e r e e x i s t s a consequence
( z ;) with v ( x i )
= V ( c ) , a n d t h u s condition (F) i s satisfied.
F o r a model having a n e x t r e m e s t a n d a r d amount, t h e r e e x i s t s amounts x
<z*
or x
> x*
r e s p e c t i v e l y in t h e half-open i n t e r v a l I . By a n argument similar to t h a t
above, condition (F) i s satisfied if and only if t h e r a n g e of v i s [ O , w ) or (-=,O].
Proof of Theorem 1. F i r s t , c o n s i d e r a model having a n intermediate s t a n d a r d
amount. Assume conditions (A)-(G). Then, t h e r e i s defined on t h e set C* a value
function V ( z 1 , z 2...)
,
as in Lemma 1 with v ( I ) = ( - = , = ) and a utility function
U ( z 1 , 2,...).
~
I t follows t h a t U ( x l r x 2,...) = f (V(Z1 , 2 ,~. . . ) )f o r some s t r i c t l y increasing function f defined on
(-w, w).
The second p a r t of condition ( G ) , i.e., t h a t e v e r y
l o t t e r y h a s a c e r t a i n t y equivalent, implies t h a t t h e r a n g e o f f i s a n interval. Since
f i s increasing, i t follows t h a t f i s continuous. Hence, U i s of t h e form ( 2 ) .
Suppose t h a t u ( z l ) denotes t h e conditional first-period utility function defined
by
u ( ~ ~ ) = U ( x ~ , z ,...
* ).
~ , zF o* r~ any consequence
CC , v -'(v(c)) i s a well-defined amount in I and c
-
c = ( z l r z,...)
2
in
( v -'(v(c)), z * ~ , z * ...).
~ , There-
. .a. v) - l ( V ( ~ ) ) and
,
t h u s U i s also of t h e
f o r e , U ( c ) = U ( v - 1 ( ~ ( ~ ) ) , ~ * 2 , ~ *=3 , u
form ( 3 ) . F o r a n y consequence c
T h e r e f o r e , v ( z )= f
= ( z l , z * 2 , z * 3 , . . . ) ,u ( z l ) = U ( C )=
f ( v (2')).
u ( z )f o r all z in I , and t h u s U i s of t h e form ( 4 ) .
Conversely, if t h e r e e x i s t s a value function V and a utility function U as
d e s c r i b e d in Theorem 1 , t h e n conditions (A)-(E) follow from t h e value function and
condition (G) follows from t h e utility function. T h e r e f o r e , i t remains to i n f e r condition (F). The form ( 3 ) of U implies t h a t t h e r a n g e of V i s contained in t h e domain
of v-'.
However, t h e domain of v-' i s by definition t h e r a n g e of v . Thus, f o r any
consequence
(z1,z2
...)
, in
CS
there
e x i s t a n amount z ;
in I
such t h a t
V ( z l , z * 2 , z * 3...)= v ( Z i ), a n d condition (F) i s established.
Second, c o n s i d e r a model having a n e x t r e m e s t a n d a r d amount. Most of t h e arguments are similar to t h o s e f o r a model having a n intermediate s t a n d a r d amount,
and t h u s i t suffices t o b e b r i e f e x c e p t f o r points of difference. Assume conditions
(A)-(G). Then, t h e r e i s defined on t h e set C* a value function V with v ( I )
o r v ( I ) = (-w,O], and t h e r e i s defined on t h e set C*
tion U . I t follows t h a t U
= LO,-)
- l ( z * ' , z L 2 , . . . ) a utility func-
= f V f o r some continuous, s t r i c t l y increasing function f
defined on (O,=) o r ( - w , O ) .
9
Thus, U i s of t h e form ( 2 ) . The forms ( 3 ) a n d ( 4 ) now
follow with t h e first-period utility function u (z l) = U(z l,z* 2,z*3,...) defined on
t h e i n t e r i o r of I. The converse arguments are similar t o those f o r a model having
a n intermediate standard amount.
Appendix B: Proofs of Results on Preference Conditions
This appendix contains t h e proofs of Lemma 2, 3 and Theorems 2-5 in Sections
4, 5, and Theorems 6-9 in Section 8. Some of these proofs use t h e following impli-
cations f o r a function g
gp
-(
1
+ ~z
,) =
1
1
g (Tz
1
+ ?Z
2) >
cave
(see, e.g.,
g(zl
+ A) =
1
-g
(z l)
2
1
-g
(z l)
2
t h a t i s defined and continuous on a n interval:
1
+ ~g
(z 2)
1
+ ?g
if and only if g i s linear,
f o r all z 1.2
(z 2) f o r all z
# z if and only if
Hardy et al. 1934), and g (zl)
1
1
T ~ ( +ZA ~
) + 2g(z3 + A) f o r all
g i s s t r i c t l y con-
1
1
= ~g
(z2) + ~g
+
Z ~ + ~ , . . . . Zh~ if
(23) implies
and only if g is
.
linear o r exponential (see, e.g , Pfanzagl1959).
Proof of Lemma 2. By Lemma A l , t h e range of t h e tradeoffs function z i s
( - w , ~ ) ,[O,w), o r (-w,O].
Thus, f o r two amounts zt
< z;
in a period t , the amount
st defined by
i s t h e tradeoffs midvalue of zt ,zi since f o r another period s t h e r e exist amounts
z
< z'
such t h a t , f o r example, asv(2')
t h e tradeoffs midvalue of z t ,zi
- asv(z) = at v (st)-
at v (zt). Moreover,
is unique since v i s s t r i c t l y increasing. The
remaining p a r t s of Lemma 2 may be established by similar arguments.
Proof of Theorem 2. Conditions (A)-(G) imply by Theorem 1 t h a t p r e f e r e n c e s can
be r e p r e s e n t e d by utility functions of t h e forms (2) and (4). First, i t will b e shown
t h a t e a c h of t h e p r e f e r e n c e conditions (a), (b) in p a r t s I, 11, i s equivalent t o t h e
corresponding p r o p e r t y (c) of t h e multiperiod r i s k function f in (2).
Consider condition (a).
vO
For a p a i r of amounts zt
= at v (zt),vl = atv (zi ) and
= at v (zt). Then,
1
= -v
2
< zi
0
in period t , let
1 1
+ -v
.
2
Thus, condi-
1 0
tion (1.a) is equivalent t o f (?v
1
+ ~v
1)
'
= Lf
(vO) + $f (v ') f o r all v 0 < v in t h e
2
interval v ( I ) which is equivalent t o f being linear on ~ ( 1 ) .Similarly, (I1.a) i s
equivalent t o f being s t r i c t l y concave on v (I).
Consider condition (b). For p a i r s of amounts z, ,z,'
let v O = a,v (2,)
= a t v ( z t ) and v1 = a,v (2;
equivalent t o f ( v O+ v l )
= $f (2v0) +
and zt , z i as described,
= atv ( z i ). Then, condition (1.b) i s
)
5f (2v1)
f o r all v 0
< v1
in v ( I ) which i s
equivalent t o f being linear on ~ ( 1 ) .Similarly, (1I.b) is equivalent t o f being
s t r i c t l y concave on v (I).
Next, i t will be shown t h a t p r o p e r t y (1.c) i s equivalent t o p r o p e r t y 0.d). If
there
exists
an
additive
utility
function
U(zl,z2, ...)
as
in
(I.c),
then
u (21) = U(zl,zS2,z* 3,...) equals v (zl), and thus t h e r e exists a n additive utility
function as in (1.d). The converse argument i s similar.
Proof of Theorem 3. First, i t will be shown t h a t e a c h of t h e conditions (a), (b) i s
equivalent t o p r o p e r t y (c) of t h e multiperiod risk function f
Consider
v&
condition
(a).
Let
v,
.
= atv ( w t ) ,..., vk =
a t v(
~ ).i
Then,
.
= vw + h , v; = v, + h and v$ = vy + h f o r some constant h . There i s ab-
1 (v, )
solute multiperiod r i s k constancy if and only if f (v, ) = ~f
+ h ) = 2.1f
f (v,
(vW + h ) +
1
yf
(up
+ h ). This
1 (vy ) implies
+ ~f
implication i s satisfied if and only if
f i s as described in (c).
Consider condition (b).
h = a, v (z ) and h '
and
only
1
$'(v,
if
For tradeoffs amounts z , z ' in a period s, let
= a, v ( z '). Period
f(v,
+h)
+ h ' ) + +(vy + h ' )
2
1
Ff(vw
t i s utility independent of the period s if
+h) +
91 ( v y
+h)
implies
f(v,
+ h ' ) ?_
f o r any h'. This implication i s satisfied if and only if f
i s as described in (c).
Next i t will b e shown t h a t (c) i s equivalent to (d) with t h e cases r
and r
< 0 in
(c) corresponding t o t h e t h r e e cases in (d). The case r
t r e a t e d in Theorem 2.
> 0,r = 0,
= 0 i s already
Proof of Theorem 4. First, i t will be shown t h a t each of the p r e f e r e n c e conditions
(a), (b) in p a r t s I, 11 is equivalent t o t h e corresponding p r o p e r t y (c) of the multiperiod risk function f' in (2).
Consider condition (a). For a p a i r of amounts zt
v 0 = a t v ( z t ) , v1
= a t v (z;
), and
not equal t o z* let
v^ = a t v ( s t ) . Then, v^ =
tion (1.a) is equivalent t o f' ((vOvl)l/ ')
v (I)
<zj
Thus, condi-
= Lf'
(vO) + Lf'
(vl) f o r all v 0 < v1 in
2
2
= (O,=) which is equivalent t o f' being logarithmic on v (I). Similarly, condi-
tion (I1.a) is equivalent t o f' oexp being s t r i c t l y concave on t h e interval (-=,=).
Consider condition (b). For a period t (actual o r imaginary) with an a r b i t r a r y
timing weight, 0
v = a,v (z,)
< at < 1,and a n amount z > z*, l e t
where z = z l
v1
= v (zl), v t = at v ( z t ) , and
= z, = z t . Any two numbers, v1 > v t > 0, can be
represented in this manner.
equivalent t o f'( ( v l v t )'I2) =
Moreover,
Af'
(vl) +
2
equivalent t o f' being logarithmic.
f = ( v l t ~ ~ ) l /Thus,
~.
condition (1.b) is
5f'( v t ) f o r all v1 > v t in (On=) which is
Similarly, condition (1I.b) is equivalent t o
f' e x p being s t r i c t l y concave on the interval (-=,=).
0
Next, i t will be shown t h a t condition (1.c) is equivalent t o condition (1.d). If
there
exists a utility function U(zl,z2, ...) as in
(I.c), then the function
6 ( z ) = logv ( z ) is a first-period utility function with 6 (z*) = -= and 6 ( z +) = +=.
Suppose
that
u(z)
f ( z ) = cu ( z ) + b
is
an
assessed
some constants c
first-period
utility
function.
Then,
> 0 and b , and thus by substitution,
Conversely, if t h e r e exists a utility function U ( Z ~ , ~ , . . .as
) in (I.d), then
V^(z) = exp(cu ( 2 ) ) is a tradeoffs function with 6 ( z * ) = 0 and V^(z +) = +=. Suppose t h a t v is a n assessed tradeoffs function with v (z*) = 0. Then,
f o r some constant a
f(z) = a v (z)
> 0, and thus by substitution
U(zl,z2, ...) = log(v (zl)
+ a 2 v (z2) + . . . ) + loga .
Proof of Theorem 5. First, i t will be shown that each of the conditions (a), (b) is
equivalent t o p r o p e r t y (c) of t h e multiperiod risk function f' .
Consider condition (a).
pv, ,v; = pv,,
and vf;
Let v,
= a t v ( w t ) , ...,vf;
= p v (vy ) f o r t h e s a m e constant
multiperiod risk constancy if and only if f (vz)
1
= J(pv,)
f(pv,)
+
= atv(y;).
p
> 0.
1
= Tf
(v,
=
There is relative
1
+
)
Then v;
Ff(vy ) implies
This implication is satisfied if and only if f i s as
$f(pvy).
described in (c).
Consider condition (b). For two actual o r imaginary periods s and
p
= at and p' = a,.
t and s.
periods
f (pv,)
2 $f (pv,)
+
< (w ), (y) >
Compare a consequence ( z ) and a l o t t e r y
The
periods
are timing
independent
$f b u y ) implies f ( p l v . ) h $f (p'v,)
+
t , let
if
and
only
in
if
$f ( p l v y ) f o r all
positive p ,pl,v, ,v, ,vy . Defining g by f ( v )
= g (Log v ) , i t follows t h a t t h e multi-
plications
replaced
p v , , ...,pl vy
can
logp
+ log v, ,..., l o g p ' + log vy .
if,
with
g (2)
= e ", r > 0 ; z , r = 0 ; o r
a
positive
by
the
additions
This modified implication i s satisfied if and only
linear
-e
be
transformation,
=,r < 0 .
g
is
of
the
form
Thus, t h e implication involving f i s sa-
tisfied if and only if f is as described in (c).
Next, i t will b e shown t h a t (c) i s equivalent to (d) with t h e cases r
< 0 in
and r
(c) corresponding t o t h e t h r e e cases in (d). The case r
> 0, r =0,
= 0 i s already
t r e a t e d in Theorem 4.
Assume t h a t t h e r e e x i s t s a utility function U(z l,~2,...) as in (c) with r
Then t h e function ; ( z )
negative values and 2 (z*)
= ( ~ ( z ) ) ' i s a first-period utility function with non= 0 . Moreover,
Suppose t h a t u i s a n assessed first-period utility function with u (z*)
;(z)
> 0.
= 0 . Then,
= a u ( z ) f o r some constant a > 0, and thus by substitution into t h e above
formula f o r U(zl,z2, ...) t h e f i r s t case of (d) follows. The argument t h a t (c) with
r
< 0 implies t h e
third case of (d) is similar.
Conversely, assume t h a t t h e r e i s a utility function U ( z l , z 2....) as in t h e f i r s t
case of (d).
v (z* )
Then, $ ( z ) = ( ~ ( 2 ) ) ' is defined f o r all z in I
= 0 . Moreover,
= [ z * , z + ) , and
T h e r e f o r e , 6 i s a t r a d e o f f s function. Suppose t h a t v i s a n a s s e s s e d t r a d e o f f s function with v ( z * )
= 0. Then, 6 ( z ) = a v ( z ) f o r some constant
into t h e above formula f o r U ( z l , z 2 , ...), t h e c a s e (c) with r
a
> 0.
By substitution
> 0 follows.
The a r g u -
ment t h a t t h e t h i r d case of (d) implies t h e t h i r d case of (c) i s similar.
Proof of Theorem 6. The functions f ' , v , a n d u in Theorem 1 are r e l a t e d by
U = f' v . Thus, if a n y two of t h e s e functions a r e l i n e a r , t h e n s o i s t h e t h i r d .
0
Proof of Theorem 7. If t h e multiperiod r i s k function f' i s l i n e a r , t h e n e x c e p t f o r
a positive l i n e a r transformation u = v . If t h e t r a d e o f f s function v i s l i n e a r , t h e n
e x c e p t f o r a positive l i n e a r transformation u
= f'. if t h e single-period r i s k func-
tion u i s l i n e a r , t h e n t h e two s t r i c t l y increasing functions f' a n d v cannot both be
s t r i c t l y concave.
Proof of Theorem 8. The p r e f e r e n c e conditions (1)-(3) are equivalent t o
f'(V)
= -elv
r
with
< 0,v ( z ) = log(z + c ) ,
and u ( z )
= -(z + c ) q with
q
< 0,
respectively (where f', v , a n d u are e a c h determined up t o a positive l i n e a r
transformation). I t may b e verified t h a t since u = f' v a n y two of t h e s e forms im8
plies t h e t h i r d a n d t h a t a l l t h r e e of t h e s e forms are equivalent to t h e form (13) of
U(z1.z2,...).
Proof of Theorem 9.
f' (V) = logV, v ( z ) = ( z
H e r e , t h e forms v ( z )
The p r e f e r e n c e conditions (1)-(3) are equivalent to
- z*)q
= log(z
with q
> 0,
and u ( z )
= log(z
- z * ) and v ( z ) = -(z - z*)q
- z * ),
with q
respectively.
< 0 are exclud-
e d s i n c e i t must b e possible t o normalize v s o t h a t v ( z * ) = 0. I t may b e verified
that since u
= f'
8
v a n y two of t h e s e forms implies t h e t h i r d and t h a t a l l t h r e e of
t h e s e forms are equivalent to t h e form (14) of U ( z l,z2,...).
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