A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Business Administration potx

The diaclerta;tion of N i b Hemming Hakaneson is appsoved, and it is acceptable In quality and Unlveraity of California, Ins Augeles.

Copyright by

N I L S H E M M I N G HAKANSSON

The diaclerta;tion of N i b Hemming Hakaneson is appsoved, and it is acceptable In quality and

Unlveraity of California, Ins Augeles

TABLE O F CONTENTS

LIST O F FIGURES

ACKNOWLEDGMENTS

VITA

ABSTRACT

1 3 The Opportunity Set

2 4 The Solution When u ( x y ) = u(x) + u ( y )

2 5 The Solution Wher, u ( x t y ) = u ( x ) ( u ( ~ ) 1 , ,

2 6 P r o p e r t i e s of the Optimal Consumption

T A B L E O F CONTENTS ( C o n t )

2 8 P r o p e r t i e s of t h e O p t i m a l B o r r o w i n g a n d

Lending S t r a t e g i e s

2 8 1 The E x i s t e n c e of a M a r k e t I n t e r e s t R a t e

2 8 2 D i f f e r e n t R a t e s f o r B o r r o w i n g a n d

Lending 2 9 P r o p e r t i e s of t h e O p t i m a l I n v e s t m e n t S t r a t e g i e s

2 1 0 G e n e r a l i z a t i o n s

2 10 1 F i n i t e H o r i z o n

2 10 2 N o n - C o n s t a n t Non- C a p i t a l I n c o m e S t r e a m

2 1 0 3 T i m e -Dependent P r o b a b i l i t y

D i s t r i b u t i o n s 2 11 I m p l i c a t i o n s with R e s p e c t to t h e T h e o r y of The F i r m

2 11 1 B a s e s f c r t h e F o r m a t i o n of F i r m s 2 11 2 The Firm's O b j e c t i v e a n d I t s O p t i m a l

C a p i t a l S t r u c t u r e 2 1 1 3 T h e Debt of t h e F i r m : L i m i t e d L i a b i l i t y

111 A P P L I C A T I O N S AND E X A M P L E S 3 1 Individual De c i s i o n - M a k i n g

3 2 The Bal.aaced Mutual F u n d

3 3 Endowed E d u c a t i o n a l a n d C h a r i t a b l e O r g a n i z a t i o n s

IV RELATION T O O T H E R MODELS

4 1 F i s h e r ' s M o d e l of t h e I n d i v i d u a l

4 2 Consumptio:? M o d e l s 4 2 1 C l a s s i c a l Models

4 2 2 P h e l p s ! .M ode1

4 3 I n v e s t m e n t Models

4 3 1 The M e a r - V a r i a n c e A p p r o a c h

4 3 2 C h a n c e - C o s s t r a i n e d M o d e l s

4 3 3 Long-Ru-2 ~ n v e s t m e n t M o d e l s

4 3 4 O t h e r I n v e s t m e n t M o d e l s

4 4 T h e S t a t e - P r e f e r e z c e A p p r o a c h : A B r i e f

C o m m e n t 4 5 S u m m a r y

BIBLIOGRAPHY

S t r e a m o n O p t i m a l A ? l o c a t i o ~ of C a p i t a l ( A t E a c h

D e c i s i o n P o i s t ) When U(C) = l o g c , a = 9 0 , a n d

r , P2, B3, P4, a r e a s i n ( 3 - 1 ) VIII N o r m a t i v e I n v e s t m e n t a n d C o n s u m p t i o n M o d e l s :

A C o m p a r a t i v e S u m m a r y

ACKNOWLEDGMENTS

While the r e s e a r c h r e p o r t e d i n t h i s study was pri.ncipally conducted during the a c a d e m i c y e a r 1965-66, many of the underlying i d e a s a r e

of a n e a r l i e r vintage T h e s e e a r l i e r i d e a s i n p a r t i c u l a r r e f l e c t the influence of s e v e r a l individuals

My g r e a t e s t debt i s to P r o f e s s o r George W Brown The high

s t a n d a r d s which h e h a s s e t have not only benefitted t h i s s t u d y but

t h e giving of h i s t i m e , P r o f e s s o r Brown h a s a l s o b e e n a continual

I a m g r a t e f u l to the F o r d Foundation f o r c a r r y i n g the fi.nancia1

b u r d e n of the l a s t t h r e e y e a r s i n the f o r m of two P r e d o c t o r a l F e l l o w -

s h i p s and one D i s s e r t a t i o n Fellowship, and to t h e W e s t e r n Data P r o -

c e s s i n g Center of the University of California a t Los Angeles f o r

e x p r e s s m y s i n c e r e a p p r e c i a t i o n to the W e s t e r n Management S,' p i e m e Institute of UCLA and t h e RAND C o r p o r a t i o n f o r typing p o r t i o n s of

t h e e a r l y d r a f t s , and to M r s Libby H Connor and M r s L a u r i e

H a r r i n g t o n f o r t h e i r excellent typi.ng of the final m a n u s c r i p t

ABSTRACT O F THE DISSERTATION

Optimal I n v e s t m e n t and Consumption S t r a t e g i e s

F o r a C l a s s of Utility Functions

Nils Hemming Hakans s o n

Doctor of Philosophy i n B u s i n e s s A d m i n i s t r a t i o n University of California, Los Angeles, 1966

i n g ) and a n a r b i t r a r y n u m b e r of productive inve s t m e a t opportu.nlties

The i n t e r e s t r a t e i s p r e s u m e d to be known and i n v a r i a n t o v e r t i m e ;

t h e c a s e when the borrowing r a t e e x c e e d s the lending r a t e i s e x a m i n e d

f o r a s p e c i a l i z e d model The r e t u r n s f r o m i t h e productive opportuni-

t i e s a r e a s s u m e d to be r a n d o m v a r i a b l e s , whose p r o b a b i l i t y d i s t r i - butions m a y d i f f e r f r o m p e r i o d to peri.od The b a s i c ( F i s h e r i a n )

c h a r a c t e r i s t i c of the a p p r o a c h taken i s that the portfolio composition

decision, the financing decision, and t h e consumption d e c i s i o n a r e

a l l analyzed s i m u l t a n e o u s l y i n - one model The vehicle of a n a l y s i s i s

The o p t i m a l consumption s t r a t e g i e s t u r n out to be l i n e a r and i n -

c r e a s i n g i n wealth and i n t h e p r e s e n t value of t h e n o n - c a p i t a l i n c o m e

s t r e a m In t h r e e of the f o u r m o d e l s studied, t h e o p t i m a l c o n s u m p - ,

hypqtheses of Modigliani a n d B r u m b e r g a n d of F r i e d m a n precise1.y The o p t i m a l lending and borrowing s t r a t e g i e s a r e faun-d to be l i n -

e a r i n wealth T h r e e of the m o d e l s always c a l l f o r borrowing when

the individual i s poor while the fourth m o d e l always c a l l s f o r ler-iding when h e i s sufficiently r i c h

The o p t i m a l i n v e s t m e n t s t r a t e g i e s have t h e s u r p r i s i n g p r o p e r t y

t h a t t h e o p t i m a l m i x of r i s k y ( p r o d u c t i v e ) i n v e s t m e n t s i n e a c h m o d e l

-"

i s independent of the individual's wealth, non- c a p i t a l i n c o m e s t r earn, and i m p a t i e n c e to consume I t i s c o n j e c t u r e d that the c l a s s of utility functions examined i s t h e only one f o r which t h i s p r o p e r t y of the op-

t i m a l i n v e s t m e n t s t r a t e g i e s holds

The p r e c e d i p g r e s u l t a p p e a r s to have significant i m p l i c a t i o n s with

r e s p e c t to t h e t h e o r y of the f i r m S t a r t i n g with a collection of h e t e r o - geneous individuals, e a c h of whom i s bent on m a x i m i z i n g ( h i s own) utility f r o m consumption o v e r t i m e , i t i s shown that t h e r e e x i s t s a

b a s i s f o r the f o r m a t i o n of f i r m s by s u b - c o l l e c t i o n s of individuals,

w h e r e e a c h s u b - c o l l e c t i o n i n t u r n p o s s e s s e s significant h e t e r o g e n e i t y Each f i r m s o f o r m e d i s found to have a well-defined ( u n i q u e ) objective function, which m a y be i n t e r p r e t e d a s imputing a p r e c i s e meaning to the t e r m " p r o f i t maximization" u n d e r r i s k and with r e s p e c t to t i m e Since t h e c a p i t a l s t r u c t u r e of the f i r m i s found to be u n i m p o r t a n t , a n unexpected t i e - i n with P r o p o s i t i o n I of Modigliani and M i l l e r i s

obtained

CHAPTER I

The objective of this r e s e a r c h i s to d e r i v e o p t i m a l i n v e s t m e n t and consumption s t r a t e g i e s f o r individuals f r o m a l t e r n a t i v e but fundamen-

t a l s t a r t i n g - p o i n t s , to e x a m i n e and c l a s s i f y t h e i r p r o p e r t i e s , and to analyze t h e i r economic i m p l i c a t i o n s , p a r t i c u l a r l y i n r e s p e c t to the

t h e o r y of the f i r m The point of view, t h e r e f o r e , i s e s s e n t i a l l y p r e -

s c r i p t i v e , placing the study i n the d o m a i n of n o r m a t i v e d e c i s i o n theor)-

In t h i s c h a p t e r , the v a r i o u s components of the economic d e c i s i o n

p r o b l e m to be s t u d i e d a r e c o n s t r u c t e d The o b j e c t i v e of the individual

i s p o s t u l a t e d to be the m a x i m i z a t i o n of e x p e c t e d utility f r o m c o n s u m p - tion o v e r t i m e w h e r e the h o r i z o n i s infinitely d i s t a n t The individual's

r e s o u r c e s a r e a s s u m e d to c o n s i s t of a n i n i t i a l c a p i t a l p o s i t i o n (which

m a y be n e g a t i v e ) a n d a n o n - c a p i t a l i n c o m e s t r e a m which i s known with

c e r t a i n t y but which m a y pos s e s s a n y t i m e - s h a p e The indi.vidua1 f a c e s both financial o p p o r t u n i t i e s ( b o r r o w i n g and lending) and a n a r b i t r a r y

The components developed i n Chapter I a r e a s s e m b l e d into a f o r m a l

m o d e l i n Chapter 11, w h e r e the m a i n r e s u l t s a r e d e r i v e d The funda-

m e n t a l c h a r a c t e r i s t i c of the a p p r o a c h taken i s t h a t the portfo1i.o c o m - position d e c i s i o n , the financing d e c i s i o n , and the consumption

d e c i s i o n a r e all analyzed simultaneously, The b a s i c m o d e l developed

Optimal consumption and i n v e s t m e n t s t r a t e g i e s a r e d e r i v e d f o r the

amount of consumption i n p e r i o d j, s u c h that e i t h e r the r i s k a v e r s i o n index - u " ( x ) / u ' ( x ) o r the r i s k a v e r s i o n index - x u " ( x ) / u ' ( x ) i s a p o s i - tive constant f o r a l l finite x > - 0 It i s shown t h a t u ( x ) belongs to t h i s

c l a s s i f and only i f u ( x ) i s s t r i c t l y concave and s a t i s f i e s one of the

t h r e e "Cauchy" e q u a t i o ~ s u ( x -t y ) = u ( x ) I U ( ~ ) I, u ( x y ) = u ( x ) -t u ( y ) ,

o r ~ ( x y ) = U ( X ) I U ( ~ ) I , i e , u ( c ) = c , 0 < y < 1, u ( c ) = - c ,

y > 0, u ( c ) = log c, o r u ( c ) = - e l Y c , y > 0

Section 2 6 i s devoted to a d i s c u s s i o n of the p r o p e r t i e s of the o p t i -

m a l consumption s t r a t e g i e s , w h i c h t u r n out to be l i n e a r and i n c r e a s i n g

i n wealth and i n t h e p r e s e n t value of the n o n - c a p i t a l i n c o m e s t r e a m

Edmund P h e l p s , " T h e Accumulation of Risky Capital: A Sequential Utility Analysis, E c o n o m e t r i c a , October 1962

r e t u r n s , the i n t e r e s t r a t e , and the i n d i v i d u a l ' s o n e - p e r i o d utility

function of consumption It i s then c o n j e c t u r e d i n 2 11 t h a t the c l a s s

of utility functions examined i s the only one f o r which t h i s p r o p e r t y of the o p t i m a l i n v e s t m e n t s t r a t e g i e s holds

The p r e c e d i n g resu1.t a p p e a r s to have significant i m p l i c a t i o n s with

r e s p e c t to the t h e o r y of t h e f i r m S t a r t i n g with a collection of h e t e r o - geneous individuals, e a c h of whom i s bent o n m a x i m i z i n g this own) utility f r o m consumption o v e r t i m e , i t i s shown i n 2 11 that t h e r e

e x i s t s a b a s i s f o r the f o r m a t i o n of f i r m s by s u b - c o l l e c t i o n s of Indi- viduals, w h e r e e a c h sub-coll.ection i n t u r n p o s s e s s e s significant

h e t e r o g e n e i t y Each f i r m s o f o r m e d i s found to have a well-defined (unique) objective function, which m a y be i n t e r p r e t e d a s imputing a

p r e c i s e meaning to the t e r m " p r o f i t maximization" u n d e r r i s k an.d with r e s p e c t to t i m e , Since the c a p i t a l s t r u c t u r e of the f i r m i s found

to be u n i m p o r t a n t , a n unexpected t i e - i n with P r o p o s i t i o n I of

Modigliani and M i l l e r i s obtained

In Chapter 111; the r e s u l t s obtained i n Chapter IX a r e i l l u s t r a t e d by

m e a n s of e x a m p l e s , and s o m e of the appl.ications to which the m o d e l

l e n d s i t s e l f a r e d i s c u s s e d It i s noted t h a t the m o d e l I s app1.icabl.e to

t h e balanced m u t u a l fund a.s w e l l a s to endowed educational and c h a r -

i t a b l e o r g a n i z a t i o n s , In the l a s t c h a p t e r , the r e l a t i o n s h i p between t h s

m o d e l developed in this study a.nd o t h e r i n v e s t m e n t and consumption

l e s s d e f e r r e d enjoyment " Turning to the m o r e popular a u t h o r s , Loeb, f o r example, w r i t e s that "the p u r p o s e of i n v e s t m e n t i s to have funds a v a i l a b l e a t a l a t e r date f o r spending 1 1 2 In a d i f f e r e n t p a s s a g e

h e s t a t e s : "Any e a r n e r who e a r n s m o r e than h e c a n spend i s a u t o m a t i -

c a l l y a n i n v e s t o r It d o e s n ' t m a t t e r i n the s l i g h t e s t whether h e r e a -

l i z e s t h a t h e i s investing l~~ While no h a r d and f a s t line c a n be d r a w n between what c o n s t i t u t e s consumption and what c o n s t i t u t e s i n v e s t m e n t , consumption i s p e r h a p s b e s t viewed a s the exchange of p r e s e n t d o l l a r s

4 By this distinction, t h e o w n e r s h i p and occupancy of a f a m i l y h o m e

i s c l e a r l y both i n v e s t m e n t and consumption The down p a y m e n t , including the d i f f e r e n c e between m o r t g a g e p a y m e n t s plus e x p e n s e s

m i n u s the r e n t a l value, if p o s i t i v e , c o n s t i t u t e s a n i n v e s t m e n t The foregone r e n t a l i n c o m e c o n s t i t u t e s consumption The r e t u r n o n the

i n v e s t m e n t i s c o m p o s e d of the r e n t a l value l e s s m o r t g a g e p a y m e n t s and e x p e n s e s , i f p o s i t i v e , plus the f i n a l p r o c e e d s f r o m the s a l e of the house

The consumption of food, for e x a m p l e , might a l s o be c a l l e d a n

i n v e s t m e n t i n that i t p r e s e r v e s the health n e c e s s a r y for s u r v i v a l However, we s h a l l not take this view h e r e

The i n v e s t m e n t d e c i s i o n - c h a r a c t e r i z a t i o n The i n v e s t m e n t

decision, l i k e m o s t p r o b l e m s of d e c i s i o n p o s e d i n a r e a l i s t i c way, h a s two fundamental c h a r a c t e r i s t i c s : i t i s s e q u e n t i a l and it i s t a k e n u n d e r

r i s k o r u n c e r t a i n t y A s e q u e n t i a l d e c i s i o n p r o b l e m i s a p r o b l e m e x - tended i n t i m e , i n which t h e consequences thus f a r of d e c i s i o n s t a k e n

i n p a s t p e r i o d s become i n i t i a l conditions f o r p r e s e n t d e c i s i o n s A

d e c i s i o n p r o b l e m u n d e r r i s k o r u n c e r t a i n t y i s one i n which t h e m o d e l employed does not a s s u m e p e r f e c t f o r e s i g h t

The i n v e s t m e n t objective Any n o r m a t i v e m o d e l p r e s u m e s t h e e x -

i s t e n c e and a v a i l a b i l i t y of a n objective function Thus, t h e d e r i v a t i o n

of "optimal" i n v e s t m e n t s t r a t e g i e s , f o r example, i s contingent upon

the f i r m , t h e r e i s wide d i s a g r e e m e n t a s to what i t s o b j e c t i v e should be,

a d i s a g r e e m e n t which shows no s i g n of n a r r o w i n g Even if one w e r e

\

to adopt t h e classi.ca1 postulate of p r o t i t maximi.zation, one i m m e d i -

a t e l y r u n s into conceptual difficulties: what does i t m e a n to m a x i m i z e

p r o f i t s under r i s k o r u n c e r t a i n t y ? &d e v e n i n the c a s e of c e r t a i n t y

o n e f a c e s t h e i n t e r t e m p o r a l question: when do we m a x i m i z e p r o f i t s - ?

Since all c l a i m s to the c a p i t a l of t h e f i r m r e s i d e i n Individuals, i t

s e e m s r e a s o n a b l e t h a t the objective of t h e f i r m should be a t l e a s t

grounded i n t h e o b j e c t i v e s of the individual i n v e s t o r s of equity capital

1

The v a r i o u s o b j e c t i v e s s u g g e s t e d i n the l i t e r a t u r e a r e too n u m e r o u s

to be d i s c u s s e d h e r e J7or-a t a s t e of the d i f f e r e n t p r o p o s a l s , t h e

r e a d e r i s r e f e r r e d to R i c h a r d E l l s , The Meaning &f Modern Busi -

n e s s , New York, Columbia u n i v e r s i t y p r e s s , 1960, pp 117-21;

C h a r l e s G r a i n g e r , " T h e H i e r a r c h y of Objectives, " H a r v a r d B u s i -

n e s s Review, May-June 1964; and P e t e r D r u c k e r , " T h e Objectives

of a B u s i n e s s , " The P r a c t i c e of Management, New York, H a r p e r

The Theory of I n t e r e s t , and the writings of H i r s h l e i f e r , s t i l l s e e m s

to be lacking This i s the m o r e s u r p r i s i n g s i n c e , when t h e p r o b l e m

is viewed i n t h i s light, one i s h a r d put to find a n a p r i o r i r e a s o n f o r -

a s s u m i n g the two d e c i s i o n s to be independent of one a n o t h e r

The m o s t significant w o r k to date o n the p r o p e r t i e s of p r e f e r e n c e

See i n p a r t i c u l a r J a c k H i r s h l e i f e r , "On t h e T h e o r y of 0ptima.l 3n-

v e s t m e n t Decision", The Jour.r\_al of Political Economy, August 1 9 5 8 , and J a c k Hir s h l e i f e r , "Investment Decision under Uncertainty:

Choice- T h e o r e t i c Approaches", Q u a r t e r l y J o u r n a l of ~ c o n o m i c s , November 1965

Tjalling Koopmans, "Stationary O r d i n a l Utility and I m p a t i e n c e ? ' ,

i m p a t i e n c e and t i m e p e r s p e c t i v e i n a b r o a d c l a s s of s u c h p r o g r a m s The notion of i m p a t i e n c e goes back to Bohm-Bawerk, who in - The

P o s i t i v e Theory of Capital advanced the i d e a of p r e f e r e n c e f o r e a r l y timing of s a t i s f a c t i o n In Koopmans' work, i m p a t i e n c e i s e s s e n t i a l l y

t a k e n to m e a n that i f i n any given p e r i o d the consumption of the c o m - modity bundle x i s p r e f e r r e d to that of bundle x ' , then the consump-

t i o n i n c o n s e c u t i v e p e r i o d s of x, x' i s p r e f e r r e d to that of x ' , x, all

o t h e r consumption being the s a m e The notion of t i m e p e r s p e c t i v e

w i l l be b r i e f l y d i s c u s s e d l a t e r

While f o r m a l l y defined in t e r m s of a utility function, i m p a t i e n c e

i s viewed a s a p r o p e r t y of t h e underlying p r e f e r e n c e o r d e r i n g This

i m p l i e s that e v e r y utility fun.ction r e p r e s e n t i n g the p r e f e r e n c e o r d e r - ing m u s t have t h e i m p a t i e n c e p r o p e r t y Consequently, i m p a t i e n c e

m u s t be e x p r e s s e d i n t e r m s of a n o r d i n a l utility function An o r d i n a l utility function i s a utility function which r e t a i n s i t s m e a n i n g u n d e r a monotonic i n c r e a s i n g t r a n s f o r m a t i o n , that i s , i f V i s a utility func- tion, s o i s U = T(V), w h e r e T i s any monotonic t r a n s f o r m a t i o n and

T f ( V ) > 0

The p o s t u l a t e s a s s e r t continuity, s e n s i t i v i t y , and s t a t i o n a r i t y of the utility function, a b s e n c e of i n t e r t e m p o r a l c o m p l e m e n t a r i t y , and

t h e e x i s t e n c e of a w o r s t ar,d a b e s t p r o g r a m Thus, the p a p e r s e s -

s e n t i a l l y constitute a study of the i m p l i c a t i o n s of a continuous and

s t a t i o n a r y o r d e r i n g of infinite consumption p r o g r a m s

Notation The bundle of n c o m m o d i t i e s consumed i n p e r i o d j ,

j = 1 , 2, 3 i s g i v e n b y

w h e r e c > 0 An infinite p r o g r a m will be w r i t t e n

J -

Statement of the pos tu.lates P1 ( E x i s t e n c e and continuity) T h e r e

e x i s t s a utility function U ( l c ) , which i s defined f o r all l c s u c h that,

f o r a l l j, c i s a point of a bounded, convex s u b s e t C of the n -

J

d i m e n s i o n a l commodity s p a c e The function U ( l c ) h a s the continuity

p r o p e r t y that, i f U i s any of the values a s s u m e d by t h a t function, and

a p o s i t i v e n u m b e r 6 s u c h t h a t t h e utility U( c ' ) of e v e r y p r o g r a m l c '

1 having a d i s t a n c e d ( l c ' , c ) 1 ' sup jc; - c 1 < 6 , w h e r e l c ' - c j 1

U, t h e continuity p r o p e r t y given i n P 1 m a y be t e r m e d uniform conti-

nuity on e a c h equivalence c l a s s In the f i r s t p a p e r , P I stipulated both u n i f o r m continuity on e a c h equivalence c l a s s and unbol~.ndednes s

of C which s e v e r e l y l i m i t s t h e choice of functions Uo Evidently

Koopmans c h o s e to s a c r i f i c e unboundedness and with i t , p e r h a p s ,

s o m e r e a l i s m The d i s t a n c e function, o r m e t r i c , a l s o t r e a t s a l l

p e r i o d s a l i k e , the p r o p r i e t y of which m a y be questioned However,

r e m e m b e r i n g t h a t t h e p r e s e n c e of i m p a t i e n c e i s the phenomenon to be

e s t a b l i s h e d , t h i s a p p r o a c h certa-lnly provides a neutral s t a r t i n g point

I

2

This p o s t u l a t e i s clea,rly s t r o n g e r than one which s i m p l y r e q u i r e s

utility function f r o m being i n s e n s i t i v e to a l l p r o g r a m changes which affect a given p e r i o d The choice of the f i r s t p e r i o d for this p u r p o s e

This p o s t u l a t e s a y s that the consumption of a p a r t i c u l a r bundle of

c o m m o d i t i e s i n one p e r i o d does not affect p r e f e r e n c e s with r e s p e c t to

f u t u r e a l t e r n a t i v e s T h i s , of c o u r s e , i s a highly q i ~ e s t l o n a b l e p r o p o - sition It would p e r h a p s be m o r e palatable if total expenditures on consumption w e r e u s e d i n s t e a d a s a m e a s u r e of s a t i s f a c t i o n , but this

i d e a i s r e j e c t e d by Koopmans However, h e r e one r u n s into the r a t -

chet p r i n c i p l e 1

The r a t c h e t p r i n c i p l e e s s e n t i a l l y s t a t e s that the utility of c o n s u m p -

t i o n i n a given p e r i o d i s strong1.y conditioned on the h i g h e s t l e v e l of consumption p r e v i o u s l y e x p e r i e n c e d , particuJ.arly if this l e v e l i s of

r e c e n t o r i g i n This point was f i r s t m a d e by J a m e s D u e s e n b e r r y ic

Income, Saving and t h e the or^ of C o n s u m e r Behavior, Cambridge,

M a s s a c h u s e t t s , H a r v a r d University P r e s s , 1949, pp 84-85,114116

As a consequence of P3, i t c a n r e a d i l y be shown that U ( l c ) m a y

p e r i o d It should be p o i ~ t e d out that the o r d e r i ~ g in question applies only to the p r e s e r t The p a s s a g e of t i m e i s c o m p l e t e l y o u t s i d e the

s c o p e of the p o s t u l a t e s e t - t h u s , the question of changes i n p r e f e r -

e n c e s a s a function of t i m e i s not c o n s i d e r e d

It c a n be shown that P3b and P4 together i m p l y

U ( c l , 2 ~ ) - > U(c c ' ) i f and only i f U( c ) > U ( 2 c ' ) f o r a l l

I

c l , C 2 ' 2 c Since Y(u, T f i s i n c r e a s i n g i n T, P 4 i s equivalent to

T ( 2 ~ ) - > T ( 2 ~ ' 1 i f and only i f U ( c ) > U ( 2 c ' )

Consequently, t h e r e e x i s t s a monotonic t r a n s f o r m a t i o n H s u c h tha.t

Thus, letting V(u, U ) = Y(u, H ( U ) ), V(u, U ) p r e s e r v e s the p r e f e r e n c e defined by U ( l c ) so t h a t - w e obtain the r e c u r r e n c e r e l a t i o n

v e c t o r s c m a y be e v a l u a t e d , j = 1, 2 ,

j

With this r e s u l t , Koopmans h a s shown that impatience, which i s usually viewed as a psychological phenomenon, i s a l s o a consequence

of quite e l e m e n t a r y p r o p e r t i e s attributed to a utility function i n which the horizon i s infinite This i s a significant accomplishment indeed

A g e o m e t r i c r e p r e s e n t a t i o n of the t h r e e impatience zones of T h e o r e m

1 i n which the s c a l e s of u and U have been equated i s given i n Fig 1 Koopmans a l s o shows t h a t when weak ( s t r o n g ) t i m e p e r s p e c t i v e (to

be d i s c u s s e d below) i s p r e s e n t , one can p r o v e that t h e r e e x i s t s weak ( s t r o n g ) impatience i n the e n t i r e (open) i n t e r v a l (u2, u l ) , which of

c o u r s e includes zone 2 ( s e e Fig 1 )

-

Concerning the outlying i n t e r v a l s 0 -< U 3 < - U and U < U3 - C 1, nothing conclusive can be s a i d about impatience in t h e m when they

a r e non-empty Both i'mpatience and s t r o n g patience m a y exist, w h e r e

s t r o n g patience i s s a i d to exist in ( u I , u,, U,) if

tion p r o g r a m s s u c h that U1 E U ( l x ) > U 2 U ( l y ) NOW postpone

e a c h p r o g r a m by one p e r i o d , i n s e r t i n g consumption v e c t o r z i n the vacated f i r s t p e r i o d Then, by P 4 and P3b, U 3 E U ( z , x l , x2, )

A c.ardina1 utility function Koopmans s u g g e s t s that a g e n e r a l d i s - count f a c t o r , cr(U), be defined by the identity

t h a t i s , a s a function of t h e o v e r a l l l e v e l of s a t i s f a c t i o n achieved and

p r o v i d e s a s a n example a utility function with a discount f a c t o r which

d e c r e a s e s i n U ( U = W(u), denoted the c o r r e s p o n d e n c e function, i s the solution to the equation Vju, U ) = U f It c a n be shown that ( 1 - 4 )

i s i n v a r i a n t under differentiable monotonic t r a n s f o r m a t i o n s

It should be o b s e r v e d that if the s c a l e s of u and U a r e equated and

t h a t i s , the discount f a c t o r given by (1 - 4 ) i s c o n s t a n t , which a g r e e s with the conventional i n t e r p r e t a t i o n Thus, i t is c l e a r that the utility function ( 1 - 5 ) i m p l i e s that i m p a t i e n c e e x i s t s i n a l l p a r t s of the p r o -

g r a m s p a c e

It was noted by Koopmans that the addition of a s t r o n g e r v e r s i o n of the n o n - c o m p l e m e n t a r i t y postulate to the s e t P 1 - P 5 l e a v e s t h i s utility function a s the only function which i s c o n s i s t e n t with the expanded

p o s t u l a t e s e t The additional p o s t u l a t e i s given by

l e a s t , c o n c e r n o u r s e l v e s with how to find the p a r t i c u l a r t r a n s f o r m a -

tions which give u s a c a r d i n a l utility function f r o m a given o r d i n a l

a v a i l a b l e consumption p r o g r a m s a r e s u b j e c t to r i s k and only o r d i n a l utility fuslctions a r e known

1 2 2 P r o p e r t i e s and Limitations of t h e [Jtility Function

the f o r m ( 1 - 5 ) The m o s t s e r i o u s drawbacks of t h i s c l a s s of f u n c -

consequence of p o s t u l a t e s 3 and 3 ' , and t h e constancy of the &scount

f a c t o r cu As s u g g e s t e d e a r l i e r , t h e a s s u m p t i o n s of n o n - c o m p l e m e n -

t a r i t y a r e p a r t i c u l a r l y l i m i t i n g when consumption i s t r e a t e d a s a

commodity v e c t o r By focusing on t o t a l ( d o l l a r ) consumption alone,

c e r t a i n t y p e s of c o m p l e m e n t a r i t y between c o m m o d i t i e s need not be

r u l e d out Consequently, we s h a l l choose to be c o n c e r n e d with t h e

l e v e l of consumption r a t h e r t h a n the composition of the consumption

a l l p o s s i b l e p r o g r a m s ( c , , c 2 , cg, ) w h e r e c , j = 2 ,

3

i s the a m o u n t of consumption i n p e r i o d j

h e r e n t i n utility functions of the f o r m ( 1 -51, l e t u s examine s o m e con-

c r e t e e x a m p l e s As a c a s e i n point, l e t u s c o n s i d e r the function

u ( c ) = log c and p o s e the p r o b l e m of finding d i f f e r e n t consumption

p r o g r a m s between which the utility function ( 1 - 5 ) r e q u i r e s t h e indi- vidual to be indifferent F o r example, we might a t t e m p t to find t h e

consumption p r o g r a m s c ' a.nd , c" which a r e equivalent to the

1

p r e f e r s t h e c e r t a i n p r o g r a m e 1 c ' + ( 1 - 8 ) 1 ~ 1 1 to t h e p r o s p e c t of o b -

t a i n i n g I c ' w i t h p r o b a b i l i t y 9 a n d c" w i t h p r o b a b i l i t y 1 1 - 6

consumption, which i m p l i e s t h a t U i s s t r i c t l y concave This a s s u m p - tion, which h a s a high degree1of acceptance, i s c r u c i a l to a l l the r e -

s u l t s which follow But if U i s monotone i n c r e a s i n g and s t r i c t l y con- cave, i t follows t r i v i a l l y t h a t u ( c ) i s likewise

J The utility function (1 - 5 ) i s defined on infinite p r o g r a m s , which

m a y s e e m to be out of s t e p with the f a c t t h a t m a n ' s l i f e s p a n i s finite However, we s h a l l a r g u e t h a t a n i n d i v i d u a l ' s p r e f e r e n c e s g e n e r a l l y extend beyond h i s own l i f e t i m e , F i r s t , h i s d e p a r t u r e point i s indefi-

n i t e ; i t t h e r e f o r e behooves h i m to be c o n s e r v a t i v e i n r e f e r e n c e t o h i s planning h o r i z o n Second, h e usually w i s h e s to p r o v i d e i n s o m e f o r m

f o r h i s h e i r s a n d s u c c e s s o r s - i t i s i n f a c t t h i s benevolence which

k e e p s m a n f r o m p e r i s h i n g f r o m the e a r t h In the f i r s t twenty y e a r s

o r s o , e a c h of u s depends on s o m e o n e e l s e f o r the economic goods he enjoys Consequently, m a n h a s , during h i s l i f e t i m e , both the m o r a l

and l e g a l r i g h t to s u p p l e m e n t h i s own p r e f e r e n c e s with r e g a r d to con-

s u m p t i o n with the p e r c e i v e d p r e f e r e n c e s of h i s s u c c e s s o r s - to infinity This i s not to s a y t h a t a n investigation of finite p r o g r a m s would be without m e r i t However, if this a p p r o a c h i s u s e d , one i s f a c e d w i t h t h e

p r o b l e m of d e t e r m i n i n g j u s t w h e r e the h o r i z o n i s F o r the r e a s o n s given, coupled with the f a c t that a utility function with a d i s t a n t (but

f i n i t e ) h o r i z o n i n which i m p a t i e n c e i s p r e s e n t i s c l o s e l y a p p r o x i m a t e d

I

by the s a m e function with the h o r i z o n extended to infinity, the i d e a

F o r e x a m p l e , i n the c a s e of function ( 1 - 5 ) with w = 9, 9 9 9 p e r c e n t

of the utility obtained f o r m a c o n s t a n t consumption l e v e l i s a s s o c i -

a t e d with t h e f i r s t 64 p e r i o d s M o r e g e n e r a l l y , i f the utility of con-

s u m p t i o n a m o u n t c i n a p a r t i c u l a r p e r i o d i s u, then the c o n t e m p o r a r y utility of c 6 4 p e r i o d s la.ter i s 001 u

of evaluating infinite p r o g r a m s a p p e a r s intuitively much m o r e

s a t i s f a c t o r y

The p r e f e r e n c e o r d e r i n g s we have d i s c u s s e d have been c o n s i d e r e d

This i s i n a g r e e m e n t with economic t r a d i t i o n , which h a s always s e p -

modification of a n individual's p r e f e r e n c e s i n the light of e x p e r i e n c e

i s r u l e d out An a t t e m p t to g r a p p l e with the q u e s t i o n of allowing f o r

flexibility of f u t u r e p r e f e r e n c e h a s been m a d e by Koopmans 1

1 2 3 Note o n t h e Boundedness of the Utilitv Function

It h a s been shown by A r r o w (who c r e d i t s the d i s c o v e r y of the proof

to M e n g e r ) that a von N e u m a n n - M o r g e n s t e r n utility function i s

c a u s e f o r questioning the r e s u l t s obtained with t h e s e functions

n e v e r l e a v e s t h a t p a r t of the d o m a i n of the (unbounded) function f o r which i t s value i s finite, the unbounded p a r t of the function m i g h t a s

w e l l be " c u t off " In the ensuing m o d e l s , by eliminating t h e p o s s i b i l i t y

of s t a r t i n g out i n the t r a p p i n g s t a t e (to be d i s c u s s e d in 2 7 ) i n Models

Tjalling Koopmans, "On Flexibility of F u t u r e P r e f e r e n c e , " Human

Kenneth A r r o w , B e r n o u l l i Utility I n d i c a t o r s f o r Distributions Over

A r b i t r a r y S p a c e s , Technical Report No, 57, Department of Econo-

11 a n d 111, t h e r e s u l t s a r e i n d e e d t h e s a m e a s they would be if only the bounded p a r t of t h e function u ( c ) h a d b e e n employed H o w e v e r , if a

bound w e r e p l a c e d o n u ( c ) i n Model I, a s o l u t i o n would a l s o e x i s t ,

though p r o b a b l y not i n c l o s e d f o r m , when t h e c o n v e r g e n c e condition

A s e c o n d a v e n u e of d e f e n s e would be to s a y t h a t t h e continuity p o s t -

u l a t e , o n which t h e boundedness of t h e u t i l i t y i n d i c a t o r d e p e n d s , i s un-

n e c e s s a r i l y r e s t r i c t i v e a n d t h a t i t should be m o d i f i e d (which would be

e a s y enough) t o p e r m i t the u t i l i t y function to be unbounded 1

1 3 THE O P P O R T U N I T Y SET

function (1 - 5 ) Thus, we h a v e i n e f f e c t equipped h i m with the power

A c r i t i q u e of t h e continuity p o s t u l a t e m a y be found i n Duncan L u c e

a n d H o w a r d Raiffa, G a m e s a n d D e c i s i o n s , New York, John Wiley,

1 9 5 7 , p 2 7 ,

e a c h opportunity i s a r a n d o m v a r i a b l e To p a r a p h r a s e , we a r e s t i p u - lating that the only thing t h a t i s c e r t a i n about the r e t u r n f r o m a n i n v e s t -

m e n t i s t h a t i s i s u n c e r t a i n This, of c o u r s e , i s e s s e n t i a l l y what J P

M o r g a n e x p r e s s e d when, a s k e d what he thought about the s t o c k m a r k e t ,

a s p o s s i b l e , while s t i l l retaining the s t o c h a s t i c n a t u r e of r e t u r n s , we

s h a l l m a k e the following s e c o n d - o r d e r a s s u m p t i o n s :

1 All i n v e s t m e n t opportunities a r e of the point-input, point-output type, i e , i n v e s t m e n t and r e a l i z a t i o n take p l a c e i n s t a n t a n e o u s l y

r a t h e r t h a n o v e r t i m e

2 All i n v e s t m e n t s a r e r e a l i z a b l e i n c a s h a t the end of e a c h p e r i o d

3 The amount i n v e s t e d i n a n opportunity m a y be any r e a l n u m b e r

6 We s h a l l a r b i t r a r i l y define a s h o r t s a l e a s the opposite of a long i n -

v e s t m e n t That i s , i f the individual s e l l s opportunity i s h o r t i n the

amount 0 , h e will r e c e i v e 8 i m m e d i a t e l y (to do with a s he p l e a s e s ) i n

r e t u r n f o r the obligation to pay the t r a n s f o r m e d v a l u e of 0 a t the end

only n e e d s to put up the m a x i m u m amount h e m a y l o s e i n making a

long i n v e s t m e n t , c a n a l s o be handled without difficulty i n the

ensuing m o d e l s

While t h e s e a s s u m e d c o n d i t i o n s undoubtedly a r e too restrictive t o